An asymptotic formula for the number of integral matrices with a fixed characteristic polynomial via orbital integrals

Seongsu Jeon and Yuchan Lee Seongsu Jeon
Department of Mathematics, POSTECH, 77, Cheongam-ro, Nam-gu, Pohang-si, Gyeongsangbuk-do, 37673, KOREA [email protected] Yuchan Lee
Department of Mathematics, POSTECH, 77, Cheongam-ro, Nam-gu, Pohang-si, Gyeongsangbuk-do, 37673, KOREA [email protected]

Abstract.

For an arbitrarily given irreducible polynomial $\chi(x)\in\mathbb{Z}[x]$ of degree $n$ with $\mathbb{Q}[x]/(\chi(x))$ totally real, let $N(X,T)$ be the number of $n\times n$ matrices over $\mathbb{Z}$ whose characteristic polynomial is $\chi(x)$ , bounded by a positive number $T$ with respect to a certain norm. In this paper, we will provide an asymptotic formula for $N(X,T)$ as $T\to\infty$ in terms of the orbital integrals of $\mathfrak{gl}_{n}$ . This generalizes the work of A. Eskin, S. Mozes, and N. Shah (1996) which assumed that $\mathbb{Z}[x]/(\chi(x))$ is the ring of integers.

In addition, we will provide an asymptotic formula for $N(X,T)$ , using the orbital integrals of $\mathfrak{gl}_{n}$ , when $\mathbb{Q}$ is generalized to a totally real number field $k$ and when $n$ is a prime number. Here we need a mild restriction on splitness of $\chi(x)$ over $k_{v}$ at $p$ -adic places $v$ of $k$ for $p\leq n$ when $k[x]/(\chi(x))$ is unramified Galois over $k$ .

Our method is based on the strong approximation property with Brauer-Manin obstruction on a variety, the formula for orbital integrals of $\mathfrak{gl}_{n}$ , and the Langlands-Shelstad fundamental lemma for $\mathfrak{sl}_{n}$ .

2020 Mathematics Subject Classification:

MSC11F72, 11G35, 11R37, 11S80, 14F22, 14G12, 20G30, 20G35

The authors are supported by Samsung Science and Technology Foundation under Project Number SSTF-BA2001-04.

1. Introduction

For an algebraic variety defined over $\mathbb{Z}$ , understanding its $\mathbb{Z}$ -points can be viewed as a classical number-theoretic problem, studying solutions of a Diophantine equation. More generally, one can expand objects to a variety $X$ defined over $\mathcal{O}_{k}$ , where $\mathcal{O}_{k}$ is the ring of integers of a number field $k$ . In this context, a Diophantine analysis investigating the distribution of $X(\mathcal{O}_{k})$ has been thoroughly studied for $k=\mathbb{Q}$ ([Bir62], [BR95], [DRS93], and [Sch85]). Here the distribution of $X(\mathcal{O}_{k})$ means the asymptotic behavior of $N(X,T)$ as $T\rightarrow\infty$ where

N(X,T)=\#\{x\in X(\mathcal{O}_{k})\mid\lVert x\rVert\leq T\}

with a certain norm $\lVert\cdot\rVert$ defined on $X(k)$ .

We concentrate on a variety $X$ which represents the set of $n\times n$ matrices whose characteristic polynomial is $\chi(x)$ , where $\chi(x)\in\mathcal{O}_{k}[x]$ is an irreducible monic polynomial of degree $n$ . We aim to formulate the constant $C\in\mathbb{R}$ and $r\in\mathbb{Z}$ such that $\frac{N(X,T)}{CT^{r}}\rightarrow 1$ as $T\rightarrow\infty$ , with respect to the norm given by

(1.1)

\lVert x\rVert=\max_{v\in\infty_{k}}\sqrt{\sum_{i,j=1}^{n}|x_{ij}|_{v}^{2}},

where $x$ embeds to $(x_{ij})\in\mathrm{M}_{n}(k)$ and $\infty_{k}$ denotes the set of Archimedean places of $k$ .

This is the case investigated in [EMS96, Theorem 1.1 and Theorem 1.16] under a certain restriction. More precisely, they established the following formula

(1.2)

N(X,T)\sim\frac{2^{n-1}h_{K}R_{K}w_{n}}{\sqrt{\Delta_{\chi}}\cdot\prod_{k=2}^{n}\Lambda(k/2)}T^{\frac{n(n-1)}{2}},

under the restrictions

(1.3)

\left\{\begin{array}[]{l}\textit{(1) $k=\mathbb{Q}$};\\ \textit{(2) $\mathbb{Z}[x]/(\chi(x))$ is the ring of integers of $\mathbb{Q}[x]/(\chi(x))$};\\ \textit{(3) $\mathbb{Q}[x]/(\chi(x))$ is totally real, equivalently $\chi(x)$ splits completely over $\mathbb{R}$}.\end{array}\right.

Here, for two functions $A(T)$ and $B(T)$ for $T$ , we denote $A(T)\sim B(T)$ if $\lim\limits_{T\rightarrow\infty}\frac{A(T)}{B(T)}=1$ . $h_{K}$ is the class number of $\mathcal{O}_{K}$ and $R_{K}$ is the regulator of $K$ , where $K=\mathbb{Q}[x]/(\chi(x))$ and $\mathcal{O}_{K}$ is its ring of integers. $\Delta_{\chi}$ is the discriminant of $\chi(x)$ , $w_{n}$ is the volume of the unit ball in $\mathbb{R}^{\frac{n(n-1)}{2}}$ , and $\Lambda(s)=\pi^{-s}\Gamma(s)\zeta(2s)$ .

The main goal of this paper is to weaken the first two restrictions in (1.3). More precisely, we will describe the asymptotic formula for $N(X,T)$ in Theorem 7.1 (which is summarized in Theorem 1.5) using orbital integrals of $\mathfrak{gl}_{n}$ , under the following restrictions

(1.4)

\left\{\begin{array}[]{l}\textit{(1) $k$ and $K\left(:=k[x]/(\chi(x))\right)$ are totally real number fields};\\ \textit{(2) if $k\neq\mathbb{Q}$, then $\chi(x)$ is of prime degree $n$};\\ \textit{(3) if $K/k$ is unramified Galois, then $\chi(x)$ splits over $k_{v}$ at $p$-adic places $v$ of $k$, for $p\leq n$}.\end{array}\right.

This largely generalizes (1.3); if $k=\mathbb{Q}$ , then the above conditions (2) and (3) are unnecessary so that we do not need the condition (2) of (1.3) (cf. Remark 7.2).

In contrast to [EMS96], our method is based on the strong approximation property with Brauer-Manin obstruction suggested in [WX16]. We will also use the formula for orbital integrals of $\mathfrak{gl}_{n}$ (cf. [CKL] and [CHL]) and the Langlands-Shelstad fundamental lemma for $\mathfrak{sl}_{n}$ (cf. [Ngfrm[o]–0, Theorem 1]).

1.1. Backgrounds

One of the important observations to understand the asymptotic behavior of $N(X,T)$ for $k=\mathbb{Q}$ is introduced in [BR95], which is called Hardy-Littlewood expectation. M. Borovoi and Z. Rudnick proved in [BR95, Theorem 5.3] that a particular homogeneous space of a semisimple group, including our case when $k=\mathbb{Q}$ , can be represented by the integration of a certain function defined on the adelic points of $X$ . However, their description of the integrand is complicated to compute directly (cf. [BR95, Section 3.5, Theorem 5.3]).

In [WX16], D. Wei and F. Xu generalized the observation of M. Borovoi and Z. Rudnick for $X$ defined over any number field $k$ . They also gave a more handleable description of the Hardy-Littlewood expectation to approximate $N(X,T)$ . They mainly used the fact that $X$ satisfies the strong approximation property with Brauer-Manin obstruction following [BD13, Theorem 0.1]. This implies that the integral points of $X$ are related to the Brauer elements (cf. [LX15, Theorem 2.10]). Combining this observation with the equidistribution property in [BO12, Theorem 1.5], they obtained Proposition 1.1.

To introduce the results in [WX16], we define the following notations;

\left\{\begin{array}[]{l}\textit{$\Omega_{k}$ is the set of places of $k$, and $\infty_{k}$ is the set of Archimedean places of $k$};\\ \textit{$k_{v}$ is a completion with respect to the place $v$};\\ \textit{$\mathcal{O}_{k_{v}}$ is its ring of integers and $\pi_{v}$ is a uniformizer of $\mathcal{O}_{k_{v}}$ for $v\in\Omega_{k}\setminus\infty_{k}$};\\ \textit{$\kappa_{v}$ is the residue field of $\mathcal{O}_{k_{v}}$ and $q_{v}=\#\kappa_{v}$ for $v\in\Omega_{k}\setminus\infty_{k}$}.\end{array}\right.

Proposition 1.1.

[WX16, Theorem 4.3] Let $\chi(x)$ be an irreducible monic polynomial over $\mathcal{O}_{k}$ . Then

(1.5)

N(X,T)\sim\lvert\Delta_{k}\rvert^{-\frac{1}{2}\dim X}\sum_{\xi\in\operatorname{Br}X/\operatorname{Br}k}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\int_{X(\mathcal{O}_{k_{v}})}\xi_{v}(x)\ \lvert\omega_{X}\rvert_{v}\prod_{v\in\infty_{k}}\int_{X(k_{v},T)}\xi_{v}(x)\ \lvert\omega_{X}\rvert_{v},

where

\left\{\begin{array}[]{l}X(k_{v},T)=\{x\in X(k_{v})\mid\lVert x\rVert_{v}\leq T\}\textit{ for $v\in\infty_{k}$ and $T>0$};\\ \lvert\omega_{X}\rvert_{v}\textit{ is the measure defined in Section \ref{subsec:measureonX} for $v\in\Omega_{k}$};\\ \Delta_{k}\textit{ is the absolute discriminant of $k$};\\ \textit{$\operatorname{Br}X/\operatorname{Br}k$ is the quotient group defined in Remark \ref{rem:normalized_eval}};\\ \xi_{v}(x)\textit{ denotes a Brauer evaluation on $x\in X(k_{v})$ which is defined in Definition \ref{def:brauer_evaluation}}.\end{array}\right.

As a special case of Proposition 1.1, they also deduced the same results with (1.2) under the assumptions in (1.3). They observed that a non-trivial element in $\operatorname{Br}X/\operatorname{Br}k$ does not affect the summation in (1.5). Indeed, $\int_{X(\mathbb{Z}_{p})}\xi_{p}(x)\ \lvert\omega_{X}\rvert_{p}$ vanishes for a ramified prime number $p$ , which always exists by the assumption $k=\mathbb{Q}$ . After that, they computed the local integration for a trivial element in $\operatorname{Br}X/\operatorname{Br}k$ under the assumption that $\mathbb{Z}[x]/(\chi(x))$ is the ring of integers of $\mathbb{Q}[x]/(\chi(x))$ .

However, this method is not applicable for general number fields since the ramified place might not exist for a number field other than $\mathbb{Q}$ (cf. [WX16, Example 6.3]). In addition, the computation of the local integration is quite a challenging problem unless $\mathbb{Z}[x]/(\chi(x))$ is the ring of integers of $\mathbb{Q}[x]/(\chi(x))$ (cf. Proposition 5.9).

1.2. Main results

Our strategy to obtain the asymptotic formula for $N(X,T)$ is based on Proposition 1.1. We first figure out the evaluation of $\xi\in\operatorname{Br}X$ in (1.5), and then investigate each local integration of $\xi_{v}$ . According to Section 2, our method depends on the $\mathrm{GL}_{n}$ -homogeneous space (resp. $\mathrm{SL}_{n}$ -homogeneous space) structure of $X$ . By Proposition 2.1, we fix $x_{0}\in X(\mathcal{O}_{k})$ and thus we have

\mathrm{T}_{x_{0}}\backslash\mathrm{GL_{n}}\xrightarrow{\sim}X\ (resp.\ \mathrm{S}_{x_{0}}\backslash\mathrm{SL}_{n}\xrightarrow{\sim}X),\ g\mapsto g^{-1}x_{0}g,

where $\mathrm{T}_{x_{0}}$ (resp. $\mathrm{S}_{x_{0}}$ ) denotes the stabilizer of $x_{0}$ under the $\mathrm{GL}_{n}$ -action (resp. the $\mathrm{SL}_{n}$ -action) on $X$ .

1.2.1. Brauer evaluation of $\xi\in\operatorname{Br}X$ on $X(k_{v})$

The Brauer evaluation of $\xi\in\operatorname{Br}X$ on local points $X(k_{v})$ is defined by using the functoriality of the Brauer group in Definition 4.3. We interpret this functoriality through the local class field theory and obtain Proposition 1.2. To establish the well-definedness of the local evaluation of $\xi\in\operatorname{Br}X$ , we consider the normalized evaluation $\tilde{\xi}_{v}$ (cf. Definition 4.5). In Remark 4.4, we show that this normalization does not affect the product in (1.5).

Proposition 1.2.

Suppose that $\chi(x)$ is of prime degree. For $\xi\in\operatorname{Br}X$ , the normalized evaluation $\tilde{\xi}_{v}$ is formulated as follows.

(1)

(Proposition 5.1) If $K/k$ is not Galois, then $\tilde{\xi}_{v}(x)=1$ .
(2)

(Proposition 5.5) If $K/k$ is Galois and $K_{v}/k_{v}$ is not a field extension, then $\tilde{\xi}_{v}(x)=1$ .

(3)

(Proposition 5.2) If $K/k$ is Galois and $K_{v}/k_{v}$ is a field extension, then

\tilde{\xi}_{v}(x)=\exp 2\pi i\Psi(\xi)(\phi_{K_{v}/k_{v}}(\det g_{x})),

where

\left\{\begin{array}[]{l}\textit{$\Psi:\operatorname{Br}X/\operatorname{Br}k\xrightarrow{\sim}\operatorname{Hom}(\operatorname{Gal}(K/k),\mathbb{Q}/\mathbb{Z})$ is an isomorphism defined in Proposition \ref{prop:evaluation}};\\ \textit{$g_{x}\in\mathrm{GL}_{n}(k_{v})$ such that $g_{x}^{-1}x_{0}g_{x}=x$};\\ \textit{$\phi_{K_{v}/k_{v}}:k_{v}^{\times}/\operatorname{Nm}_{K_{v}/k_{v}}K_{v}^{\times}\xrightarrow{\sim}\operatorname{Gal}(K_{v}/k_{v})$ is the Artin reciprocity isomorphism }\\ (\textit{cf. \cite[cite]{[\@@bibref{}{Ser}{}{}, Chapter XI]}}).\end{array}\right.

1.2.2. Computation of each local integration

To investigate the integration of $\tilde{\xi}_{v}$ for each $v\in\Omega_{k}$ , we will use the more handlealbe measure $|\omega_{X_{k_{v}}}^{can}|_{v}$ defined in Section 3.2.2. Here, in the equation (1.5), Proposition 3.9 enables us to interchange the measure $|\omega_{X}|_{v}$ with $|\omega_{X_{k_{v}}}^{can}|_{v}$ .

In the case of an Archimedean place, it is enough to consider the case that $\tilde{\xi}_{v}$ is trivial by the assumption (1) in (1.4). We formulate the integration as the following proposition by using the Iwasawa decomposition on $\mathrm{GL}_{n}(\mathbb{R})$ .

Proposition 1.3.

(Proposition 5.6) Suppose that $k$ and $K$ are totally real number fields. Then we have

\prod_{v\in\infty_{k}}\int_{X(k_{v},T)}\tilde{\xi}_{v}(x)\ \lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}\sim\left(w_{n}\pi^{\frac{n(n+1)}{4}}T^{\frac{n(n-1)}{2}}\prod_{i=1}^{n}\Gamma(\frac{i}{2})^{-1}\right)^{[k:\mathbb{Q}]}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\lvert\Delta_{\chi}\rvert_{v}^{\frac{1}{2}},

where $w_{n}$ is the volume of the unit ball in $\mathbb{R}^{\frac{n(n-1)}{2}}$ and $\Delta_{\chi}$ is the discriminant of $\chi(x)$ .

In the case of a non-Archimedean place, Proposition 1.2 implies that $\tilde{\xi}_{v}$ is non-trivial only if $K/k$ is Galois and $K_{v}$ is a Galois field extension over $k_{v}$ . In this case, we address the following cases separately; when $K_{v}/k_{v}$ is ramified and when $K_{v}/k_{v}$ is unramified.

Proposition 1.4.

Suppose $v\in\Omega_{k}\setminus\infty_{k}$ .

(1)

(Proposition 5.9) If $\tilde{\xi}_{v}$ is trivial, then we have

\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ |\omega_{X_{k_{v}}}^{can}|_{v}=\int_{\mathrm{T}_{x_{0},k_{v}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v})}\mathbbm{1}_{\mathfrak{gl}_{n}(\mathcal{O}_{k_{v}})}(g^{-1}x_{0}g)\ |\omega_{X_{k_{v}}}^{can}|_{v}.

Here, the right hand side is the orbital integral of $x_{0}$ for $\mathfrak{gl}_{n}$ with respect to the measure $|\omega_{X_{k_{v}}}^{can}|_{v}$ .

(2)

If $\tilde{\xi}_{v}$ is non-trivial (and so $K_{v}/k_{v}$ is a Galois field extension), then we have the following equations.

•

(Proposition 5.11) If $K_{v}/k_{v}$ is a ramified Galois extension, then we have

$\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ \lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}=0.$

•

(Proposition 6.8) If $K_{v}/k_{v}$ is an unramified Galois extension and the characteristic of the residue field of $k_{v}$ is bigger than $n$ , then we have

\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ |\omega_{X_{k_{v}}}^{can}|_{v}=\lvert\Delta_{\chi}\rvert_{v}^{-\frac{1}{2}}\frac{\#\mathrm{SL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{n^{2}-1}}{\#\mathrm{S}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{n-1}},

where $\mathrm{S}_{x_{0},\mathcal{O}_{k_{v}}}$ , is an integral model of the stabilizer $\mathrm{S}_{x_{0}}$ of $x_{0}$ in $\mathrm{SL}_{n,k_{v}}$ , defined in Section 6.1.1.

When $\tilde{\xi}_{v}$ is trivial, the formula directly follows that of the orbital integral for $\mathfrak{gl}_{n}$ . Although the orbital integral, which appears in the geometric side of the Arthur-Selberg trace formula, has been studied extensively in the literature, to obtain its closed formula is quite involved.

In this paper, for the closed formula of the orbital integral for $\mathfrak{gl}_{n}$ , we will refer to the results in [CKL] for the case that $n=2,3$ and the results in [CHL] for the case that $\mathcal{O}_{k}[x]/(\chi(x))$ is a Bass order. Here a Bass order is an order of a number field whose fractional ideals are generated by two elements (cf. [CHL, Definition 6.1]). We note that majority of number fields contain infinitely many Bass orders. For example, any order of a number field which contains the maximal order of a subfield with degree 2 or whose discriminant is fourth-power-free in $\mathbb{Z}$ , is a Bass order.

In the case that $\tilde{\xi}_{v}$ is non-trivial and $K_{v}/k_{v}$ is an unramified Galois extension, we use the Langlands-Shelstad fundamental lemma in [Ngfrm[o]–0, Theorem 1]. This is the reason why we need the assumption (3) in (1.4), involving the characteristic of $\kappa_{v}$ .

1.2.3. Conclusion

Combining Propositions 1.2-1.4, we conclude our main theorem.

Theorem 1.5 (Theorem 7.1).

Let $k$ be a number field and $\chi(x)\in\mathcal{O}_{k}[x]$ be an irreducible monic polynomial of degree $n$ . Let $X$ be an $\mathcal{O}_{k}$ -scheme representing the set of $n\times n$ matrices whose characteristic polynomial is $\chi(x)$ . We define

N(X,T)=\#\{x\in X(\mathcal{O}_{k})\mid\lVert x\rVert\leq T\},

for $T>0$ , where the norm $\lVert\cdot\rVert$ is defined in (1.1).

Suppose that $k$ and $K=k[x]/(\chi(x))$ are totally real, and if $k\neq\mathbb{Q}$ , we further assume that $n$ is a prime number. Then we have the following asymptotic formulas.

(1)

If $K/k$ is not Galois or ramified Galois, then

$N(X,T)\sim C_{T}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{\mathcal{SO}_{v}(\chi)}{q_{v}^{S_{v}(\chi)}}.$

(2)

If $K/k$ is unramified Galois and splits over all $p$ -adic places for $p\leq n$ , then

N(X,T)\sim C_{T}\left(\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{\mathcal{SO}_{v}(\chi)}{q_{v}^{S_{v}(\chi)}}+n-1\right).

Here, $C_{T}$ is formulated as follows

C_{T}=\lvert\Delta_{k}\rvert^{\frac{-n^{2}+n}{2}}\frac{R_{K}h_{K}\sqrt{\lvert\Delta_{K}\rvert}^{-1}}{R_{k}h_{k}\sqrt{\lvert\Delta_{k}\rvert}^{-1}}\prod_{i=2}^{n}\zeta_{k}(i)^{-1}\left(\frac{2^{n-1}\pi^{\frac{n(n+1)}{4}}w_{n}}{\prod_{i=1}^{n}\Gamma(\frac{i}{2})}T^{\frac{n(n-1)}{2}}\right)^{[k:\mathbb{Q}]},

and we use the following notations

\left\{\begin{array}[]{l}\textit{$R_{F}$ is the regulator of $F$ for $F=k$ or $K$};\\ \textit{$h_{F}$ is the class number of $\mathcal{O}_{F}$ for $F=k$ or $K$};\\ \textit{$\Delta_{F}$ is the discriminant of $F/\mathbb{Q}$ for $F=k$ or $K$};\\ \textit{$w_{n}$ is the volume of the unit ball in $\mathbb{R}^{\frac{n(n-1)}{2}}$};\\ \textit{$\zeta_{k}$ is the dedekind zeta function of $k$};\\ \textit{$S_{v}(\chi)$ is the $\mathcal{O}_{k_{v}}$-module length between $\mathcal{O}_{K_{v}}$ and $\mathcal{O}_{k_{v}}[x]/(\chi(x))$};\\ \textit{$\mathcal{SO}_{v}(\chi)$ is the orbital integral for $\mathfrak{gl}_{n,k_{v}}$ associated with $\chi(x)$ with respect to $\frac{dg_{v}}{dt_{v}}$}\\ \textit{(cf. Proposition \ref{prop:result_case:trivial} and Lemma \ref{cor:compareclassic})}.\end{array}\right.

In Proposition 7.3 (respectively, Proposition 7.4), we will provide a closed formula for the product $\prod_{v\in\Omega_{k}\setminus\infty_{k}}\mathcal{SO}_{v}(\chi)$ when $n=2$ or $n=3$ (respectively, when $\mathcal{O}_{k}[x]/(\chi(x))$ is a Bass order).

Remark 1.6.

According to [Yun13, Theorem 1.5], $\mathcal{SO}_{v}(\chi)$ is an integer, which is a $\mathbb{Z}$ -polynomial in $q_{v}$ , whose leading term is $q_{v}^{S_{v}(\chi)}$ . Since $S_{v}(\chi)=0$ for all but finitely many places $v\in\Omega_{k}\setminus\infty_{k}$ , the product $\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{\mathcal{SO}_{v}(\chi)}{q_{v}^{S_{v}(\chi)}}$ is a finite product.

Remark 1.7.

In this remark, we explain how the assumptions in (1.4) affect the main steps of our argument.

(1)

Since we assume that $k$ is totally real, for the integration on $X(k_{v},T)$ where $v\in\infty_{k}$ , we can apply the theory of $\mathbb{R}$ -manifolds (cf. Propsition 5.6-5.8).

On the other hand, by virtue of the assumption that $K$ is totally real, Proposition 5.5 yields that the Brauer evaluations for Archimedean places are trivial. Moreover, this condition enables us to describe the stabilizer $\mathrm{T}_{x_{0},k_{v}}$ , for $v\in\infty_{k}$ , in Section 3.2.2. This description affects the computations in Proposition 5.2.
(2)

The assumption (2) in (1.4) implies that the Brauer evaluation is non-trivial only if $K/k$ is an abelian extension, as explained in Proposition 5.1, which enables us to use Proposion 5.2.

This assumption also facilitates figuring out an endoscopic group associated with a local integration for $k_{v}$ , in Lemma 6.6, when $K_{v}/k_{v}$ is unramified.
(3)

As we explained in Section 1.2.2, we assumed (3) in (1.4) in order to use the Langlands-Shelstad fundamental fundamental lemma for the Lie algebra $\mathfrak{sl}_{n}$ of $\mathrm{SL}_{n}$ over $k_{v}$ , where $K_{v}/k_{v}$ is unramified.

Organizations. We organize this paper as follows. In Section 2, we observe the structure of $X$ as a homogeneous space and investigate the related objects. In Section 3, we introduce the Tamagawa measures on homogeneous spaces and the canonical measure on $X$ which is used in our local calculations. To apply Proposition 1.3, we define the notion of Brauer evaluation in Section 4. The description of Brauer evaluation and the computation of local integrations are in Section 5-6. In Section 7, we provide the asymptotic formula for $N(X,T)$ , in terms of the orbital integrals of $\mathfrak{gl}_{n}$ , and we refer to the closed formula of orbital integrals of $\mathfrak{gl}_{n}$ in [CKL] and [CHL] for certain cases.

Acknowledgments. We would like to thank our advisor Sungmun Cho for suggesting this problem and encouraging us, and Sug Woo Shin for helpful comments on Section 6. We also thank Seewoo Lee for meaningful discussions on the exterior product of differential forms and Fei Xu for his lecture note on Brauer-Manin obstruction, which was given at POSTECH in 2019.

1.3. Notations

•

Let $k$ be a number field with ring of integers $\mathcal{O}_{k}$ , and $\bar{k}$ be an algebraic closure of $k$ .

•

Suppose that $A$ is a commutative $\mathcal{O}_{k}$ -algebra. We define the following schemes over $A$ .

\left\{\begin{array}[]{l}\textit{$\mathrm{GL}_{n,A}:$ the general linear group scheme over $A$, $\mathfrak{gl}_{n,A}:$ the Lie algebra of $\mathrm{GL}_{n,A}$};\\ \textit{$\mathrm{SL}_{n,A}:$ the special linear group scheme over $A$, $\mathfrak{sl}_{n,A}:$ the Lie algebra of $\mathrm{SL}_{n,A}$};\\ \textit{$\mathrm{M}_{n,A}:$ the scheme over $A$ representing the set of $n\times n$ matrices.}\end{array}\right.

If there is no confusion, we sometimes omit $A$ in the subscript to express schemes over $k$ .

