On monic abelian trace-one cubic polynomials

Shubhrajit Bhattacharya Department of Mathematics University of British Columbia Vancouver, BC Canada [email protected] and Andrew O’Desky Department of Mathematics Princeton University Princeton, NJ USA [email protected]

(Date: October 15, 2023)

Abstract.

We compute the asymptotic number of monic trace-one integral polynomials with Galois group $C_{3}$ and bounded height. For such polynomials we compute a height function coming from toric geometry and introduce a parametrization using the quadratic cyclotomic field $\mathbb{Q}(\sqrt{-3})$ . We also give a formula for the number of polynomials of the form $t^{3}-t^{2}+at+b\in\mathbb{Z}[t]$ with Galois group $C_{3}$ for a fixed integer $a$ .

1. Introduction

Let $F$ denote the set of polynomials of the form $t^{3}-t^{2}+at+b\in\mathbb{Z}[t]$ which have Galois group $C_{3}$ , the cyclic group of order three. The primary aim of this paper is to prove the following asymptotic formula.

Theorem 1.

Let $\varepsilon>0$ . The number of polynomials $t^{3}-t^{2}+at+b\in F$ with $\max(|a|^{1/2},|b|^{1/3})\leq H$ is equal to

CH^{2}\log H+\left(C\log\sqrt{3}+D-\frac{\pi}{3\sqrt{3}}\right)H^{2}+O_{\varepsilon}(H^{1+\varepsilon})

as $H\to\infty$ , where

(1)

C=\frac{4\pi^{2}}{81}\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2}}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2}}+\frac{2}{p^{3}}\right)

and

(2)

\frac{D}{C}=2\gamma+\log(2\pi)-3\log\left(\frac{\Gamma(1/3)}{\Gamma(2/3)}\right)+\frac{9}{8}\log 3+\frac{9}{4}\sum_{q\equiv 2\ (\mathrm{mod}\ 3)}\frac{\log q}{q^{2}-1}+\frac{27}{4}\sum_{p\equiv 1\ (\mathrm{mod}\ 3)}\frac{(p+1)\log p}{p^{3}-3p+2}.

This may be qualitatively compared with [14, Theorem 1.1] which asserts that the number $N(H)$ of monic integral cubic polynomials $t^{3}+at^{2}+bt+c$ with Galois group $C_{3}$ and $\max(|a|,|b|,|c|)\leq H$ satisfies $2H\leq N(H)\ll H(\log H)^{2}$ , however their height function is inequivalent to the height in Theorem 1 and there is no trace-one condition.

We also prove a formula of sorts for the number of $f\in F$ with specified nonconstant coefficients.

Theorem 2.

For any $H\geq 1$ let $E_{H}\subset\mathbb{R}^{2}$ be the ellipse defined by

(3)

E_{H}:x^{2}+y^{2}+xy-x-y=\tfrac{1}{3}(H^{2}-1).

If $t^{3}-t^{2}+at+b\in F$ then $a\leq 0$ . Fix $a\in\mathbb{Z}_{\leq 0}$ . The number of polynomials of the form $t^{3}-t^{2}+at+b\in F$ for any $b\in\mathbb{Z}$ is equal to

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\frac{1}{2}\sum_{d|(1-3a)}3^{\omega(P_{1}(d))}(-1)^{\Omega(P_{2}(d))}-\frac{1}{6}\#E_{\sqrt{1-3a}}(\mathbb{Z})

where $P_{j}(d)$ denotes the largest divisor of $d$ only divisible by primes $\equiv j\ (\mathrm{mod}\ 3)$ , and $\omega(n)$ (resp. $\Omega(n)$ ) denotes the number of prime factors of a positive integer $n$ counted without (resp. with) multiplicity.

An integral Diophantine problem

To prove these theorems we relate the polynomial counting problem to an integral Diophantine problem on a certain singular toric surface $S$ and then solve the Diophantine problem. Let $\mathbb{A}^{3}=\mathrm{Spec}\,\mathbb{Q}[X,Y,Z]$ and $\mathbb{P}_{2}=\mathbb{P}(\mathbb{A}^{3})=\mathrm{Proj}\,\mathbb{Q}[X,Y,Z]$ be equipped with the regular action of $C_{3}$ . Consider the quotient surface

(5)

S=\mathbb{P}_{2}/C_{3}.

Let $T\subset S$ denote the image of the unit group in the group algebra $\mathbb{A}^{3}$ of $C_{3}$ under $\mathbb{A}^{3}-\{0\}\to\mathbb{P}_{2}\to S$ . One can show that $T$ is a rank-two torus and $S$ is a toric compactification of $T$ . The set of rational points $S(\mathbb{Q})$ is thus equipped with a family of toric height functions $H(-,s)$ constructed in [1], where $s$ is a parameter in the complexified Picard group $\mathrm{Pic}(S)\otimes\mathbb{C}$ . The surface $S$ has Picard rank one [12, Corollary 3.6], so we may regard $s$ as a complex number where $s=3$ corresponds to the ample generator. Let $D_{0}$ be the divisor $\{\varepsilon\coloneqq X+Y+Z=0\}\subset S$ . A rational point $P$ of $S-D_{0}$ is $D_{0}$ -integral if every regular function in $\mathcal{O}(S_{\mathbb{Z}}-D_{0})=\mathbb{\mathbb{Z}}[X/\varepsilon,Y/\varepsilon]^{C_{3}}$ is $\mathbb{Z}$ -valued on $P$ .

Our third result is an explicit formula for the height zeta function for $D_{0}$ -integral rational points on the torus $T\subset S$ .

Theorem 3.

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\sum_{\begin{subarray}{c}P\in T(\mathbb{Q}),\\ \text{$D_{0}$-\emph{integral}}\end{subarray}}H(P,s)^{-1}=\left(1-\frac{1}{3^{z}}\right)^{2}\zeta_{\mathbb{Q}(\sqrt{-3})}(z)^{2}\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2z}}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2z}}+\frac{2}{p^{3z}}\right)

where $z=\tfrac{s}{2}$ and $\zeta_{\mathbb{Q}(\sqrt{-3})}$ is the Dedekind zeta function of $\mathbb{Q}(\sqrt{-3})$ . This height zeta function can be meromorphically continued to the half-plane $\mathrm{Re}(s)>1$ and its only pole in this region is at $s=2$ with order $2$ . If $n\in\mathbb{Z}_{\geq 1}$ is not divisible by $3$ , then the number of $D_{0}$ -integral rational points on $T$ with toric height $\sqrt{n}$ is equal to

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\sum_{d|n}3^{\omega(P_{1}(d))}(-1)^{\Omega(P_{2}(d))}

Relation between the problems

In [12] it was shown that the torus $T$ is the moduli space for $C_{3}$ -algebras with a given trace-one normal element. In particular,

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T(\mathbb{Q})\cong\{(K/\mathbb{Q}\text{ $C_{3}$-algebra},\,\,x\text{ trace-one normal})\}

where a $C_{3}$ -algebra $K/\mathbb{Q}$ is a $\mathbb{Q}$ -algebra equipped with an action of $C_{3}$ for which there is a $C_{3}$ -linear $\mathbb{Q}$ -algebra isomorphism from $K$ to either a cubic abelian number field or the split algebra $\mathbb{Q}^{3}$ , and an element $x\in K$ is normal if its Galois conjugates are linearly independent over $\mathbb{Q}$ . Using this bijection we consider the function

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T(\mathbb{Q})\,\longrightarrow\{t^{3}-t^{2}+at+b\in\mathbb{Q}[t]\}

taking a rational point $(K/\mathbb{Q},x)$ to the characteristic polynomial of $x$ . We prove that the image of this function is the subset of polynomials which either have Galois group $C_{3}$ or split into three linear factors over $\mathbb{Q}$ with at most two being the same, and if $f$ is such a polynomial, then the number of rational points of $T$ with characteristic polynomial $f$ is given by

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w_{f}=\begin{cases}1&\text{if $f$ has a double root,}\\ 2&\text{otherwise.}\\ \end{cases}

Moreover we show that a rational point $P$ of $T$ is $D_{0}$ -integral if and only if the associated characteristic polynomial $t^{3}-t^{2}+at+b$ is integral, and we also prove that

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H(P,1)=\sqrt{1-3a}

for $D_{0}$ -integral points. This toric height is equivalent to the height used in Theorem 1.

Further remarks

The restriction to trace-one normal elements was made out of convenience in [12] and should not be essential for the method. In place of $S$ , there is a three-fold with a similar construction and an open subset which parametrizes all normal elements of $C_{3}$ -algebras. In forthcoming work [11] the method presented here will be extended to count monic integral polynomials with bounded height and any given abelian Galois group.

Acknowledgements

A.O. is very grateful to Timothy Browning, Vesselin Dimitrov, Jef Laga, Peter Sarnak, Sameera Vemulapalli, Victor Wang, and Shou-Wu Zhang for helpful discussions and comments on an earlier draft. A.O. would also like to thank Alexandra Pevzner for pointing out the reference [2]. A.O. was supported by NSF grant DMS-2103361.

2. The orbit parametrization

In this section we recall some facts from [12] and describe the orbit parametrization. Let $\sigma$ be a generator of $C_{3}$ . Let $\Delta=3XYZ-X^{3}-Y^{3}-Z^{3}$ , the determinant of multiplication by an element $Xe+Y\sigma+Z\sigma^{2}$ of the group algebra. We set

\mathcal{G}=\mathbb{P}_{2}[\Delta^{-1}]\quad\text{and}\quad T=\mathcal{G}/C_{3}.

Then $\mathcal{G}$ is an algebraic torus over $\mathbb{Q}$ which may be identified with the units of the group algebra of $C_{3}$ with augmentation one, i.e.

\mathcal{G}=\{(x,y,z)\in\mathbb{A}^{3}:\Delta(x,y,z)\in\mathbb{G}_{m}\text{ and }x+y+z=1\}.

Since $C_{3}$ is abelian, the homogeneous space $T=\mathcal{G}/C_{3}$ is itself an algebraic torus over $\mathbb{Q}$ . The action of $\mathcal{G}$ on the regular representation induces an action of $T$ on $S$ extending the regular action of $T$ on itself. Let $\mathbb{A}^{2}=\mathrm{Spec}\,\mathbb{Q}[X/\varepsilon,Y/\varepsilon]$ denote the open affine plane in $\mathbb{P}_{2}$ where the augmentation map $\varepsilon=X+Y+Z$ is nonvanishing. A rational (or adelic) point $P$ of $\mathbb{A}^{2}/C_{3}$ is $D_{0}$ -integral if every regular function in $\mathcal{O}(\mathbb{A}_{\mathbb{Z}}^{2}/C_{3})=\mathbb{\mathbb{Z}}[X/\varepsilon,Y/\varepsilon]^{C_{3}}$ is $\mathbb{Z}$ -valued (resp. $\widehat{\mathbb{Z}}\times\mathbb{R}$ -valued) on $P$ .

2.1. $T$ as a moduli space

Let $K/\mathbb{Q}$ be a separable $\mathbb{Q}$ -algebra equipped with the action of a finite group $G$ of $\mathbb{Q}$ -algebra automorphisms of $K$ . We say that $K/\mathbb{Q}$ regarded with its $G$ -action is a (Galois) $G$ -algebra if the subset of $K$ fixed by $G$ is equal to $\mathbb{Q}$ . Geometrically, a $G$ -algebra is the ring of functions on a principal $G$ -bundle, equipped with its natural $G$ -action.

Warning 1.

Since we regard the $G$ -action as part of the data of a $G$ -algebra, a $G$ -algebra is not generally determined by the isolated data of the underlying $\mathbb{Q}$ -algebra $K$ and the abstract finite group $G$ . The $G$ -action on a $G$ -algebra may be twisted by any outer automorphism of $G$ , and the twisted $G$ -algebra will not generally be isomorphic to the original $G$ -algebra.

Two pairs $(K/\mathbb{Q},x)$ , $(K^{\prime}/\mathbb{Q},x^{\prime})$ are regarded as equivalent if there is a $G$ -equivariant $\mathbb{Q}$ -algebra isomorphism $K\to K^{\prime}$ sending $x$ to $x^{\prime}$ . We make use of the following modular interpretation for $T$ .

Theorem 4 ([12, §2]).

The homogeneous variety $T$ is the moduli space for $C_{3}$ -algebras with a given trace-one normal element. In particular, there is a bijection between rational points of $T$ and equivalence classes of $C_{3}$ -algebras $K/\mathbb{Q}$ equipped with a trace one normal element $x\in K$ .

Example 1.