•

Let $\chi(x)\in\mathcal{O}_{k}[x]$ be an irreducible monic polynomial of degree $n$ . We define $X$ to be the closed subscheme of $\mathrm{M}_{n,\mathcal{O}_{k}}$ representing the set of $n\times n$ matrices whose characteristic polynomial is $\chi(x)$ .
•

Let $\Omega_{k}$ be the set of places and $\infty_{k}$ be the set of Archimedean places. For $v\in\Omega_{k}$ , we denote the normalized absolute value associated with $v$ by $|\cdot|_{v}$ and the completion with respect to $|\cdot|_{v}$ by $k_{v}$ .

•

For $v\in\Omega_{k}\setminus\infty_{k}$ , we define the following notations

\left\{\begin{array}[]{l}\mathcal{O}_{k_{v}}:\textit{ the ring of integers of $k_{v}$};\\ \pi_{v}:\textit{ a uniformizer of $\mathcal{O}_{k_{v}}$};\\ \kappa_{v}:\textit{ the residue field of $\mathcal{O}_{k_{v}}$};\\ q_{v}:\textit{ the cardinality of the residue field $\kappa_{v}$}.\end{array}\right.

For an element $x\in k_{v}$ , the exponential order of $x$ with respect to the maximal ideal in $\mathcal{O}_{k_{v}}$ is written by $\operatorname{ord}_{v}(x)$ . We then have that $|x|_{v}=q_{v}^{-\operatorname{ord}_{v}(x)}$ .

•

We define $K=k[x]/(\chi(x))$ with ring of integers $\mathcal{O}_{K}$ . For $v\in\Omega_{k}$ , we define $K_{v}=k_{v}[x]/(\chi(x))$ with ring of integers $\mathcal{O}_{K_{v}}$ .

•

For $v\in\Omega_{k}$ , let $B_{v}(\chi)$ be an index set in bijection with the irreducible factors $\chi_{v,i}$ of $\chi$ over $k_{v}$ . For $i\in B_{v}(\chi)$ , we define the following notations

\left\{\begin{array}[]{l}\textit{$K_{v,i}=k_{v}[x]/(\chi_{v,i}(x))$ with ring of integers $\mathcal{O}_{K_{v,i}}$};\\ \kappa_{K_{v,i}}:\textit{ the residue field of $\mathcal{O}_{K_{v,i}}$}.\end{array}\right.

Here, we note that $K_{v}\cong\prod_{i\in B_{v}(\chi)}K_{v,i}$ and $\mathcal{O}_{K_{v}}\cong\prod_{i\in B_{v}(\chi)}\mathcal{O}_{K_{v,i}}$ .

•
For a finite field extension $E/F$ where $F$ is $\mathbb{Q}$ or $k_{v}$ , let $\Delta_{E/F}$ be the discriminant ideal of $E/F$ .
- –
  
  If $F=\mathbb{Q}$ , then we denote $\Delta_{E/\mathbb{Q}}$ by $\Delta_{E}$ , and $\lvert\Delta_{E}\rvert$ by the absolute value of a genearator of $\Delta_{E}$ as an ideal in $\mathbb{Z}$ .
- –
  
  If $F=k_{v}$ , then $\operatorname{ord}_{v}(\Delta_{E/k_{v}})$ (resp. $|\Delta_{E/k_{v}}|_{v}$ ) denotes the exponential order (resp. the normalized absolute value) of a generator of $\Delta_{E/k_{v}}$ as an ideal in $\mathcal{O}_{k_{v}}$ .
•

For a field $F$ , let $\Delta_{f}\in F$ be the discriminant of a polynomial $f(x)\in F[x]$ .

•

We define norms for $x\in X(k)$ by

\lVert x\rVert_{v}=\sqrt{\sum_{i,j=1}^{n}|x_{ij}|_{v}^{2}}\ \ \text{ and }\ \lVert x\rVert=\max_{v\in\infty_{k}}\lVert x\rVert_{v},

where $x$ embeds to $(x_{ij})\in\mathrm{M}_{n}(k)$ . For $T>0$ , we define

N(X,T)=\#\{x\in X(\mathcal{O}_{k})\mid\lVert x\rVert\leq T\}.

We note that $N(X,T)$ is finite since $\mathcal{O}_{k}$ is discrete in $k_{v}$ for each Archimendian place $v\in\infty_{k}$ .

•

For two functions $A(T)$ and $B(T)$ for $T$ , we denote $A(T)\sim B(T)$ if $\lim\limits_{T\rightarrow\infty}\frac{A(T)}{B(T)}=1$ .

2. Homogeneous space $X$

2.1. Homogeneous space

In this section, we observe that $X$ is a homogeneous space of both $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ . We write $\chi(x)=x^{n}+\sum_{i=0}^{n-1}a_{i}x^{i}$ where $a_{i}\in\mathcal{O}_{k}$ for $0\leq i\leq n-1$ . To regard $X$ as a homogeneous space of $\mathrm{GL}_{n}$ (resp. $\mathrm{SL}_{n}$ ), we fix an integral matrix $x_{0}\in X(\mathcal{O}_{k})$ whose characteristic polynomial is $\chi(x)$ . For example, we can choose $x_{0}$ as the companion matrix of $\chi(x)$ . We define a map

(2.1)

\varphi_{\mathrm{GL}_{n}}:\mathrm{GL}_{n}\rightarrow X\ (\text{resp. }\varphi_{\mathrm{SL}_{n}}:\mathrm{SL}_{n}\rightarrow X),\ g\mapsto g^{-1}x_{0}g.

We define a right $\mathrm{GL}_{n}$ -action (resp. $\mathrm{SL}_{n}$ -action) on $X$ by the conjugation $x\cdot g=g^{-1}xg$ , so that $\varphi_{\mathrm{GL}_{n}}$ (resp. $\varphi_{\mathrm{SL}_{n}}$ ) is an equivariant map under the right action of $\mathrm{GL}_{n}$ (resp. $\mathrm{SL}_{n}$ ).

Proposition 2.1.

(1)

$X$ is a homogeneous space of $\mathrm{GL}_{n}$ (resp. $\mathrm{SL}_{n}$ ) with respect to conjugation by $\mathrm{GL}_{n}$ (resp. $\mathrm{SL}_{n}$ ) and $\varphi_{\mathrm{GL}_{n}}$ (resp. $\varphi_{\mathrm{SL}_{n}}$ ) induces an following isomorphism

$X\cong\mathrm{T}_{x_{0}}\backslash\mathrm{GL_{n}}\ (resp.\ X\cong\mathrm{S}_{x_{0}}\backslash\mathrm{SL}_{n}),$

where $\mathrm{T}_{x_{0}}$ (resp. $\mathrm{S}_{x_{0}}$ ) denotes the stabilizer of $x_{0}$ under the $\mathrm{GL}_{n}$ -action (resp. the $\mathrm{SL}_{n}$ -action) on $X$ .

(2)

We have the following isomorphism for the stabilizer of $x_{0}$

\mathrm{T}_{x_{0}}\cong\mathrm{R}_{K/k}(\mathbb{G}_{m,K})\textit{ $($resp. $\mathrm{S}_{x_{0}}\cong\mathrm{R}^{(1)}_{K/k}(\mathbb{G}_{m,K}))$},

where we use the following notations:

\left\{\begin{array}[]{l}\textit{$\mathrm{R}_{K/k}(\mathbb{G}_{m,K})$ is the Weil restriction of $\mathbb{G}_{m,K}$ and};\\ \textit{$\mathrm{R}^{(1)}_{K/k}(\mathbb{G}_{m,K}):=\ker(\operatorname{Nm}_{K/k}:\mathrm{R}_{K/k}(\mathbb{G}_{m,K})\to\mathbb{G}_{m,k})$ is the norm-1 torus of $K/k$}.\end{array}\right.

Proof.

Since every square matrix over a field has a rational canonical form determined by its characteristic polynomial, for $x\in X(\bar{k})$ , there exists $g\in\mathrm{GL}_{n}(\bar{k})$ such that $x=g^{-1}x_{0}g$ . Therefore, $X$ is a homogeneous space of $\mathrm{GL}_{n}$ . Similarly, we can show that $X$ is also a homogeneous space of $\mathrm{SL}_{n}$ by choosing $h=\frac{g}{\sqrt[n]{\det g}}\in\mathrm{SL}_{n}(\bar{k})$ . We then obtain the following identifications induced by $\varphi_{\mathrm{GL}_{n}}$ and $\varphi_{\mathrm{SL}_{n}}$ defined in (2.1),

X\cong\mathrm{T}_{x_{0}}\backslash\mathrm{GL}_{n}\cong\mathrm{S}_{x_{0}}\backslash\mathrm{SL}_{n}.

We now prove the second statement. Since $x_{0}$ has a rational canonical form, there exists an invertible matrix $Q\in\mathrm{GL}_{n}(k)$ such that

(2.2)

x_{0}=Q^{-1}\begin{pmatrix}0&\cdots&0&-a_{0}\\ 1&\cdots&0&-a_{1}\\ \vdots&\ddots&\vdots&\vdots\\ 0&\cdots&1&-a_{n-1}\\ \end{pmatrix}Q.

For each $1\leq i\leq n$ , we define $f_{i}\in K=k[x]/(\chi(x))$ by

f_{i}=\sum_{k=1}^{n}Q_{ki}x^{k-1}.

Since $\{1,x,\ldots,x^{n-1}\}$ is a $k$ -basis of $K$ , $\{f_{1},f_{2},\ldots,f_{n}\}$ is also a basis.

For each $k$ -algebra $R$ , we now construct an explicit bijection $\mathrm{T}_{x_{0}}(R)\cong\mathrm{R}_{K/k}(\mathbb{G}_{m,K})(R)$ . Let $V$ be a free $R$ -module $R[x]/(\chi(x))$ . With respect to the basis $\{f_{1},\ldots,f_{n}\}$ of $V$ , we have the following identification

\Phi:\mathrm{M}_{n}(R)\xrightarrow{\sim}\mathrm{End}_{R}(V).

Since the companion matrix in (2.2) represents the left-multiplication by $x$ with respect to the basis $\{1,x,\ldots,x^{n-1}\}$ , we have that $\Phi(x_{0})$ equals to $(f(x)\mapsto xf(x))\in\operatorname{End}_{R}(V)$ by the change of basis.

For $g\in\mathrm{M}_{n}(R)$ , $gx_{0}=x_{0}g$ in $\mathrm{M}_{n}(R)$ if and only if $\Phi(g)(xv)=x\Phi(g)(v)$ for each $v\in V$ . We then identify the stabilizer $\mathrm{T}_{x_{0}}(R)\subset\mathrm{M}_{n}(R)$ with the set of $R$ -linear automorphisms on $V$ that commute with $x$ , under the isomorphism $\Phi$ . By the $R$ -linearity, we also have

(2.3)

\mathrm{T}_{x_{0}}(R)=\{g\in\mathrm{M}_{n}(R)\mid g\text{ is invertible and }\Phi(g)(v_{1}v_{2})=v_{1}\Phi(g)(v_{2})\text{ for all }v_{1},v_{2}\in V\}.

We now define the following $R$ -linear map

\Phi_{R}:\mathrm{T}_{x_{0}}(R)\to(R[x]/(\chi(x)))^{\times},\ g\mapsto\Phi(g)(1_{R}),

where $1_{R}$ is the identity element in $R$ . The image of $\Phi_{R}$ is contained in $(R[x]/(\chi(x)))^{\times}$ since, for $g\in\mathrm{T}_{x_{0}}(R)$ ,

\Phi_{R}(g)\Phi_{R}(g^{-1})=\Phi(g)(\Phi(g)^{-1}(1_{R}))=1_{R}.

We assert that $\Phi_{R}$ is a bijection by constructing the inverse. For $v\in(R[x]/(\chi(x)))^{\times}$ , let $g_{v}\in\mathrm{M}_{n}(R)$ be a matrix whose $i$ -th column consists of the coefficients of $f_{i}v\in V$ with respect to the basis $\{f_{1},\ldots,f_{n}\}$ . Then we have $\Phi(g_{v})(f)=fv$ for all $f\in V$ . Since $v$ is invertible, $g_{v}$ is contained in $\mathrm{T}_{x_{0}}(R)$ using the identification (2.3). Thus, $(R[x]/(\chi(x)))^{\times}\to\mathrm{T}_{x_{0}}(R),\ v\mapsto g_{v}$ defines an inverse of $\Phi_{R}$ . One can deduce that $\Phi_{R}$ is functorial in $R$ and this concludes that $\mathrm{T}_{x_{0}}\cong\mathrm{R}_{K/k}(\mathbb{G}_{m,K})$ as $k$ -schemes.

Moreover, since $\mathrm{Nm}_{K/k}:\mathrm{R}_{K/k}(\mathbb{G}_{m,K})\to\mathbb{G}_{m,k}$ commutes with $\det:\mathrm{GL}_{n}\to\mathbb{G}_{m,k}$ , it follows that $\mathrm{S}_{x_{0}}\cong\ker\mathrm{Nm}_{K/k}=\mathrm{R}^{(1)}_{K/k}(\mathbb{G}_{m,K})$ . ∎

2.2. Conjugacy class of $\mathrm{GL}_{n}(k_{v})$ and $\mathrm{SL}_{n}(k_{v})$ in $X(k_{v})$

In this subsection, we will recall useful properties of a homogeneous space $X$ . In the previous subsection, we proved that $X$ over $k$ is identified as follows.

X\cong\mathrm{T}_{x_{0}}\backslash\mathrm{GL}_{n}\cong\mathrm{S}_{x_{0}}\backslash\mathrm{SL}_{n}.

By [EvdGM, 4.30], the homogeneous space structure is preserved under base change and thus we have

X_{k_{v}}\cong\mathrm{T}_{x_{0},k_{v}}\backslash\mathrm{GL}_{n,k_{v}}\cong\mathrm{S}_{x_{0},k_{v}}\backslash\mathrm{SL}_{n,k_{v}},

for any place $v\in\Omega_{k}$ . The structures of $\mathrm{T}_{x_{0},k_{v}}$ and $\mathrm{S}_{x_{0},k_{v}}$ are described in the following lemma.

Lemma 2.2.

We have the following isomorphisms

\left\{\begin{array}[]{l}\mathrm{T}_{x_{0},k_{v}}\cong\prod\limits_{i\in B_{v}(\chi)}\mathrm{R}_{K_{v,i}/k_{v}}(\mathbb{G}_{m,K_{v,i}});\\ \mathrm{S}_{x_{0},k_{v}}\cong\ker\Big{(}\prod\limits_{i\in B_{v}(\chi)}\operatorname{Nm}_{K_{v,i}/k_{v}}:\prod\limits_{i\in B_{v}(\chi)}\mathrm{R}_{K_{v,i}/k_{v}}(\mathbb{G}_{m,K_{v,i}})\rightarrow\mathbb{G}_{m,k_{v}}\Big{)},\end{array}\right.

where

\left\{\begin{array}[]{l}\textit{$B_{v}(\chi)$ is an index set in bijection with the irreducible factors $\chi_{i}$ of $\chi$ over $k_{v}$};\\ \textit{$K_{v}=k_{v}[x]/(\chi(x))$ and $K_{v,i}=k_{v}[x]/(\chi_{i}(x))$.}\end{array}\right.

Proof.

The describtion for $\mathrm{T}_{x_{0},k_{v}}$ follows from [Vos98, 3.12]. For the second description, we consider the following exact sequence of algebraic groups

(2.4)

1\to\mathrm{S}_{x_{0},k_{v}}\to\mathrm{T}_{x_{0},k_{v}}\to{\mathbb{G}}_{m,k_{v}}\to 1.

We claim that the norm map for an étale algebra $\prod_{i\in B_{v}(\chi)}K_{v,i}$ over $k_{v}$ is the product of norms for $K_{v,i}$ for $i\in B_{v}(\chi)$ .

We verify the statements for $\mathrm{S}_{x_{0},k_{v}}$ on $A$ -points for each $k_{v}$ -algebra $A$ . We fix an element $x\in A$ and define $\phi:A\otimes_{k}K\to A\otimes_{k}K$ to be the left-multiplication by $x\otimes 1$ . Under the following composition of isomorphisms

A\otimes_{k}K=(A\otimes_{k_{v}}k_{v})\otimes_{k}K\cong A\otimes_{k_{v}}(k_{v}\otimes_{k}K)\cong A\otimes_{k_{v}}\prod_{i\in B_{v}(\chi)}K_{v,i}\cong\prod_{i\in B_{v}(\chi)}A\otimes_{k_{v}}K_{v,i},

the transfer of $\phi$ from $\prod_{i\in B_{v}(\chi)}A\otimes_{k_{v}}K_{v,i}$ to itself is also the left-multiplication by $x\otimes 1$ . We then have

\operatorname{Nm}_{A\otimes_{k}K/A}(x)=\det\phi=\prod_{i\in B_{v}(\chi)}\det\phi_{i}=\prod_{i\in B_{v}(\chi)}\operatorname{Nm}_{A\otimes_{k_{v}}K_{v,i}/A}(x),

where $\phi_{i}:A\otimes_{k_{v}}K_{v,i}\to A\otimes_{k_{v}}K_{v,i}$ is the left-multiplication on $A\otimes_{k_{v}}K_{v,i}$ by $x\otimes 1$ . This proves the claim and concludes the lemma. ∎

We now describe the orbits of $X(k_{v})$ under the $\mathrm{GL}_{n}(k_{v})$ -action and the $\mathrm{SL}_{n}(k_{v})$ -action, respectively.

Proposition 2.3.

There is only one $\mathrm{GL}_{n}(k_{v})$ -orbit in $X(k_{v})$ and thus $X(k_{v})\cong\mathrm{T}_{x_{0}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v})$ .

Proof.

We consider the following exact sequence for the homogeneous space $X_{k_{v}}$ ,

1\to\mathrm{T}_{x_{0},k_{v}}\to\mathrm{GL}_{n,k_{v}}\to X_{k_{v}}\to 1.

This induces the following long exact sequence

\cdots\to\mathrm{GL}_{n}(k_{v})\to X(k_{v})\to\mathrm{H}^{1}(k_{v},\mathrm{T}_{x_{0},k_{v}})\to\cdots.

Using Lemma 2.2, by [Vos98, 3.12] and Hilbert’s Theorem 90, we have

\mathrm{H}^{1}(k_{v},\mathrm{T}_{x_{0},k_{v}})\cong\prod_{i\in B_{v}(\chi)}\mathrm{H}^{1}(k_{v},\mathrm{R}_{K_{v,i}/k_{v}}(\mathbb{G}_{m,K_{v,i}}))\cong\prod_{i\in B_{v}(\chi)}\mathrm{H}^{1}(K_{v,i},\mathbb{G}_{m,K_{v,i}})=1,

where $K\otimes_{k}k_{v}\cong\prod_{i\in B_{v}(\chi)}K_{v,i}$ . ∎

Proposition 2.4.

The set of $\mathrm{SL}_{n}(k_{v})$ -orbits within $X(k_{v})$ is in bijection with $\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ .

Proof.

We consider the following exact sequence for the homogeneous space $X_{k_{v}}$ ,

1\to\mathrm{S}_{x_{0},k_{v}}\to\mathrm{SL}_{n,k_{v}}\to X_{k_{v}}\to 1.

This induces the following long exact sequence

\cdots\to\mathrm{SL}_{n}(k_{v})\to X(k_{v})\to\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})\to\mathrm{H}^{1}(k_{v},\mathrm{SL}_{n,k_{v}})\to\cdots.

Since $\mathrm{H}^{1}(k_{v},\mathrm{SL}_{n,k_{v}})=1$ , it follows that the set of $\mathrm{SL}_{n}(k_{v})$ -orbits is in bijection with $\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ . ∎

Remark 2.5.

If $K_{v}$ is a field extension of $k_{v}$ , then $\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ is isomorphic to $\operatorname{Gal}(K_{v}/k_{v})^{ab}$ . Indeed, the following exact sequence is induced by (2.4),

K_{v}^{\times}\xrightarrow{\operatorname{Nm}_{K_{v}/k_{v}}}k_{v}^{\times}\to\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})\to\mathrm{H}^{1}(k_{v},\mathrm{T}_{x_{0},k_{v}}).

As shown in the proof of Proposition 2.3, we have $\mathrm{H}^{1}(k_{v},\mathrm{T}_{x_{0},k_{v}})=0$ . Thus, $\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ is isomorphic to $k_{v}^{\times}/\operatorname{Nm}_{K_{v}/k_{v}}K_{v}^{\times}$ by (2.4) and this quotient is isomorphic to $\operatorname{Gal}(K_{v}/k_{v})^{ab}$ by [Ser79, Section 3, XI].

Definition 2.6.

We define $\mathcal{R}_{x_{0},v}$ to be the set of representatives of each $\mathrm{SL}_{n}(k_{v})$ -orbits in $X(k_{v})$ . For each $x_{0}^{\prime}\in\mathcal{R}_{x_{0},v}$ , we define $z_{x_{0}^{\prime}}$ to be the element in $\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ corresponding to the orbit $x_{0}^{\prime}\cdot\mathrm{SL}_{n}(k_{v})$ according to Proposition 2.4.

Remark 2.7.

By Proposition 2.4, we can describe $X(k_{v})$ as the following disjoint union.

X(k_{v})=\bigsqcup_{x_{0}^{\prime}\in\mathcal{R}_{x_{0},v}}x_{0}^{\prime}\cdot\mathrm{SL}_{n}(k_{v})\cong\bigsqcup_{x_{0}^{\prime}\in\mathcal{R}_{x_{0},v}}\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v}),

where $\mathrm{S}_{x_{0}^{\prime}}$ is the stabilizer of $x_{0}^{\prime}$ under the $\mathrm{SL}_{n}$ -action.

Remark 2.8.

We remark that each $\mathrm{SL}_{n}(k_{v})$ -orbit is open in $X(k_{v})$ . For each $x\in X(k_{v})$ , we define the equivariant map over $k_{v}$ , $\varphi_{x}:\mathrm{SL}_{n}\to X$ by $g\to x\cdot g$ . By the same argument in Lemma 2.2, one can deduce that the stabilizer $\mathrm{Stab}_{\mathrm{SL}_{n}(k_{v})}(x)$ is isomorphic to $\ker(\prod_{i\in B_{v}(\chi)}\operatorname{Nm}_{K_{v,i}/k_{v}})$ and it is smooth over $k_{v}$ . Thus, $\varphi_{x}$ is also smooth over $k_{v}$ by [EvdGM, Corollary 4.33] and $\varphi_{x}(k_{v})$ is an open map by [Poo17, Proposition 3.5.73]. Therefore, the image of $\mathrm{SL}_{n}(k_{v})$ via $\varphi_{x}$ , that is the $\mathrm{SL}_{n}(k_{v})$ -orbit of $x$ , is open in $X(k_{v})$ .

3. Measures on $X$

In this section, we introduce the Tamagawa measure on $X(\mathbb{A}_{k})$ , which is used to explain the Hardy-Littlewood expectation in Proposition 1.5. The Tamagawa measure is induced from a gauge form on $X$ (cf. Definition 3.1), and so we provide various concepts about volume forms.

In Section 3.1, we define the Tamagawa measure on $V(\mathbb{A}_{k})$ in the general setting: $V$ is a homogeneous space $V\cong H\backslash G$ over $k$ where $G$ and $H$ are assumed to be connected reductive $k$ -groups. In Section 3.2, we return to $X$ in Section 2.1, and describe the Tamgawa measure on $X({\mathbb{A}}_{k})$ using another handleable measure $\lvert\omega_{X}^{can}\rvert_{v}$ , which is defined in Section 3.2.2.

3.1. Tamagawa Measure on a homogeneous space

Let $V\cong H\backslash G$ be a homogeneous space of $G$ where $G$ and $H$ are connected reductive $k$ -groups. To define the Tamagawa measure on $V(\mathbb{A}_{k})$ , we will use a top-degree volume form on $V$ that is "compatible" with non-vanishing invariant top-degree volume forms on $G$ and $H$ (cf. Definition 3.2). In our case where $X\cong\mathrm{T}_{x_{0}}\backslash\mathrm{GL}_{n}\cong\mathrm{S}_{x_{0}}\backslash\mathrm{SL}_{n}$ , we will prove that the Tamagawa measure on $X$ is unique. We mainly follow concepts and backgrounds in [Wei82, Section 2] and [BR95, Section 1].

3.1.1. Gauge forms

Definition 3.1.

For a geometrically irreducible non-singular algebraic variety $V$ over a field $k$ , a gauge form on $V$ is a nowhere-zero and regular differential form in $\bigwedge^{\dim V}\Omega_{V/k}(V)$ where $\Omega_{V/k}$ denotes a cotangent sheaf of $V$ .

From now on, we will simply denote $\bigwedge^{d}\Omega_{V/k}$ by $\Omega_{V/k}^{d}$ . We assume that $V$ has a $k$ -point $x_{0}$ whose stabilizer is $H$ . This gives the canonical map from $G$ into $V$ ,

(3.1)

\varphi_{G}:G\rightarrow V,\ g\mapsto x_{0}\cdot g,

which induces an isomorphism $H\backslash G\cong V$ . Using this identification, we introduce the notion of algebraically matching for gauge forms on homogeneous spaces.

Definition 3.2 ([Wei82, Section 2.4, p24]).

Let $\omega_{G}$ , $\omega_{H}$ , and $\omega_{V}$ be an invariant gauge form on $G$ , an invariant gauge form on $H$ , and a $G$ -invariant gauge form on $V$ , respectively. Let $\varphi_{G}^{*}\omega_{V}$ be the pullback of $\omega_{V}$ along the morphism $\varphi_{G}$ in (3.1), and let a differential form $\widetilde{\omega}_{H}$ on $G$ be a lifting of $\omega_{H}$ such that, for every $g\in G$ , $\widetilde{\omega}_{H}(gh)$ induces on $H$ the form $\omega_{H}(h)$ for $h\in H$ . Then the form $\varphi_{G}^{*}\omega_{V}\wedge\widetilde{\omega}_{H}$ is a gauge form which is independent of the choice of a lifting $\widetilde{\omega}_{H}$ . We say that $\omega_{G}$ , $\omega_{H}$ , and $\omega_{V}$ match together algebraically, if $\omega_{G}=\varphi_{G}^{*}\omega_{V}\wedge\tilde{\omega}_{H}$ and we denote it by $\omega_{G}=\omega_{V}\cdot\omega_{H}$ .

We note that $G$ and $H$ are unimodular since they are connected reductive $k$ -groups. As shown in [Wei82, Corollary of Theorem 2.2.2], $G$ and $H$ admit translation-invariant gauge forms. Moreover, $V$ admits a $G$ -invariant gauge form $\omega_{V}$ by [BR95, Section 1.4]. Following the arguments in [Wei82, p. 24], by multiplying a constant in $k^{\times}$ on $\omega_{V}$ , we can assume that $\omega_{G}=\omega_{V}\cdot\omega_{H}$ .

For a gauge form $\omega$ on a variety $V$ over $k$ , let $|\omega|_{v}$ denote the measure on $V(k_{v})$ induced from the canonical Haar measure on $k_{v}$ , as in [Wei82, Section 2.2.1], for each $v\in\Omega_{k}$ . We then have the following proposition in the sentence right below [BR95, Lemma 1.6.4].

Proposition 3.3.

Let $\omega_{G},\omega_{H}$ , and $\omega_{V}$ be a translation-invariant gauge form on $G$ , a translation-invariant gauge form on $H$ , and $G$ -invariant gauge form on $V$ , respectively, such that $\omega_{G}=\omega_{V}\cdot\omega_{H}$ . Then we have

(3.2)

\int_{G(k_{v})}f(g)|\omega_{G}|_{v}=\int_{H(k_{v})\backslash G(k_{v})}|\omega_{V}|_{v}\int_{H(k_{v})}f(hg)|\omega_{H}|_{v}.

Here we note that $H(k_{v})\backslash G(k_{v})$ is an open set in $V(k_{v})$ . For triples of measures $(|\omega_{G}|_{v},|\omega_{H}|_{v},|\omega_{V}|_{v})$ satisfying (3.2), we say that they match together topologically.

3.1.2. Tamagawa measure

From the previous section, we choose a $G$ -invariant gauge form $\omega_{V}$ on $V\cong H\backslash G$ . Using this gauge form, we define the Tamagawa measure on $V({\mathbb{A}}_{k})$ as follows.

Definition 3.4.

We define the Tamagawa measure $m_{V}^{Tam}$ on $V(\mathbb{A}_{k})$ with respect to $\omega_{V}$ to be

m_{V}^{Tam}=\lvert\Delta_{k}\rvert^{-\frac{1}{2}\dim V}\prod_{v\in\Omega_{k}}\lvert\omega_{V}\rvert_{v}.

Remark 3.5.

In general, the volume of a compact set $\prod_{v\in\Omega_{k}\setminus\infty_{k}}V(\mathcal{O}_{k_{v}})$ with respect to the non-Archimedean part $\prod_{v\in\Omega_{k}\setminus\infty_{k}}\lvert\omega_{V}\rvert_{v}$ of the Tamagawa measure does not converge. To resolve this problem, Ono introduced the convergence factors using the Artin L-function in [Ono65]. However, in the case of $X$ in Section 2.1, the product $\prod_{v\in\Omega_{k}\setminus\infty_{k}}vol(X(\mathcal{O}_{k_{v}}),|\omega_{X}|_{v})$ already converges and thus we use the notion for the Tamagawa measure without convergence factors.

The definition of the Tamagawa measure $m_{V}^{Tam}$ might depend on the choice of a $G$ -invariant gauge form $\omega_{V}$ . However, the following proposition yields that the Tamgawa measure on $X$ , in Section 2.1, is uniquely defined regardless of the choice of a gauge form.

Proposition 3.6.

The Tamagawa measure on $X$ is independent of the choice of a gauge form $\omega_{X}$ .

Proof.

We claim that any gauge form on $X$ is unique up to a constant factor in $k^{\times}$ . A gauge form $\omega\in\Omega_{X/k}^{\dim X}(X)$ defines an isomorphism $\mathcal{O}_{X}\cong\Omega_{X/k}^{\dim X}(X)$ of locally free $\mathcal{O}_{X}$ -modules, by mapping $1\mapsto\omega$ . For another gauge form $\omega^{\prime}$ on $X$ , $\omega\circ\omega^{\prime-1}$ defines a nowhere-zero regular automorphism on $\mathcal{O}_{X}$ which corresponds to an element in the unit group of the coordinate ring $k[X]^{\times}$ . Thus, we have $\omega^{\prime}=c\omega$ for some $c\in k[X]^{\times}$ . Since $X$ is a homogeneous space of $\mathrm{SL}_{n}$ and $\operatorname{Hom}(\mathrm{SL}_{n},\mathbb{G}_{m})=1$ , we have $k[X]^{\times}=k^{\times}$ by [BR95, Lemma 1.5.1].

For $c\in k^{\times}$ , we have $\lvert c\omega\rvert_{v}=\lvert c\rvert_{v}\lvert\omega\rvert_{v}$ , and the product formula yields that $\prod_{v\in\Omega_{k}}|c|_{v}=1$ . Therefore, the Tamagawa measure is independent of the choice of a gauge form on $X$ . ∎

3.2. Measures on $X(\mathbb{A}_{k})$ for $X\cong\mathrm{T}_{x_{0}}\backslash\mathrm{GL}_{n}$

We now return to our main task. We note that $X\cong\mathrm{T}_{x_{0}}\backslash\mathrm{GL}_{n}$ where both $\mathrm{GL}_{n}$ and $\mathrm{T}_{x_{0}}$ are connected reductive groups. The Tamagawa measure used in (1.5) is therefore given by $|\Delta_{k}|^{-\frac{1}{2}\dim X}\prod_{v\in\Omega_{k}}|\omega_{X}|_{v}$ , which is independent of the choice of a gauge form $\omega_{X}$ on $X$ by Proposition 3.6. To clarify the subsequent statements in this section, we fix a translation-invariant gauge form $\omega_{\mathrm{GL}_{n}}$ on $\mathrm{GL}_{n}$ , a translation-invariant gauge form $\omega_{\mathrm{T}_{x_{0}}}$ on $\mathrm{T}_{x_{0}}$ , and let $\omega_{X}$ be a $\mathrm{GL}_{n}$ -invariant gauge form on $X$ such that $\omega_{\mathrm{GL}_{n}}=\omega_{X}\cdot\omega_{\mathrm{T}_{x_{0}}}$ .

To compute each integration in (1.5), we need an explicit description of the measure $|\omega_{X}|_{v}$ on $X(k_{v})$ . To address this, we define a translation-invariant gauge form $\omega_{X_{k_{v}}}^{can}$ on $X_{k_{v}}$ such that the induced measure $\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}$ is suitable to investigate the integration of $\xi_{v}$ for each $v\in\Omega_{k}$ . After that, we will establish a relation between the Tamagawa measure and $\prod_{v\in\Omega_{k}}\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}$ .

3.2.1. The standard integral model for $\mathrm{T}_{x_{0},k_{v}}$ for $v\in\Omega_{k}\setminus\infty_{k}$

To define a translation-invariant volume form on ${\mathrm{T}_{x_{0},k_{v}}}$ for $v\in\Omega_{k}\setminus\infty_{k}$ , we need a precise description of an integral model for $\mathrm{T}_{x_{0},k_{v}}$ .

For $v\in\Omega_{k}\setminus\infty_{k}$ , we recall that the following description of $\mathrm{T}_{x_{0},k_{v}}$ given in Lemma 2.2,

\mathrm{T}_{x_{0},k_{v}}\cong\prod_{i\in B_{v}(\chi)}\mathrm{R}_{K_{v,i}/k_{v}}\mathbb{G}_{m,K_{v,i}}.

For each $i\in B_{v}(\chi)$ , the component $\mathrm{R}_{K_{v,i}/k_{v}}\mathbb{G}_{m,K_{v,i}}$ has an integral model, given by the smooth group scheme $\mathrm{R}_{\mathcal{O}_{K_{v,i}}/\mathcal{O}_{k_{v}}}\mathbb{G}_{m,\mathcal{O}_{K_{v,i}}}$ over $\mathcal{O}_{k_{v}}$ (cf. [KP23, B.3.(2)]). Thus, $\mathrm{T}_{x_{0},k_{v}}$ has a smooth integral structure $\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}$ , given by

\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}:=\prod_{i\in B_{v}(\chi)}\mathrm{R}_{\mathcal{O}_{K_{v,i}}/\mathcal{O}_{k_{v}}}\mathbb{G}_{m,\mathcal{O}_{K_{v,i}}}.

This is refered to the standard integral model for $\mathrm{T}_{x_{0},k_{v}}$ and we note that $\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})=\prod_{i\in B_{v}(\chi)}\mathcal{O}_{K_{v,i}}^{\times}$ is the maximal compact open subgroup of $\mathrm{T}_{x_{0}}(k_{v})$ . Moreover, the identity component $\mathcal{T}^{0}_{x_{0}}$ of the Néron-Raynaud model $\mathcal{T}_{x_{0}}$ (cf. [BLR90, Chapter 6]) for $\mathrm{T}_{x_{0},k_{v}}$ coincides with the standard integral model by the description right below [Bit11, Remark 1.2].

3.2.2. Volume forms on $X_{k_{v}}\cong\mathrm{T}_{x_{0},k_{v}}\backslash\mathrm{GL}_{n,k_{v}}$

To define $\omega_{X_{k_{v}}}^{can}$ , we construct the translation-invariant volume forms $\omega_{\mathrm{GL}_{n,k_{v}}}^{can}$ on $\mathrm{GL}_{n,k_{v}}$ and $\omega_{\mathrm{T}_{x_{0},k_{v}}}^{can}$ on $\mathrm{T}_{x_{0},k_{v}}$ . Since our main computation on Archimdean places requires the assumption (1) in (1.4), we assume that both $k$ and $K$ are totally real.

For an Archimedean place $v\in\infty_{k}$ , by the assumption, we have $k_{v}\cong\mathbb{R}$ and $\chi(x)$ splits completely over $k_{v}$ for all $v\in\infty_{k}$ . In this case we define the volume forms on $\mathrm{GL}_{n,k_{v}}$ and $\mathrm{T}_{x_{0},k_{v}}$ as follows.

(1)

On $\mathrm{GL}_{n,k_{v}}$ , we define $\omega_{\mathrm{GL}_{n,k_{v}}}^{can}$ to be a translation-invariant volume form in $\Omega^{n^{2}}_{\mathrm{GL}_{n,k_{v}}/k_{v}}(\mathrm{GL}_{n})$ which generates the free $\mathbb{Z}$ -module $\operatorname{Hom}(\bigwedge^{n^{2}}\mathfrak{gl}_{n,\mathbb{Z}}(\mathbb{Z}),\mathbb{Z})$ , following [GG99, Section 9]. We note that a translation-invariant volume form on $\mathrm{GL}_{n,k_{v}}$ is a form of

$\frac{c}{\det^{n}}\bigwedge_{1\leq i,j\leq n}dx_{ij}$

for some $c\in k_{v}^{\times}$ . To generate $\operatorname{Hom}(\bigwedge^{n^{2}}\mathfrak{gl}_{n,\mathbb{Z}}(\mathbb{Z}),\mathbb{Z})$ , the constant $c$ must be $\pm 1$ . We note that both choices induce the same measure on $\mathrm{GL}_{n}(k_{v})$ .
(2)

Since $k$ and $K$ are totally real, we have

$\mathrm{T}_{x_{0},k_{v}}\cong\mathrm{R}_{K_{v}/k_{v}}(\mathbb{G}_{m,K_{v}})\cong\mathbb{G}_{m,k_{v}}^{n}.$

On $\mathrm{T}_{x_{0},k_{v}}$ , we define $\omega_{\mathrm{T}_{x_{0},k_{v}}}^{can}$ to be a pullback of a translation-invariant volume form in $\Omega^{n}_{\mathbb{G}_{m,k_{v}}^{n}/k_{v}}(\mathbb{G}_{m,k_{v}}^{n})$ which generates the free $\mathbb{Z}$ -module $\operatorname{Hom}(\bigwedge^{n}\mathbb{G}_{a,\mathbb{Z}}^{n}(\mathbb{Z}),\mathbb{Z})$ , following [GG99, Section 9]. We note that a translation-invariant volume form on $\mathbb{G}^{n}_{m,k_{v}}$ is a form of

$c\bigwedge_{i=1}^{n}\frac{dx_{i}}{x_{i}}$

for some $c\in k_{v}^{\times}$ . To generate $\operatorname{Hom}(\bigwedge^{n^{2}}\mathbb{G}^{n}_{a,\mathbb{Z}}(\mathbb{Z}),\mathbb{Z})$ , the constant $c$ must be $\pm 1$ . We note that both choices induce the same measure on $\mathrm{T}_{x_{0}}(k_{v})$ .

On the other hand, for a non-Archimedean place $v\in\Omega_{k}\setminus\infty_{k}$ , we define the volume forms on $\mathrm{GL}_{n,k_{v}}$ and $\mathrm{T}_{x_{0},k_{v}}$ as follows.

(1)

On $\mathrm{GL}_{n,k_{v}}$ , we define $\omega_{\mathrm{GL}_{n,k_{v}}}^{can}$ to be a translation-invariant volume form in $\Omega^{n^{2}}_{\mathrm{GL}_{n,k_{v}}/k_{v}}(\mathrm{GL}_{n,k_{v}})$ which generates the $\mathcal{O}_{k_{v}}$ -submodule $\operatorname{Hom}(\bigwedge^{n^{2}}\mathfrak{gl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}),\mathcal{O}_{k_{v}})$ . Then the volume form $\omega_{\mathrm{GL}_{n,k_{v}}}^{can}$ has good reduction (mod $\pi_{v}$ ) in the sense of [Gro97, Section 4]. We then have the following equation by [Gro97, Proposition 4.7],

vol(\mathrm{GL}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}),|\omega_{\mathrm{GL}_{n,k_{v}}}^{can}|_{v})=\frac{\#\mathrm{GL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})}{q_{v}^{n^{2}}}.

(2)

On $\mathrm{T}_{x_{0},k_{v}}$ , we define $\omega_{\mathrm{T}_{x_{0},k_{v}}}^{can}$ to be a translation-invariant volume form in $\Omega^{n}_{\mathrm{T}_{x_{0},k_{v}}/k_{v}}(\mathrm{T}_{x_{0},k_{v}})$ which generates the $\mathcal{O}_{k_{v}}$ -submodule $\mathrm{Hom}(\bigwedge^{n}\mathrm{Lie}(\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}})(\mathcal{O}_{k_{v}}),\mathcal{O}_{k_{v}})$ . Since $\mathcal{T}_{x_{0}}^{0}\cong\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}$ , the volume form $\omega_{\mathrm{T}_{x_{0},k_{v}}^{can}}$ has good redcution (mod $\pi_{v}$ ) in the sense of [Gro97, Section 4]. We then have the following equation by [Bit11, Proposition 2.14],

vol(\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}),|\omega_{\mathrm{T}_{x_{0},k_{v}}}^{can}|_{v})=\frac{\#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})}{q_{v}^{n}}.

In both two cases, for $v\in\Omega_{k}$ , we define $\omega_{X_{k_{v}}}^{can}$ on $X_{k_{v}}\cong\mathrm{T}_{x_{0},k_{v}}\backslash\mathrm{GL}_{n,k_{v}}$ to be a translation-invariant volume form such that $\omega_{\mathrm{GL}_{n,k_{v}}}^{can}=\omega_{X_{k_{v}}}^{can}\cdot\omega_{\mathrm{T}_{x_{0},k_{v}}}^{can}$ . Then by Proposition 3.3, $|\omega_{X_{k_{v}}}^{can}|_{v}$ defines a measure on $\mathrm{T}_{x_{0}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v})$ which matches topologically together with $|\omega_{\mathrm{GL}_{n,k_{v}}}^{can}|_{v}$ and $|\omega_{\mathrm{T}_{x_{0},k_{v}}}^{can}|_{v}$ .

For $v\in\Omega_{k}\setminus\infty_{k}$ , comparing with the quotient measure $\frac{dg_{v}}{dt_{v}}$ induced from the canonically normalized Haar measures $dg_{v}$ on $\mathrm{GL}_{n}(k_{v})$ and $dt_{v}$ on $\mathrm{T}_{x_{0}}(k_{v})$ , we have the following relation.

Lemma 3.7.

Suppose that $v\in\Omega_{k}\setminus\infty_{k}$ . Let $\frac{dg_{v}}{dt_{v}}$ be the quotient measure on $X(k_{v})$ , which is defined via the isomorphism $X(k_{v})\cong\mathrm{T}_{x_{0}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v})$ , where

\left\{\begin{array}[]{l}\textit{$dg_{v}$ be the Haar measure on $\mathrm{GL}_{n}(k_{v})$ such that $\mathrm{vol}(dg_{v},\mathrm{GL}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}))=1$};\\ \textit{$dt_{v}$ be the Haar measure on $\mathrm{T}_{x_{0}}(k_{v})$ such that $\mathrm{vol}(dt_{v},\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}))=1$}.\end{array}\right.

We have

|\omega_{X_{k_{v}}}^{can}|_{v}=\frac{\#\mathrm{GL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n^{2}}}{\#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n}}\cdot\frac{dg_{v}}{dt_{v}}.

Lemma 3.8.

We have the following equations which are independent of the choice of gauge forms $\omega_{\mathrm{GL}_{n}}$ and $\omega_{\mathrm{T}_{x_{0}}}$ , respectively,

\prod_{v\in\Omega_{k}}\frac{|\omega_{\mathrm{GL}_{n,k_{v}}}^{can}|_{v}}{|\omega_{\mathrm{GL}_{n}}|_{v}}=1\textit{ and }\prod_{v\in\Omega_{k}}\frac{|\omega_{\mathrm{T}_{x_{0},k_{v}}}^{can}|_{v}}{|\omega_{\mathrm{T}_{x_{0}}}|_{v}}=\prod_{v\in\Omega_{k}\setminus\infty_{k}}\prod_{i\in B_{v}(\chi)}|\Delta_{K_{v,i}/k_{v}}|^{-\frac{1}{2}}.

Proof.

For $G=\mathrm{GL}_{n}$ or $\mathrm{T}_{x_{0},k_{v}}$ , by [GG99, Corollary 7.3 and Proposition 9.3] we have that

\prod_{v\in\Omega_{k}}\frac{|\omega_{G_{k_{v}}}^{can}|_{v}}{|\omega_{G}|_{v}}=f(M_{G})^{\frac{1}{2}},

which is independent of the choice of a gauge form $\omega_{G}$ where $f(M_{G})$ is the global conductor of the motive $M_{G}$ of $G$ (cf. [GG99, Equation (9.1)]).

(1)

Since $\mathrm{GL}_{n}$ is split over $k$ , we take a gauge form $\omega_{\mathrm{GL}_{n}}$ to generate the Chevalley differetials over $\mathbb{Z}$ . Then by the proof of [GG99, Proposition 9.3], we have $|\omega_{\mathrm{GL}_{n,k_{v}}}^{can}|_{v}=|\omega_{\mathrm{GL}_{n}}|_{v}$ for all $v\in\Omega_{k}$ and so we have $f(M_{\mathrm{GL}_{n}})^{\frac{1}{2}}=1$ .
(2)

On the other hand, in the case of $\mathrm{T}_{x_{0}}$ , the global conductor of the motive $M_{\mathrm{T}_{x_{0}}}$ is given by $f(M_{\mathrm{T}_{x_{0}}})=\prod_{v\in\Omega_{k}\setminus\infty_{k}}q_{v}^{a_{p}(X^{*}(\mathrm{T}_{x_{0}})\otimes\mathbb{Q})}$ where $a_{p}(X^{*}(\mathrm{T}_{x_{0}}\otimes\mathbb{Q}))$ is the local Artin conductor of $X^{*}(\mathrm{T}_{x_{0}}\otimes\mathbb{Q})$ following [Gro97, Section 2]. By [Neu99, Proposition VII.11.7] and the argument in [Lee21, Section 1.2], we have that

$f(M_{\mathrm{T}_{x_{0}}})=\prod_{v\in\Omega_{k}\setminus\infty_{k}}\prod_{i\in B_{v}(\chi)}|\Delta_{K_{v,i}/k_{v}}|^{-1}.$

∎

Then Proposition 3.3 and Lemma 3.8 directly yield the following relation between two global measures $\prod_{v\in\Omega_{k}}\lvert\omega_{X}\rvert_{v}$ and $\prod_{v\in\Omega_{k}}\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}$ .

Proposition 3.9.

We have

\prod_{v\in\Omega_{k}}\lvert\omega_{X}\rvert_{v}=\prod_{v\in\Omega_{k}\setminus\infty_{k}}\prod_{i\in B_{v}(\chi)}|\Delta_{K_{v,i}/k_{v}}|^{-\frac{1}{2}}\prod_{v\in\Omega_{k}}\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}.

4. Brauer groups and evaluations

In this section, we will explain various notions, used in the observation of D. Wei and F. Xu in Proposition 1.1. For the reader’s convenience, we restate this proposition as follows.

Proposition 4.1.

[WX16, Theorem 4.3] Suppose that $\chi(x)$ is an irreducible monic polynomial over $\mathcal{O}_{k}$ . Then we have

(4.1)

N(X,T)\sim\lvert\Delta_{k}\rvert^{-\frac{1}{2}\dim X}\sum_{\xi\in\operatorname{Br}X/\operatorname{Br}k}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\int_{X(\mathcal{O}_{k_{v}})}\xi_{v}(x)\ \lvert\omega_{X}\rvert_{v}\prod_{v\in\infty_{k}}\int_{X(k_{v},T)}\xi_{v}(x)\ \lvert\omega_{X}\rvert_{v},

where

\left\{\begin{array}[]{l}X(k_{v},T)=\{x\in X(k_{v})\mid\lVert x\rVert_{v}\leq T\}\textit{ for $v\in\infty_{k}$ and $T>0$};\\ \lvert\omega_{X}\rvert_{v}\textit{ is the measure defined in \ref{subsec:gauge_form}};\\ \xi_{v}(x)\textit{ denotes a Brauer evaluation for $x\in X(k_{v})$ which is defined in Definition \ref{def:brauer_evaluation}}.\end{array}\right.

We mainly follow [Poo17, Section 8] to define the Brauer group and the evaluation of each element in the Brauer group.

Definition 4.2.

For any $k$ -scheme $Y$ , the Brauer group is defined by

\operatorname{Br}Y=\mathrm{H}_{\acute{e}t}^{2}(Y,\mathbb{G}_{m}),

where $\mathbb{G}_{m}$ stands for the étale sheaf associated with the multiplicative group $\mathbb{G}_{m}$ over $Y$ (cf. [Poo17, Proposition 6.3.19]). To ease the notation, we denote $\operatorname{Br}(\operatorname{Spec}A)$ by $\operatorname{Br}A$ for affine schemes.

Since $\mathrm{H}_{\acute{e}t}^{2}(-,\mathbb{G}_{m})$ gives a contravariant functor from $\mathrm{Sch}_{k}$ to $\mathrm{Ab}$ , each element of $\operatorname{Br}Y$ induces an evaluation on the set of $A$ -points $Y(A)$ , for any $k$ -algebra $A$ , as follows.

Definition 4.3.

Let $Y$ be a $k$ -scheme and $A$ be a $k$ -algebra.

(1)

For any $x\in Y(A)$ and $\xi\in\operatorname{Br}Y$ , we define the Brauer evaluation $\xi(x)$ to be the image of $\xi$ under the morphism $\operatorname{Br}(\operatorname{Spec}A\to Y)$ .

(2)

In particular, for a local field $A=k_{v}$ , [Poo17, Theorem 1.5.34] yields that the invariant map $\operatorname{inv}:\operatorname{Br}k_{v}\to\mathbb{Q}/\mathbb{Z}$ is injective. In this case, for any $x\in Y(k_{v})$ and $\xi\in\operatorname{Br}Y$ , we define the Brauer evaluation $\xi_{v}(x)$ to be the image of $\xi$ under the following composition,

\operatorname{Br}Y\xrightarrow{\operatorname{Br}(x)}\operatorname{Br}k_{v}\xhookrightarrow{\operatorname{inv}}\mathbb{Q}/\mathbb{Z}\xrightarrow{x\mapsto\exp(2\pi ix)}\mathbb{C}^{\times}.

Remark 4.4.

We note that $X$ in Section 2.1 has a $k$ -point $x_{0}\in X(\mathcal{O}_{k})\subset X(k)$ . Since $X$ is a $k$ -scheme, the structure morphism induces a morphism $\operatorname{Br}k\to\operatorname{Br}X$ . Let $\operatorname{Br}X/\operatorname{Br}k$ be the quotient group of $\operatorname{Br}X$ by the image of $\operatorname{Br}k$ under this induced morphism. We remark that the summation in (4.1) is well-defined. More precisely, each infinite product over $v\in\Omega_{k}$ is independent of the choice of representatives of $\operatorname{Br}X/\operatorname{Br}k$ .

Since the Brauer evaluation $\operatorname{Br}X\to\mathbb{C}^{\times}$ is a group homomorphism, it suffices to show that the evaluation $\xi_{v}$ is constant on $X(k_{v})$ and the product of these constant values over $v\in\Omega_{k}$ is trivial. For a fixed embedding $k\hookrightarrow k_{v}$ , the composition $\operatorname{Br}k\rightarrow\operatorname{Br}X\xrightarrow{\operatorname{Br}(x)}\operatorname{Br}k_{v}$ is independenet of $x\in X(k_{v})$ . This concludes that $\xi_{v}(x)$ is constant on $X(k_{v})$ by Definition 4.3.(2). Therefore it suffices to show that $\prod_{v\in\Omega_{k}}\xi_{v}(x_{0})$ is trivial for $x_{0}\in X(k)\subset X(k_{v})$ , and this follows from [Poo17, Proposition 8.2.2].

For $X$ in Section 2.1, we define the normalized Brauer evaluation as follows to remove the vagueness.

Definition 4.5.

For each $\xi\in\operatorname{Br}X$ , we define the normalized Brauer evaluation $\tilde{\xi}_{v}$ on $x\in X(k_{v})$ to be

\tilde{\xi}_{v}(x)=\frac{\xi_{v}(x)}{\xi_{v}(x_{0})},

where $v\in\Omega_{k}$ and $x_{0}$ is the point in $X(\mathcal{O}_{k})$ chosen in Section 2.1.