Let $K$ be a cubic abelian number field. Then $K$ , equipped with its canonical Galois action, is a $C_{3}$ -algebra. The twist $K^{\prime}$ of the $C_{3}$ -algebra $K$ by the outer automorphism $g\mapsto g^{-1}$ of $C_{3}$ (with twisted action $g\ast x=g^{-1}x$ ) is not isomorphic to $K$ as a $C_{3}$ -algebra.¹¹1 In terms of Galois cohomology, the non-cohomologous $1$ -cocycles in $H^{1}(\mathbb{Q},C_{3})$ corresponding to the $C_{3}$ -algebras $K$ and $K^{\prime}$ have the same image under the canonical map $H^{1}(\mathbb{Q},C_{3})\to H^{1}(\mathbb{Q},S_{3})$ because the outer automorphism of $C_{3}$ is realized by $S_{3}$ -conjugation.

Example 2.

Let $K_{\text{spl}}=\mathbb{Q}^{3}$ , the split cubic algebra. Then $C_{3}\subset S_{3}=\mathrm{Aut}_{\text{$\mathbb{Q}$-alg}}(K_{\text{spl}})$ and $K_{\text{spl}}$ , equipped with its canonical $C_{3}$ -action, is a $C_{3}$ -algebra. Any transposition gives an isomorphism of $C_{3}$ -algebras from $K_{\text{spl}}$ to its twist $K_{\text{spl}}^{\prime}$ .

Example 3.

An element $x$ of the split $C_{3}$ -algebra $K_{\text{spl}}$ is normal if and only if $x$ either has distinct coordinates or exactly two identical coordinates. The pairs $(K_{\text{spl}},x)$ and $(K_{\text{spl}}^{\prime},x)$ are equivalent if and only if $x$ has exactly two identical coordinates (swapping the identical coordinates gives the required isomorphism); in particular, if $x$ has distinct coordinates then $(K_{\text{spl}},x)$ and $(K_{\text{spl}}^{\prime},x)$ determine different rational points of $\mathcal{G}/C_{3}$ , even though $K_{\text{spl}}$ and $K_{\text{spl}}^{\prime}$ are isomorphic as $C_{3}$ -algebras.

2.2. $T$ as a torus

Here we describe some of the toric data associated with $T$ which will be needed later. For more details see e.g. [3, p. 202]. Let $E=\mathbb{Q}(\zeta)$ where $\zeta$ is a primitive cube root of unity, and let $\gamma$ denote the generator of the Galois group $\Gamma$ of $E$ over $\mathbb{Q}$ . Let $Pl_{E}$ denote the set of places of $E$ . The group of units $U$ in the group algebra is a three-dimensional algebraic torus defined over $\mathbb{Q}$ which canonically factors as $U=\mathbb{G}_{m}\times\mathcal{G}$ . The characters and cocharacters of $T$ may be described as follows. The larger torus $U$ is diagonalized over $E$ by the three elementary idempotents in the group algebra:

\displaystyle v_{0}^{\prime}=\tfrac{1}{3}(1+\sigma+\sigma^{2}),\quad v_{1}^{\prime}=\tfrac{1}{3}(1+\zeta^{2}\sigma+\zeta\sigma^{2}),\quad v_{2}^{\prime}=\tfrac{1}{3}(1+\zeta\sigma+\zeta^{2}\sigma^{2}).

Each idempotent is associated with a character $\chi_{i}\colon U(E)\to E^{\times}$ for $i=0,1,2$ determined by $uv_{i}^{\prime}=\chi_{i}(u)v_{i}^{\prime}$ , corresponding to the action of $U$ on the $i$ th irreducible representation of $C_{3}$ . The character $\chi_{0}$ is trivial on $\mathcal{G}$ , so the lattice of characters of $\mathcal{G}_{E}$ is generated by $\chi_{1}$ and $\chi_{2}$ . We denote this lattice by $M_{E}^{\prime}$ and let $N_{E}^{\prime}$ denote the dual lattice to $M_{E}^{\prime}$ . To describe the fans it is more symmetric to work with the isomorphic image of $N_{E}^{\prime}$ in the quotient of $\mathbb{C}C_{3}$ by the line spanned by $v_{0}^{\prime}+v_{1}^{\prime}+v_{2}^{\prime}$ , and we write $v_{i}$ for the image of $v_{i}^{\prime}$ ( $i=0,1,2$ ). The Galois group $\Gamma$ of $E$ acts on $M_{E}^{\prime}$ by swapping $\chi_{1}$ and $\chi_{2}$ , and on $N_{E}^{\prime}$ via the dual action.

To pass from $\mathcal{G}$ to $T$ , consider the element

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\omega=\tfrac{1}{3}(2v_{1}+v_{2})\in N_{E,\mathbb{Q}}^{\prime}

and set

(13)

N_{E}=N_{E}^{\prime}+\omega\quad\text{and}\quad M_{E}=N_{E}^{\vee}=\{m\in M_{E,\mathbb{Q}}^{\prime}:m(n)\in\mathbb{Z}\text{ for all $n\in N_{E}$}\}.

The character lattice (resp. cocharacter lattice) of $T_{E}$ is $M_{E}$ (resp. $N_{E}$ ). The cocharacters $\omega$ and $\gamma\omega$ span $N_{E}$ so the dual basis $(a^{\prime},b^{\prime})=(\omega,\gamma\omega)^{\vee}$ spans $M_{E}$ .

Figure 1. The dual lattice

M_{E}=\mathbb{Z}\langle a^{\prime},b^{\prime}\rangle

(left) and the fan

\Sigma

S

N_{E,\mathbb{R}}

(right).

The fan $\Sigma$ of $S$ is the same as the fan for $\mathbb{P}_{2}$ and has three generators $\Sigma(1)=\{v_{0},v_{1},v_{2}\}$ .

We also make use of the following formulas for the characters of $\mathcal{G}$ . Let $(v_{1}^{\vee},v_{2}^{\vee})\in M_{E}^{\prime}$ be the dual basis to $(v_{1},v_{2})\in N_{E}^{\prime}$ . The characters of $\mathcal{G}$ associated to $v_{1}^{\vee}$ and $v_{2}^{\vee}$ are given on $E$ -points of $U$ by

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\chi^{v_{1}^{\vee}}(uv_{0}^{\prime}+vv_{1}^{\prime}+wv_{2}^{\prime})=\frac{v}{u}\quad\text{and}\quad\chi^{v_{2}^{\vee}}(uv_{0}^{\prime}+vv_{1}^{\prime}+wv_{2}^{\prime})=\frac{w}{u}.

This explicit description of the character lattices leads to an (unexpected) isomorphism between $\mathcal{G}$ and its quotient $T=\mathcal{G}/C_{3}$ . On character lattices it is given by the $\Gamma$ -equivariant isomorphism

(15)

N_{E}=\mathbb{Z}\langle\omega,\gamma\omega\rangle\to N_{E}^{\prime}=\mathbb{Z}\langle v_{1},v_{2}\rangle

taking $\omega$ to $v_{1}$ and $\gamma\omega$ to $v_{2}$ . This implies that the multiplicative group of the cyclotomic field $\mathbb{Q}(\sqrt{-3})$ naturally parametrizes cubic trace-one polynomials.

Proposition 1.

The tori $T$ and $\mathcal{G}=R^{E}_{\mathbb{Q}}\mathcal{\mathbb{G}}_{m}$ are isomorphic as algebraic groups over $\mathbb{Q}$ . Every rational point $(K/\mathbb{Q},x)$ of $T$ thereby determines an element of $\mathbb{Q}(\sqrt{-3})^{\times}$ which is canonically determined up to the action of $\mathrm{Aut}(\mathcal{G})$ . The toric height $H(f)\coloneqq\sqrt{1-3a}$ on $T(\mathbb{Q})$ is identified with the square-root of the norm on $\mathbb{Q}(\sqrt{-3})^{\times}$ . Let $\zeta$ be a primitive cube root of unity. If $u+v\zeta\in\mathbb{Q}(\sqrt{-3})^{\times}$ has norm $N$ and trace $T$ , then the characteristic polynomial of the corresponding rational point $(K/\mathbb{Q},x)$ is

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f=t^{3}-t^{2}+\tfrac{1}{3}(1-N)t+\tfrac{1}{27}(1+N(T-3))\in\mathbb{Q}[t].

Such a polynomial either has Galois group $C_{3}$ or splits into three linear factors over $\mathbb{Q}$ , with at most two linear factors being the same. Conversely, a monic trace-one polynomial $f=t^{3}-t^{2}+at+b\in\mathbb{Q}[t]$ which either has Galois group $C_{3}$ or splits into three linear factors over $\mathbb{Q}$ , with at most two linear factors being the same, can be expressed in this way for precisely two rational points of $T$ if $f$ has no repeated roots, or for precisely one rational point of $T$ if $f$ has a double root which is not a triple root. The elements $u+v\zeta\in\mathbb{Q}(\sqrt{-3})^{\times}$ corresponding to $f$ will be the roots of the quadratic polynomial

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g=t^{2}-\left(3-\frac{1-27b}{1-3a}\right)t+1-3a\in\mathbb{Q}[t].

The polynomial $f$ will have integral coefficients if and only if

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\begin{cases}u^{2}+v^{2}-uv\in 1+3\mathbb{Z}\text{ and }\\ (u^{2}+v^{2}-uv)(3-2u+v)\in 1+27\mathbb{Z}.\end{cases}

Proof.

The character lattice of a torus over $\mathbb{Q}$ as a Galois representation determines the torus as an algebraic group up to isomorphism, cf. e.g. [9, Theorem 12.23]. Equation (30) below identifies the toric height with the square-root of the norm. The formulas for $a$ and $b$ follow from expressing $a$ and $b$ in terms of characters of $T$ and then using (15) to reexpress these using characters on $\mathcal{G}$ . ∎

Cubic $f$	Quadratic $g$	$\mathrm{disc}(f)$	$\mathrm{disc}(g)$	$H(f)^{2}$
$t^{3}-t^{2}$	$t^{2}-2t+1$	$0$	$0$	$1$
$t^{3}-t^{2}-t+1$	$t^{2}+4t+4$	$0$	$0$	$4$
$t^{3}-t^{2}-2t+1$	$t^{2}+t+7$	$7^{2}$	$-1\cdot 3^{3}$	$7$
$t^{3}-t^{2}-2t$	$t^{2}-\frac{20}{7}t+7$	$2^{2}\cdot 3^{2}$	$-1\cdot 2^{2}\cdot 3^{5}\cdot 7^{-2}$	$7$
$t^{3}-t^{2}-4t+4$	$t^{2}+\frac{70}{13}t+13$	$2^{4}\cdot 3^{2}$	$-1\cdot 2^{4}\cdot 3^{5}\cdot 13^{-2}$	$13$
$t^{3}-t^{2}-4t-1$	$t^{2}-5t+13$	$13^{2}$	$-1\cdot 3^{3}$	$13$
$t^{3}-t^{2}-5t-3$	$t^{2}-8t+16$	$0$	$0$	$16$
$t^{3}-t^{2}-6t+7$	$t^{2}+7t+19$	$19^{2}$	$-1\cdot 3^{3}$	$19$
$t^{3}-t^{2}-6t$	$t^{2}-\frac{56}{19}t+19$	$2^{2}\cdot 3^{2}\cdot 5^{2}$	$-1\cdot 2^{2}\cdot 3^{5}\cdot 5^{2}\cdot 19^{-2}$	$19$
$t^{3}-t^{2}-8t+12$	$t^{2}+10t+25$	$0$	$0$	$25$
$\quad\vdots$	$\quad\vdots$	$\quad\vdots$	$\quad\vdots$	$\vdots$
$t^{3}-t^{2}-190t+719$	$t^{2}+31t+571$	$7^{2}\cdot 571^{2}$	$-1\cdot 3^{3}\cdot 7^{2}$	$571$
$t^{3}-t^{2}-190t-800$	$t^{2}-\frac{23312}{571}t+571$	$2^{2}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2}\cdot 13^{2}$	$-1\cdot 2^{2}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2}\cdot 13^{2}\cdot 571^{-2}$	$571$
$t^{3}-t^{2}-192t+720$	$t^{2}+\frac{17710}{577}t+577$	$2^{6}\cdot 3^{6}\cdot 19^{2}$	$-1\cdot 2^{6}\cdot 3^{9}\cdot 19^{2}\cdot 577^{-2}$	$577$
$t^{3}-t^{2}-192t-171$	$t^{2}-11t+577$	$3^{4}\cdot 577^{2}$	$-1\cdot 3^{7}$	$577$
$t^{3}-t^{2}-196t+1124$	$t^{2}+\frac{922}{19}t+589$	$2^{4}\cdot 31^{2}$	$-1\cdot 2^{4}\cdot 3^{3}\cdot 19^{-2}$	$589$
$t^{3}-t^{2}-196t+1109$	$t^{2}+\frac{1483}{31}t+589$	$7^{4}\cdot 19^{2}$	$-1\cdot 3^{3}\cdot 7^{4}\cdot 31^{-2}$	$589$
$t^{3}-t^{2}-196t+539$	$t^{2}+\frac{673}{31}t+589$	$7^{2}\cdot 19^{2}\cdot 37^{2}$	$-1\cdot 3^{3}\cdot 7^{2}\cdot 31^{-2}\cdot 37^{2}$	$589$
$t^{3}-t^{2}-196t+349$	$t^{2}+13t+589$	$3^{4}\cdot 19^{2}\cdot 31^{2}$	$-1\cdot 3^{7}$	$589$
$t^{3}-t^{2}-196t+196$	$t^{2}+\frac{3526}{589}t+589$	$2^{4}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2}\cdot 13^{2}$	$-1\cdot 2^{4}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2}\cdot 13^{2}\cdot 19^{-2}\cdot 31^{-2}$	$589$
$t^{3}-t^{2}-196t-704$	$t^{2}-\frac{20774}{589}t+589$	$2^{4}\cdot 3^{6}\cdot 5^{2}\cdot 7^{2}$	$-1\cdot 2^{4}\cdot 3^{9}\cdot 5^{2}\cdot 7^{2}\cdot 19^{-2}\cdot 31^{-2}$	$589$

Figure 2. Some

f\in\mathbb{Z}[t]

with Galois group

C_{3}

and the characteristic polynomials

g\in\mathbb{Q}[t]

of their corresponding elements in

\mathbb{Q}(\sqrt{-3})

3. Toric heights

In this section we show that the toric height $H(-,1)$ of a $D_{0}$ -integral point $(K,x)$ of $T$ in the sense of [1] is equal to $H(f)=\sqrt{1-3a}$ where $f$ is the characteristic polynomial of $x$ .