Remark 4.4 indicates that the evaluation $\tilde{\xi}_{v}\equiv\tilde{\xi}_{v}^{\prime}$ on $X(k_{v})$ if $\xi$ and $\xi^{\prime}$ belong to the same coset in $\operatorname{Br}X/\operatorname{Br}k$ . We also have that for $\xi\in\operatorname{Br}X$

(4.2)		$\displaystyle\prod_{v\in\Omega_{k}\setminus\infty_{k}}\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)$	$\displaystyle\ \lvert\omega_{X}\rvert_{v}\prod_{v\in\infty_{k}}\int_{X(k_{v},T)}\tilde{\xi}_{v}(x)\ \lvert\omega_{X}\rvert_{v}$
(4.2)			$\displaystyle=\prod_{v\in\Omega_{k}\setminus\infty_{k}}\int_{X(\mathcal{O}_{k_{v}})}\xi_{v}(x)\ \lvert\omega_{X}\rvert_{v}\prod_{v\in\infty_{k}}\int_{X(k_{v},T)}\xi_{v}(x)\ \lvert\omega_{X}\rvert_{v}$

by [Poo17, Proposition 8.2.2] since $x_{0}$ is in $X(k)$ . Therefore, we can use the normalized Brauer evaluation $\tilde{\xi}_{v}$ on (4.1).

Remark 4.6.

If $\xi$ is trivial in $\operatorname{Br}X/\operatorname{Br}k$ , then the normalized Brauer evaluation is trivial on $X(k_{v})$ by Remark 4.4. However, the normalized Brauer evaluation can be trivial even if $\xi$ is non-trivial in $\operatorname{Br}X/\operatorname{Br}k$ .

5. The asymptotic formula for $N(X,T)$

From now on, throughout the remaining part of this paper, we will work on the following assumptions in (1.4),

\left\{\begin{array}[]{l}\textit{(1) $k$ and $K$ are totally real number fields};\\ \textit{(2) if $k\neq\mathbb{Q}$, then $\chi(x)$ is of prime degree $n$};\\ \textit{(3) if $K/k$ is unramified Galois, then $\chi(x)$ splits over $k_{v}$ at $p$-adic places $v$ of $k$, for $p\leq n$}.\end{array}\right.

We mainly use the result in Proposition 4.1. According to the formula (4.1), we will investigate the Brauer evaluation $\xi_{v}$ , using the local class field theory, in Section 5.1. Based on this observation, we will investigate the integration for each place $v\in\Omega_{k}$ throughout Section 5.2- 5.3 and Section 6.

5.1. Evaluation of $\xi\in\operatorname{Br}X/\operatorname{Br}k$

In Definition 4.3, we described a concrete definition of the Brauer evaluation using the functor $\operatorname{Br}$ and the invariant map. Alternatively, we can explicitly formulate the Brauer evaluation using local class field theory in our case, where $X$ is a homogeneous space of $\mathrm{SL}_{n}$ . The key fact is that $\mathrm{SL}_{n}$ is a semisimple and simply connected algebraic group (cf. [CTX09, Section 2]). We divide into cases based on whether $K/k$ is Galois or not.

5.1.1. The case that $K/k$ is not a Galois extension

If $K/k$ is not Galois, the lemma below states that $\operatorname{Br}X/\operatorname{Br}k$ is trivial. Thus, by Remark 4.6, we do not need to consider any nontrivial Brauer evaluation in this case.

Proposition 5.1.

If $K/k$ is of prime degree and not Galois, then $\operatorname{Br}X/\operatorname{Br}k$ is trivial.

Proof.

Let $L$ be the Galois closure of $K/k$ , with $\Lambda=\operatorname{Gal}(L/k)$ and $\Upsilon=\operatorname{Gal}(L/K)$ as the corresponding Galois groups. By the proof of [WX16, Theorem 6.1] and [CTX09, Proposition 2.10], we have

\operatorname{Br}X/\operatorname{Br}k\cong\operatorname{Pic}\mathrm{S}_{x_{0}}\cong\ker(\operatorname{Hom}(\Lambda,\mathbb{Q}/\mathbb{Z})\to\operatorname{Hom}(\Upsilon,\mathbb{Q}/\mathbb{Z})).

Suppose that there is a non-trivial element $\xi\in\operatorname{Br}X/\operatorname{Br}k$ . Let $f\in\operatorname{Hom}(\Lambda,\mathbb{Q}/\mathbb{Z})$ be the image of $\xi$ via the above isomorphism, and let $A$ be the fixed field of $\ker f$ . We claim that $A=K$ . Then $K$ is Galois over $k$ since $\ker f$ is normal in $\Lambda$ , which contradicts the hypothesis. Since $f$ vanishes on $\Upsilon$ , we have $\Upsilon\subset\ker f\subset\Lambda$ . The non-trivial $\xi$ yields that $\ker f$ is not equal to $\Lambda$ . We then have that $\ker f=\Upsilon$ since $\deg(K/k)$ is a prime number, which implies that $A=K$ . ∎

5.1.2. The case that $K/k$ is a Galois extension

Since we assume $\deg(K/k)$ is a prime number, the field extension $K/k$ is an abelian extension if it is Galois. For any non-Archimedean place $v\in\Omega_{k}\setminus\infty_{k}$ , if $K_{v}=K\otimes_{k}k_{v}$ is a field, we can explictly describe the normalized Brauer evaluation.

Proposition 5.2.

Suppose that $K/k$ is an abelian extension. Let $\Lambda=\operatorname{Gal}(K/k)$ . Then there is an isomorphism

\Psi:\operatorname{Br}X/\operatorname{Br}k\xrightarrow{\sim}\operatorname{Pic}\mathrm{S}_{x_{0}}\xrightarrow{\sim}\operatorname{Hom}(\Lambda,\mathbb{Q}/\mathbb{Z}).

In particular, if $K_{v}=K\otimes_{k}k_{v}$ is a field for $v\in\Omega_{k}\setminus\infty_{k}$ , then the normalized Brauer evaluation $\tilde{\xi}_{v}$ for $\xi\in\operatorname{Br}X$ is given by

\tilde{\xi}_{v}(x)=\exp 2\pi i\Psi(\xi)(\phi_{K_{v}/k_{v}}(\det g_{x}))

for $x\in X(k_{v})$ . Here, $\phi_{K_{v}/k_{v}}:k_{v}^{\times}/\operatorname{Nm}_{K_{v}/k_{v}}K_{v}^{\times}\to\operatorname{Gal}(K_{v}/k_{v})\subset\Lambda$ is the Artin reciprocity isomorphism (cf. [Ser79, Chapter XI]) and $g_{x}\in\mathrm{GL}_{n}(k_{v})$ such that $g_{x}^{-1}x_{0}g_{x}=x$ .

Proof.

The first isomorphism, $\operatorname{Br}X/\operatorname{Br}k\xrightarrow{\sim}\operatorname{Pic}\mathrm{S}_{x_{0}}$ in $\Psi$ , is proved in [CTX09, Proposition 2.10]. On the other hand, the isomorphism $\operatorname{Pic}\mathrm{S}_{x_{0}}\xrightarrow{\sim}\operatorname{Hom}(\Lambda,\mathbb{Q}/\mathbb{Z})$ in $\Psi$ is proved in [WX16, Theorem 6.1]. The same arguments apply for $\Psi_{v}:\operatorname{Br}X/\operatorname{Br}k_{v}\xrightarrow{\sim}\operatorname{Pic}\mathrm{S}_{x_{0},k_{v}}\xrightarrow{\sim}\operatorname{Hom}(\Lambda_{v},\mathbb{Q}/\mathbb{Z})$ where $\Lambda_{v}=\operatorname{Gal}(K_{v}/k_{v})\subset\Lambda$ , and it satisfies the following commutative diagram,

Therefore it suffices to consider the evaluation for the base change $X_{k_{v}}\cong\mathrm{S}_{x_{0},k_{v}}\backslash\mathrm{SL}_{n,k_{v}}$ .

The Brauer evaluation of a homogeneous space is induced from the following commutative diagram, as shown in [CTX09, Proposition 2.9-2.10],

(5.1)

Here, $\operatorname{Br}_{*}X\subset\operatorname{Br}X$ is the subgroup of the Brauer group consisting of the elements whose evaluation vanishes at $x_{0}\in X(k_{v})$ . The pairing in the first row is the Brauer evaluation for $\operatorname{Br}_{*}X$ , and the second row is the cup product $\cup$ on the group cohomology. We will explain the first vertical map later. The second vertical map is defined by

\operatorname{Br}_{*}X\cong\operatorname{Br}X/\operatorname{Br}k_{v}\to\operatorname{Pic}\mathrm{S}_{x_{0},k_{v}}\cong\mathrm{H}^{1}(k_{v},\widehat{\mathrm{S}_{x_{0},k_{v}}}).

First, we explain how to operate the cup product in the second row of (5.1). Since the cup product is compatible with inflation homomorphisms, and both $\mathrm{S}_{x_{0},k_{v}}$ and $\widehat{\mathrm{S}_{x_{0},k_{v}}}$ split over $K_{v}$ , we consider the cup product $\cup$ on the group cohomology for $\Lambda_{v}$ -modules as follows,

\mathrm{H}^{1}(\Lambda_{v},\mathrm{S}_{x_{0},k_{v}})\times\mathrm{H}^{1}(\Lambda_{v},\widehat{\mathrm{S}_{x_{0},k_{v}}})\xrightarrow{\cup}\mathrm{H}^{2}(\Lambda_{v},\mathrm{S}_{x_{0},k_{v}}\otimes_{\mathbb{Z}}\widehat{\mathrm{S}_{x_{0},k_{v}}})\to\mathrm{H}^{2}(\Lambda_{v},K_{v}^{\times})\xrightarrow{\operatorname{inv}}\mathbb{Q}/\mathbb{Z}.

We claim that the following diagram commutes, allowing us to use the cup product in the second row.

(5.2)

Here, the right vertical map is obtained from the canonical pairing. $\delta_{1}$ and $\delta_{2}$ are connecting homomorphisms derived from the following exact sequences of $\Lambda_{v}$ -modules,

(5.3)

One can show that two exact sequences in (5.3) split by choosing splitting homomorphisms $K_{v}^{\times}\to(K_{v}\otimes_{k_{v}}K_{v})^{\times},\ a\mapsto 1\otimes a$ and $\mathbb{Z}[\Lambda_{v}]\to\mathbb{Z},\ \sum_{\sigma}n_{\sigma}\sigma\mapsto n_{1}$ , respectively. We have that $\delta_{1}$ is surjective since the proof of Proposition 2.3 yields that $\mathrm{H}^{1}(\Lambda_{v},(K_{v}\otimes_{k_{v}}K_{v})^{\times})$ is trivial. Since $\mathbb{Z}[\Lambda_{v}]=\operatorname{Ind}_{1}^{\Lambda_{v}}\mathbb{Z}$ , we have

\mathrm{H}^{i}(\Lambda_{v},\mathbb{Z}[\Lambda_{v}])\cong\mathrm{H}^{i}(1,\mathbb{Z})=0

for $i=1$ and $2$ , by Shapiro’s lemma, and thus $\delta_{2}$ is an isomorphism. From the exact sequences in (5.3), we consider the diagram obtained by applying $(-)\otimes_{\mathbb{Z}}\widehat{\mathrm{S}_{x_{0},k_{v}}}(K_{v})$ at the first row, and by applying $K_{v}^{\times}\otimes_{\mathbb{Z}}(-)$ at the second row as follows,

The first vertical map is defined by the canonical pairing of the character modules. Since $(-)\otimes_{\mathbb{Z}}\widehat{\mathrm{S}_{x_{0},k_{v}}}(K_{v})$ and $K_{v}^{\times}\otimes_{\mathbb{Z}}(-)$ preserve split exactness, both two rows are also split exact. Thus, there is an induced central map that such that the diagram commutes, through the splitting homomorphisms. Let $\delta_{3}$ and $\delta_{4}$ be the connecting homomorphisms from the above two. Then the diagram below commutes.

For $a\in\mathrm{H}^{0}(\Lambda_{v},K_{v}^{\times})$ and $b\in\mathrm{H}^{1}(\Lambda_{v},\widehat{\mathrm{S}_{x_{0},k_{v}}})$ , it follows from [Ser79, Proposition 5, VIII] that $\delta_{3}(a\cup b)=\delta_{1}a\cup b$ and $\delta_{4}(a\cup b)=a\cup\delta_{2}b$ . By the commutativity of $\delta_{3}$ and $\delta_{4}$ , $\delta_{1}a\cup b$ and $a\cup\delta_{2}b$ coincide in $\mathrm{H}^{2}(\Lambda_{v},K_{v}^{\times})$ , so that the diagram (5.2) commutes.

Now we describe the map $X(k_{v})\to\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ in the commutative diagram (5.1) explicitly. By Proposition 2.1, we have

(5.4)

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To identify two $\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ in the diagram, we choose $g_{x}\in\mathrm{GL}_{n}(k_{v})$ for $x\in X(k_{v})$ such that $g_{x}^{-1}x_{0}g_{x}=x$ . Thus, $\delta_{1}(\det g_{x})$ represents the image of $x$ in $\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ . For $\xi\in\operatorname{Br}X$ , we can choose $\xi_{0}\in\operatorname{Br}k_{v}$ such that $\operatorname{inv}\xi_{0}=\xi_{v}(x_{0})\in\mathbb{Q}/\mathbb{Z}$ , which implies that $\xi-\xi_{0}\in\operatorname{Br}_{*}X$ . Let $\chi\in\operatorname{Hom}(\Lambda_{v},\mathbb{Q}/\mathbb{Z})$ be the image of $\xi-\xi_{0}$ under

\operatorname{Br}_{*}X\cong\mathrm{H}^{1}(k_{v},\widehat{\mathrm{S}_{x_{0},k_{v}}})\cong\operatorname{Pic}\mathrm{S}_{x_{0},k_{v}}\cong\operatorname{Hom}(\Lambda_{v},\mathbb{Q}/\mathbb{Z}).

We note that the evaluation of $\xi-\xi_{0}\in\operatorname{Br}_{*}X\subset\operatorname{Br}X$ equals the normalized evaluation $\tilde{\xi}_{v}$ . By the proof of [WX16, Theorem 6.1] that $\operatorname{Pic}\mathrm{S}_{x_{0},k_{v}}\cong\operatorname{Hom}(\Lambda_{v},\mathbb{Q}/\mathbb{Z})$ , the image of $\xi-\xi_{0}$ under

\operatorname{Br}_{*}X\cong\mathrm{H}^{1}(k_{v},\widehat{\mathrm{S}_{x_{0},k_{v}}})\xrightarrow{\delta_{2}}\mathrm{H}^{2}(\Lambda_{v},\mathbb{Z})

coincides with $d\chi$ . Here, $d:\operatorname{Hom}(\Lambda_{v},\mathbb{Q}/\mathbb{Z})=\mathrm{H}^{1}(\Lambda_{v},\mathbb{Q}/\mathbb{Z})\to\mathrm{H}^{2}(\Lambda,\mathbb{Z})$ is a connecting homomorphism derived from the exact sequence

0\to\mathbb{Z}\to\mathbb{Q}\to\mathbb{Q}/\mathbb{Z}\to 0.

Therefore, the evaluation of $\xi-\xi_{0}$ in (5.1) is

(\xi-\xi_{0})_{v}(x)=\tilde{\xi}_{v}(x)=\operatorname{inv}(\det g_{x}\cup d\chi)=\Psi(\xi)(\phi_{K_{v}/k_{v}}(\det g_{x}))

by [Ser79, Proposition 2, XI]. ∎

Corollary 5.3.

Suppose that $K/k$ is an abelian extension. For $v\in\Omega_{k}\setminus\infty_{k}$ , if $K_{v}$ is a field, then the normalized Brauer evaluation $\tilde{\xi}_{v}$ on $X(k_{v})$ is constant on each $\mathrm{SL}_{n}(k_{v})$ -orbits in $X(k_{v})$ , for any $\xi\in\operatorname{Br}X$ .

Proof.

For $x,x^{\prime}\in X(k_{v})$ on the same $\mathrm{SL}_{n}(k_{v})$ -orbit, we have $x=x^{\prime}\cdot g$ for some $g\in\mathrm{SL}_{n}(k_{v})$ . Then, by Proposition 5.2, we have $\tilde{\xi}_{v}(x)=\tilde{\xi}_{v}(x^{\prime})$ for any $\xi\in\operatorname{Br}X$ . ∎

Corollary 5.4.

If $K/k$ is a Galois extension of prime degree and $K_{v}/k_{v}$ is also a field extension, then for a non-trivial $\xi\in\operatorname{Br}X/\operatorname{Br}k$ , the evaluation $\tilde{\xi}_{v}$ on $X(k_{v})$ is non-trivial.

Proof.

The hypothesis on $K$ and $K_{v}$ yields that $\operatorname{Gal}(K/k)=\operatorname{Gal}(K_{v}/k_{v})$ and it is abelian. By Proposition 5.2, $\Psi(\xi)\in\operatorname{Hom}(\operatorname{Gal}(K/k),\mathbb{Q}/\mathbb{Z})$ is non-trivial. In other words, there exists $\sigma\in\operatorname{Gal}(K/k)=\operatorname{Gal}(K_{v}/k_{v})$ such that $\Psi(\xi)(\sigma)\neq 0$ . Let $c\in k_{v}^{\times}$ be any lift of $\phi_{K_{v}/k_{v}}^{-1}(\sigma)\in k_{v}^{\times}/\operatorname{Nm}_{K_{v}/k_{v}}K_{v}^{\times}$ and we define $g=\operatorname{diag}(c,1,\ldots,1)\in\mathrm{GL}_{n}(k_{v})$ . Then, $\tilde{\xi}_{v}(x_{0}\cdot g)$ is nontrivial by Proposition 5.2. ∎

For the case that $K_{v}$ is not a field, the normalized Brauer evaluation is always trivial.

Proposition 5.5.

If $K/k$ is a Galois extension of prime degree and $K_{v}$ is not a field, then $\tilde{\xi}_{v}\equiv 1$ on $X(k_{v})$ .

Proof.

Since $K$ is a Galois extension of $k$ of prime degree, if $K_{v}$ is not a field, then $\chi(x)$ splits completely over $k_{v}$ . By [CTX09, Propositoin 2.10], we have

\operatorname{Br}X/\operatorname{Br}k_{v}\cong\operatorname{Pic}\mathrm{S}_{x_{0},k_{v}}\cong\mathrm{H}^{1}(k_{v},\widehat{\mathrm{S}_{x_{0},k_{v}}}).

By Lemma 2.2, $\mathrm{S}_{x_{0},k_{v}}$ is the kernel of the multiplication map $\mathbb{G}_{m,k_{v}}^{n}\to\mathbb{G}_{m,k_{v}}$ , which is isomorphic to $\mathbb{G}_{m,k_{v}}^{n-1}$ . Since $\mathrm{S}_{x_{0},k_{v}}$ splits over $k_{v}$ , $\widehat{\mathrm{S}_{x_{0},k_{v}}}$ is also splits over $k_{v}$ . We conclude that $\operatorname{Br}X/\operatorname{Br}k_{v}$ is trivial. If $\xi\in\operatorname{Br}X$ is contained in the image of $\operatorname{Br}k_{v}\to\operatorname{Br}X$ , then for $x\in X(k_{v})$ , $\xi_{v}(x)$ is defined by the image of $\xi$ via

\operatorname{Br}k_{v}\to\operatorname{Br}X\xrightarrow{\operatorname{Br}(x)}\operatorname{Br}k_{v}\xrightarrow{\operatorname{inv}}\mathbb{Q}/\mathbb{Z}\xrightarrow{x\mapsto\exp(2\pi ix)}\mathbb{C}^{\times}.

Since $\xi_{v}$ is constant on $X(k_{v})$ , $\tilde{\xi}_{v}$ is trivial on $X(k_{v})$ . ∎

5.2. Computation on Archimedean places

We now investigate the integration in (4.1) for Archimedean places. Since we assume that $k$ and $K$ are totally real number fields, $\chi(x)$ splits completely over $k_{v}$ for any Archimedean place $v\in\infty_{k}$ . Thus, by Proposition 5.5, the Brauer evaluation $\tilde{\xi}_{v}$ is trivial. It suffices to consider the volume of $X(k_{v},T)$ defined in (4.1), with respect to $\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}$ .

Proposition 5.6.

Suppose that $k$ and $K$ are totally real number fields. Then we have

\prod_{v\in\infty_{k}}\int_{X(k_{v},T)}\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}\sim\left(w_{n}\pi^{\frac{n(n+1)}{4}}T^{\frac{n(n-1)}{2}}\prod_{i=1}^{n}\Gamma(\frac{i}{2})^{-1}\right)^{[k:\mathbb{Q}]}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\lvert\Delta_{\chi}\rvert_{v}^{\frac{1}{2}},

where $w_{n}$ is the volume of the unit ball in $\mathbb{R}^{\frac{n(n-1)}{2}}$ .

Proof.

We fix an Archimedean place $v\in\infty_{k}$ . Since $k$ and $K$ are totally real, we have $k_{v}\cong\mathbb{R}$ and $\chi(x)$ splits completely over $k_{v}$ . Let $\lambda_{1},\ldots,\lambda_{n}\in k_{v}$ be the distinct roots of $\chi(x)$ , and $\lambda:=\operatorname{diag}(\lambda_{1},\ldots,\lambda_{n})\in X(k_{v})$ be the diagonal matrix. We denote by $\mathrm{T}_{\lambda,k_{v}}$ the stabilizer of $\lambda$ under the conjugation of $\mathrm{GL}_{n,k_{v}}$ . Since the roots $\lambda_{i}$ are distinct, the stabilizer $\mathrm{T}_{\lambda,k_{v}}(k_{v})$ is the set of diagonal matrices in $\mathrm{GL}_{n}(k_{v})$ .

To obtain the volume $\int_{X(k_{v},T)}|\omega_{X_{k_{v}}}^{can}|_{v}$ , we define a test function $\mathbbm{1}_{G_{T}}$ where $G_{T}\subset\mathrm{GL}_{n}(k_{v})$ denotes

G_{T}=\{g\in\mathrm{GL}_{n}(k_{v})\mid\lVert\lambda\cdot g\rVert_{v}\leq T\text{ and }1<r_{ii}<2\text{ for }g=rq\}.

Here, $g=rq$ represents the unique RQ-decomposition of $g\in\mathrm{GL}_{n}(k_{v})$ , where $r$ is an upper triangular matrix with positive diagonal entries, and $q$ is an orthogonal matrix in $\mathrm{GL}_{n}(k_{v})$ .

We note that

(5.5)

X(k_{v})=\mathrm{T}_{x_{0},k_{v}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v})=\mathrm{T}_{\lambda,k_{v}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v}).

As explained in Section 3.2.2, we have $(\mathbb{G}_{m,k_{v}})^{n}\cong\mathrm{T}_{x_{0},k_{v}}$ . Comparing this with $\mathrm{T}_{\lambda,k_{v}}$ , we have $(\mathbb{G}_{m,k_{v}})^{n}\cong\mathrm{T}_{x_{0},k_{v}}\cong\mathrm{T}_{\lambda,k_{v}}$ by $(t_{i})\mapsto\operatorname{diag}(t_{i})$ . Thus, the pullback of $\omega_{\mathrm{T}_{x_{0},k_{v}}}^{can}$ under the isomorphism $\mathrm{T}_{x_{0},k_{v}}\cong\mathrm{T}_{\lambda,k_{v}}$ induces the Haar measure $\bigwedge_{i=1}^{n}\frac{dt_{i}}{\lvert t_{i}\rvert}$ on $\mathrm{T}_{\lambda,k_{v}}(k_{v})$ . Therefore, through the identification (5.5), the triple of measures $(\lvert\omega_{\mathrm{GL}_{n,k_{v}}}^{can}\rvert_{v},\bigwedge_{i=1}^{n}\frac{dt_{i}}{\lvert t_{i}\rvert},\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v})$ match together topologically by Proposition 3.3, we have

(5.6)

\int_{\mathrm{GL}_{n}(k_{v})}\mathbbm{1}_{G_{T}}(g)\lvert\omega_{\mathrm{GL}_{n,k_{v}}}^{can}\rvert_{v}=\int_{X(k_{v})}\left(\int_{\mathrm{T}_{\lambda,k_{v}}(k_{v})}\mathbbm{1}_{G_{T}}(tg)\bigwedge_{i=1}^{n}\frac{dt_{i}}{\lvert t_{i}\rvert}\right)\ \lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}.

First, we express the inner integration in the right hand side of (5.6) in terms of $\mathbbm{1}_{X(k_{v},T)}$ . We fix an element $g\in\mathrm{GL}_{n}(k_{v})$ , with the unique RQ-decomposition $g=rq$ . For any $t=\operatorname{diag}(t_{1},\ldots,t_{n})\in\mathrm{T}_{\lambda}(k_{v})$ , the condition $tg\in G_{T}$ is equivalent that $\lambda\cdot g\in X(k_{v},T)$ and $1<\lvert t_{i}\rvert r_{ii}<2$ . Thus, we have

(5.7)

\int_{\mathrm{T}_{\lambda,k_{v}}(k_{v})}\mathbbm{1}_{G_{T}}(tg)\bigwedge_{i=1}^{n}\frac{dt_{i}}{\lvert t_{i}\rvert}=\left(\prod_{i=1}^{n}2\int_{1/r_{ii}}^{2/r_{ii}}\frac{dt_{i}}{t_{i}}\right)\mathbbm{1}_{X(k_{v},T)}(\lambda\cdot g)=(2\ln 2)^{n}\mathbbm{1}_{X(k_{v},T)}(\lambda\cdot g).

Next, we compute the left hand side of (5.6) by applying the Iwasawa decomposition

(5.8)

\mathrm{GL}_{n}(k_{v})=ANK,

where

\left\{\begin{array}[]{l}\textit{$A$ is the subgroup of $\mathrm{GL}_{n}(k_{v})$ consisting of positive diagonal matrices};\\ \textit{$N$ is the subgroup of $\mathrm{GL}_{n}(k_{v})$ consisting of unipotent upper triangular matrices};\\ \textit{$K$ is the orthogonal group $\mathrm{O}_{n}(k_{v})\subset\mathrm{GL}_{n}(k_{v})$}.\\ \end{array}\right.

By decomposing $G_{T}\subset\mathrm{GL}_{n}(k_{v})$ , we have

G_{T}=(1,2)^{n}N_{T}K,

where $(1,2)^{n}$ represents the set of diagonal matrices with entries in $(1,2)$ and $N_{T}:=\{n\in N\mid\lVert n^{-1}\lambda n\rVert\leq T\}$ . Indeed, for $g=ank\in\mathrm{GL}_{n}(k_{v})$ , we have $\lVert g^{-1}\lambda g\rVert=\lVert n^{-1}\lambda n\rVert$ .

We decompose the Haar measure $|\omega_{\mathrm{GL}_{n,k_{v}}}^{can}|_{v}$ following [Kna02, Section 8] with respect to the decomposition (5.8). Let $da$ and $dn$ be Haar measures on $A$ and $N$ defined as follows,

(5.9)

\left\{\begin{array}[]{ll}da=\bigwedge_{i=1}^{n}\frac{dt_{i}}{\lvert t_{i}\rvert}&\text{on }A;\\ dn=\bigwedge_{1\leq i<j\leq n}dx_{ij}&\text{on }N.\end{array}\right.

Then [Kna02, Proposition 8.43] implies that there exists the Haar measure $dk$ on $K$ such that

(5.10)

|\omega_{\mathrm{GL}_{n,k_{v}}}^{can}|_{v}=\frac{1}{\lvert\det\rvert^{n}}\bigwedge_{1\leq i,j\leq n}dx_{ij}=da\ dn\ dk.

To obtain the volume of $G_{T}$ with respect to $|\omega_{\mathrm{GL}_{n,k_{v}}}^{can}|_{v}$ , it suffices to compute each factor in the right hand side of the following equation,

vol(G_{T},|\omega_{\mathrm{GL}_{n,k_{v}}}^{can}|_{v})=vol((1,2)^{n},da)\cdot vol(N_{T},dn)\cdot vol(K,dk).

By definition of $da$ , we have $vol((1,2)^{n},da)=(\ln 2)^{n}$ . The formulas in [EMS96, p. 282] yield that

vol(N_{T},dn)\sim w_{n}\lvert\Delta_{\chi}\rvert_{v}^{-\frac{1}{2}}T^{\frac{n(n-1)}{2}}.

Lemma 5.7, provided below, proves that

vol(K,dk)=2^{n}\pi^{\frac{n(n+1)}{4}}\prod_{i=1}^{n}\Gamma(\frac{i}{2})^{-1}.

Combining these results with (5.7), we have

	$\displaystyle vol(X(k_{v},T),\|\omega_{X_{k_{v}}}^{can}\|_{v})$	$\displaystyle=(2\ln 2)^{-n}vol(G_{T},\|\omega_{\mathrm{GL}_{n,k_{v}}}^{can}\|_{v})$
		$\displaystyle=(2\ln 2)^{-n}\cdot vol((1,2)^{n},da)\cdot vol(N_{T},dn)\cdot vol(K,dk)$
		$\displaystyle\sim\pi^{\frac{n(n+1)}{4}}\left(\prod_{i=1}^{n}\Gamma(\frac{i}{2})^{-1}\right)w_{n}\lvert\Delta_{\chi}\rvert_{v}^{-\frac{1}{2}}T^{\frac{n(n-1)}{2}}.$

By the product formula for $\Delta_{\chi}\in k$ , we obtain

\prod_{v\in\infty_{k}}\int_{X(k_{v},T)}\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}\sim\left(w_{n}\pi^{\frac{n(n+1)}{4}}T^{\frac{n(n-1)}{2}}\prod_{i=1}^{n}\Gamma(\frac{i}{2})^{-1}\right)^{[k:\mathbb{Q}]}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\lvert\Delta_{\chi}\rvert_{v}^{\frac{1}{2}}.

∎

Lemma 5.7.

Let $dk$ be the Haar measure on $K=\mathrm{O}_{n}(k_{v})$ satisfying (5.10). Then we have

vol(K,dk)=2^{n}\pi^{\frac{n(n+1)}{4}}\prod_{i=1}^{n}\Gamma(\frac{i}{2})^{-1}.

Proof.

To obtain the volume of $K$ , we introduce an alternative Haar measure $dk^{\prime}$ on $K$ using another decomposition

(5.11)

\mathrm{GL}_{n}(k_{v})=KMAN,

where $M$ denotes the set of signature matrices, and $K$ , $A$ , and $N$ are as defined in (5.8). Like defining $dk$ in (5.10), we should fix the other Haar measures on $M$ , $A$ , and $N$ . We define $da$ on $A$ and $dn$ on $N$ as (5.9).

We choose a Haar measure $dm$ on $M$ such that

(5.12)

|\omega_{\mathrm{GL}_{n,k_{v}}}^{can}|_{v}=e^{2\rho\log a}d\bar{n}\ dm\ da\ dn,

where $2\rho$ denotes the sum of positive roots in the root system of $\operatorname{GL}_{n}$ (cf. [Kna02, Proposition 8.45]). Here, $d\bar{n}=\bigwedge_{1\leq j<i\leq n}dx_{ij}$ denotes a Haar measure on the subgroup $\bar{N}$ of $\mathrm{GL}_{n}(k_{v})$ consisting of unipotent lower triangular matrices. We claim that $dm$ is the counting measure on $M$ so that $vol(M,dm)=2^{n}$ . To express the Haar measure $\lvert\omega_{\mathrm{GL}_{n,k_{v}}}^{can}\rvert_{v}$ in terms of $da$ , $dn$ , and $d\bar{n}$ , we apply Lemma 5.8, provided below, which describe the Jacobian of the LDU-decomposition. Let $a=\operatorname{diag}(t_{1},\ldots,t_{n})$ be the diagonal matrix, and $\bar{n}$ (resp. $n$ ) be a unipotent lower (resp. upper) triangular matrix with entries $x_{ij}$ for $1\leq j<i\leq n$ (resp. $1\leq i<j\leq n$ ). By a change of variables, we obtain

	$\displaystyle\frac{1}{\lvert\det(x_{ij})\rvert^{n}}\bigwedge_{1\leq i,j\leq n}dx_{ij}$	$\displaystyle=\frac{\lvert t_{1}^{2n-2}t_{2}^{2n-4}\cdots t_{n-1}\rvert}{\lvert t_{1}t_{2}\cdots t_{n}\rvert^{n}}\bigwedge_{1\leq j<i\leq n}dx_{ij}\wedge\bigwedge_{1\leq i\leq n}dt_{i}\wedge\bigwedge_{1\leq i<j\leq n}dx_{ij}$
		$\displaystyle=\left\|t_{1}^{n-1}t_{2}^{n-3}\cdots t_{n}^{-n+1}\right\|\bigwedge_{1\leq j<i\leq n}dl_{ij}\wedge\frac{1}{\lvert t_{1}t_{2}\cdots t_{n}\rvert}\bigwedge_{1\leq i\leq n}dt_{i}\wedge\bigwedge_{1\leq i<j\leq n}dx_{ij}$
		$\displaystyle=e^{2\rho\log a}d\bar{n}\ da\ dn.$

The last equality follows from the root system of $\operatorname{GL}_{n}$ , for $a=\operatorname{diag}(t_{1},\cdots,t_{n})\in A$ ,

e^{2\rho\log a}=\left|\prod_{1\leq i<j\leq n}\frac{t_{i}}{t_{j}}\right|=\lvert t_{1}^{n-1}t_{2}^{n-3}\cdots t_{n}^{-n+1}\rvert.

The decomposition (5.12) then implies that each element in $M$ has measure 1.

[Kna02, Proposition 8.44] implies that there exists a Haar measure $dk^{\prime}$ on $K$ such that

(5.13)

|\omega_{\mathrm{GL}_{n,k_{v}}}^{can}|_{v}=e^{2\rho\log a}dk^{\prime}\ dm\ da\ dn.

By [Kna02, Proposition 8.43], exchanging $dk$ and $da\ dn$ in (5.10) yields

da\ dn\ dk=e^{2\rho\log a}dk\ da\ dn.

Combining this with (5.13), we have $dk=vol(M,dm)dk^{\prime}=2^{n}dk^{\prime}$ . [Kna02, Proposition 8.46] determines the normalization of $dk^{\prime}$ ,

\int_{K}dk^{\prime}=\int_{\bar{N}}e^{-2\rho\log a(\bar{n})}d\bar{n}=\pi^{\frac{n(n+1)}{4}}\prod_{i=1}^{n}\Gamma(\frac{i}{2})^{-1},

where $a(\bar{n})$ is the diagonal matrix in the decomposition of $\bar{n}$ with respect to (5.11) and the second equality follows from [Vos98, Theorem 1, 14.10]. Therefore, we conclude that

vol(K,dk)=2^{n}vol(K,dk^{\prime})=2^{n}\pi^{\frac{n(n+1)}{4}}\prod_{i=1}^{n}\Gamma(\frac{i}{2})^{-1}.

∎

Lemma 5.8.

Let

\phi_{n}:\mathbb{R}^{\frac{n(n-1)}{2}}\times(\mathbb{R}_{>0})^{n}\times\mathbb{R}^{\frac{n(n-1)}{2}}\to\mathrm{M}_{n}(\mathbb{R})\cong\mathbb{R}^{n^{2}}

defined by

(x_{ij})_{i>j},(t_{i}),(x_{ij})_{i<j}\mapsto\bar{n}an

where $\bar{n}$ (resp. $n$ ) is the unipotent lower (resp. upper) triangular matrix with the entries $x_{ij}$ for $1\leq j<i\leq n$ (resp. $1\leq i<j\leq n$ ), and $a=\operatorname{diag}(t_{1},\ldots,t_{n})$ . Then the absolute value of the Jacobian of $\phi_{n}$ is $|\det J_{\phi_{n}}|=\lvert t_{1}^{2n-2}t_{2}^{2n-4}\cdots t_{n-1}\rvert$ .

Proof.

We use an induction on $n$ . For $n=1$ and $\phi=\operatorname{id}$ , we have the Jacobian of $\phi_{n}$ is 1. Suppose that the lemma holds for $n=k-1$ . We denote the blocks of $l$ , $d$ , and $u$ by

l=\left(\begin{array}[]{c|ccc}1&&0&\\ \hline\cr x_{21}&&&\\ \vdots&&\bar{n}^{\prime}&\\ x_{n1}&&&\end{array}\right),\ \ d=\left(\begin{array}[]{c|ccc}t_{1}&&0&\\ \hline\cr&&&\\ 0&&a^{\prime}&\\ &&&\end{array}\right),\ \ u=\left(\begin{array}[]{c|ccc}1&x_{12}&\cdots&x_{1n}\\ \hline\cr&&&\\ 0&&n^{\prime}&\\ &&&\end{array}\right).

Then, we have

\bar{n}an=\left(\begin{array}[]{c|ccc}t_{1}&t_{1}x_{12}&\cdots&t_{1}x_{1n}\\ \hline\cr t_{1}x_{21}&&&\\ \vdots&&\bar{n}^{\prime}a^{\prime}n^{\prime}&\\ t_{1}x_{n1}&&&\end{array}\right).

Since the entries of $\bar{n}^{\prime}a^{\prime}n^{\prime}$ are independent of $t_{1}$ , $x_{i1}$ , and $x_{1i}$ for $2\leq i\leq n$ , the Jacobian matrix $J_{\phi_{n}}$ equals $\operatorname{diag}(1,t_{1},\ldots,t_{1})\oplus J_{\phi_{n-1}}$ . This completes the proof by the inductive assumption. ∎

5.3. Computation on non-Archimedean places

Now we consider the integration $\tilde{\xi}_{v}$ on $X(\mathcal{O}_{k_{v}})$ for $v\in\Omega_{k}\setminus\infty_{k}$ . In the case that the evaluation $\tilde{\xi}_{v}$ is trivial, we find that the integration is associated with the orbital integral for $\mathfrak{gl}_{n}$ (cf. Proposition 5.9). Otherwise, $K_{v}/k_{v}$ is a Galois field extension by Proposition 5.1 and Proposition 5.5. In this case, we will address the two cases separately; when $K_{v}/k_{v}$ is ramified and when $K_{v}/k_{v}$ is unramified.

In particular, when $K_{v}/k_{v}$ is unramified, we use the Langlands-Shelstad fundamental lemma for $\mathfrak{sl}_{n}$ , proven by Ngô in [Ngfrm[o]–0]. Because of the complexity and the various involved concepts, we will provide it in Section 6.

5.3.1. The case that the evaluation $\tilde{\xi}_{v}$ is trivial

By Proposition 5.1 and Proposition 5.5, the evaluation $\tilde{\xi}_{v}$ is trivial if the following cases hold.

\left\{\begin{array}[]{l}K/k\textit{ is not a Galois field extension};\\ K/k\textit{ is a Galois field extension and $K_{v}/k_{v}$ splits};\\ K/k\textit{ is a Galois field extension, $K_{v}/k_{v}$ is a field extension, and $\xi\in\operatorname{Br}X/\operatorname{Br}k$ is trivial}.\end{array}\right.

Otherwise, $K_{v}/k_{v}$ is a Galois field extension and $\xi\in\operatorname{Br}X/\operatorname{Br}k$ is non-trivial, then the evaluation $\tilde{\xi}_{v}$ is non-trivial by Proposition 5.4.

Proposition 5.9.

For $v\in\Omega_{k}\setminus\infty_{k}$ , we identify $\mathrm{M}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})$ with $\mathfrak{gl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})$ so that $x_{0}\in\mathfrak{gl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})$ . Then, we have the following equation.

\int_{X(\mathcal{O}_{k_{v}})}|\omega_{X_{k_{v}}}^{can}|_{v}=\int_{\mathrm{T}_{x_{0}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v})}\mathbbm{1}_{\mathfrak{gl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})}(g^{-1}x_{0}g)\ |\omega_{X_{k_{v}}}^{can}|_{v}.

Here, the right hand side is the orbital integral of $x_{0}$ for $\mathfrak{gl}_{n}$ with respect to the measure $|\omega_{X_{k_{v}}}^{can}|_{v}$ (See [Yun13, Section 1.3] for the definition of the orbital integral for $\mathfrak{gl}_{n}$ with respect to the measure $\frac{dg_{v}}{dt_{v}}$ ).

Proof.

Via the identification $X\cong\mathrm{T}_{x_{0}}\backslash\mathrm{GL}_{n}$ in Proposition 2.1.(1) and Proposition 2.3, we have

X(k_{v})\cong\mathrm{T}_{x_{0}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v}).

Under this identification, we have $X(\mathcal{O}_{k_{v}})\cong\{g\in\mathrm{T}_{x_{0}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v})\mid g^{-1}x_{0}g\in\mathfrak{gl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})\}$ and this yields the formula. ∎

In general, obtaining the closed formula of the orbital integral for $\mathfrak{gl}_{n}$ is a challenging problem, and the difficulty increases as the rank $n$ grows. In certain cases, including $n=2,3$ , we will summarize the closed formula of the orbital integrals for $\mathfrak{gl}_{n}$ in Proposition 7.3-7.4.

Remark 5.10.

Suppose that $\chi(x)$ is a irreducible polynomial such that $\mathcal{O}_{K}=\mathcal{O}_{k}[x]/(\chi(x))$ (cf. assumption (2) in (1.3)). By [Yun13, Theorem 1.5], we have

(5.14)

\int_{X(\mathcal{O}_{k_{v}})}\frac{dg_{v}}{dt_{v}}=\int_{\mathrm{T}_{x_{0}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v})}\mathbbm{1}_{\mathfrak{gl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})}(g^{-1}x_{0}g)\ \frac{dg_{v}}{dt_{v}}=1,

where the measure $\frac{dg_{v}}{dt_{v}}$ is defined in Lemma 3.7. Indeed, according to loc. cit. the orbtal integral of $x_{0}$ for $\mathfrak{gl}_{n}$ with respect to $\frac{dg_{v}}{dt_{v}}$ is an integer formulated by a $\mathbb{Z}$ -polynomial in $q_{v}$ whose leading term is $q_{v}^{S_{v}(\chi)}$ . Here $S_{v}(\chi)$ is the $\mathcal{O}_{k_{v}}$ -module length between $\mathcal{O}_{K_{v}}$ and $\mathcal{O}_{k_{v}}[x]/(\chi(x))$ . Therefore the assumption yields that $S_{v}(\chi)=0$ and thus (5.14) holds.

5.3.2. The case that the evaluation $\tilde{\xi}_{v}$ is non-trivial and $K_{v}/k_{v}$ is ramified

Proposition 5.11.

Suppose that $K_{v}/k_{v}$ is a ramified Galois extension of prime degree $n$ . For $\xi\in\mathrm{Br}(X)$ whose evaluation $\tilde{\xi}_{v}$ is non-trivial, we have

\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ \lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}=0.

Proof.

[Neu99, (1.7) Proposition, Chapter 5] states that $K_{v}/k_{v}$ is ramified if and only if there exists an element $u\in\mathcal{O}_{k_{v}}^{\times}$ such that $u\notin\operatorname{Nm}_{K_{v}/k_{v}}K_{v}^{\times}$ . Therefore the image of $u$ under the Artin reciprocity isomorphism $\phi_{K_{v}/k_{v}}$ , defined in Proposition 5.2, is non-trivial in $\operatorname{Gal}(K_{v}/k_{v})$ . On the other hand, by the assumption that $\mathrm{deg}(K/k)$ is a prime number, we have that $\operatorname{Gal}(K/k)=\operatorname{Gal}(K_{v}/k_{v})$ . Then the kernel of $\Psi(\xi)\in\operatorname{Hom}(\operatorname{Gal}(K_{v}/k_{v}),\mathbb{Q}/\mathbb{Z})$ is trivial, where $\Psi$ is defined in Proposition 5.2. We then have that $\exp 2\pi i\Psi(\xi)(\phi_{K_{v}/k_{v}}(u))\neq 1$ .

Let $g:=\operatorname{diag}(u,1,\ldots,1)\in\mathrm{GL}_{n,\mathcal{O}_{k}}(\mathcal{O}_{k})$ . Since $\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}$ is a $\mathrm{GL}_{n}(k_{v})$ -invariant measure and $X(\mathcal{O}_{k})$ is stable under the $\mathrm{GL}_{n,\mathcal{O}_{k}}(\mathcal{O}_{k})$ -action, we have

	$\displaystyle\int_{X(k_{v})}\tilde{\xi}_{v}(x)\ \mathbbm{1}_{X(\mathcal{O}_{k})}\ \lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}(x)$	$\displaystyle=\int_{X(k_{v})}\tilde{\xi}_{v}(x\cdot g)\ \mathbbm{1}_{X(\mathcal{O}_{k})}(x\cdot g)\ \lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}$
		$\displaystyle=\exp 2\pi i\Psi(\xi)(\phi_{K_{v}/k_{v}}(u))\int_{X(k_{v})}\tilde{\xi}_{v}(x)\ \mathbbm{1}_{X(\mathcal{O}_{k})}(x)\ \lvert\omega_{X_{k_{v}}}^{can}\rvert$

by Proposition 5.2. Therefore, we have

\int_{X(\mathcal{O}_{k})}\tilde{\xi}_{v}(x)\ \lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}=0.

∎

6. Computation on non-Archimedean places: the case that the evaluation $\tilde{\xi}_{v}$ is non-trivial and $K_{v}/k_{v}$ is unramified

By Proposition 5.3, the integration of $\tilde{\xi}_{v}$ on $X(\mathcal{O}_{k_{v}})$ turns to be the weighted summation for each $\mathrm{SL}_{n}(k_{v})$ -orbit. In the case that $K_{v}/k_{v}$ is unramified and $\tilde{\xi}_{v}$ is non-trivial, we will show that the enumeration of this summation follows from the Langlands-Shelstad fundamental lemma, which is proven in [Ngfrm[o]–0].

We first collect the necessary materials to state the Langlands-Shelstad fundamental lemma for $\mathfrak{sl}_{n}$ in Section 6.1. Summing up our observations, we will provide the conclusion in Section 6.2. Throughout this section, we fix $v\in\Omega_{k}\setminus\infty_{k}$ such that $K_{v}/k_{v}$ is an unramified Galois field extension.

6.1. Measures on each $\mathrm{SL}_{n}(k_{v})$ -orbit in $X({k_{v}})$

In the Langlands-Shelstad fundamental lemma, the measure $\frac{dh_{v}}{ds^{\prime}_{v}}$ defined in Lemma 6.1 is used, whereas our integration is defined with respect to the measure $|\omega_{X_{k_{v}}}^{can}|_{v}$ . Therefore, it is necessary to describe the difference between two measures $|\omega_{X_{k_{v}}}^{can}|_{v}$ and $\frac{dh_{v}}{ds^{\prime}_{v}}$ on $x_{0}^{\prime}\cdot\mathrm{SL}_{n}(k_{v})\cong\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v})$ , for each $x_{0}^{\prime}\in\mathcal{R}_{x_{0},v}$ .

To address this, we will define a measure $|\omega_{X_{k_{v}},x_{0}^{\prime}}^{can}|_{v}$ on each $x_{0}^{\prime}\cdot\mathrm{SL}_{n}(k_{v})$ , which behaves as a bridge between two measures $|\omega_{X_{k_{v}}}^{can}|_{v}$ and $\frac{dh_{v}}{ds^{\prime}_{v}}$ (cf. Lemma 6.1-6.2). The construction of this measure will follow the method used in Section 3.2.2.

6.1.1. Integral models for $\mathrm{S}_{x_{0}^{\prime},k_{v}}$ when $K_{v}/k_{v}$ is an unramified field extension

As in Subsection 3.2.1, we first construct an integral model for $\mathrm{S}_{x_{0}^{\prime},k_{v}}$ . Since Lemma 2.2 is independent of the choice of $x_{0}$ , we have the following description of $\mathrm{S}_{x_{0}^{\prime},k_{v}}\cong\mathrm{S}_{x_{0},k_{v}}$ ,

\mathrm{S}_{x_{0}^{\prime},k_{v}}\cong\ker\bigg{(}\operatorname{Nm}_{K_{v}/k_{v}}:\mathrm{R}_{K_{v}/k_{v}}(\mathbb{G}_{m,K_{v}})\rightarrow\mathbb{G}_{m,k_{v}}\bigg{)}.

By [Vos98, Section 10.5], the standard integral model for $\mathrm{S}_{x_{0}^{\prime},k_{v}}$ is given by $\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}:=\mathrm{R}^{(1)}_{\mathcal{O}_{K_{v}}/\mathcal{O}_{k_{v}}}(\mathbb{G}_{m,\mathcal{O}_{K_{v}}})$ which is the kernel of the following morphism,

\operatorname{Nm}_{\mathcal{O}_{K_{v}}/\mathcal{O}_{k_{v}}}:\mathrm{R}_{\mathcal{O}_{K_{v}}/\mathcal{O}_{k_{v}}}(\mathbb{G}_{m,\mathcal{O}_{K_{v}}})\rightarrow\mathbb{G}_{m,\mathcal{O}_{k_{v}}},\ g\mapsto\prod_{\sigma\in\operatorname{Gal}(K_{v}/k_{v})}\sigma(g).

Since $K_{v}/k_{v}$ is unramified, [Vos98, Theorem 2, 10.3] yields that $\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}$ is an $\mathcal{O}_{k_{v}}$ -torus, with $\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}\otimes_{\mathcal{O}_{k_{v}}}\mathcal{O}_{K_{v}}\cong\mathbb{G}_{m,\mathcal{O}_{K_{v}}}^{n-1}$ which is smooth over $\mathcal{O}_{K_{v}}$ and has a connected fiber over each point in $\mathrm{Spec}(\mathcal{O}_{K_{v}})$ . Then by [Poo17, Theorem 4.3.7], these properties descend to $\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}$ as well and thus $\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}$ is also smooth over $\mathcal{O}_{k_{v}}$ and its fiber over each point in $\mathrm{Spec}(\mathcal{O}_{k_{v}})$ is connected. Indeed, $\mathrm{Spec}(\mathcal{O}_{K_{v}})\rightarrow\mathrm{Spec}(\mathcal{O}_{k_{v}})$ is a fpqc morphism. Moreover, the identity component $\mathcal{S}_{x_{0}^{\prime}}^{0}$ of the Néron-Raynaud model $\mathcal{S}_{x_{0}^{\prime}}$ (cf. [BLR90, Chapter 10]) for $\mathrm{S}_{x_{0}^{\prime},k_{v}}$ coincides with the standard integral model by [Bit11, Corollary 1.4].

6.1.2. Volume forms on $X_{k_{v}}\cong\mathrm{S}_{x_{0}^{\prime},k_{v}}\backslash\mathrm{SL}_{n,k_{v}}$

We construct translation-invariant volume forms $\omega_{\mathrm{SL}_{n,k_{v}}}^{can}$ on $\mathrm{SL}_{n,k_{v}}$ , $\omega_{\mathrm{S}_{x_{0}^{\prime},k_{v}}}^{can}$ on $\mathrm{S}_{x_{0}^{\prime},k_{v}}$ , and the corresponding $\mathrm{SL}_{n,k_{v}}$ -invariant volume form $\omega_{X_{k_{v}},x_{0}^{\prime}}^{can}$ on $X_{k_{v}}$ as follows.

(1)

On $\mathrm{SL}_{n,k_{v}}$ , we define $\omega_{\mathrm{SL}_{n,k_{v}}}^{can}$ to be a translation-invariant volume form in $\Omega^{n^{2}-1}_{\mathrm{SL}_{n,k_{v}}/k_{v}}(\mathrm{SL}_{n,k_{v}})$ which generates the $\mathcal{O}_{k_{v}}$ -submodule $\operatorname{Hom}(\bigwedge^{n^{2}-1}\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}),\mathcal{O}_{k_{v}})$ . Then the volume form $\omega_{\mathrm{SL}_{n,k_{v}}}^{can}$ has good reduction (mod $\pi_{v}$ ) in the sense of [Gro97, Section 4]. We then have the following equation by [Gro97, Proposition 4.7],

\mathrm{vol}(|\omega_{\mathrm{SL}_{n,k_{v}}}^{can}|_{v},\mathrm{SL}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}))=\frac{\#\mathrm{SL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})}{q_{v}^{n^{2}-1}}.

(2)

On $\mathrm{S}_{x_{0}^{\prime},k_{v}}$ , we define $\omega_{\mathrm{S}_{x_{0}^{\prime},k_{v}}}^{can}$ to be a translation-invariant volume form in $\Omega^{n-1}_{\mathrm{S}_{x_{0}^{\prime},k_{v}}/k_{v}}(\mathrm{S}_{x_{0}^{\prime},k_{v}})$ which generates the $\mathcal{O}_{k_{v}}$ -submodule $\mathrm{Hom}(\bigwedge^{n-1}\mathrm{Lie}(\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}})(\mathcal{O}_{k_{v}}),\mathcal{O}_{k_{v}})$ . Since $\mathcal{S}_{x_{0}^{\prime}}^{0}\cong\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}$ , the volume form $\omega_{\mathrm{S}_{x_{0}^{\prime},k_{v}}}^{can}$ has good reduction (mod $\pi_{v}$ ) in the sense of [Gro97, Section 4]. We then have the following equation by [Bit11, Proposition 2.14],

\mathrm{vol}(|\omega_{\mathrm{S}_{x_{0}^{\prime},k_{v}}}^{can}|_{v},\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}))=\frac{\#\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}(\kappa_{v})}{q_{v}^{n-1}}.