Definition 1.

Let $w$ be a place of $E$ . For any $x\in T(E_{w})$ the function $\chi\mapsto\mathrm{ord}_{w}(\chi(x))$ on characters $\chi\in X^{\ast}(T_{E_{w}})$ determines an element of $X_{\ast}(T_{E_{w}})_{\mathbb{R}}$ . Let

n_{w}(x)\in X_{\ast}(T_{E})_{\mathbb{R}}

be the cocharacter corresponding to this element under the canonical isomorphism $X_{\ast}(T_{E_{w}})_{\mathbb{R}}\cong X_{\ast}(T_{E})_{\mathbb{R}}$ induced by base change of the split torus $T_{E}$ along $E\to E_{w}$ .

For any place $v$ of $\mathbb{Q}_{v}$ let $K_{v}$ denote the maximal compact subgroup of $T(\mathbb{Q}_{v})$ . Evaluating characters of $T_{E}$ on $\mathbb{Q}_{v}$ -points gives a canonical bijection

T(\mathbb{Q}_{v})=\mathrm{Hom}_{\Gamma(w/v)}(M_{E},E_{w}^{\times})

where $w$ is any place of $E$ over $v$ . When $v$ is finite, $K_{v}$ may be identified with the subset of $O_{w}^{\times}$ -valued homomorphisms $K_{v}=\mathrm{Hom}_{\Gamma(w/v)}(M_{E},O_{w}^{\times})\subset T(\mathbb{Q}_{v})$ .

Proposition 2.

Let $w$ be a place of $E$ lying over a place $v$ of $\mathbb{Q}$ . There is an exact sequence

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1\longrightarrow K_{v}\longrightarrow T(\mathbb{Q}_{v})\xlongrightarrow{n_{w}}X_{\ast}(T_{E})_{\mathbb{R}}^{\Gamma(w/v)}.

If $w$ is infinite then $n_{w}$ is surjective, and if $w$ is finite then the image of $n_{w}$ is the lattice $X_{\ast}(T_{E})^{\Gamma(w/v)}$ .

Proof.

[4, (1.3), p. 449] nearly proves the claim but at the ramified place $w$ over $v=3$ only ensures that the image of $n_{w}$ is a finite index subgroup of $X_{\ast}(T_{E})^{\Gamma(w/v)}$ . To see that the image of $n_{w}$ is all of $X_{\ast}(T_{E})^{\Gamma(w/v)}$ recall that the cocharacter lattices of $T_{E}$ and $\mathcal{G}_{E}$ are isomorphic as Galois representations via (15). Since $T(\mathbb{Q}_{v})$ and $K_{v}$ are determined by the dual modules $M_{E}$ and $M_{E}^{\prime}$ , it suffices to show that $n_{w}$ is surjective when defined relative to $\mathcal{G}$ ; in more detail, there is a diagram

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where the vertical arrows are isomorphisms of abelian groups induced by the transpose of the $\Gamma$ -isomorphism $N_{E}\to N_{E}^{\prime}$ , and the homomorphisms $n_{w}$ correspond to post-composing with $\mathrm{ord}_{w}$ . The diagram commutes so surjectivity of the upper $n_{w}$ implies surjectivity of the lower $n_{w}$ .

To see that the upper $n_{w}$ is surjective, observe that the upper row of the diagram is the $\Gamma(w/v)$ -invariants of the short exact sequence of $\Gamma(w/v)$ -modules

(21)

(here the exactness on the right follows from $\mathrm{Ext}^{1}(M_{E}^{\prime},O_{w}^{\times})=0$ since $M_{E}^{\prime}$ is free); thus the upper row of the diagram continues to the first cohomology group $H^{1}(\Gamma(w/v),\mathrm{Hom}(M_{E}^{\prime},O_{w}^{\times}))$ . Now recall that the group of units $U$ in the group algebra of $C_{3}$ is $\mathbb{G}_{m}\times R^{E}_{\mathbb{Q}}\mathbb{G}_{m}$ where the first projection is the augmentation character, so the torus $\mathcal{G}$ is isomorphic to $R^{E}_{\mathbb{Q}}\mathbb{G}_{m}$ . This implies that $M_{E}^{\prime}$ is a free $\mathbb{Z}\Gamma(w/v)$ -module, $\mathrm{Hom}(M_{E}^{\prime},O_{w}^{\times})$ is coinduced, and therefore $H^{1}(\Gamma(w/v),\mathrm{Hom}(M_{E}^{\prime},O_{w}^{\times}))=0$ so $n_{w}$ is surjective. ∎

The toric variety $S$ has at worst cyclic quotient singularities since its fan is simplicial so every Weil divisor on $S$ is $\mathbb{Q}$ -Cartier. The toric height with respect to a Weil divisor $D$ for which $nD$ is Cartier is defined as $H(-,\mathcal{O}(D))$ as $H(-,\mathcal{O}(nD))^{1/n}$ . Let $D_{0},D_{1},D_{2}$ be the three irreducible $T$ -stable divisors corresponding respectively to the three generators $v_{0},v_{1},v_{2}$ in $\Sigma$ of the fan of $S$ (cf. [6, §3.1]). We call any formal $\mathbb{C}$ -linear combination $s_{0}D_{0}+s_{1}D_{1}+s_{2}D_{2}$ a toric divisor of $S$ . A support function is a continuous $\Gamma$ -invariant function $\varphi\colon N_{E,\mathbb{R}}\to\mathbb{C}$ whose restriction to any cone of $\Sigma$ is linear. Support functions and $\Gamma$ -invariant toric divisors are in bijection under

\varphi\leftrightarrow(s_{0},s_{1},s_{2})=(-\varphi(v_{0}),-\varphi(v_{1}),-\varphi(v_{2}))

where $s_{1}=s_{2}$ to ensure $\Gamma$ -invariance. Any Cartier toric divisor $\sum_{e}s_{e}D_{e}$ corresponds to a $T_{E}$ -linearized line bundle $\mathcal{O}(\sum_{e}s_{e}D_{e})$ whose corresponding support function $\varphi$ satisfies $\varphi(e)=-s_{e}$ for each $e\in{\Sigma}(1)$ .

Definition 2 ([1]).

For $x=(x_{w})_{w}\in T(\mathbb{A}_{E})$ and $\varphi$ a support function let

(22)

H(x,\varphi)=\prod_{w\in Pl_{E}}\left(q_{w}^{-\varphi(n_{w}(x_{w}))}\right)^{\frac{1}{[E:\mathbb{Q}]}}

where $\varphi(n_{w}(x_{w}))$ is evaluated using the canonical isomorphism $X_{\ast}(T_{E})\cong X_{\ast}(T_{E_{w}})$ .

The following simplified form is often useful. If $x=(x_{v})_{v}\in T(\mathbb{A})$ , embedded diagonally in $T(\mathbb{A}_{E})$ , then the quantity $\varphi(n_{w}(x_{v}))$ is independent of the choice of $w$ over $v$ , and

(23)

H(x,\varphi)=\prod_{v\in M_{\mathbb{Q}}}q_{v}^{-\frac{1}{e_{v}}\varphi(n_{w}(x_{v}))}

where $e_{v}$ is the ramification index of any prime of $E$ lying over $v$ ( $1$ by definition if $v=\infty$ ).

3.1. Computing the local toric height

Let $L$ be a globally generated line bundle on $S$ and let $\{v_{1},\ldots,v_{N}\}\subset H^{0}(S_{E},L)$ be a generating set of global sections. The standard height function on $S$ associated to $L$ and the generating set $\{v_{1},\ldots,v_{N}\}$ is

(24)

H(x,L,(v_{i})_{i=1}^{N})=\prod_{w\in Pl_{E}}\max\left(\left|\frac{v_{1}(x)}{s(x)}\right|_{w},\ldots,\left|\frac{v_{N}(x)}{s(x)}\right|_{w}\right)^{\frac{1}{[E:\mathbb{Q}]}}\qquad(x\in S(E)).

where $s$ is any local nonvanishing section at $x$ , and $|\cdot|_{w}=q_{w}^{-\mathrm{ord}_{w}(\cdot)}$ if $w$ is nonarchimedean and $|\cdot|_{w}=|\cdot|^{d_{w}}$ otherwise. The quantity $H(x,L,(v_{i})_{i=1}^{N})$ does not depend on the local section $s$ or the choice of splitting field.

If the line bundle $L$ is linearized by the open torus $T$ of $S$ in the sense of [10, §1.3], then the space of sections of $L$ on any $T$ -stable open subset of $S$ carries a linear action of $T$ and may therefore be diagonalized. The toric height on $S$ associated to a $T$ -line bundle $L$ is the standard height function on $S$ defined using a basis of weight vectors for $H^{0}(S,L)$ . The advantage of this height is that its local height functions are amenable to harmonic analysis — namely their Fourier transforms have a simple form.

The next lemma computes the weight vectors we need to express the toric height relative to the toric divisor $D_{0}$ .

Lemma 1.

Let $\mathbf{1}$ denote the canonical nowhere-vanishing global section in $H^{0}(S_{E},\mathcal{O}(3D_{0}))$ . The space $H^{0}(S_{E},\mathcal{O}(3D_{0}))$ is spanned over $E$ by the following four weight vectors:

(25)

\mathbf{1},\quad(1-3e_{2}e_{1}^{-2})\mathbf{1},\quad(e_{1}^{3}-\tfrac{9}{2}e_{1}e_{2}+\tfrac{27}{2}e_{3}+\tfrac{\sqrt{-27}}{2}\sqrt{\mathrm{disc}})e_{1}^{-3}\mathbf{1},\quad(e_{1}^{3}-\tfrac{9}{2}e_{1}e_{2}+\tfrac{27}{2}e_{3}-\tfrac{\sqrt{-27}}{2}\sqrt{\mathrm{disc}})e_{1}^{-3}\mathbf{1}

where $\sqrt{\mathrm{disc}}=(X-Z)(Y-X)(Z-Y)$ . The associated characters of $T_{E}$ are, respectively,

(26)

1,\quad\chi^{a^{\prime}+b^{\prime}},\quad\chi^{2a^{\prime}+b^{\prime}},\quad\chi^{2b^{\prime}+a^{\prime}}

where $a^{\prime}=2v_{1}^{\vee}-v_{2}^{\vee}$ and $b^{\prime}=2v_{2}^{\vee}-v_{1}^{\vee}$ in the character lattice $M_{E}=X^{\ast}T_{E}$ and $(v_{1}^{\vee},v_{2}^{\vee})$ is the dual basis to $(v_{1},v_{2})$ .

Proof.

Let $\varphi_{0}$ be the support function corresponding to $-D_{0}$ .

Figure 3. The support function

\varphi_{0}

N_{E,\mathbb{R}}

On $S_{E}=S\otimes E$ we have the weight decomposition [6, p. 66, §3.4]

(27)

H^{0}(S_{E},\mathcal{O}(3D_{0}))\cong\bigoplus_{u\in P\cap M_{E}}E\chi^{u}

where $P$ is the polyhedron in $M_{E,\mathbb{R}}$ defined by

P=\{u\in M_{E,\mathbb{R}}:u\geq 3\varphi_{0}\text{ on $N_{E,\mathbb{R}}$}\}.