(3)

On $X_{k_{v}}\cong\mathrm{S}_{x_{0}^{\prime},k_{v}}\backslash\mathrm{SL}_{n,k_{v}}$ , we define $\omega_{X_{k_{v}},x_{0}^{\prime}}^{can}$ to be a $\mathrm{SL}_{n,k_{v}}$ -invariant volume form such that $\omega_{\mathrm{SL}_{n,k_{v}}}^{can}=\omega_{X_{k_{v}},x_{0}^{\prime}}^{can}\cdot\omega_{\mathrm{S}_{x_{0}^{\prime},k_{v}}}^{can}$ . Then by Proposition 3.3, $|\omega_{X_{k_{v}},x_{0}^{\prime}}^{can}|_{v}$ defines a measure on $x_{0}^{\prime}\cdot\mathrm{SL}_{n}(k_{v})\cong\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v})$ which matches topologically together with $|\omega_{\mathrm{SL}_{n,k_{v}}}^{can}|_{v}$ and $|\omega_{\mathrm{S}_{x_{0}^{\prime},k_{v}}}^{can}|_{v}$ .

Comparing with the quotient measure $\frac{dh_{v}}{ds_{v}^{\prime}}$ induced from the canonically normalized Haar measures $dh_{v}$ on $\mathrm{SL}_{n}(k_{v})$ and $ds_{v}^{\prime}$ on $\mathrm{S}_{x_{0}^{\prime}}(k_{v})$ , we have the following relation.

Lemma 6.1.

Let $\frac{dh_{v}}{ds^{\prime}_{v}}$ be the quotient measure on $x_{0}^{\prime}\cdot\mathrm{SL}_{n}(k_{v})$ , which is defined via the isomorphism $x_{0}^{\prime}\cdot\mathrm{SL}_{n}(k_{v})\cong\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v})$ , where

\left\{\begin{array}[]{l}\textit{$dh_{v}$ be the Haar measure on $\mathrm{SL}_{n}(k_{v})$ such that $\mathrm{vol}(dh_{v},\mathrm{SL}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}))=1$};\\ \textit{$ds^{\prime}_{v}$ be the Haar measure on $\mathrm{S}_{x_{0}^{\prime}}(k_{v})$ such that $\mathrm{vol}(ds^{\prime}_{v},\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}))=1$}.\end{array}\right.

We have

|\omega_{X_{k_{v}},x_{0}^{\prime}}^{can}|_{v}=\frac{\#\mathrm{SL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-(n^{2}-1)}}{\#\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-(n-1)}}\cdot\frac{dh_{v}}{ds^{\prime}_{v}}.

On the other hand, the measure $|\omega_{X_{k_{v}},x_{0}^{\prime}}^{can}|_{v}$ coincides with the restriction of $|\omega_{X_{k_{v}}}^{can}|_{v}$ on $x_{0}^{\prime}\cdot\mathrm{SL}_{n}(k_{v})$ , by the following lemma.

Lemma 6.2.

We have

|\omega_{X_{k_{v}}}^{can}|_{v}\Big{|}_{x_{0}^{\prime}\cdot\mathrm{SL}_{n}(k_{v})}=|\omega_{X_{k_{v}},x_{0}^{\prime}}^{can}|_{v}.

Proof.

In this proof, to ease the notations, we will omit the subscript $k_{v}$ for each scheme over $k_{v}$ if there is no confusion. For the centralizer $\mathrm{T}_{x_{0}^{\prime}}$ of $x_{0}^{\prime}$ in $\mathrm{GL}_{n}$ , we define $\omega_{\mathrm{T}_{x_{0}^{\prime}}}^{can}$ following Section 3.2.2. Since there exists $g\in\mathrm{GL}_{n}(k_{v})$ such that $g^{-1}x_{0}g=x_{0}^{\prime}$ , we have $\mathrm{T}_{x_{0}}(k_{v})\cong\mathrm{T}_{x_{0}^{\prime}}(k_{v})$ and $X(k_{v})\cong\mathrm{T}_{x_{0}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v})\cong\mathrm{T}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{GL}_{n}(k_{v})$ . Therefore $|\omega_{\mathrm{T}_{x_{0}^{\prime}}}\backslash\omega_{\mathrm{GL}_{n}}|_{v}$ and $|\omega_{\mathrm{T}_{x_{0}}}\backslash\omega_{\mathrm{GL}_{n}}|_{v}$ are the same measure on $X(k_{v})$ by Lemma 3.7. Here we note that $\mathrm{S}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}$ is embedded in $\mathrm{T}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}$ as the kernel of $\mathrm{Nm}_{\mathcal{O}_{K_{v}}/\mathcal{O}_{k_{v}}}:{\mathrm{T}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}}\to{\mathbb{G}_{m,\mathcal{O}_{k_{v}}}}$ . We consider the following commutative diagram defined over $\mathcal{O}_{k_{v}}$ ,

(6.1)

Here the first horizontal morphism $\operatorname{Nm}_{\mathcal{O}_{K_{v}}/\mathcal{O}_{k_{v}}}$ is smooth over $\mathcal{O}_{k_{v}}$ since $K_{v}/k_{v}$ is unramified. On the other hand, the second horizontal morphism $\mathrm{det}$ is also smooth over $\mathcal{O}_{k_{v}}$ since its differential $d(\mathrm{det})=\mathrm{tr}:\mathfrak{gl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})\rightarrow\mathbb{G}_{a,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})$ is surjective.

We first claim that there exists a translation-invariant gauge form $\omega_{\mathbb{G}_{m}}$ on $\mathbb{G}_{m}\cong\mathrm{SL}_{n}\backslash\mathrm{GL}_{n}\cong\mathrm{S}_{x_{0}^{\prime}}\backslash\mathrm{T}_{x_{0}^{\prime}}$ satisfying the following two equations, up to multiplication of a unit in $\mathcal{O}_{k_{v}}$ ,

\omega_{\mathrm{GL}_{n}}^{can}=\omega_{\mathbb{G}_{m}}\cdot\omega_{\mathrm{SL}_{n}}^{can}\textit{ and }\omega_{\mathrm{T}_{x_{0}^{\prime}}}^{can}=\omega_{\mathbb{G}_{m}}^{\prime}\cdot\omega_{\mathrm{S}_{x_{0}^{\prime}}}^{can}.

Since $\omega_{\mathrm{SL}_{n}}^{can}$ generates an $\mathcal{O}_{k_{v}}$ -module $\mathrm{Hom}(\bigwedge^{n^{2}-1}\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}),\mathcal{O}_{k_{v}})$ , its lifting $\widetilde{\omega}_{\mathrm{SL}_{n}}^{can}$ on $\mathrm{GL}_{n}$ (cf. Definition 3.2), which is in $\mathrm{Hom}(\bigwedge^{n^{2}-1}\mathfrak{gl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}),\mathcal{O}_{k_{v}})$ , does not vanish under the modulo $\pi_{v}$ reduction. Since $\omega_{\mathrm{GL}_{n}}^{can}$ generates an $\mathcal{O}_{k_{v}}$ -module $\mathrm{Hom}(\bigwedge^{n^{2}}\mathfrak{gl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}),\mathcal{O}_{k_{v}})$ and $\det:\mathrm{GL}_{n,\mathcal{O}_{k_{v}}}\to\mathbb{G}_{m,\mathcal{O}_{k_{v}}}$ is smooth over $\mathcal{O}_{k_{v}}$ , a translation-invariant volume form $\omega_{\mathbb{G}_{m}}$ on $\mathbb{G}_{m}$ , such that $\omega_{\mathrm{GL}_{n}}^{can}=\det^{*}\omega_{\mathbb{G}_{m}}\wedge\tilde{\omega}_{\mathrm{SL}_{n}}^{can}$ , has good reduction (mod $\pi$ ). Since $\mathrm{Nm}_{\mathcal{O}_{K_{v}}/\mathcal{O}_{k_{v}}}:{\mathrm{T}_{x_{0}^{\prime},\mathcal{O}_{k_{v}}}}\to{\mathbb{G}_{m,\mathcal{O}_{k_{v}}}}$ is smooth over $\mathcal{O}_{k_{v}}$ , a translation-invariant gauge form $\omega_{\mathbb{G}_{m}}^{\prime}$ such that $\omega_{\mathrm{T}_{x_{0}^{\prime}}}^{can}=(\operatorname{Nm}_{\mathcal{O}_{K_{v}}/\mathcal{O}_{k_{v}}})^{*}\omega_{\mathbb{G}_{m}}^{\prime}\wedge\tilde{\omega}_{\mathrm{S}_{x_{0}^{\prime}}}^{can}$ also has good reduction (mod $\pi$ ) by the same argument. By [Gro97, p. 293], $\omega_{\mathbb{G}_{m}}^{\prime}$ differs from $\omega_{\mathbb{G}_{m}}$ up to multiplication of a unit in $\mathcal{O}_{k_{v}}$ . This yields the claim.

Now we denote a lifting of a volume form $\omega$ on $\mathrm{S}_{x_{0}^{\prime}},\mathrm{T}_{x_{0}^{\prime}},$ or $\mathrm{SL}_{n}$ in the sense of Definition 3.2, along the embedding into $\mathrm{GL}_{n}$ , by $\tilde{\omega}^{\mathrm{GL}_{n}}$ . Since $\omega_{\mathrm{GL}_{n}}^{can}=\omega_{X}^{can}\cdot\omega_{\mathrm{T}_{x_{0}^{\prime}}}^{can}$ , we have

\omega_{\mathrm{GL}_{n}}^{can}=\varphi_{\mathrm{GL}_{n}}^{*}\omega^{can}_{X}\wedge\tilde{\omega}_{\mathrm{T}_{x_{0}^{\prime}}}^{can,\mathrm{GL}_{n}},

where $\varphi_{\mathrm{GL}_{n}}:\mathrm{GL}_{n}\rightarrow X$ represents the mapping defined by $g\rightarrow g^{-1}x_{0}^{\prime}g$ . The above claim yields that

(6.2)

\mathrm{det}^{*}\omega_{\mathbb{G}_{m}}\wedge\tilde{\omega}_{\mathrm{SL}_{n}}^{can,\mathrm{GL}_{n}}=\varphi_{\mathrm{GL}_{n}}^{*}\omega^{can}_{X}\wedge\tilde{\omega}_{\mathrm{T}_{x_{0}^{\prime}}}^{can,\mathrm{GL}_{n}},

where $\tilde{\omega}_{\mathrm{T}_{x_{0}^{\prime}}}^{can,\mathrm{GL}_{n}}$ is equal to a lifting of $\operatorname{Nm}_{\mathcal{O}_{K_{v}}/\mathcal{O}_{k_{v}}}^{*}\omega_{\mathbb{G}_{m}}\wedge\tilde{\omega}_{\mathrm{S}_{x_{0}^{\prime}}}^{can}$ , along $\mathrm{T}_{x_{0}^{\prime}}\hookrightarrow\mathrm{GL}_{n}$ . Since the right hand side of (6.2) is independent of the choice of a lifting $\tilde{\omega}_{\mathrm{T}_{x_{0}^{\prime}}}^{can,\mathrm{GL}_{n}}$ (cf. Definition 3.2), we choose it to be the exterior product of liftings of $\operatorname{Nm}_{\mathcal{O}_{K_{v}}/\mathcal{O}_{k_{v}}}^{*}\omega_{\mathbb{G}_{m}}$ and $\tilde{\omega}_{\mathrm{S}_{x_{0}^{\prime}}}^{can,\mathrm{GL}_{n}}$ on $\mathrm{GL}_{n}$ . By the commutative diagram (6.1), $\det^{*}\omega_{\mathbb{G}_{m}}$ is a lifting of $\mathrm{Nm}^{*}_{\mathcal{O}_{K_{v}}/\mathcal{O}_{k_{v}}}\omega_{\mathbb{G}_{m}}$ along $\mathrm{T}_{x_{0}^{\prime}}\hookrightarrow\mathrm{GL}_{n}$ . We then have

\mathrm{det}^{*}\omega_{\mathbb{G}_{m}}\wedge\tilde{\omega}^{can,\mathrm{GL}_{n}}_{\mathrm{SL}_{n}}=\varphi_{\mathrm{GL}_{n}}^{*}\omega_{X}^{can}\wedge\mathrm{det}^{*}\omega_{\mathbb{G}_{m}}\wedge\tilde{\omega}_{\mathrm{S}_{x_{0}^{\prime}}}^{can,\mathrm{GL}_{n}}.

Therefore the $(n^{2}-1)$ -form $\tilde{\omega}^{can,\mathrm{GL}_{n}}_{\mathrm{SL}_{n}}+\varphi_{\mathrm{GL}_{n}}^{*}\omega_{X}^{can}\wedge\tilde{\omega}_{\mathrm{S}_{x_{0}^{\prime}}}^{can,\mathrm{GL}_{n}}$ vanishes under the exterior product with $\mathrm{det}^{*}\omega_{\mathbb{G}_{m}}$ . Since $\det^{*}\omega_{\mathbb{G}_{m}}$ is of degree 1, there exists an $(n^{2}-2)$ -differential form $\omega^{\prime}$ on $\mathrm{GL}_{n}$ such that

\tilde{\omega}^{can,\mathrm{GL}_{n}}_{\mathrm{SL}_{n}}+\varphi_{\mathrm{GL}_{n}}^{*}\omega_{X}^{can}\wedge\tilde{\omega}_{\mathrm{S}_{x_{0}^{\prime}}}^{can,\mathrm{GL}_{n}}=\mathrm{det}^{*}\omega_{\mathbb{G}_{m}}\wedge\omega^{\prime}.

Restricting this equation on $\mathrm{SL}_{n}$ , by plugging $\det=1$ with $d(\det)=0$ , we then have the following equation up to sign

\omega_{\mathrm{SL}_{n}}^{can}=\varphi_{\mathrm{SL}_{n}}^{*}\omega_{X}^{can}\wedge\tilde{\omega}_{\mathrm{S}_{x_{0}^{\prime}}}^{can,\mathrm{SL}_{n}},

where $\varphi_{\mathrm{SL}_{n}}:\mathrm{SL}_{n}\rightarrow X$ represents the mapping defined by $g\mapsto g^{-1}x_{0}^{\prime}g$ . This yields that $|\omega_{X_{k_{v}}}^{can}|_{v}=|\omega_{X_{k_{v}},x_{0}^{\prime}}^{can}|_{v}$ on $\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v})\cong x_{0}^{\prime}\cdot\mathrm{SL}_{n}(k_{v})$ by Proposition 3.3. ∎

6.2. Computation using the Langlands-Shelstad fundamental lemma

The Langlands-Shelstad fundamental lemma, stated in [Ngfrm[o]–0, Theorem 1], represents the equation between two orbital integrals encoded by an endoscopic data (cf. [Hal05, Section 2-3] and [Kot84, Section 7.1]). We find that the evaluation $\tilde{\xi}_{v}$ on $X(\mathcal{O}_{k_{v}})$ is related to an endoscopic data for $\mathrm{SL}_{n}$ , and one side of the fundamental lemma for $\mathfrak{sl}_{n}$ coincides with the integration of $\tilde{\xi}_{v}$ on $X(\mathcal{O}_{k_{v}})$ .

In Section 6.2.1, we will provide the explicit relation between the evaluation $\tilde{\xi}_{v}$ and an endoscopic data for $\mathrm{SL}_{n}$ , and demonstrate that $\mathrm{S}_{x_{0},k_{v}}$ is an endoscopic group associated with the endoscopic data derived from $\tilde{\xi}_{v}$ . This connection provides the information on the other side of the fundamental lemma for $\mathfrak{sl}_{n}$ . Using this argument, in Section 6.2.2, we will formulate the integration of $\tilde{\xi}_{v}$ on $X(\mathcal{O}_{k_{v}})$ .

6.2.1. Endoscopic data and endoscopic group for $\mathrm{SL}_{n}$

Since $\mathrm{SL}_{n,k_{v}}$ is a split group over $k_{v}$ , we use the definition for the endoscopic data and the endoscopic group, in terms of the root data following [Hal05, Section 2-3].

Definition 6.3 ([Hal05, Section 2]).

Let $G$ be a reductive group over $k_{v}$ . If $G$ is a quasi-split group which splits over an unramified extension over $k_{v}$ , then $G$ is said to be an unramified reductive group.

An unramified reductive group $G$ is classified by the following root data

(X^{*}(T),X_{*}(T),\Phi_{G},\Phi_{G}^{\vee},\sigma_{G})

where

\left\{\begin{array}[]{l}X^{*}(T)\textit{ is the character group of a Cartan subgroup $T$ of $G$};\\ X_{*}(T)\textit{ is the cocharacter group of a Cartan subgroup $T$ of $G$};\\ \Phi_{G}\textit{ is the set of roots};\\ \Phi_{G}^{\vee}\textit{ is the set of coroots};\\ \sigma_{G}\textit{ is an automorphism of finite order of $X^{*}(T)$ sending a set of simple roots in $\Phi_{G}$ to itself.}\end{array}\right.

In general, $\sigma_{G}$ is obtained from the action on $X^{*}(T)$ induced from the Frobenius automorphism of $\mathrm{Gal}(k_{v}^{un}/k_{v})$ on the maximally split Cartan subgroup in $G$ . Here, $k_{v}^{un}$ is the maximal unramified extension over $k_{v}$ , in a fixed algebraically closure $\overline{k}_{v}$ of $k_{v}$ containing $\overline{k}$ .

Definition 6.4 ([Kot84, Section 7.1] and [Hal05, Section 3]).

Let $G$ be an unramified reductive group over $k_{v}$ with the root data $(X^{*}(T),X_{*}(T),\Phi_{G},\Phi_{G}^{\vee},\sigma_{G})$ . $H$ is an endoscopic group of $G$ if it is an unramified reductive group over $k_{v}$ whose classifying data has the form

(X^{*}(T),X_{*}(T),\Phi_{H},\Phi_{H}^{\vee},\sigma_{H}).

The data for $H$ is subject to the constraints that there exists an element $\kappa\in\mathrm{Hom}(X_{*}(T),\mathbb{C}^{\times})$ and a Weyl group element $w\in W(\Phi_{G})$ such that $\Phi_{H}^{\vee}=\{\alpha\in\Phi_{G}^{\vee}\mid\kappa(\alpha)=1\}$ , $\sigma_{H}=w\circ\sigma_{G}$ , and $\sigma_{H}(\kappa)=\kappa$ .

Now, we return to our context, plugging $G=\mathrm{SL}_{n,k_{v}}$ and $T=\mathrm{S}_{x_{0},k_{v}}$ . The following two lemmas explain an endoscopic data of $\mathrm{SL}_{n,k_{v}}$ associated with the evaluation $\tilde{\xi}_{v}$ and the corresponding endoscopic group.

Lemma 6.5.

The following morphism $\kappa_{\tilde{\xi}_{v}}$ is a character of $\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ ,

\kappa_{\tilde{\xi}_{v}}:\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})\rightarrow\mathbb{C}^{\times},\ z_{x_{0}^{\prime}}\mapsto\tilde{\xi}_{v}(x_{0}^{\prime})

where $x_{0}^{\prime}\in\mathcal{R}_{x_{0},v}$ and $z_{x_{0}^{\prime}}\in\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ are in Definition 2.6.

Proof.

By Proposition 2.4 and Corollary 5.3, the morphism $\kappa_{\tilde{\xi}_{v}}$ is well-defined and independent of the choice of $\mathcal{R}_{x_{0},v}$ . For $g_{x_{0}^{\prime}}\in\mathrm{GL}_{n}(k_{v})$ such that $g_{x_{0}^{\prime}}^{-1}x_{0}g_{x_{0}^{\prime}}=x_{0}^{\prime}$ , we have

\kappa_{\tilde{\xi}_{v}}(z_{x_{0}^{\prime}})=\tilde{\xi}_{v}(x_{0}^{\prime})=\exp 2\pi i\Psi(\xi)(\phi_{K_{v}/k_{v}}(\det g_{x_{0}^{\prime}}))=\exp 2\pi i\Psi(\xi)(\phi_{K_{v}/k_{v}}(\delta_{1}^{-1}(z_{x_{0}^{\prime}})))

by Proposition 5.2, where the notations $\Psi$ and $\phi_{K_{v}/k_{v}}$ are defined in loc. cit. and $\delta_{1}$ is defined in the diagram (5.4). Since the morphisms $\exp$ , $\Psi(\xi)$ , $\phi_{K_{v}/k_{v}}$ , and $\delta_{1}^{-1}$ are group homomorphisms, the above equation concludes the desired result. ∎

By the Tate-Nakayama isomorphism (cf. [Lab08, Theorem 6.5.1]) for $\mathrm{S}_{x_{0},k_{v}}$ , we can identify the character $\kappa$ of $\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ as an element in $\mathrm{Hom}(X_{*}(\mathrm{S}_{x_{0},k_{v}}),\mathbb{C}^{\times})$ . Since $\mathrm{S}_{x_{0},k_{v}}\cong\mathrm{R}_{K_{v}/k_{v}}^{(1)}(\mathbb{G}_{m,K_{v}})$ by Section 6.1.1, we have

\mathrm{S}_{x_{0},K_{v}}\cong\mathbb{G}_{m,K_{v}}^{n-1}\textit{ and }X_{*}(\mathrm{S}_{x_{0},k_{v}})\cong(\mathbb{Z}^{n})_{\Sigma=0}

where $(\mathbb{Z}^{n})_{\Sigma=0}=\{(z_{1},\cdots,z_{n})\in\mathbb{Z}^{n}\mid\sum_{i}z_{i}=0\}$ . Here, the action of $\Lambda_{v}=\mathrm{Gal}(K_{v}/k_{v})\cong\mathbb{Z}/n\mathbb{Z}$ on $X_{*}(\mathrm{S}_{x_{0},k_{v}})$ , induced from that on $\mathrm{S}_{x_{0},K_{v}}$ , is given by cyclic permutations of the factors of $(\mathbb{Z}^{n})_{\Sigma=0}$ . On the other hand, by [Lab08, Proposition 6.5.2], we have the following isomorphism

\mathrm{ker}(N_{\Lambda_{v}})/I_{\Lambda_{v}}X_{*}(\mathrm{S}_{x_{0},k_{v}})\cong\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})

where

\left\{\begin{array}[]{l}N_{\Lambda_{v}}:X_{*}(\mathrm{S}_{x_{0},k_{v}})\rightarrow X_{*}(\mathrm{S}_{x_{0},k_{v}}),\ x\mapsto\sum_{\sigma\in\Lambda_{v}}\sigma(x);\\ I_{\Lambda_{v}}:=\{\tau=\sum_{\sigma\in\Lambda_{v}}n_{\sigma}\sigma\mid\sum_{\sigma\in\Lambda_{v}}n_{\sigma}=0\}.\end{array}\right.

Since $N_{\Lambda_{v}}$ is a trivial morphism in $X_{*}(\mathrm{S}_{x_{0},k_{v}})\cong(\mathbb{Z}^{n})_{\Sigma=0}$ , the following composition is surjective

X_{*}(\mathrm{S}_{x_{0},k_{v}})\rightarrow X_{*}(\mathrm{S}_{x_{0},k_{v}})/I_{\Lambda_{v}}X_{*}(\mathrm{S}_{x_{0},k_{v}})\cong\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}}).

Hence, a character $\kappa$ on $\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ assigns an element in $\mathrm{Hom}(X_{*}(\mathrm{S}_{x_{0},k_{v}}),\mathbb{C}^{\times})$ .

Lemma 6.6.

Under the above identifiaction of $\kappa_{\tilde{\xi}_{v}}$ as an element in $\operatorname{Hom}(X_{*}(\mathrm{S}_{x_{0},k_{v}}),\mathbb{C}^{\times})$ , an unramified reductive group $\mathrm{S}_{x_{0},k_{v}}$ is an endoscopic group of $\mathrm{SL}_{n,k_{v}}$ , associated with $\kappa_{\tilde{\xi}_{v}}$ (cf. Definition 6.4).

Proof.

Let $\sigma_{\mathrm{S}_{x_{0},k_{v}}}$ be the action on $X^{*}(\mathrm{S}_{x_{0},k_{v}})$ induced from the Frobenius automorphism of $\mathrm{Gal}(K_{v}/k_{v})$ on $\mathrm{S}_{x_{0},K_{v}}$ . We claim that there exists an element $w$ of Weyl group for $\Phi_{\mathrm{SL}_{n,k_{v}}}$ such that $\sigma_{\mathrm{S}_{x_{0},k_{v}}}=w\circ\sigma_{\mathrm{SL}_{n,k_{v}}}$ and $\sigma_{\mathrm{S}_{x_{0},k_{v}}}(\kappa_{\tilde{\xi}_{v}})=\kappa_{\tilde{\xi}_{v}}$ . Since $\mathrm{SL}_{n,k_{v}}$ is split over $k_{v}$ , we have $\sigma_{\mathrm{SL}_{n,k_{v}}}=id$ . The Weyl group $W(\Phi_{\mathrm{SL}_{n,k_{v}}})$ is isomorphic to the symmetric group $S_{n}$ which acts on $X_{*}(\mathrm{S}_{x_{0},k_{v}})\cong\mathbb{Z}^{n}_{\Sigma=0}$ by permuting the factors. Thus we can find $w\in S_{n}$ such that $\sigma_{\mathrm{S}_{x_{0},k_{v}}}=w\circ\sigma_{\mathrm{SL}_{n,k_{v}}}=w$ . Indeed, $\sigma_{\mathrm{S}_{x_{0},k_{v}}}$ acts on $X_{*}(\mathrm{S}_{x_{0},k_{v}})\cong\mathbb{Z}^{n}_{\Sigma=0}$ by a cyclic permuation of the factors. On the other hand, since $id-\sigma_{\mathrm{S}_{x_{0},k_{v}}}\in I_{\Lambda_{v}}$ , we have $\sigma_{\mathrm{S}_{x_{0},k_{v}}}(\kappa_{\tilde{\xi}_{v}})=\kappa_{\tilde{\xi}_{v}}$ .