Let $(v_{1}^{\vee},v_{2}^{\vee})\in M_{E,\mathbb{Q}}$ be the dual basis to $(v_{1},v_{2})\in N_{E}$ . Write $u=u_{1}v_{1}^{\vee}+u_{2}v_{2}^{\vee}\in M_{E}$ . The polyhedron $P$ is cut out by the inequalities

(28)

\begin{cases}u\geq 0\,\,\text{on $\sigma_{12}$}\\ u\geq 3v_{2}^{\vee}\,\,\text{on $\sigma_{10}$}\\ u\geq 3v_{1}^{\vee}\,\,\text{on $\sigma_{20}$}.\end{cases}

Figure 4 depicts the polyhedron $P$ when $N_{E}$ is identified with the lattice in $\mathbb{R}^{2}$ generated by $\omega=(1,0)$ and $\gamma\omega=(\frac{1}{2},\frac{\sqrt{3}}{2})$ . Then the character lattice $M_{E}$ is generated by $a^{\prime}=(1,-\frac{\sqrt{3}}{3})$ and $b^{\prime}=(0,\frac{2\sqrt{3}}{3})$ . We have that $u_{1}=\frac{3}{2}x-\frac{\sqrt{3}}{2}y$ and $u_{2}=\sqrt{3}y$ where $x,y$ are the standard coordinates on $\mathbb{R}^{2}$ , and the polyhedron $P$ is cut out by the inequalities

(29)

\begin{cases}\frac{3}{2}x-\frac{\sqrt{3}}{2}y\geq 0\\ {\sqrt{3}}y\geq 0\\ 3\geq\frac{3}{2}x+\frac{\sqrt{3}}{2}y.\end{cases}

Figure 4. The polyhedron

P\subset M_{E,\mathbb{R}}

for the

T_{E}

-line bundle

\mathcal{O}(3D_{0})

We conclude that $h^{0}(S_{E},\mathcal{O}(3D_{0}))=4$ .

The global section $\mathbf{1}$ is clearly the weight vector in $H^{0}(S_{E},\mathcal{O}(3D_{0}))$ with trivial $T_{E}$ -action. We may find the other three weight vectors in $H^{0}(S_{E},\mathcal{O}(3D_{0}))$ by twisting $\mathbf{1}$ by the three nontrivial characters in $P$ . Using the formulas from §2.2, one finds that

(30)

\chi^{a^{\prime}+b^{\prime}}(uv_{0}^{\prime}+vv_{1}^{\prime}+wv_{2}^{\prime})=\frac{vw}{u^{2}}=\frac{(X+\zeta Y+\zeta^{2}Z)(X+\zeta^{2}Y+\zeta Z)}{(X+Y+Z)^{2}}=\frac{e_{1}^{2}-3e_{2}}{e_{1}^{2}}

with associated weight vector $\chi^{a^{\prime}+b^{\prime}}\mathbf{1}$ . Similarly,

(31)

\chi^{2b^{\prime}+a^{\prime}}(uv_{0}^{\prime}+vv_{1}^{\prime}+wv_{2}^{\prime})=\frac{w^{3}}{u^{3}}=\frac{e_{1}^{3}-\tfrac{9}{2}e_{1}e_{2}+\tfrac{27}{2}e_{3}-\tfrac{\sqrt{-27}}{2}\sqrt{\mathrm{disc}}}{e_{1}^{3}}

and $\chi^{2a^{\prime}+b^{\prime}}=\gamma\chi^{2b^{\prime}+a^{\prime}}$ is the conjugate character. ∎

3.2. Completing the orbit parametrization

Consider the function

(32)		$\displaystyle T(\mathbb{Q})$	$\displaystyle\to\mathbb{P}_{3}(E)$
(33)		$\displaystyle(K/\mathbb{Q},x)$	$\displaystyle\mapsto[w_{1}:w_{2}:w_{3}:w_{4}]$

where $w_{1},\ldots,w_{4}$ are the weight vectors in $H^{0}(S_{E},\mathcal{O}(3D_{0}))$ given by (25).

Proposition 3.

The characteristic polynomial $f=t^{3}-t^{2}+at+b\in\mathbb{Q}[t]$ of a rational point $(K/\mathbb{Q},x)\in T(\mathbb{Q})$ has integer coefficients if and only if $(K/\mathbb{Q},x)$ is $D_{0}$ -integral. For any $D_{0}$ -integral rational point $(K/\mathbb{Q},x)$ on $T$ ,

(34)

H((K/\mathbb{Q},x),\mathcal{O}(D_{0}))=H(f)=\sqrt{1-3a}.

Proof.

First we verify that the $C_{3}$ -invariant functions $e_{1},e_{2},e_{3},\sqrt{\mathrm{disc}}$ of $X,Y,Z$ appearing in the formulas (25) for the weight vectors are polynomial functions of the coefficients of the characteristic polynomial of $x$ . By [12, Prop. 2.5] the unit

(35)

u=\sum_{g\in C_{3}}g(x)[g^{-1}]\in\mathcal{G}(K)

maps to $(K/\mathbb{Q},x)$ under $\mathcal{G}\to\mathcal{G}/C_{3}$ . Thus the three rational functions $X/\varepsilon,Y/\varepsilon,Z/\varepsilon$ on $\mathbb{P}$ evaluate on $u$ to the Galois conjugates of $x$ , and therefore any $C_{3}$ -invariant polynomial in $X/\varepsilon,Y/\varepsilon,Z/\varepsilon$ is a polynomial function in the coefficients of the characteristic polynomial of $x$ .

This proves the ‘if’ direction of the first assertion, since $a$ and $b$ are the values at $(K/\mathbb{Q},x)$ of the $C_{3}$ -invariant polynomials $e_{2}(X/\varepsilon,Y/\varepsilon,1-X/\varepsilon-Y/\varepsilon)$ and $-e_{3}(X/\varepsilon,Y/\varepsilon,1-X/\varepsilon-Y/\varepsilon)$ in $\mathbb{Z}[X/\varepsilon,Y/\varepsilon]^{C_{3}}$ . For the ‘only if’ direction, first we use that

(36)

\mathbb{Z}[X,Y,Z]^{C_{3}}=\mathbb{Z}[e_{1},e_{2},e_{3},X^{2}Y+Y^{2}Z+Z^{2}X]

(see e.g. [2, Example 4.6]). For any integer $d\geq 1$ , dehomogenizing with respect to $\varepsilon$ induces an isomorphism of $C_{3}$ -modules $\mathbb{Z}[X,Y,Z]_{d}\cong\mathbb{Z}[X/\varepsilon,Y/\varepsilon]_{\leq d}$ where $(-)_{d}$ (resp. $(-)_{\leq d}$ ) denotes the submodule of homogeneous degree $d$ elements (resp. degree $\leq d$ elements). In particular,

\mathbb{Z}[X,Y,Z]_{d}^{C_{3}}\cong\mathbb{Z}[X/\varepsilon,Y/\varepsilon]_{\leq d}^{C_{3}},

and so $(K/\mathbb{Q},x)$ is $D_{0}$ -integral if and only if the four generators of $\mathbb{Z}[X,Y,Z]^{C_{3}}$ are integral on $(K/\mathbb{Q},x)$ . In fact, it already suffices for $e_{2}$ and $e_{3}$ to be integral: if $e_{2}$ and $e_{3}$ evaluate to integers on $(K/\mathbb{Q},x)$ , then $X^{2}Y$ will evaluate to an integral element of $K$ and its trace will be an integer, equal to the value of the last generator. This proves the first assertion.

To compute the toric height, we use [6, p. 68] to express the support function $\varphi_{0}$ associated to $D_{0}$ using the weight vectors in $H^{0}(S_{E},\mathcal{O}(3D_{0}))$ found in Lemma 1. The local toric height $H_{v}$ with respect to $\mathcal{O}(3D_{0})$ of any point $(K,x)\in T(\mathbb{Q})$ is

(37)			$\displaystyle\max\left(\left\|\frac{w_{1}(x)}{\mathbf{1}(x)}\right\|_{w},\ldots,\left\|\frac{w_{4}(x)}{\mathbf{1}(x)}\right\|_{w}\right)^{\frac{1}{[E:\mathbb{Q}]}}$
		$\displaystyle=\max\left(1,\|1-3e_{2}\|_{w},\left\|1-\tfrac{9}{2}e_{2}+\tfrac{27}{2}e_{3}-\tfrac{\sqrt{-27}}{2}\sqrt{\mathrm{disc}}\right\|_{w},\left\|1-\tfrac{9}{2}e_{2}+\tfrac{27}{2}e_{3}+\tfrac{\sqrt{-27}}{2}\sqrt{\mathrm{disc}}\right\|_{w}\right)^{\frac{1}{[E:\mathbb{Q}]}}$

where $|\cdot|_{w}=q_{w}^{-\mathrm{ord}_{w}(\cdot)}$ if $w$ is nonarchimedean and $|\cdot|_{w}=|\cdot|^{d_{w}}$ otherwise. When $(K,x)$ is $D_{0}$ -integral, the only contribution to the height is the local contribution from the complex place $w$ of $E$ at infinity, which is

(38)

\displaystyle\max\left(1,|1-3e_{2}|^{2},\left|1-\tfrac{9}{2}e_{2}+\tfrac{27}{2}e_{3}+\tfrac{\sqrt{-27}}{2}\sqrt{\mathrm{disc}}\right|^{2}\right)^{1/2}.

A short computation shows that

(39)

\left|1-\tfrac{9}{2}e_{2}+\tfrac{27}{2}e_{3}+\tfrac{\sqrt{-27}}{2}\sqrt{\mathrm{disc}}\right|^{2}=(1-3e_{2})^{3}.

Thus $1-3e_{2}>0$ and $(1-3e_{2})^{3}\geq(1-3e_{2})^{2}$ which shows that

(40)

H((K,x),\mathcal{O}(D_{0}))=H((K,x),\mathcal{O}(3D_{0}))^{1/3}=\sqrt{1-3e_{2}}=H(f).\qed

Remark 1.

As a function of characteristic polynomials $t^{3}-t^{2}+at+b$ of rational points on $T$ , the quotient

(41)

\frac{\sqrt{3}\max(|a|^{1/2},|b|^{1/3})}{\sqrt{1-3a}}

is bounded and tends to $1$ as $a,b\to\infty$ . This shows that the toric height is equivalent to the “root height” in Theorem 1.

4. The Poisson summation formula

In this section we prove the following formula for the height zeta function for $D_{0}$ -integral rational points on the open torus of $S$ .

Theorem 5.

Fix any $s\in(\mathbb{C}^{\Sigma(1)})^{\Gamma}$ with $\mathrm{Re}(s_{e})\gg 0$ for every $e\in\Sigma_{w}(1)$ . Then the multivariate Dirichlet series

(42)

Z(s)=\sum_{\begin{subarray}{c}P\in T(\mathbb{Q})\\ \text{$D_{0}$-\emph{integral}}\end{subarray}}H(P,s)^{-1}

is absolutely convergent and equals

(43)

\displaystyle\left({1-3^{-z}}\right)\zeta(z)\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1+\frac{1}{q^{z}}\right)^{-1}\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1+\frac{3}{p^{z}}\left(1-\frac{1}{p^{z}}\right)^{-1}\right)

where $z=\frac{1}{2}(s_{0}+s_{1}+s_{2})$ . This multivariate Dirichlet series admits a meromorphic continuation to $\{s\in(\mathbb{C}^{\Sigma(1)})^{\Gamma}:\mathrm{Re}(s_{0}+s_{1}+s_{2})>1\}$ .

For the proof, we recall some well-known facts from harmonic analysis. For any finite place $v$ of $\mathbb{Q}$ let $d^{\times}x_{v}$ be the Haar measure on $T(\mathbb{Q}_{v})$ for which the maximal compact subgroup has measure one, and at the infinite place choose the Haar measure $d^{\times}x_{\infty}$ on $T(\mathbb{R})$ for which $\mathbb{Z}v_{0}\subset N_{\mathbb{R}}$ is a unimodular lattice with respect to the pushforward to $N_{\mathbb{R}}$ under $n_{w}$ of $d^{\times}x_{\infty}$ . For any finite set $S$ of places of $\mathbb{Q}$ containing $v=\infty$ let $\mathbb{A}_{S}$ denote the subring of adeles which are integral at places not in $S$ . There is a unique Haar measure on $T(\mathbb{A})$ , denoted $d^{\times}x$ , whose restriction to $T(\mathbb{A}_{S})=\prod_{v\in S}T(\mathbb{Q}_{v})\times\prod_{v\not\in S}K_{v}$ is the product measure $\prod_{v\not\in S}d^{\times}x_{v}$ for all $S$ . The Fourier transform of any factorizable integrable function $f=\otimes_{v}f_{v}\in L^{1}(T(\mathbb{A}))$ is defined by

(44)

\widehat{f}(\chi)=\int_{T(\mathbb{A})}f(x)\chi(x)^{-1}\,d^{\times}x=\prod_{v}\int_{T(\mathbb{Q}_{v})}f_{v}(x)\chi_{v}(x)^{-1}\,d^{\times}x_{v}.