We now verify that

(X^{*}(\mathrm{S}_{x_{0},k_{v}}),X_{*}(\mathrm{S}_{x_{0},k_{v}}),\Phi_{H},\Phi_{H}^{\vee},\sigma_{\mathrm{S}_{x_{0},k_{v}}})

is the root data classifying an unramified reductive group $\mathrm{S}_{x_{0},k_{v}}$ , where $\Phi^{\vee}_{H}=\{\alpha\in\Phi_{\mathrm{SL}_{n,k_{v}}}^{\vee}\mid\kappa_{\tilde{\xi}_{v}}(\alpha)=1\}$ and $\Phi_{H}$ is its dual set. Since the evaluation $\tilde{\xi}_{v}$ is non-trivial, $\kappa_{\tilde{\xi}_{v}}$ is also non-trivial by Lemma 6.5. Moreover since $\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})\cong\mathbb{Z}/n\mathbb{Z}$ and $n$ is a prime number, the induced morphism in $\mathrm{Hom}(X_{*}(\mathrm{S}_{x_{0},k_{v}}),\mathbb{C}^{\times})$ from $\kappa_{\tilde{\xi}_{v}}$ , via the above identification, maps $\epsilon_{1,2}\in\mathbb{Z}_{\Sigma=0}\cong X_{*}(\mathrm{S}_{x_{0},k_{v}})$ to a non-trivial $n$ -th root of unity $\zeta_{n}$ in $S^{1}\subset\mathbb{C}^{\times}$ where $\epsilon_{i,j}=(\{\underbrace{0,\cdots,0,1}_{i},\underbrace{0,\cdots,0,-1}_{j-i},\underbrace{0,\cdots,0}_{n-j})$ . Indeed, $\epsilon_{1,2}$ represents a non-trivial element in $X_{*}(\mathrm{S}_{x_{0},k_{v}})/I_{\Lambda_{v}}X_{*}(\mathrm{S}_{x_{0},k_{v}})$ . For any coroot $\epsilon_{i,j}\in\Phi_{\mathrm{SL}_{n,k_{v}}}^{\vee}$ where $1\leq i<j\leq n$ , we then have

(6.3)

\kappa_{\tilde{\xi}_{v}}(\epsilon_{i,j})=\kappa_{\tilde{\xi}_{v}}(\sum_{k=0}^{j-i-1}\sigma^{i+k-1}(\epsilon_{1,2}))=\prod_{k=0}^{j-i-1}\kappa_{\tilde{\xi}_{v}}(\sigma^{i+k-1}(\epsilon_{1,2}))=\zeta_{n}^{j-i},

where $\sigma\in\Lambda_{v}$ such that $\sigma((\alpha_{1},\cdots,\alpha_{n}))=(\alpha_{n},\alpha_{1},\cdots,\alpha_{n-1})$ . Here, the last equality in (6.3) follows from the fact that $id-\sigma^{i+k-1}\in I_{\Lambda_{v}}$ . Since $1\leq j-i\leq n$ , we then have

\Phi^{\vee}_{\mathrm{S}_{x_{0},k_{v}}}=\{\alpha\in\Phi_{\mathrm{SL}_{n,k_{v}}}^{\vee}\mid\kappa_{\tilde{\xi}_{v}}(\alpha)=1\}=\emptyset,

and the dual set $\Phi_{\mathrm{S}_{x_{0},k_{v}}}$ is also empty. In conculusion, the root data $(X^{*}(\mathrm{S}_{x_{0},k_{v}}),X_{*}(\mathrm{S}_{x_{0},k_{v}}),\emptyset,\emptyset,\sigma_{\mathrm{S}_{x_{0},k_{v}}})$ corresponds to $\mathrm{S}_{x_{0},k_{v}}$ . ∎

6.2.2. Computation

We first introduce one lemma which will be used in our main formulation.

Lemma 6.7.

For $c\in\mathcal{O}_{k}$ , let $X_{c}$ be the closed subscheme of $\mathrm{M}_{n,\mathcal{O}_{k}}$ representing the set of $n\times n$ matrices whose characteristic polynomial is $\chi(x-c)$ .

(1)

Then, $X_{c}$ is a homogeneous space of $\mathrm{GL}_{n}$ (resp. $\mathrm{SL}_{n}$ ), in the sence of Section 2.1 with a point $x_{c}:=x_{0}+cI_{n}$ , and we have the following isomorphism

$X_{c}\cong\mathrm{T}_{x_{c}}\backslash\mathrm{GL_{n}}\ (resp.\ X_{c}\cong\mathrm{S}_{x_{c}}\backslash\mathrm{SL}_{n}),$

where $\mathrm{T}_{x_{c}}$ (resp. $\mathrm{S}_{x_{c}}$ ) is the stabilizer of $x_{c}$ under the $\mathrm{GL}_{n}$ -action (resp. the $\mathrm{SL}_{n}$ -action) on $X$ .

(2)

There is an isomorphism $\operatorname{Br}X\cong\operatorname{Br}X_{c},\ \xi\mapsto\xi_{c}$ such that $\tilde{\xi}_{v}(x)=\tilde{\xi}_{c,v}(x+cI_{n})$ and the following equation holds

\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ |\omega_{X_{k_{v}}}^{can}|_{v}=\int_{{X_{c}}(\mathcal{O}_{k_{v}})}\tilde{\xi}_{c,v}(x)\ |\omega_{X_{c,k_{v}}}^{can}|_{v}.

Proof.

Since the map $\iota_{c}:X\rightarrow X_{c},\ x\mapsto x+cI_{n}$ is an isomorphism which is compatible with conjugation of $\mathrm{GL}_{n}$ (resp. $\mathrm{SL}_{n}$ ), we obtain the first argument from the homogeneous space structure of $X$ with a point $x_{0}$ , in Proposition 2.1. This directly yields the following commutative diagram.

Here, $\mathrm{T}_{x_{0},k_{v}}=\mathrm{T}_{x_{c},k_{v}}$ . Therefore the measure $|\omega_{X_{k_{v}}}^{can}|_{v}$ transports to the measure $|\omega_{X_{c,k_{v}}}^{can}|_{v}$ along the isomorphism $\iota_{c}(k_{v}):X(k_{v})\rightarrow X_{c}(k_{v})$ , according to Section 3.2.2.

On the other hand, applying the functor $\operatorname{Br}=\mathrm{H}_{\acute{e}t}^{2}(-,\mathbb{G}_{m})$ on the isomorphism $\iota_{c}$ (cf. Definition 4.2), we have the induced isomorphism $\operatorname{Br}(\iota_{c}):\operatorname{Br}X_{c}\xrightarrow{\sim}\operatorname{Br}X$ . We then have the following commutative diagram for $x\in X(k_{v})$ ,

For $\xi\in\operatorname{Br}X$ , this yields that $\xi_{v}(x)=\xi_{c,v}(x+cI_{n})$ where $\xi_{c}:=\operatorname{Br}(\iota_{c})^{-1}\xi$ . By Definition 4.5, we have

\tilde{\xi}_{v}(x)=\frac{\xi_{v}(x)}{\xi_{v}(x_{0})}=\frac{\xi_{c,v}(x+cI_{n})}{\xi_{c,v}(x_{c})}=\tilde{\xi}_{c,v}(x+cI_{n})=\tilde{\xi}_{c,v}(\iota_{c}(k_{v})(x))

for $x\in X(k_{v})$ . Since the measure $|\omega_{X_{c,k_{v}}}^{can}|_{v}$ and the evaluation $\tilde{\xi}_{c,v}$ on $X_{c}(k_{v})$ are compatible with the measure $|\omega_{X_{k_{v}}}^{can}|_{v}$ and the evaluation $\tilde{\xi}_{v}$ on $X(k_{v})$ along the isomorphism $\iota_{c}:X_{k_{v}}\rightarrow X_{c,k_{v}}$ , we conclude that

\int_{X_{c}(\mathcal{O}_{k_{v}})}\tilde{\xi}_{c,v}(x)\ |\omega_{X_{c,k_{v}}}^{can}|_{v}=\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{c,v}(x+cI_{n})\ |\omega_{X_{k_{v}}}^{can}|_{v}=\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ |\omega_{X_{k_{v}}}^{can}|_{v}.

∎

Proposition 6.8.

Suppose that $K_{v}/k_{v}$ is an unramified Galois extension of prime degree $n$ . For $\xi\in\mathrm{Br}(X)$ whose evaluation $\tilde{\xi}_{v}$ is non-trivial, we have

\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ |\omega_{X_{k_{v}}}^{can}|_{v}=q_{v}^{\frac{\operatorname{ord}_{v}(\Delta_{\chi})}{2}}\frac{\#\mathrm{SL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-(n^{2}-1)}}{\#\mathrm{S}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-(n-1)}},

where $\mathrm{char}(\kappa_{v})>n$ .

Proof.

Since $\mathrm{char}(\kappa_{v})>n$ , we can always choose $c\in\mathcal{O}_{k}$ for $x_{0}$ such that $x_{c}:=x_{0}+cI_{n}\in\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})$ . Accordingly, we can take $\mathcal{R}_{x_{c},v}$ to be $\{x_{0}^{\prime}+cI_{n}\mid x_{0}^{\prime}\in\mathcal{R}_{x_{0},v}\}\subset\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})$ as well. Since the discriminant $\Delta_{\chi}$ is invaraint under the constant translation on $\chi(x)$ and $\mathrm{S}_{x_{0},k_{v}}=\mathrm{S}_{x_{c},k_{v}}$ , we may and do assume that $x_{0}\in\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})$ with $\mathcal{R}_{x_{0},v}\subset\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})$ by Lemma 6.7.

By Lemma 5.3 and Lemma 6.5, we have the folowing equation where $z_{x_{0}^{\prime}}\in\mathrm{H}^{1}(k_{v},\mathrm{S}_{x_{0},k_{v}})$ which corresponds to $x_{0}^{\prime}\in\mathcal{R}_{x_{0},v}$ (cf. Definition 2.6),

(6.4)

\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ |\omega_{X_{k_{v}}}^{can}|_{v}=\sum_{x_{0}^{\prime}\in\mathcal{R}_{x_{0},v}}\kappa_{\tilde{\xi}_{v}}(z_{x_{0}^{\prime}})\cdot\int_{\left(\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v})\right)\cap X(\mathcal{O}_{k_{v}})}\ |\omega_{X_{k_{v}}}^{can}|_{v},

with the identification $x_{0}^{\prime}\cdot\mathrm{SL}_{n}(k_{v})\cong\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v})$ in Remark 2.7. By the similar argument in the proof of Proposition 5.9 and the assumption that $\mathcal{R}_{x_{0},v}\subset\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})$ , we have

\int_{\left(\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v})\right)\cap X(\mathcal{O}_{k_{v}})}\ |\omega_{X_{k_{v}}}^{can}|_{v}=\int_{\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v})}\mathbbm{1}_{\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})}(g^{-1}x_{0}^{\prime}g)\ |\omega_{X_{k_{v}}}^{can}|_{v}.

Combining the above two equations, we have

(6.5)

\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ |\omega_{X_{k_{v}}}^{can}|_{v}=\sum_{x_{0}^{\prime}\in\mathcal{R}_{x_{0},v}}\kappa_{\tilde{\xi}_{v}}(z_{x_{0}^{\prime}})\int_{\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v})}\mathbbm{1}_{\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})}(g^{-1}x_{0}^{\prime}g)\ |\omega_{X_{k_{v}}}^{can}|_{v},

for $x_{0}\in\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})$ .

To compute the right hand side of (6.5), we apply the formula in [Ngfrm[o]–0, Theorem 1] by plugging $G=\mathrm{SL}_{n,k_{v}},\ H=\mathrm{S}_{x_{0},k_{v}}$ , and $\kappa=\kappa_{\tilde{\xi}_{v}}$ according to Lemma 6.6. We note that the condition $\mathrm{char}(\kappa_{v})>n$ is necessary to apply this formula. For the measure $ds_{v}$ on $\mathrm{S}_{x_{0}}(k_{v})$ such that $\mathrm{vol}(ds_{v},\mathrm{S}_{x_{0},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}}))=1$ , we have

(6.6)

\mathcal{O}_{a(x_{0})}^{\kappa_{\tilde{\xi}_{v}}}(\mathbbm{1}_{\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})},ds_{v})=q_{v}^{\frac{\mathrm{ord}(\mathcal{D}_{\mathrm{SL}_{n,k_{v}}}(a(x_{0})))-\mathrm{ord}(\mathcal{D}_{\mathrm{S}_{x_{0},k_{v}}}(a(x_{0,H})))}{2}}\cdot\mathcal{SO}_{a(x_{0,H})}(\mathbbm{1}_{\mathfrak{s}_{x_{0},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})},ds_{v}).

We explain the notations in the following (1)-(3), and further compute the right hand side of (6.6) in (2)-(3).

(1)

The $\kappa_{\tilde{\xi}_{v}}$ -orbital integral $\mathcal{O}_{a(x_{0})}^{\kappa_{\tilde{\xi}_{v}}}(\mathbbm{1}_{\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})},ds_{v})$ in the left hand side of (6.6) is formulated as follows

\mathcal{O}_{a(x_{0})}^{\kappa_{\tilde{\xi}_{v}}}(\mathbbm{1}_{\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})},ds_{v})=\sum_{x_{0}^{\prime}\in\mathcal{R}_{x_{0},v}}\kappa_{\tilde{\xi}_{v}}(z_{x_{0}^{\prime}})\int_{\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v})}\mathbbm{1}_{\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})}(g^{-1}x_{0}^{\prime}g)\ \frac{dh_{v}}{ds^{\prime}_{v}},

where $a(x_{0})$ is the stable conjugacy class of $x_{0}$ and the measure $\frac{dh_{v}}{ds^{\prime}_{v}}$ is defined in Lemma 6.1.

(2)

We denote the Lie algebra of $\mathrm{S}_{x_{0},k_{v}}$ by $\mathfrak{s}_{x_{0},k_{v}}$ , and correspondingly the Lie algebra of $\mathrm{S}_{x_{0},\mathcal{O}_{k_{v}}}$ by $\mathfrak{s}_{x_{0},\mathcal{O}_{k_{v}}}$ . Let $x_{0,H}$ be an element in $\mathfrak{s}_{x_{0},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})$ , whose stable conjugacy class $a(x_{0,H})$ transfers to $a(x_{0})$ via the map given by [Ngfrm[o]–0, Section 1.9]. By [Ngfrm[o]–0, Lemma 1.4.3 and Lemma 1.9.2], the centralizer of $x_{0,H}$ in $\mathrm{S}_{x_{0},k_{v}}$ coincides with $\mathrm{S}_{x_{0},k_{v}}$ . Then the stable orbital integral $\mathcal{SO}_{a(x_{0,H})}(\mathbbm{1}_{\mathfrak{s}_{x_{0},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})},ds_{v})$ in the right hand side of (6.6) is formulated as follows

\mathcal{SO}_{a(x_{0,H})}(\mathbbm{1}_{\mathfrak{s}_{x_{0},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})},ds_{v})=\int_{\mathrm{S}_{x_{0}}(k_{v})\backslash{\mathrm{S}_{x_{0}}(k_{v})}}\mathbbm{1}_{\mathfrak{s}_{x_{0},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})}(g^{-1}x_{0,H}g)\frac{ds_{v}}{ds_{v}},

where $a(x_{0,H})\text{ is the stable conjugacy class of $x_{0,H}$}$ . Indeed, the adjoint action of $\mathrm{S}_{x_{0},k_{v}}$ on $\mathfrak{s}_{x_{0},k_{v}}$ is trivial and so $a(x_{0,H})=\{x_{0,H}\}$ . Therefore we have

\mathcal{SO}_{a(x_{0,H})}(\mathbbm{1}_{\mathfrak{s}_{x_{0},\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})},ds_{v})=1.

(3)

In the right hand side of (6.6), $\mathcal{D}_{\mathrm{SL}_{n,k_{v}}}$ and $\mathcal{D}_{\mathrm{S}_{x_{0},k_{v}}}$ are discriminant functions of $\mathrm{SL}_{n,k_{v}}$ and $\mathrm{S}_{x_{0},k_{v}}$ defined in [Ngfrm[o]–0, Section 1.10], respectively. Since $\mathrm{S}_{x_{0},k_{v}}$ is torus, $\mathcal{D}_{\mathrm{S}_{x_{0},k_{v}}}$ is trivial and so

$\operatorname{ord}_{v}(\mathcal{D}_{\mathrm{S}_{x_{0},k_{v}}}(a(x_{0,H})))=0.$

On the other hand, by [Gor22, (20) and Example 3.6], we have

$\operatorname{ord}_{v}(\mathcal{D}_{\mathrm{SL}_{n,k_{v}}}(a(x_{0})))=\operatorname{ord}_{v}(\Delta_{\chi}).$

To sum up, we have

\sum_{x_{0}^{\prime}\in\mathcal{R}_{x_{0},v}}\kappa_{\tilde{\xi}_{v}}(z_{x_{0}^{\prime}})\int_{\mathrm{S}_{x_{0}^{\prime}}(k_{v})\backslash\mathrm{SL}_{n}(k_{v})}\mathbbm{1}_{\mathfrak{sl}_{n,\mathcal{O}_{k_{v}}}(\mathcal{O}_{k_{v}})}(g^{-1}x_{0}g)\ \frac{dh_{v}}{ds^{\prime}_{v}}=q_{v}^{\frac{\operatorname{ord}_{v}(\Delta_{\chi})}{2}}.

Finally, by the measure difference between $\frac{dh_{v}}{ds_{v}^{\prime}}$ and $|\omega_{X_{k_{v}}}^{can}|_{v}$ obtained in Lemma 6.1-6.2, we obtain the desired conclusion. ∎

7. Main Result

We provide our main theorem on the asymptotic formula for $N(X,T)$ based on the results in Section 5-6. We note that the product $\prod_{v\in\Omega_{k}\setminus\infty_{k}}\mathcal{SO}_{v}(\chi)$ which appears in the following theorem is a finite product by Remark 1.6.

Theorem 7.1.

N(X,T)=\#\{x\in X(\mathcal{O}_{k})\mid\lVert x\rVert\leq T\},

for $T>0$ , where the norm $\lVert\cdot\rVert$ is defined in (1.1).

Suppose that $k$ and $K=k[x]/(\chi(x))$ are totally real, and if $k\neq\mathbb{Q}$ , we further assume that $n$ is a prime number. Then we have the following asymptotic formulas.

(1)

If $K/k$ is not Galois or ramified Galois, then

$N(X,T)\sim C_{T}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{\mathcal{SO}_{v}(\chi)}{q_{v}^{S_{v}(\chi)}}.$

(2)

If $K/k$ is unramified Galois and splits over all $p$ -adic places for $p\leq n$ , then

N(X,T)\sim C_{T}\left(\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{\mathcal{SO}_{v}(\chi)}{q_{v}^{S_{v}(\chi)}}+n-1\right).

Here, $C_{T}$ is formulated as follows

C_{T}=\lvert\Delta_{k}\rvert^{\frac{-n^{2}+n}{2}}\frac{R_{K}h_{K}\sqrt{\lvert\Delta_{K}\rvert}^{-1}}{R_{k}h_{k}\sqrt{\lvert\Delta_{k}\rvert}^{-1}}\prod_{i=2}^{n}\zeta_{k}(i)^{-1}\left(\frac{2^{n-1}\pi^{\frac{n(n+1)}{4}}w_{n}}{\prod_{i=1}^{n}\Gamma(\frac{i}{2})}T^{\frac{n(n-1)}{2}}\right)^{[k:\mathbb{Q}]},

where we use the following notations,

\left\{\begin{array}[]{l}\textit{$\Delta_{F}$ is the discriminant of $F/\mathbb{Q}$ for $F=K$ or $k$};\\ \textit{$R_{F}$ is the regulator of $F$ for $F=K$ or $k$};\\ \textit{$h_{F}$ is the class number of $\mathcal{O}_{F}$ for $F=K$ or $k$};\\ \textit{$w_{n}$ is the volume of the unit ball in $\mathbb{R}^{\frac{n(n-1)}{2}}$};\\ \textit{$\zeta_{k}$ is the dedekind zeta function of $k$};\\ \textit{$S_{v}(\chi)$ is the $\mathcal{O}_{k_{v}}$-module length between $\mathcal{O}_{K_{v}}$ and $\mathcal{O}_{k_{v}}[x]/(\chi(x))$};\\ \textit{$\mathcal{SO}_{v}(\chi)$ is the orbital integral for $\mathfrak{gl}_{n,k_{v}}$ associated with $\chi(x)$ with respect to $\frac{dg_{v}}{dt_{v}}$}\\ \textit{(cf. Proposition \ref{prop:result_case:trivial} and Lemma \ref{cor:compareclassic})}.\end{array}\right.

Proof.

We recall the formula in Proposition 4.1, by the equation (4.2), we have

(7.1)

N(X,T)\sim\lvert\Delta_{k}\rvert^{-\frac{1}{2}\dim X}\sum_{\xi\in\operatorname{Br}X/\operatorname{Br}k}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ \lvert\omega_{X}\rvert_{v}\prod_{v\in\infty_{k}}\int_{X(k_{v},T)}\tilde{\xi}_{v}(x)\ \lvert\omega_{X}\rvert_{v}.

By Proposition 3.9, we have

\prod_{v\in\Omega_{k}}|\omega_{X}|_{v}=\prod_{v\in\Omega_{k}\setminus\infty_{k}}\prod_{i\in B_{v}(\chi)}|\Delta_{K_{v,i}/k_{v}}|^{-\frac{1}{2}}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}.

For Archimedean place $v\in\infty_{k}$ , the assumption yields that $\tilde{\xi}_{v}$ is trivial. By Proposition 5.6, we have

\prod_{v\in\infty_{k}}\int_{X(k_{v},T)}\ \lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}\sim\left(\frac{\pi^{\frac{n(n+1)}{4}}w_{n}}{\prod_{i=1}^{n}\Gamma(\frac{i}{2})}T^{\frac{n(n-1)}{2}}\right)^{[k:\mathbb{Q}]}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\lvert\Delta_{\chi}\rvert_{v}^{\frac{1}{2}}.

We denote the $\mathcal{O}_{k_{v}}$ -module length between $\mathcal{O}_{K_{v,i}}$ and $\mathcal{O}_{k_{v}}[x]/(\chi_{v,i}(x))$ by $S_{v}(\chi_{v,i})$ . Then by [CKL, Proposition 2.5], we have

\lvert\Delta_{\chi}\rvert_{v}^{\frac{1}{2}}\prod_{i\in B_{v}(\chi)}|\Delta_{K_{v,i}/k_{v}}|_{v}^{-\frac{1}{2}}=\Big{(}\lvert\Delta_{\chi}\rvert_{v}^{\frac{1}{2}}\prod_{i\in B_{v}(\chi)}\lvert\Delta_{\chi_{v,i}}\rvert_{v}^{-\frac{1}{2}}\Big{)}\prod_{i\in B_{v}(\chi)}q_{v}^{-S_{v}(\chi_{v,i})}=q_{v}^{-S_{v}(\chi)},

where the last equality follows from [Yun13, Section 4.1] and [Bou03, Corollary 1 of Prosition 11 in Chapter 4.6]. Therefore we have

N(X,T)\sim|\Delta_{k}|^{\frac{-n^{2}+n}{2}}\left(\frac{\pi^{\frac{n(n+1)}{4}}w_{n}}{\prod_{i=1}^{n}\Gamma(\frac{i}{2})}T^{\frac{n(n-1)}{2}}\right)^{[k:\mathbb{Q}]}\left(\sum_{\xi\in\operatorname{Br}X/\operatorname{Br}k}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{1}{q_{v}^{S_{v}(\chi)}}\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ \lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}\right).

We now compute the integration of $\tilde{\xi}_{v}(x)$ on $X(\mathcal{O}_{k_{v}})$ with respect to $\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}$ for each $v\in\Omega_{k}\setminus\infty_{k}$ . For $k=\mathbb{Q}$ , we note that the only case (1) is possible since every extension of $\mathbb{Q}$ is ramified.

(1)

We claim that the summand in (7.1) for a non-trivial element $\xi\in\operatorname{Br}X/\operatorname{Br}k$ vanishes. Therefore, it suffices to obtain the volume of $X(\mathcal{O}_{k_{v}})$ with respect to $\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}$ by Remark 4.6. In the case that $k\neq\mathbb{Q}$ and $K/k$ is not Galois, $\operatorname{Br}X/\operatorname{Br}k$ is trivial by Proposition 5.1. On the other hand, if $k\neq\mathbb{Q}$ and $K/k$ is ramified, then Corollary 5.4 and Proposition 5.11 yields the claim. In the case that $k=\mathbb{Q}$ , without the assumption that $n$ is a prime number, the claim also holds by the proof of [WX16, Theorem 6.1].

By Lemma 3.7 and Proposition 5.9, we then have

\int_{X(\mathcal{O}_{k_{v}})}\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}=\frac{\#\mathrm{GL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n^{2}}}{\#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n}}\mathcal{SO}_{v}(\chi).

Here, we note that the following equalities hold,

\left\{\begin{array}[]{l}\#\mathrm{GL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n^{2}}=(1-\frac{1}{q_{v}^{n}})(1-\frac{1}{q_{v}^{n-1}})\cdots(1-\frac{1}{q_{v}});\\ \#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n}=\prod_{i\in B_{v}(\chi)}(1-\frac{1}{\#\kappa_{K_{v,i}}}),\end{array}\right.

where the formula for $\#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})$ is induced from Lemma 2.2 and the fact that

\#\mathrm{R}_{\mathcal{O}_{K_{v,i}}/\mathcal{O}_{k_{v}}}(\mathbb{G}_{m,\mathcal{O}_{K_{v,i}}})(\kappa_{v})=q_{v}^{[K_{v,i}:k_{v}]}(1-\frac{1}{\#\kappa_{K_{v,i}}}).

Therefore, we have

\displaystyle\begin{split}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{1}{q_{v}^{S_{v}(\chi)}}\int_{X(\mathcal{O}_{k_{v}})}\ \lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}&=\left.\frac{\zeta_{K}(s)}{\zeta_{k}(s)}\right|_{s=1}\prod_{i=2}^{n}\zeta_{k}(i)^{-1}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{\mathcal{SO}_{v}(\chi)}{{q_{v}^{S_{v}(\chi)}}}\\ &=2^{(n-1)[k:\mathbb{Q}]}\frac{R_{K}h_{K}\sqrt{\lvert\Delta_{k}\rvert}^{-1}}{R_{k}h_{k}\sqrt{\lvert\Delta_{k}\rvert}^{-1}}\prod_{i=2}^{n}\zeta_{k}(i)^{-1}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{\mathcal{SO}_{v}(\chi)}{q_{v}^{S_{v}(\chi)}},\end{split}

where the last equality follows from the class number formula for $k$ and $K$ . In conclusion, we have

N(X,T)\sim C_{T}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{\mathcal{SO}_{v}(\chi)}{q_{v}^{S_{v}(\chi)}}.

(2)

In the case that $\xi\in\operatorname{Br}X/\operatorname{Br}k$ is trivial, the computation of the integration of $\tilde{\xi}_{v}$ on $X(\mathcal{O}_{k_{v}})$ is exactly same with that of (1), and so we will omit this. For a non-trivial element $\xi\in\operatorname{Br}X/\operatorname{Br}k$ , we address the following two cases separately; $K_{v}$ splits completely over $k_{v}$ and $K_{v}$ is unramified over $k_{v}$ .