The subgroup $E^{\times}=T(\mathbb{Q})$ is discrete in $\mathbb{A}_{E}^{\times}=T(\mathbb{A})$ . We equip $T(\mathbb{Q})$ with its counting measure and the quotient group $T(\mathbb{Q})\backslash T(\mathbb{A})$ with the quotient measure (also denoted $d^{\times}x$ ) of $d^{\times}x$ by the counting measure. The dual measure $d\chi$ of this quotient measure is by definition the unique Haar measure on $(T(\mathbb{Q})\backslash T(\mathbb{A}))^{\vee}$ with the property that for all $F\in L^{1}(T(\mathbb{Q})\backslash T(\mathbb{A}))$ satisfying $\widehat{F}\in L^{1}((T(\mathbb{Q})\backslash T(\mathbb{A}))^{\vee})$ , the Fourier inversion formula holds:

(45)

F(x)=\int_{(T(\mathbb{Q})\backslash T(\mathbb{A}))^{\vee}}\widehat{F}(\chi)\chi(x)\,d\chi.

Let $T(\mathbb{Q})^{\perp}$ denote the the subgroup of characters on $T(\mathbb{A})$ that are trivial on $T(\mathbb{Q})$ ; this subgroup is canonically isomorphic to $(T(\mathbb{Q})\backslash T(\mathbb{A}))^{\vee}$ . Let $f\in L^{1}(T(\mathbb{A}))$ . The general Poisson summation formula — following from the classical proof for $\mathbb{Z}\subset\mathbb{R}$ — says that if $\widehat{f}\,|_{T(\mathbb{Q})^{\perp}}\in L^{1}(T(\mathbb{Q})^{\perp})$ then

(46)

\int_{T(\mathbb{Q})}f(xy)\,dx=\int_{T(\mathbb{Q})^{\perp}}\widehat{f}(\chi)\chi(y)\,d\chi

for a.e. $y\in T(\mathbb{Q})$ and suitably normalized Haar measure $d\chi$ on $T(\mathbb{Q})^{\perp}$ [5, Theorem 4.4.2, p. 105].

To apply the Poisson summation formula we will compute the Fourier transform of

(47)

x\mapsto H(x,-s,D_{0})=H(x,-s)1_{D_{0}}(x)\qquad(x\in T(\mathbb{A}))

where $1_{D_{0}}\colon T(\mathbb{A})\to\{0,1\}$ is the characteristic function on $D_{0}$ -integral points. The function $H(x,-s,D_{0})$ is factorizable so its Fourier transform is equal to the product of the transforms of its local factors:

(48)

\widehat{H}(\chi,-s,D_{0})=\prod_{v\in M_{\mathbb{Q}}}\widehat{H_{v}}(\chi_{v},-s,D_{0}).

As usual, we say that a character $\chi$ on $T(\mathbb{Q}_{v})$ is ramified if its restriction to the maximal compact subgroup is nontrivial, and otherwise it is unramified.

Proposition 4.

Let $s\in(\mathbb{C}^{\Sigma(1)})^{\Gamma}$ and assume $\mathrm{Re}(s_{e})>0$ for each $e\in\{0,1,2\}$ . Let $w$ be the infinite place of $E$ . Let $\chi\in T(\mathbb{R})^{\vee}$ be a unitary character. If $\chi$ is ramified then $\widehat{H_{\infty}}(\chi,-s)$ is identically zero. If $\chi$ is unramified, then $\chi(x)=e(\langle n_{w}(x),m\rangle)$ for all $x\in T(\mathbb{R})$ for a unique $m\in M_{\mathbb{R}}$ , and

(49)

\widehat{H_{\infty}}(m,-s)=\left(\frac{-1}{2\pi i}\right)\frac{s_{0}+s_{1}+s_{2}}{2\pi i}\frac{1}{(m(v_{0})+\frac{s_{0}}{2\pi i})(m(v_{0})-\frac{s_{1}+s_{2}}{2\pi i})}.

Next let $v$ be a finite place of $\mathbb{Q}$ . For any unitary character $\chi\in T(\mathbb{Q}_{v})^{\vee}$ , the integral defining $\widehat{H_{v}}(\chi,-s,D_{0})$ converges absolutely to a holomorphic function of $s$ in the region

\{s\in(\mathbb{C}^{\Sigma(1)})^{\Gamma}:\mathrm{Re}(s_{1}),\mathrm{Re}(s_{2})>0\}.

Assume $v\neq 3$ . Let $w$ be any place of $E$ lying over $v$ . The local characteristic function $1_{D_{0},v}$ is $K_{v}$ -invariant. If $\chi$ is ramified, then $\widehat{H_{v}}(\chi,-s,D_{0})$ is identically zero. If $\chi$ is unramified then we may regard $\chi$ as a character on $X_{\ast}(T_{E})^{\Gamma(w/v)}$ (Proposition 2) and

(50)

\widehat{H_{v}}(\chi,-s,D_{0})=\sum_{\begin{subarray}{c}n\in X_{\ast}(T_{E})^{\Gamma(w/v)}\\ n\in\mathbb{R}_{\geq 0}v_{1}+\mathbb{R}_{\geq 0}v_{2}\end{subarray}}\chi(n)^{-1}q_{v}^{\varphi(n)}.

If $v=3$ then the support of $x\mapsto H_{3}(x,-s,D_{0})$ is the unique subgroup $K_{3,2}$ of $K_{3}$ of index six. Under the isomorphism $T(\mathbb{Q}_{3})\to E_{3}^{\times}$ the support corresponds to the subgroup $1+3O_{E,w}$ of $O_{E,w}^{\times}$ where $w$ is the unique place of $E$ lying over $3$ .

Remark 2.

The local Fourier transforms — and therefore the entire Poisson summation argument — must be computed before restricting to the line in $\mathrm{Pic}^{T}(S)\otimes\mathbb{C}$ spanned by the $T_{E}$ -line bundle $\mathcal{O}(D_{0})$ of interest since $x\mapsto H_{v}(x,-s,\mathcal{O}(D_{0}))$ will not be integrable for any place $v\neq 3,\infty$ once either of $s_{1}$ or $s_{2}$ vanishes, no matter how large and positive $\mathrm{Re}(s_{0})$ is.

Proof.

Note that $1_{D_{0},\infty}$ is identically one since integrality conditions are only imposed at finite places, and also observe that the integrand is $K_{\infty}$ -invariant. If $\chi$ is ramified then $\widehat{H_{\infty}}(\chi,-s,D_{0})$ vanishes by Schur’s lemma, so suppose $\chi$ is unramified. Then

(51)		$\displaystyle\widehat{H_{\infty}}(\chi,-s,D_{0})=\int_{T(\mathbb{R})}H_{\infty}(x,-s)1_{D_{0},\infty}(x)\chi(x)^{-1}\,d^{\times}x_{\infty}$	$\displaystyle=\int_{N_{w,\mathbb{R}}}H_{\infty}(y,-s)e(-\langle y,m\rangle)\,d\mu(y)$
(52)			$\displaystyle=\int_{N_{w,\mathbb{R}}}e^{\varphi(y)}e(-\langle y,m\rangle)\,d\mu(y).$

Next we compute that

(53)	$\displaystyle\int_{N_{w,\mathbb{R}}}e^{\varphi(y)}e(-\langle y,m\rangle)\,d\mu(y)$	$\displaystyle=\int_{\mathbb{R}_{\geq 0}}e^{\varphi(yv_{0})}e(-\langle yv_{0},m\rangle)\,d\mu(y)+\int_{\mathbb{R}_{\geq 0}}e^{\varphi(-yv_{0})}e(\langle yv_{0},m\rangle)\,d\mu(y)$
(54)		$\displaystyle=\int_{\mathbb{R}_{\geq 0}}e^{-y(s_{0}+2\pi im(v_{0}))}\,d\mu(y)+\int_{\mathbb{R}_{\geq 0}}e^{-y(s_{1}+s_{2}-2\pi im(v_{0}))}\,d\mu(y)$
(55)		$\displaystyle=(s_{0}+2\pi im(v_{0}))^{-1}+(s_{1}+s_{2}-2\pi im(v_{0}))^{-1}$
(56)		$\displaystyle=\left(\frac{-1}{2\pi i}\right)\left(\left(-m(v_{0})-\frac{s_{0}}{2\pi i}\right)^{-1}+\left(m(v_{0})-\frac{s_{1}+s_{2}}{2\pi i}\right)^{-1}\right)$
(57)		$\displaystyle=\left(\frac{-1}{2\pi i}\right)\frac{-(s_{0}+s_{1}+s_{2})}{2\pi i}\frac{1}{(-m(v_{0})-\frac{s_{0}}{2\pi i})(m(v_{0})-\frac{s_{1}+s_{2}}{2\pi i})}$

which proves the claimed formula.

Next let $v$ be a finite place of $\mathbb{Q}$ and let $w$ be any place of $E$ lying over $v$ . Let $N_{w}=X_{\ast}(T_{E})^{\Gamma(w/v)}$ . The weight vectors in $H^{0}(S_{E},\mathcal{O}(3D_{0}))$ correspond to the characters $0,v_{1}^{\vee},v_{2}^{\vee},3v_{1}^{\vee},3v_{2}^{\vee}$ in $M_{E}$ , so from (37) we see that the local height $H_{v}(x,D_{0})$ is $\leq 1$ if and only if $n_{w}(x)\in\mathbb{R}_{\geq 0}v_{1}+\mathbb{R}_{\geq 0}v_{2}$ .

Now consider the sub- $O_{E}$ -module

(58)

O_{E}\langle w_{1},w_{2},w_{3},w_{4}\rangle\subset O_{E}[X,Y,Z]^{C_{3}}_{3}=O_{E}\langle e_{1}^{3},e_{1}e_{2},e_{3},\delta\rangle

where $\delta=X^{2}Y+Y^{2}Z+Z^{2}X$ (cf. (36)). From the formulas for the weight vectors, one computes that the homomorphism taking the basis vectors $e_{1}^{3},e_{1}e_{2},e_{3},\delta$ to the weight vectors $w_{1},w_{2},w_{3},w_{4}$ , respectively, has the matrix

(59)

\begin{pmatrix}1&1&1&1\\ 0&-3&-3(2+\zeta)&-3(2+\zeta^{2})\\ 0&0&9(2+\zeta)&9(2+\zeta^{2})\\ 0&0&3(1+2\zeta)&3(1+2\zeta^{2})\\ \end{pmatrix}

which has determinant $243\sqrt{-3}$ .

Assume $v\neq 3$ . The cokernel of (58) is a $3$ -group, so this inclusion becomes an isomorphism after tensoring with $\mathbb{Z}_{v}$ . Thus $x\in T(\mathbb{Q}_{v})$ is $D_{0}$ -integral $\iff e_{2}(x),e_{3}(x),\delta(x)\in\mathbb{Z}_{v}\iff w_{1}(x),\ldots,w_{4}(x)\in O_{E}\otimes\mathbb{Z}_{v}\iff H_{v}(x,D_{0})\leq 1\iff n_{w}(x)\in\mathbb{R}_{\geq 0}v_{1}+\mathbb{R}_{\geq 0}v_{2}$ . This also shows that $1_{D_{0},v}$ is $K_{v}$ -invariant since the $w$ -adic size of each weight vector is unchanged under the action of $K_{v}$ . If $\chi$ is ramified then the Fourier transform of $H_{v}(x,s)^{-1}1_{D_{0},v}(x)$ vanishes by Schur’s lemma, so suppose $\chi$ is unramified at $v$ . The integrand is $K_{v}$ -invariant and $d^{\times}x_{v}(K_{v})=1$ so

(60)

\int_{T(\mathbb{Q}_{v})}H_{v}(x,s)^{-1}1_{D_{0},v}(x)\chi(x)^{-1}\,d^{\times}x_{v}=\sum_{n\in N_{w}}q_{v}^{\frac{1}{e_{v}}\varphi(n)}1_{D_{0},v}(n)\chi(n)^{-1}=\sum_{\begin{subarray}{c}n\in X_{\ast}(T_{E})^{\Gamma(w/v)}\\ n\in\mathbb{R}_{\geq 0}v_{1}+\mathbb{R}_{\geq 0}v_{2}\end{subarray}}\chi(n)^{-1}q_{v}^{\varphi(n)}.

For $v=3$ we use the integrality conditions (18) rephrased in terms of cyclotomic numbers from Proposition 1, which in this local context take the form

(61)

e_{2}(x),e_{3}(x),\delta(x)\in\mathbb{Z}_{3}\iff\begin{cases}u^{2}+v^{2}-uv\in 1+3\mathbb{Z}_{3}\text{ and }\\ (u^{2}+v^{2}-uv)(3-2u+v)\in 1+27\mathbb{Z}_{3}\end{cases}

where $x\leftrightarrow u+v\zeta\in\mathbb{Q}(\zeta)\otimes\mathbb{Q}_{3}$ . These conditions imply that $u+v\zeta$ is a $3$ -adic unit, so the support of $x\mapsto H_{3}(x,-s,D_{0})$ is contained in $K_{3}$ . Suppose that $z=u+v\zeta\in K_{3}$ is in the support. Let $N=u^{2}+v^{2}-uv$ and $T=2u-v$ . Define $n,\tau\in\mathbb{Z}_{3}$ by $N=(1+3n)^{-1}$ , $N(3-T)=1+27\tau$ . One easily sees from these equations that

(62)

T-2=3n+O(3^{2})\quad\text{and}\quad 1-T+N=3^{2}n^{2}+O(3^{3})

and therefore from the Newton polygon of the characteristic polynomial of $z$ ,

(63)

t^{2}-Tt+N=(t-1)^{2}-(T-2)(t-1)+1-T+N,

one concludes that $z\in 1+3O_{w}$ . Conversely if $z=1+3x$ with $x\in O_{w}$ then clearly $N(z)\in 1+3\mathbb{Z}_{3}$ while $N(3-T)=1+9(N-T^{2})-27NT=1+9(-3+9n)+O(3^{3})\in 1+27\mathbb{Z}_{3}$ . ∎

To compute the quantities arising in the Poisson summation formula, we need to parameterize the continuous part of the automorphic spectrum of the torus $T$ . For any $x=(x_{w})_{w}\in T(\mathbb{A}_{E})$ let

L(x)=\tfrac{1}{2}\sum_{w\in Pl_{E}}n_{w}(x_{w})\log q_{w}\in N_{E,\mathbb{R}}.

We can give a simpler expression for $L$ using the isomorphism $T\cong R^{E}_{\mathbb{Q}}\mathbb{G}_{m}$ . It is easy to check that

(64)

L(x)(\mathrm{N})=\log|\mathrm{N}(x)|_{\mathbb{A}}

where $\mathrm{N}\colon\mathbb{A}_{E}^{\times}\to\mathbb{A}^{\times}$ is the norm character. The norm character generates the rational character lattice $M_{E}^{\Gamma}$ so $N_{E}^{\Gamma}$ is generated by the unique $\Gamma$ -invariant cocharacter in $N_{E}$ which takes the norm character to $1$ . Thus for any $x\in\mathbb{A}_{E}^{\times}=T(\mathbb{Q})$ , $L(x)=\tfrac{1}{2}\log|\mathrm{N}(x)|_{\mathbb{A}}(v_{1}+v_{2})\in N_{\mathbb{R}}.$

Proposition 5.

There is an exact sequence

(65)

1\longrightarrow K/\mu\longrightarrow T(\mathbb{Q})\backslash T(\mathbb{A})\xlongrightarrow{L}N_{\mathbb{R}}\longrightarrow 0

where $K$ is the maximal compact subgroup of $T(\mathbb{A})$ and $\mu=T(\mathbb{Q})\cap K$ .

Proof.

From (64) we see the kernel of $L$ is the norm-one subgroup of the idèle class group $T(\mathbb{Q})\backslash T(\mathbb{A})$ of $E$ . The rank of the group of units is zero and the class group is trivial so the norm-one subgroup of the idèle class group is generated by $K/\mu$ . Finally $L$ is surjective since $n_{w}$ is already surjective for the complex place $w$ of $E$ (Proposition 2). ∎

Lemma 2.

Let $K^{\prime}\subset K$ denote the subgroup which fixes the characteristic function $1_{D_{0}}=\otimes_{v}1_{D_{0},v}$ for $D_{0}$ -integral points in $T(\mathbb{A})$ . Then

K^{\prime}=K_{3,2}\times\prod_{v\neq 3}K_{v}

where $K_{3,2}\subset K_{3}$ is the unique subgroup with index $6$ . There is an exact sequence

(66)

1\longrightarrow K/(K^{\prime}\cdot\mu)\longrightarrow T(\mathbb{Q})\backslash T(\mathbb{A})/K^{\prime}\xlongrightarrow{L}N_{\mathbb{R}}\longrightarrow 0.

Restriction to the connected component of the identity in $T(\mathbb{Q})\backslash T(\mathbb{A})/K^{\prime}$ gives a canonical splitting $s\colon N_{\mathbb{R}}\to T(\mathbb{Q})\backslash T(\mathbb{A})/K^{\prime}$ of $L$ , inducing the isomorphisms

(67)		$\displaystyle T(\mathbb{Q})\backslash T(\mathbb{A})/K^{\prime}$	$\displaystyle\xlongrightarrow{\sim}T(\mathbb{Q})\backslash T(\mathbb{Q})K/K^{\prime}\times N_{\mathbb{R}}\xlongrightarrow{\sim}K/(K^{\prime}\cdot\mu)\times N_{\mathbb{R}}$
(68)		$\displaystyle T(\mathbb{Q})xK^{\prime}$	$\displaystyle\mapsto(T(\mathbb{Q})xs(L(x))^{-1}K^{\prime},L(x))$

where the second map is defined using the natural isomorphism $T(\mathbb{Q})\backslash T(\mathbb{Q})K/K^{\prime}\cong K/(K^{\prime}\cdot\mu)$ .

Proof.

The equality $K^{\prime}=K_{3,2}\times\prod_{v\neq 3}K_{v}$ follows from $K_{v}$ -invariance of the local characteristic functions $1_{D_{0},v}$ when $v\neq 3$ and the computation of the support when $v=3$ from Proposition 4. The short exact sequence is obtained by taking the quotient by $K^{\prime}$ of the first two groups in the short exact sequence of Proposition 5. The group $K/(K^{\prime}\cdot\mu)$ is finite so the natural quotient map $T(\mathbb{Q})\backslash T(\mathbb{A})/K^{\prime}\to T(\mathbb{Q})\backslash T(\mathbb{A})/K$ identifies the connected component of the identity of $T(\mathbb{Q})\backslash T(\mathbb{A})/K^{\prime}$ with $T(\mathbb{Q})\backslash T(\mathbb{A})/K$ . Thus the restriction of $L$ to the connected component of the identity of $T(\mathbb{Q})\backslash T(\mathbb{A})/K^{\prime}$ is an isomorphism onto $N_{\mathbb{R}}$ , so its inverse gives the canonical splitting map $s$ . ∎

Now we may prove Theorem 5.

Proof of Theorem 5.

Let $1_{D_{0}}\colon T(\mathbb{A})\to\{0,1\}$ be the characteristic function on $D_{0}$ -integral points. By the definition of $D_{0}$ -integrality, $1_{D_{0}}=\otimes_{v}1_{D_{0},v}$ is a factorizable function. Take $f=H(\cdot,-s)1_{D_{0}}$ . To apply the Poisson formula we verify that $f$ is in $L^{1}(T(\mathbb{A}))$ and the restriction of $\widehat{f}$ is in $L^{1}(T(\mathbb{Q})^{\perp})$ . From (23) we have

(69)

H(x,-s)1_{D_{0}}=\prod_{v\in M_{\mathbb{Q}}}1_{D_{0},v}(x_{v})q_{v}^{\frac{1}{e_{v}}\varphi(n_{w}(x_{v}))},\quad x=(x_{v})_{v}\in T(\mathbb{A})\subset T(\mathbb{A}_{E}).

For any finite set $S$ of places of $\mathbb{Q}$ containing $v=\infty$ let $\mathbb{A}_{S}$ denote the subring of adeles which are integral at places not in $S$ . The chain of inequalities

\int_{T(\mathbb{A})}f(x)\,d^{\times}x=\lim_{C\text{ compact}}\int_{C}f(x)\,d^{\times}x\leq\lim_{S\text{ finite}}\int_{T(\mathbb{A}_{S})}f(x)\,d^{\times}x\leq\int_{T(\mathbb{A})}f(x)\,d^{\times}x

in the limits of larger $C$ and $S$ shows that

\int_{T(\mathbb{A})}f(x)\,d^{\times}x=\lim_{S}\int_{T(\mathbb{A}_{S})}f(x)\,d^{\times}x\leq\lim_{S}\prod_{\begin{subarray}{c}v\in S\\ v\neq 3\end{subarray}}\int_{T(\mathbb{Q}_{v})}1_{D_{0},v}(x_{w})q_{v}^{\varphi(n_{w}(x_{w}))}\,d^{\times}x_{v}

(recall that $H_{3}(x,-s,D_{0})$ is supported in $K_{3}$ by Proposition 4). Let $|{\cdot}|$ be any norm on $N_{\mathbb{R}}$ . There is a constant $\rho>0$ such that for any finite place $v\neq 3$ , any place $w$ of $E$ lying over $v$ , and $n\in N_{E}^{\Gamma(w/v)}$ ,

\left|1_{D_{0},v}(n)q_{v}^{\varphi(n)}\right|\leq\begin{cases}0&\text{if $n$ is not in $\mathbb{R}_{\geq 0}v_{1}+\mathbb{R}_{\geq 0}v_{2}$,}\\ q_{v}^{-\rho|n|\min\{\mathrm{Re}(s_{1}),\mathrm{Re}(s_{2})\}}&\text{otherwise.}\end{cases}

Set $t=\min\{\mathrm{Re}(s_{1}),\mathrm{Re}(s_{2})\}$ . Then for $v\neq 3$ we have

(70)

|\int_{T(\mathbb{Q}_{v})}1_{D_{0},v}(x_{w})q_{v}^{\varphi(n_{w}(x_{w}))}\,d^{\times}x_{v}|\leq\sum_{n\in N_{E}^{\Gamma(w/v)}\cap(\mathbb{R}_{\geq 0}v_{1}+\mathbb{R}_{\geq 0}v_{2})}q_{v}^{-\rho t|{n}|}\ll\left(1-q_{v}^{-\rho t}\right)^{-\mathrm{rk}N_{E}^{\Gamma(w/v)}}

where the implied constant is independent of $v$ . For $v=\infty$ we have already seen that $x\mapsto H_{\infty}(x,-s,D)$ is integrable once $\mathrm{Re}(s_{e})>0$ for all $e\in\Sigma_{\Gamma}(1)$ (Proposition 4). Thus for any finite set of places $S$ ,

(71)

|\int_{T(\mathbb{A}_{S})}f(x)\,d^{\times}x|\ll\prod_{\begin{subarray}{c}v\in S\\ v\neq\infty\end{subarray}}\left(1-q_{v}^{-\rho t}\right)^{-1}\leq\zeta(\rho t)

which is finite for $t>1/\rho$ . Taking the limit over $S$ shows $f$ is integrable.

Next we prove that the restriction of $\widehat{f}$ to $T(\mathbb{Q})^{\perp}\cong(T(\mathbb{A})/T(\mathbb{Q}))^{\vee}$ is integrable by evaluating the integral. By Schur’s lemma, this function is supported on $(T(\mathbb{Q})\backslash T(\mathbb{A})/K^{\prime})^{\vee}$ where $K^{\prime}\subset K$ is the subgroup which fixes the characteristic function $1_{D_{0}}$ . We will use the isomorphism in Lemma 2 to perform the integral over the automorphic spectrum of $T$ . Let $C$ denote the finite group $K/(K^{\prime}\cdot\mu)$ . For any $\chi\in(T(\mathbb{Q})\backslash T(\mathbb{A})/K)^{\vee}$ there is a unique $m\in M_{\mathbb{R}}$ such that $\chi(x)=e(\langle m,L(x)\rangle)$ for all $x\in T(\mathbb{A})$ . Set $t=m(v_{0})$ for $m\in M_{\mathbb{R}}$ and let $\chi_{t}$ be the corresponding character. Any $K^{\prime}$ -unramified automorphic character of $T$ is of the form $\psi\chi_{t}$ for a unique $\psi\in C^{\vee}$ and $t\in\mathbb{R}$ . The Haar measure on $T(\mathbb{R})$ was chosen so that $\mathbb{Z}v_{0}\subset N_{\mathbb{R}}$ was unimodular for the pushforward measure to $N_{\mathbb{R}}$ , and so

(72)

\int_{(T(\mathbb{A})/T(\mathbb{Q}))^{\vee}}\widehat{f}(\chi)\,d\chi=\kappa\sum_{\psi\in C^{\vee}}\int_{\mathbb{R}}\widehat{f}(\psi\chi_{t})\,dt

where $\kappa$ is a positive constant yet to be determined. The local $v$ -adic component $\psi_{v}\in T(\mathbb{Q}_{v})^{\vee}$ of $\psi$ is $T(\mathbb{Q}_{v})\to T(\mathbb{Q})\backslash T(\mathbb{A})/K^{\prime}\twoheadrightarrow C\xrightarrow{\psi}\mathbb{C}^{\times}$ . Because the group $N_{\mathbb{R}}=T(\mathbb{R})/K_{\infty}$ has no nontrivial finite quotients, the local component $\psi_{\infty}$ is trivial, and the infinite factor of $\widehat{f}$ is (49). With the help of (50) we find the product over the finite factors besides $v=3$ is

(73)

\prod_{{v\neq 3,\infty}}\widehat{f_{v}}(\psi_{v}\chi_{t,v})=\prod_{{v\neq 3,\infty}}\sum_{\begin{subarray}{c}m\in X_{\ast}(T_{E})^{\Gamma(w/v)}\\ m\in\mathbb{R}_{\geq 0}v_{1}+\mathbb{R}_{\geq 0}v_{2}\end{subarray}}\xi_{v}(m)^{-1}q_{v}^{\varphi(m)}=\sum_{\eta}\xi(\eta)^{-1}\eta^{-s}

where $\xi=\psi\chi_{t}$ , $\eta=(m_{w})_{v}\in\prod_{v\neq 3,\infty}X_{\ast}(T_{E})^{\Gamma(w/v)}$ satisfies certain conditions, $\xi(\eta)\coloneqq\prod_{v\neq 3,\infty}\xi_{v}(m_{w})$ , and $\eta^{-s}\coloneqq\prod_{v\neq 3,\infty}q_{v}^{\varphi(m_{w})}$ . This is a multivariate Dirichlet series in $s_{1}$ and $s_{2}$ with summands indexed by $\eta$ which is absolutely convergent when $s_{1}$ and $s_{2}$ have sufficiently large and positive real parts, so the integral in (72) may be distributed into the sum over $\eta$ . With the help of (49), we see that (72) equals

(74)

\kappa\left(\frac{-1}{2\pi i}\right)\frac{s_{0}+s_{1}+s_{2}}{2\pi i}\sum_{\eta}\eta^{-s}\int_{\mathbb{R}}\frac{1}{(t+\frac{s_{0}}{2\pi i})(t-\frac{s_{1}+s_{2}}{2\pi i})}\sum_{\psi\in C^{\vee}}\widehat{f_{3}}(\xi_{3})\xi(\eta)^{-1}\,dt.