•

If $K_{v}$ splits completely over $k_{v}$ , then the evaluation $\tilde{\xi}_{v}$ is trivial by Proposition 5.5. We then apply the computation in (1),

\frac{1}{q_{v}^{S_{v}(\chi)}}\int_{X(\mathcal{O}_{k_{v}})}\lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}=\frac{\#\mathrm{GL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n^{2}}}{\#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n}}\frac{\mathcal{SO}_{v}(\chi)}{q_{v}^{S_{v}(\chi)}}.

Here, $\frac{\mathcal{SO}_{v}(\chi)}{q_{v}^{S_{v}(\chi)}}=1$ by the following argument. Since $\chi_{v,i}$ is a linear polynomial for each $i\in B_{v}(\chi)$ where $\chi=\prod_{i\in B_{v}}\chi_{v,i}$ over $k_{v}$ , the orbital integral $\mathcal{SO}_{v}(\chi_{v,i})$ for $\mathfrak{gl}_{n}$ associated with $\chi_{v,i}(x)$ is equal to 1. In addition, the $\mathcal{O}_{k_{v}}$ -module length $S_{v}(\chi_{v,i})$ between $\mathcal{O}_{K_{v,i}}=\mathcal{O}_{k_{v}}$ and $\mathcal{O}_{k_{v}}[x]/(\chi_{v,i}(x))$ equals to 0. By [Yun13, Corollary 4.10], we then have

\mathcal{SO}_{v}(\chi)=q_{v}^{S_{v}(\chi)-\sum\limits_{i\in B_{v}(\chi)}S_{v}(\chi_{v,i})}\prod_{i\in B_{v}(\chi)}\mathcal{SO}_{v}(\chi_{v,i})=q_{v}^{S_{v}(\chi)}.

•

If $K_{v}$ is unramified over $k_{v}$ and so the place $v$ is $p$ -adic for $p>n$ by the assumption, then the evaluation $\tilde{\xi}_{v}$ is non-trivial by Corollary 5.4. By Proposition 6.8, we then have

\frac{1}{q_{v}^{S_{v}(\chi)}}\int_{X(\mathcal{O}_{k_{v}})}\tilde{\xi}_{v}(x)\ |\omega_{X_{k_{v}}}^{can}|_{v}=\frac{\#\mathrm{SL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-(n^{2}-1)}}{\#\mathrm{S}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-(n-1)}}=\frac{\#\mathrm{GL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n^{2}}}{\#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n}}.

Indeed, $\frac{\operatorname{ord}_{v}(\Delta_{\chi})}{2}=S_{v}(\chi)+\frac{\operatorname{ord}_{v}(\Delta_{K_{v}/k_{v}})}{2}=S_{v}(\chi)$ by [CKL, Proposition 2.5] and [Neu99, Theorem 2.6 in Chapter III].

To sum up, for a non-trivial $\xi\in\operatorname{Br}X/\operatorname{Br}k$ , we have

	$\displaystyle\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{1}{q_{v}^{S_{v}(\chi)}}\int_{X(\mathcal{O}_{k_{v}})}\ \lvert\omega_{X_{k_{v}}}^{can}\rvert_{v}$	$\displaystyle=\prod_{\begin{subarray}{c}v\in\Omega_{k}\setminus\infty_{k}\\ v:splits\end{subarray}}\frac{\#\mathrm{GL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n^{2}}}{\#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n}}\prod_{\begin{subarray}{c}v\in\Omega_{k}\setminus\infty_{k}\\ v:unram\end{subarray}}\frac{\#\mathrm{GL}_{n,\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n^{2}}}{\#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cdot q_{v}^{-n}}$
		$\displaystyle=\left(2^{(n-1)[k:\mathbb{Q}]}\frac{R_{K}h_{K}\sqrt{\lvert\Delta_{k}\rvert}^{-1}}{R_{k}h_{k}\sqrt{\lvert\Delta_{k}\rvert}^{-1}}\prod_{i=2}^{n}\zeta_{k}(i)^{-1}\right).$

Proposition 5.2 yields that $\operatorname{Br}X/\operatorname{Br}k\cong\operatorname{Hom}(\operatorname{Gal}(K/k),\mathbb{Q}/\mathbb{Z})$ . Since $\operatorname{Gal}(K/k)$ is a cyclic group of prime order $n$ , we have $n-1$ nontrivial elements in $\operatorname{Br}X/\operatorname{Br}k$ . In conclusion, we have

N(X,T)\sim C_{T}\left(\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{\mathcal{SO}_{v}(\chi)}{q_{v}^{S_{v}(\chi)}}+n-1\right).

∎

Remark 7.2.

Suppose that $k=\mathbb{Q}$ and $\chi(x)$ splits completely over $\mathbb{R}$ (assumptions (1.3) without the assumption (2) that $\mathbb{Z}[x]/(\chi(x))=\mathcal{O}_{K}$ ). Since every extension of $\mathbb{Q}$ is ramified, the asymptotic formula for $N(X,T)$ follows from Theorem 7.1.(1). We then have

N(X,T)\sim\left(\frac{2^{n-1}h_{K}R_{K}w_{n}}{\sqrt{\Delta_{K}}\prod_{k=2}^{n}\Lambda(k/2)}\prod_{v\in\Omega_{k}\setminus\infty_{k}}\frac{\mathcal{SO}_{v}(\chi)}{q_{v}^{S_{v}(\chi)}}\right)T^{\frac{n(n-1)}{2}},

where $\Lambda(k/2)=\pi^{-k/2}\Gamma(k/2)\zeta(k)$ .

We verify that the above formula coincides with (1.2) when $\mathbb{Z}[x]/(\chi(x))=\mathcal{O}_{K}$ . Indeed, since $\mathbb{Z}[x]/(\chi(x))=\mathcal{O}_{K}$ , equivalently $S_{v}(\chi)=0$ for all $v\in\Omega_{k}\setminus\infty_{k}$ , we have $\mathcal{SO}_{v}(\chi)=1$ by Remark 5.10 and $\Delta_{K}=\Delta_{\chi}$ . Therefore we have

N(X,T)\sim\frac{2^{n-1}h_{K}R_{K}w_{n}}{\sqrt{\Delta_{\chi}}\cdot\prod_{k=2}^{n}\Lambda(k/2)}T^{\frac{n(n-1)}{2}}.

In the case that $n=2$ and $3$ , we refer to the results in [CKL], for the closed formula of $\prod_{v\in\Omega_{k}\setminus\infty_{k}}\mathcal{SO}_{v}(\chi)$ .

Proposition 7.3.

(1)

For $n=2$ , the product of orbital integrals $\mathcal{SO}_{v}(\chi)$ is given as follows

\prod_{v\in\Omega_{k}\setminus\infty_{k}}\mathcal{SO}_{v}(\chi)=\prod_{v\in\Omega_{k}\setminus\infty_{k}}\left(1+\frac{\#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})}{q_{v}-1}\frac{q_{v}^{S_{v}(\chi)}-1}{q_{v}-1}\right).

(2)

For $n=3$ , the product of orbital integrals $\mathcal{SO}_{v}(\chi)$ is given as follows

\prod_{v\in\Omega_{k}\setminus\infty_{k}}\mathcal{SO}_{v}(\chi)=\prod_{v\in\Omega_{k}\setminus\infty_{k}}q_{v}^{\rho_{v}(\chi)}\left(1+\frac{\#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})}{(q_{v}-1)^{2}}\Phi_{v}(\chi)\right).

Here,

\left\{\begin{array}[]{l}\#\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})=\prod\limits_{i\in B_{v}(\chi)}q_{v}^{[K_{v,i}:k_{v}]-[\kappa_{K_{v,i}}:\kappa_{v}]}(q_{v}^{[\kappa_{K_{v,i}}:\kappa_{v}]}-1),\\ \rho_{v}(\chi):=\sum\limits_{{\{i,j\}\subset B_{v}(\chi),i\neq j}}\operatorname{ord}_{v}(\mathrm{Res}(\chi_{v,i},\chi_{v,j}))=\left\{\begin{array}[]{l l}0&\textit{if }|B_{v}(\chi)|=1;\\ S_{v}(\chi)-\delta_{v}&\textit{if }|B_{v}(\chi)|=2;\\ S_{v}(\chi)&\textit{if }|B_{v}(\chi)|=3,\end{array}\right.\\ \Phi_{v}(\chi)=\left\{\begin{array}[]{l l}\frac{q_{v}^{\delta_{v}}-1}{q_{v}-1}&\textit{if $K_{v}$ splits over $k_{v}$};\\ \frac{q_{v}^{\delta_{v}}-1}{q_{v}-1}-\frac{3(q_{v}^{\delta_{v}-d_{v}}-1)}{q_{v}^{2}-1}&\textit{if $K_{v}$ is an unramified field extension of $k_{v}$};\\ \frac{q_{v}^{\delta_{v}}-1}{q_{v}-1}-\frac{3(q_{v}^{\delta_{v}-d_{v}}-1)}{q_{v}^{2}-1}+\frac{(1+\delta_{v}-3d_{v})q_{v}^{\delta_{v}-d_{v}}-1}{q_{v}(q_{v}+1)}&\textit{if $K_{v}$ is a ramified field extension of $k_{v}$},\end{array}\right.\end{array}\right.

where $\left\{\begin{array}[]{l}\textit{$B_{v}(\chi)$ is an index set in bijection with irreducible factors $\chi_{v,i}$ of $\chi$ over $\mathcal{O}_{k_{v}}$};\\ \textit{$\mathrm{Res}(\chi_{v,i},\chi_{v,j})$ denotes the resultant of two polynomials $\chi_{v,i}$ and $\chi_{v,j}$};\\ \delta_{v}:=\max\limits_{i\in B_{v}(\chi)}\{S_{v}(\chi_{v,i})\}\textit{ for the $\mathcal{O}_{k_{v}}$-module length $S_{v}(\chi_{v,i})$ between $\mathcal{O}_{K_{v,i}}$ and $\mathcal{O}_{k_{v}}[x]/(\chi_{v,i}(x))$};\\ d_{v}:=\lfloor\frac{\delta_{v}}{3}\rfloor\textit{ for the floor function $\lfloor\cdot\rfloor$}.\end{array}\right.$

Proof.

(1)

By [CKL, Remark 5.7], we have

\mathcal{SO}_{v}(\chi)=\left\{\begin{array}[]{l l}q_{v}^{S_{v}(\chi)}&\textit{if $K_{v}$ splits over $k_{v}$};\\ 1+(q_{v}+1)\frac{q_{v}^{S_{v}(\chi)}-1}{q_{v}-1}&\textit{if $K_{v}$ is an unramified field extension over $k_{v}$};\\ \frac{q_{v}^{S_{v}(\chi)+1}-1}{q_{v}-1}&\textit{if $K_{v}$ is a ramified field extension over $k_{v}$}.\end{array}\right.

In order to write the above in a uniform way, we describe $\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})$ explicitly as follows.

\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cong\left\{\begin{array}[]{l l}(\kappa_{v}\times\kappa_{v})^{\times}&\textit{if $K_{v}$ splits over $k_{v}$};\\ \kappa_{K_{v}}^{\times}&\textit{if $K_{v}$ is an unramified field extension over $k_{v}$};\\ (\kappa_{v}[x]/(x^{2}))^{\times}&\textit{if $K_{v}$ is a ramified field extension over $k_{v}$},\end{array}\right.

where in the second case, $\kappa_{K_{v}}$ is a quadratic field extension over $\kappa_{v}$ . Plugging it into the above formula for $\mathcal{SO}_{v}(\chi)$ , we obtain the desired formula.

(2)

By [CKL, Remark 6.7], we have

(a)

If $\chi(x)$ is an irreducible monic polynomial $\mathcal{O}_{k_{v}}$ , then we have

{\small\mathcal{SO}_{v}(\chi)=\left\{\begin{array}[]{l l}(q_{v}^{2}+q_{v}+1)\sum\limits_{i=1}^{d_{v}}(q_{v}^{3i-2}+2q_{v}^{3i-3}+3q_{v}^{2i-2}\frac{q_{v}^{i-1}-1}{q_{v}-1})+1&\textit{if $K_{v}/k_{v}$ is unramified and $S_{v}(\chi)=3d_{v}$};\\ \sum\limits_{i=1}^{d_{v}}(q_{v}^{3i}+2q_{v}^{3i-1}+q_{v}^{2i-1}+3q_{v}^{2i}\frac{q_{v}^{i-1}-1}{q_{v}-1})+1&\textit{if $K_{v}/k_{v}$ is ramified and $S_{v}(\chi)=3d_{v}$};\\ \sum\limits_{i=1}^{d_{v}}(q_{v}^{3i+1}+2q_{v}^{3i}+2q_{v}^{2i}+3q_{v}^{2i+1}\frac{q_{v}^{i-1}-1}{q_{v}-1})+q_{v}+1&\textit{if $K_{v}/k_{v}$ is ramified and $S_{v}(\chi)=3d_{v}+1$}.\\ \end{array}\right.}

Here, we note that $S_{v}(\chi)$ cannot be of the form $3d_{v}+2$ (cf. [CKL, Proposition 6.2]).

(b)

If $\chi(x)=\chi_{v,1}(x)\chi_{v,2}(x)$ over $\mathcal{O}_{k_{v}}$ where $\chi_{v,1}(x)\in\mathcal{O}_{k_{v}}[x]$ is an irreducible monic quadratic polynomial, we have

\mathcal{SO}_{v}(\chi)=\left\{\begin{array}[]{l l}q_{v}^{S_{v}(\chi)-S_{v}(\chi_{v,1})}(1+(q_{v}+1)\frac{q_{v}^{S_{v}(\chi_{v,1})}-1}{q_{v}-1})&\textit{if $K_{v,1}/k_{v}$ is unramified};\\ q_{v}^{S_{v}(\chi)-S_{v}(\chi_{v,1})}\frac{q_{v}^{S_{v}(\chi_{v,1})+1}-1}{q_{v}-1}&\textit{if $K_{v,1}/k_{v}$ is ramified}.\end{array}\right.

(c)

If $\chi(x)$ splits into linear terms completely over $\mathcal{O}_{k_{v}}$ , we have

$\mathcal{SO}_{v}(\chi)=q_{v}^{S_{v}(\chi)}.$

On the other hand, by [Yun13, Section 4.1], we have $\rho_{v}(\chi)=S_{v}(\chi)-\sum\limits_{i\in B_{v}(\chi)}S_{v}(\chi_{v,i})$ . Since $S_{v}(\chi_{v,i})=0$ for a linear factor $\chi_{v,i}(x)\in\mathcal{O}_{k_{v}}[x]$ of $\chi(x)$ , we verify that

\rho_{v}(\chi)=\left\{\begin{array}[]{l l}0&\textit{if }|B_{v}(\chi)|=1;\\ S_{v}(\chi)-\delta_{v}&\textit{if }|B_{v}(\chi)|=2;\\ S_{v}(\chi)&\textit{if }|B_{v}(\chi)|=3.\end{array}\right.

In order to write the above in a uniform way, we describe $\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})$ explicitly as follows.

\mathrm{T}_{x_{0},\mathcal{O}_{k_{v}}}(\kappa_{v})\cong\left\{\begin{array}[]{l l}\kappa_{K_{v}}^{\times}&\textit{in the case (a), if $K_{v}/k_{v}$ is unramified;}\\ (\kappa_{v}[x]/(x^{3}))^{\times}&\textit{in the case (a), if $K_{v}/k_{v}$ is ramified;}\\ (\kappa_{K_{v,1}}\times\kappa_{v})^{\times}&\textit{in the case (b), if $K_{v,1}/k_{v}$ is unramified;}\\ (\kappa_{v}[x]/(x^{2})\times\kappa_{v})^{\times}&\textit{in the case (b), if $K_{v,1}/k_{v}$ is ramified;}\\ (\kappa_{v}^{3})^{\times}&\textit{in the case (c)},\end{array}\right.

where $\kappa_{K_{v}}$ is a cubic field extension over $\kappa_{v}$ in the first case, and $\kappa_{K_{v,1}}$ is a quadratic field extension over $\kappa_{v}$ in the third case. Plugging it into the above formula for $\mathcal{SO}_{v}(\chi)$ , we obtain the desired formula.

∎

In the case that an $\mathcal{O}_{k}$ -order $\mathcal{O}_{k}[x]/(\chi(x))$ is a Bass order, we refer to the results in [CHL], for the closed formula of $\prod_{v\in\Omega_{k}\setminus\infty_{k}}\mathcal{SO}_{v}(\chi)$ . A Bass order is an order of a number field whose fractional ideals are generated by two elements (cf. [CHL, Definition 6.1]). For example, any order of a number field which contains the maximal order of a subfield with degree 2 or whose discriminant is fourth-power-free in $\mathbb{Z}$ , is a Bass order.

To state the closed formula of $\prod_{v\in\Omega_{k}\setminus\infty_{k}}\mathcal{SO}_{v}(\chi)$ when $R:=\mathcal{O}_{k}[x]/(\chi(x))$ is a Bass order, we define $|R|$ to be the set of maximal ideals of $R$ and define the following notations for $w\in|R|$ ;

\left\{\begin{array}[]{l}R_{w}\textit{ is a $w$-adic completion of $R$};\\ K_{w}\textit{ is the ring of total fractions of $R_{w}$};\\ \mathcal{O}_{K_{w}}\textit{ is the ring of integers of $K_{w}$}.\end{array}\right.

Let $|R|^{irred}$ and $|R|^{split}$ be subsets of $|R|$ such that

\left\{\begin{array}[]{l}|R|^{irred}\subset\{w\in|R|:\textit{ $R_{w}$ is an integral domain}\};\\ |R|^{split}\subset\{w\in|R|:\textit{ $R_{w}$ is not an integral domain}\}\end{array}\right.\textit{and thus }~{}~{}~{}~{}~{}~{}~{}|R|=|R|^{irred}\sqcup|R|^{split}.

We denote the residue field of $R_{w}$ by $\kappa_{R_{w}}$ and the cardinality of $\kappa_{R_{w}}$ by $q_{R_{w}}$ . For $w\in|R|^{irred}$ , we denote the residue field of $\mathcal{O}_{K_{w}}$ by $\kappa_{K_{w}}$ .

Proposition 7.4.

Suppose that $R=\mathcal{O}_{k}[x]/(\chi(x))$ is a Bass order. The product of orbital integrals $\mathcal{SO}_{v}(\chi)$ is given as follows

\prod_{v\in\Omega_{k}\setminus\infty_{k}}\mathcal{SO}_{v}(\chi)=\left(\prod_{w\in|R|^{split}}q_{R_{w}}^{S_{w}(R_{w})}\cdot\prod_{w\in|R|^{irred}}\bigg{(}q_{R_{w}}^{S_{w}(R_{w})}+[\kappa_{K_{w}}:\kappa_{R_{w}}]\frac{q_{R_{w}}^{S_{w}(R_{w})}-1}{q_{R_{w}}-1}\bigg{)}\right),

where $S_{w}(R_{w}):=\{\textit{$\mathcal{O}_{k_{v}}$-module length between $\mathcal{O}_{K_{w}}$ and $R_{w}$}\}/[\kappa_{R_{w}}:\kappa_{v}]$ .

Proof.

To ease the notation, we denote $\mathcal{O}_{k_{v}}[x]/(\chi(x))$ by $R_{v}$ . For $(\mathcal{O},E)=(R_{v},K_{v})$ or $(R_{w},K_{w})$ respectively, we define

\left\{\begin{array}[]{l}X_{\mathcal{O}}\textit{ to be the set of frational $\mathcal{O}$-ideals and};\\ \Lambda_{E}\textit{ to be complementary to $\mathcal{O}_{E}^{\times}$ inside $E^{\times}$, which acts on $X_{\mathcal{O}}$ by multiplication.}\end{array}\right.

By [CHL, Definition 2.3], we have the following identification

\mathcal{SO}_{v}(\chi)=\#(\Lambda_{K_{v}}\backslash X_{R_{v}}).

The isomorphism $R_{v}\cong\bigoplus\limits_{w|v,\ w\in|R|}R_{w}$ in [CHL, Remark B.(1)] yields that $X_{R_{v}}\cong\prod\limits_{w|v,\ w\in|R|}X_{R_{w}}$ and thus $\Lambda_{K_{v}}\backslash X_{R_{v}}\cong\prod\limits_{w|v,\ w\in|R|}\Lambda_{K_{w}}\backslash X_{R_{w}}$ . Since the map from $\mathrm{Spec}\ R$ to $\mathrm{Spec}\ \mathcal{O}_{k}$ is surjective, we then have

(7.2)

\prod_{v\in\Omega_{k}\setminus\infty_{k}}\mathcal{SO}_{v}(\chi)=\prod_{w\in|R|}\#(\Lambda_{K_{w}}\backslash X_{R_{w}}).

For $w\in|R|^{irred}$ , by [CHL, Theorem 3.7.(1)], we have

\#(\Lambda_{E_{w}}\backslash X_{R_{w}})=q_{R_{w}}^{S_{w}(R_{w})}+[\kappa_{K_{w}}:\kappa_{R_{w}}]\frac{q_{R_{w}}^{S_{w}(R_{w})}-1}{q_{R_{w}}-1}.

For $w\in|R|^{split}$ , by [CHL, Theorem 6.11], we have

\#(\Lambda_{E_{w}}\backslash X_{R_{w}})=q_{R_{w}}^{S_{w}(R_{w})}.

Plugging these closed formula for $\#(\Lambda_{E_{w}}\backslash X_{R_{w}})$ into (7.2), we have the desired result. ∎

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An asymptotic formula for the number of integral matrices with a fixed characteristic polynomial via orbital integrals

Abstract.

2020 Mathematics Subject Classification:

1. Introduction

1.1. Backgrounds

Proposition 1.1.

1.2. Main results

1.2.1. Brauer evaluation of ξ∈Br⁡X\xi\in\operatorname{Br}X on X​(kv)X(k_{v})

Proposition 1.2.

1.2.2. Computation of each local integration

Proposition 1.3.

Proposition 1.4.

1.2.3. Conclusion

Theorem 1.5 (Theorem 7.1).

Remark 1.6.

Remark 1.7.

1.3. Notations

2. Homogeneous space XX

2.1. Homogeneous space

Proposition 2.1.

Proof.

2.2. Conjugacy class of GLn​(kv)\mathrm{GL}_{n}(k_{v}) and SLn​(kv)\mathrm{SL}_{n}(k_{v}) in X​(kv)X(k_{v})

Lemma 2.2.

Proof.

Proposition 2.3.

Proof.

Proposition 2.4.

Proof.

Remark 2.5.

Definition 2.6.

Remark 2.7.

Remark 2.8.

3. Measures on XX

3.1. Tamagawa Measure on a homogeneous space

3.1.1. Gauge forms

Definition 3.1.

Definition 3.2 ([Wei82, Section 2.4, p24]).

Proposition 3.3.

3.1.2. Tamagawa measure

Definition 3.4.

Remark 3.5.

Proposition 3.6.

Proof.

3.2. Measures on X​(𝔸k)X(\mathbb{A}_{k}) for X≅Tx0\GLnX\cong\mathrm{T}_{x_{0}}\backslash\mathrm{GL}_{n}

3.2.1. The standard integral model for Tx0,kv\mathrm{T}_{x_{0},k_{v}} for v∈Ωk∖∞kv\in\Omega_{k}\setminus\infty_{k}

3.2.2. Volume forms on Xkv≅Tx0,kv\GLn,kvX_{k_{v}}\cong\mathrm{T}_{x_{0},k_{v}}\backslash\mathrm{GL}_{n,k_{v}}

Lemma 3.7.

Lemma 3.8.

Proof.

Proposition 3.9.

4. Brauer groups and evaluations

Proposition 4.1.

Definition 4.2.

Definition 4.3.

Remark 4.4.

Definition 4.5.

Remark 4.6.

5. The asymptotic formula for N​(X,T)N(X,T)

5.1. Evaluation of ξ∈Br⁡X/Br⁡k\xi\in\operatorname{Br}X/\operatorname{Br}k

5.1.1. The case that K/kK/k is not a Galois extension

Proposition 5.1.

Proof.

5.1.2. The case that K/kK/k is a Galois extension

Proposition 5.2.

Proof.

Corollary 5.3.

Proof.

Corollary 5.4.

Proof.

Proposition 5.5.

Proof.

5.2. Computation on Archimedean places

Proposition 5.6.

Proof.

Lemma 5.7.

Proof.

Lemma 5.8.

Proof.

5.3. Computation on non-Archimedean places

5.3.1. The case that the evaluation ξ~v\tilde{\xi}_{v} is trivial

1.2.1. Brauer evaluation of $\xi\in\operatorname{Br}X$ on $X(k_{v})$

2. Homogeneous space $X$

2.2. Conjugacy class of $\mathrm{GL}_{n}(k_{v})$ and $\mathrm{SL}_{n}(k_{v})$ in $X(k_{v})$

3. Measures on $X$

3.2. Measures on $X(\mathbb{A}_{k})$ for $X\cong\mathrm{T}_{x_{0}}\backslash\mathrm{GL}_{n}$

3.2.1. The standard integral model for $\mathrm{T}_{x_{0},k_{v}}$ for $v\in\Omega_{k}\setminus\infty_{k}$

3.2.2. Volume forms on $X_{k_{v}}\cong\mathrm{T}_{x_{0},k_{v}}\backslash\mathrm{GL}_{n,k_{v}}$

5. The asymptotic formula for $N(X,T)$

5.1. Evaluation of $\xi\in\operatorname{Br}X/\operatorname{Br}k$

5.1.1. The case that $K/k$ is not a Galois extension

5.1.2. The case that $K/k$ is a Galois extension

5.3.1. The case that the evaluation $\tilde{\xi}_{v}$ is trivial

5.3.2. The case that the evaluation $\tilde{\xi}_{v}$ is non-trivial and $K_{v}/k_{v}$ is ramified

6. Computation on non-Archimedean places: the case that the evaluation $\tilde{\xi}_{v}$ is non-trivial and $K_{v}/k_{v}$ is unramified

6.1. Measures on each $\mathrm{SL}_{n}(k_{v})$ -orbit in $X({k_{v}})$

6.1.1. Integral models for $\mathrm{S}_{x_{0}^{\prime},k_{v}}$ when $K_{v}/k_{v}$ is an unramified field extension

6.1.2. Volume forms on $X_{k_{v}}\cong\mathrm{S}_{x_{0}^{\prime},k_{v}}\backslash\mathrm{SL}_{n,k_{v}}$

6.2.1. Endoscopic data and endoscopic group for $\mathrm{SL}_{n}$