Let $K_{3,2}$ denote the support of the local characteristic function $1_{D_{0},3}$ (cf. Proposition 4). Since $\chi_{t,3}$ is trivial on $K_{3}$ , we have $\widehat{f_{3}}(\xi_{3})=\widehat{f_{3}}(\psi_{3})$ . Recall that $n_{w}\colon T(\mathbb{Q}_{v})\to X_{\ast}(T_{E})^{\Gamma(w/v)}$ is surjective for any finite $w$ (Proposition 2). Since $T(\mathbb{Q})\subset T(\mathbb{Q}_{v})$ is dense for any $v$ ( $E^{\times}$ is obviously dense in $E^{\times}_{v}=(E\otimes\mathbb{Q}_{v})^{\times}$ ), there is a $y_{v}\in T(\mathbb{Q})\subset T(\mathbb{Q}_{v})$ which is a $n_{w}$ -preimage of $m_{w}$ where $\eta=(m_{w})_{v}$ . Then

(75)		$\displaystyle\sum_{\psi\in C^{\vee}}\widehat{f_{3}}(\xi_{3})\xi(\eta)^{-1}$	$\displaystyle=\sum_{\psi\in C^{\vee}}\int_{K_{3,2}}\psi_{3}(x_{3})^{-1}\,d^{\times}x_{3}\,\xi(\eta)^{-1}$
		$\displaystyle=\chi_{t}(\eta)^{-1}\sum_{\psi\in C^{\vee}}\int_{K_{3,2}}\psi_{3}(x_{3})^{-1}\prod_{v\neq 3,\infty}\psi_{v}(y_{v})^{-1}\,d^{\times}x_{3}.$

Since $K_{v}=K_{v}^{\prime}$ for all $v\neq 3,\infty$ (Lemma 2), the $3$ -adic projection map $\mathrm{pr}_{3}$ induces an isomorphism $C\xrightarrow{\sim}K_{3}/(K_{3,2}\cdot\mathrm{pr}_{3}(\mu))$ . Let $k_{\eta}\in K_{3}/(K_{3,2}\cdot\mathrm{pr}_{3}(\mu))$ be the image of $\prod_{v\neq 3,\infty}y_{v}$ under $T(\mathbb{Q})\backslash T(\mathbb{A})/K^{\prime}\to C\to K_{3}/(K_{3,2}\cdot\mathrm{pr}_{3}(\mu))$ so that $\prod_{v\neq 3,\infty}\psi_{v}(y_{v})=\psi_{3}(k_{\eta})$ . Explicitly, $k_{\eta}$ is $\prod_{v\neq 3,\infty}k_{v}$ where $k_{v}=(k_{v,v^{\prime}})_{v^{\prime}}\in K$ is the idèle with components

(76)

k_{v,v^{\prime}}=\begin{cases}1&\text{if $v^{\prime}=v$,}\\ y_{v}^{-1}|y_{v}|_{v}^{-1/2}&\text{if $v^{\prime}=\infty$,}\\ y_{v}^{-1}&\text{otherwise.}\end{cases}

In particular, $\psi_{v}(y_{v})=\psi_{3}(\mathrm{pr}_{3}(y_{v})^{-1})$ .

We claim that $\mathrm{pr}_{3}(y_{v})\in K_{3,2}$ for all $v\neq 3,\infty$ (a priori it is only in $K_{3}$ ). This amounts to the assertion that every prime ideal in $O_{E}$ not dividing $3$ admits a generator that is congruent to $1\ (\mathrm{mod}\ 3O_{E})$ . In other words, we claim that the ray class group $C_{\mathfrak{m}}$ of $O_{E}$ with modulus $\mathfrak{m}=3O_{E}$ is trivial. This follows from the short exact sequence [8, Ch. V, Theorem 1.7, p. 146] (with notation defined there)

(77)

0\longrightarrow O_{E}^{\times}/O_{E,1}^{\times}\longrightarrow E_{\mathfrak{m}}^{\times}/E_{\mathfrak{m},1}^{\times}\longrightarrow C_{\mathfrak{m}}\longrightarrow C\longrightarrow 0

which implies that

(78)

h_{\mathfrak{m}}=h\cdot\#(O_{E}^{\times}/O_{E,1}^{\times})^{-1}\cdot 2^{r_{0}}\cdot N(\mathfrak{m}_{0})\cdot\prod_{\mathfrak{p}|\mathfrak{m}_{0}}\left(1-N(\mathfrak{p})^{-1}\right)=1\cdot 1^{-1}\cdot 2^{0}\cdot 3^{2}\cdot(1-3^{-1})=1.

Thus the integral in (75) simplifies down to

(79)

\sum_{\psi\in C^{\vee}}\int_{K_{3,2}}\psi_{3}(x_{3})^{-1}\prod_{v\neq 3,\infty}\psi_{v}(y_{v})^{-1}\,d^{\times}x_{3}.=\sum_{\psi\in C^{\vee}}\int_{K_{3,2}}\psi_{3}(x_{3}k_{\eta})^{-1}\,d^{\times}x_{3}=\sum_{\psi\in C^{\vee}}\int_{K_{3,2}}\psi_{3}(x_{3})^{-1}\,d^{\times}x_{3}

by absorbing $k_{\eta}$ into the Haar measure. Since $K_{3,2}\subset\ker\psi_{3}$ for any $\psi\in C^{\vee}$ , and recalling that $d^{\times}x_{3}(K_{3})=1$ , this is equal to

(80)

\sum_{\psi\in C^{\vee}}\int_{K_{3,2}}\psi_{3}(x_{3})^{-1}\,d^{\times}x_{3}=|C|\cdot d^{\times}x_{3}(K_{3,2})=|C|\cdot[K_{3}:K_{3,2}]^{-1}=[K_{3,2}\mathrm{pr}_{3}(\mu):K_{3,2}]=6.

Returning to (74), we see that $Z(s)$ is equal to

(81)

\int_{(T(\mathbb{A})/T(\mathbb{Q}))^{\vee}}\widehat{f}(\chi)\,d\chi=6\kappa\left(\frac{-1}{2\pi i}\right)\frac{s_{0}+s_{1}+s_{2}}{2\pi i}\sum_{\eta}\eta^{-s}\int_{\mathbb{R}}\frac{\chi_{t}(\eta)^{-1}\,dt}{(t+\frac{s_{0}}{2\pi i})(t-\frac{s_{1}+s_{2}}{2\pi i})}.

This can be evaluated using Cauchy’s residue formula. The numerator of the integrand in (74) is bounded in the upper half-plane and the denominator is $\ll t^{-2}$ so we may deform the path of integration along $\mathbb{R}$ to the upper half-plane and obtain

(82)		$\displaystyle\left(\frac{-1}{2\pi i}\right)({s_{0}+s_{1}+s_{2}})$	$\displaystyle\sum_{\eta}\eta^{-s}\mathrm{Res}\left[\frac{\chi_{t}(\eta)^{-1}}{(t-\frac{s_{1}+s_{2}}{2\pi i})}\,;\,t=\frac{-s_{0}}{2\pi i}\right]$
(83)		$\displaystyle=$	$\displaystyle\sum_{\eta}\eta^{-s}\chi_{{\frac{s_{0}}{2\pi i}}}(\eta).$

We now describe the conditions determining $\eta=(m_{w})_{v}$ . A tuple $(m_{w})_{v}\in\prod_{v}N_{w}$ corresponds to a summand of (73) if and only if $m_{w}\in\mathrm{im}\,n_{w}\cap(\mathbb{R}_{\geq 0}v_{1}+\mathbb{R}_{\geq 0}v_{2})$ for all $w$ . By Proposition 2,

(84)

\mathrm{im}\,n_{w}=\begin{cases}\mathbb{Z}\langle v_{1},\omega\rangle&\text{if $w$ split,}\\ N_{E}^{\Gamma}=\mathbb{Z}\langle v_{0}\rangle&\text{otherwise.}\end{cases}

Any element of $N_{E}$ may be expressed as $av_{1}+b\omega=av_{1}+b(\tfrac{1}{3}(2v_{1}+v_{2}))=(a+\tfrac{2}{3}b)v_{1}+\tfrac{1}{3}bv_{2}$ for integers $a,b$ . Then

(85)

n_{1}=\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}q^{c_{q}}\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}p^{a_{p}+\frac{2}{3}b_{p}}

and

(86)

n_{2}=\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}q^{c_{q}}\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}p^{\frac{1}{3}b_{p}}

for integer exponents $a_{p},b_{p},c_{q}$ almost all zero and satisfying

(87)

\begin{cases}a_{p}+\tfrac{2}{3}b_{p}\text{ and }\tfrac{1}{3}b_{p}\geq 0&\text{if $p\equiv 1\ (\mathrm{mod}\ 3)$,}\\ c_{q}\geq 0&\text{if $q\equiv 2\ (\mathrm{mod}\ 3)$.}\end{cases}

For a given $\eta=(m_{w})_{v}$ , let $n_{1},n_{2}\in\mathbb{R}_{\geq 1}$ be determined by the equality

v_{1}\log n_{1}+v_{2}\log n_{2}=\sum_{v\neq 3,\infty}m_{w}\log q_{v}.

The $v$ -adic component of $\chi_{t}\in M_{\mathbb{R}}$ ( $t\in\mathbb{R}$ ) is given by

(88)

\chi_{t,v}(m_{w})=\chi_{t,v}(m_{w,1}v_{1}+m_{w,2}v_{2})=q_{v}^{-\pi i(m_{w,1}+m_{w,2})t}

and so

\chi_{t}(\eta)=\prod_{v\neq 3,\infty}\chi_{t,v}(m_{w})=\prod_{v\neq 3,\infty}q_{v}^{-\pi i(m_{w,1}+m_{w,2})t}=(n_{1}n_{2})^{-\pi it}.

We have that

(89)

\eta^{-s}=\prod_{v\neq 3,\infty}q_{v}^{\varphi(m_{w})}=(n_{1}n_{2})^{-s_{1}}

and finally

(90)

\eta^{-s}\chi_{{\frac{s_{0}}{2\pi i}}}(\eta)=(n_{1}n_{2})^{-\left(\frac{s_{0}}{2}+s_{1}\right)}.

Set $z=\frac{s_{0}}{2}+s_{1}$ (the unique $M_{\mathbb{R}}$ -invariant linear form on $(\mathbb{C}^{\Sigma(1)})^{\Gamma}$ up to scaling). Then

(91)

\int_{(T(\mathbb{A})/T(\mathbb{Q}))^{\vee}}\widehat{f}(\chi)\,d\chi=6\kappa\left(\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\sum_{c_{q}}q^{-2c_{q}z}\right)\left(\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\sum_{a_{p},b_{p}}p^{-(a_{p}+b_{p})z}\right).

Fix $b_{p}\geq 0$ and sum over all compatible $a_{p}$ in the right-most sum:

(92)

\sum_{a_{p}\geq-\frac{2}{3}b_{p}}p^{-(a_{p}+b_{p})z}=p^{-b_{p}z}{\sum_{a_{p}\geq-\lfloor\frac{2}{3}b_{p}\rfloor}p^{-a_{p}z}}={p^{-(b_{p}-\lfloor\frac{2}{3}b_{p}\rfloor)z}}{\left(1-\frac{1}{p^{z}}\right)^{-1}}.

Let $b=3k+j$ for $j\in\{0,1,2\}$ and $k\in\mathbb{Z}_{\geq 0}$ . Observe that

(93)

\displaystyle\lfloor\tfrac{2}{3}b\rfloor=\begin{cases}2k&\text{if $b=3k$ or $3k+1$,}\\ 2k+1&\text{if $b=3k+2$.}\end{cases}

Now summing (92) over $b_{p}\geq 0$ obtains

	$\displaystyle\sum_{\begin{subarray}{c}b_{p}\geq 0\\ a_{p}\geq-\frac{2}{3}b_{p}\end{subarray}}p^{-(a_{p}+b_{p})z}$	$\displaystyle=\left(1-\frac{1}{p^{z}}\right)^{-1}\left(\sum_{b=3k}p^{-kz}+\sum_{b=3k+1}p^{-(k+1)z}+\sum_{b=3k+2}p^{-(k+1)z}\right)$
		$\displaystyle=\left(1-\frac{1}{p^{z}}\right)^{-1}\left((1-p^{-z})^{-1}+2p^{-z}(1-p^{-z})^{-1}\right)$
(94)			$\displaystyle=\left(1-\frac{1}{p^{z}}\right)^{-1}\left(1+\frac{3}{p^{z}}\left(1-\frac{1}{p^{z}}\right)^{-1}\right).$

Finally we return to finish computing the zeta function. Combining (4) and (91) obtains

(95)

\displaystyle Z(s)=6\kappa\left({1-3^{-z}}\right)\zeta(z)\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1+\frac{1}{q^{z}}\right)^{-1}\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1+\frac{3}{p^{z}}\left(1-\frac{1}{p^{z}}\right)^{-1}\right).

This shows that the restriction of $\widehat{f}$ to $T(\mathbb{Q})^{\perp}\cong(T(\mathbb{A})/T(\mathbb{Q}))^{\vee}$ is integrable and given by this multivariate Dirichlet series for $\mathrm{Re}(z)=\frac{1}{2}\mathrm{Re}(s_{0}+s_{1}+s_{2})\gg 0$ . The precise region of convergence claimed in the theorem statement will be computed in the lemma below.

To compute the constant $\kappa$ , note there is only one monic trace-one cubic polynomial of toric height equal to $1$ which either has Galois group $C_{3}$ or splits into linear factors over $\mathbb{Q}$ , with at most two being the same, and it is $f=t^{3}-t^{2}$ . This polynomial corresponds to a unique rational point of $T$ since it has repeated factors (Proposition 1). This means the coefficient of $1$ in this Dirichlet series is $1$ and $\kappa=\frac{1}{6}$ . ∎

In the next lemma we reexpress $Z(s)$ in a form better suited for determining the poles and leading constants.

Lemma 3.

The height zeta function is also given by

(96)

\displaystyle Z(s)=\left(1-\frac{1}{3^{z}}\right)^{2}\zeta_{\mathbb{Q}(\sqrt{-3})}(z)^{2}\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2z}}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2z}}+\frac{2}{p^{3z}}\right)

where $\zeta_{\mathbb{Q}(\sqrt{-3})}$ is the Dedekind zeta function of the cyclotomic field $\mathbb{Q}(\sqrt{-3})$ . The height zeta function has meromorphic continuation to the region $\{s\in(\mathbb{C}^{\Sigma(1)})^{\Gamma}:\mathrm{Re}(s_{0}+s_{1}+s_{2})>1\}$ .

Proof.

Since $(1+3x(1-x)^{-1})(1-x)^{3}=1-3x^{2}+2x^{3}$ we have

(97)

\displaystyle\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1+\frac{3}{p^{z}}\left(1-\frac{1}{p^{z}}\right)^{-1}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{p^{z}}\right)^{3}=\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2z}}+\frac{2}{p^{3z}}\right).

Let $\chi=\left(\frac{-3}{\cdot}\right)=\left(\frac{\cdot}{3}\right)$ be the nontrivial quadratic character of modulus $3$ . Multiplying both sides of (97) by $L(z,\chi)$ obtains

(98)

\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1+\frac{1}{q^{z}}\right)^{-1}\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1+\frac{3}{p^{z}}\left(1-\frac{1}{p^{z}}\right)^{-1}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{p^{z}}\right)^{2}\\ =L(z,\chi)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2z}}+\frac{2}{p^{3z}}\right).

Now

		$\displaystyle\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{p^{z}}\right)^{2}$
	$\displaystyle=$	$\displaystyle\frac{\displaystyle\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{p^{z}}\right)\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{z}}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{p^{z}}\right)\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1+\frac{1}{q^{z}}\right)}{\displaystyle\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2z}}\right)}$
	$\displaystyle=$	$\displaystyle\left({\left(1-\frac{1}{3^{z}}\right)\zeta(z)L(z,\chi)}\right)^{-1}{\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2z}}\right)^{-1}}.$

Putting this into the previous equation obtains

(99)		$\displaystyle L(z,\chi)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2z}}+\frac{2}{p^{3z}}\right)=$	$\displaystyle\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1+\frac{1}{q^{z}}\right)^{-1}\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1+\frac{3}{p^{z}}\left(1-\frac{1}{p^{z}}\right)^{-1}\right)$
(100)			$\displaystyle\times\left({\left(1-\frac{1}{3^{z}}\right)\zeta(z)L(z,\chi)}\right)^{-1}{\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2z}}\right)^{-1}}$

which shows that $Z(s)$ is equal to

(101)		$\displaystyle(1-3^{-z})\zeta(z)\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1+\frac{1}{q^{z}}\right)^{-1}\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1+\frac{3}{p^{z}}\left(1-\frac{1}{p^{z}}\right)^{-1}\right)$
(102)		$\displaystyle=\left(\left(1-\frac{1}{3^{z}}\right)\zeta(z)L(z,\chi)\right)^{2}\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2z}}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2z}}+\frac{2}{p^{3z}}\right)$
(103)		$\displaystyle=\left(\left(1-\frac{1}{3^{z}}\right)\zeta_{\mathbb{Q}(\sqrt{-3})}(z)\right)^{2}\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2z}}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2z}}+\frac{2}{p^{3z}}\right).$

The Dedekind zeta function has meromorphic continuation to the entire complex plane, so the meromorphic continuation of the height zeta function is determined by the remaining Euler product:

(104)

\displaystyle\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2z}}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2z}}+\frac{2}{p^{3z}}\right)

We have

(105)

1-x^{2}=(1+x^{2})^{-1}(1-x^{4})

and

(106)

1-3x^{2}+2x^{3}=(1-x^{2})^{3}(1+2x^{3}-3x^{4}+O(x^{5})).

This shows that the Euler product in question is

(107)

L(2z,\chi)\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{4z}}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{p^{2z}}\right)^{4}\left(1+\frac{2}{p^{3z}}-\frac{3}{p^{4z}}+\cdots\right).

The Dirichlet $L$ -function is entire. The Euler product over ${q\equiv 2\ (\mathrm{mod}\ 3)}$ is absolutely convergent in the region $\mathrm{Re}(z)>1/4$ . The Euler product $\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{p^{2z}}\right)^{-4}$ has meromorphic continuation to the region $\mathrm{Re}(z)\geq 1/2$ with a pole of order $2$ when $z=1/2$ and is nonvanishing on the line $\mathrm{Re}(z)=1/2$ , so $\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{p^{2z}}\right)^{4}$ is holomorphic in the region $\mathrm{Re}(z)>1/2$ . The remaining Euler product $\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1+\frac{2}{p^{3z}}-\frac{3}{p^{4z}}+\cdots\right)$ is absolutely convergent in the region $\mathrm{Re}(z)>1/3$ . ∎

We specialize to the line spanned by $D_{0}$ in the vector space of toric divisors, and write

Z_{0}(s)=Z(sD_{0})

where $s$ now denotes a single complex variable.

Proposition 6.

The height zeta function $Z_{0}(s)=Z(sD_{0})$ can be meromorphically continued to the half-plane $\mathrm{Re}(s)>1$ and its only pole in this region is at $s=2$ with order $2$ . Let

E(s)=\left(1-\frac{1}{3^{z}}\right)^{2}\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2z}}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2z}}+\frac{2}{p^{3z}}\right).

Then the Laurent expansion of $Z_{0}(s)$ at $s=2$ has the form

(108)

c_{2}(s-2)^{-2}+c_{1}(s-2)^{-1}+\cdots\\ =4L(1,\chi)^{2}E(2)(s-2)^{-2}+\bigg{(}4L(1,\chi)\big{(}\gamma L(1,\chi)+L^{\prime}(1,\chi)\big{)}E(2)+4L(1,\chi)^{2}E^{\prime}(2)\bigg{)}(s-2)^{-1}+\cdots.

Explicitly,

(109)

c_{2}=\frac{16\pi^{2}}{243}\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2}}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2}}+\frac{2}{p^{3}}\right)

and

(110)

\frac{c_{1}}{c_{2}}=2\gamma+\log(2\pi)-3\log\left(\frac{\Gamma(1/3)}{\Gamma(2/3)}\right)+\frac{9}{8}\log 3+\frac{9}{4}\sum_{q\equiv 2\ (\mathrm{mod}\ 3)}\frac{\log q}{q^{2}-1}+\frac{27}{4}\sum_{p\equiv 1\ (\mathrm{mod}\ 3)}\frac{(p+1)\log p}{p^{3}-3p+2}.

Proof.

The infinite product for $E(s)$ converges to an analytic function on the half-plane $\mathrm{Re}(s)\geq 2$ so $E(2)$ and $E^{\prime}(2)$ are well-defined. The class number formula gives

(111)

\lim_{s\to 2}(s-2)\zeta_{\mathbb{Q}(\sqrt{-3})}(s/2)=2\cdot\frac{2^{r_{1}}\cdot(2\pi)^{r_{2}}\cdot R\cdot h}{w\cdot\sqrt{|D|}}=2\cdot\frac{2^{0}\cdot(2\pi)^{1}\cdot 1\cdot 1}{6\cdot\sqrt{3}}=\frac{2\pi}{3\sqrt{3}}.

Thus the coefficient of the leading term is

(112)

c_{2}=\left(1-\frac{1}{3}\right)^{2}\left(\frac{2\pi}{3\sqrt{3}}\right)^{2}\prod_{q\equiv 2\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{q^{2}}\right)\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{3}{p^{2}}+\frac{2}{p^{3}}\right).

The coefficient $c_{1}$ can be computed using the factorization $\zeta_{\mathbb{Q}(\sqrt{-3})}(z)=\zeta(z)L(z,\chi)$ and [13, (3.8)]

(113)

-L^{\prime}(1,\chi)=\sum_{n=2}^{\infty}\frac{\chi(n)\log n}{n}=\frac{\pi}{\sqrt{3}}\left(\log\left(\frac{\Gamma(1/3)}{\Gamma(2/3)}\right)-\frac{1}{3}(\gamma+\log(2\pi))\right).\qed

Proof of Theorem 3.

The expression for $Z_{0}(s)$ follows from combining Lemma 3 and Proposition 6. Fix the isomorphism $\mathrm{Pic}(S)\otimes\mathbb{Q}\to\mathbb{Q}$ taking the ample generator to $3$ . It remains to be seen that the image of the line spanned by $D_{0}$ in the vector space of toric divisors is identified with $\mathrm{Pic}(S)\otimes\mathbb{C}$ such that $D_{0}$ corresponds to $s=1$ . The canonical divisor $K$ is $D_{0}+D_{1}+D_{2}$ . The surface $S$ has Picard rank one [12, Corollary 3.6] and the unique ample generator is equivalent up to torsion in the divisor class group to $-K$ by [12, Theorem 3.5] and [12, Theorem 3.7]. One computes that $3D_{0}$ is linearly equivalent to $K$ so $\mathcal{O}(D_{0})=\frac{1}{3}\mathcal{O}(K)\leftrightarrow s=1$ . ∎

5. Proofs of Theorem 1 and Theorem 2

Lemma 4.

Let $x\in\mathbb{C}$ be a root of an irreducible polynomial with rational coefficients with Galois group $C_{3}$ and $t^{2}$ -coefficient $-1$ . Then $x$ is a normal element in the Galois extension $\mathbb{Q}(x)/\mathbb{Q}$ .

Proof.

Let $\sigma$ be a generator for $C_{3}$ and set $y=\sigma x$ , $z=\sigma^{2}x$ . Suppose for the sake of contradiction that the points $x,y$ and $z$ lie on a plane $P$ in $\mathbb{Q}(x)\otimes\mathbb{R}$ containing $0$ . Then $x,y$ and $z$ lie on a line $L$ , namely the intersection of $P$ with the trace-one affine hyperplane $\{\mathrm{tr}^{\mathbb{Q}(x)}_{\mathbb{Q}}=1\}$ . This implies that $z-y=\sigma(y-x)$ is proportional to $y-x$ , and thus $y-x$ is an eigenvector of $\sigma$ . The only real eigenvalue of $\sigma$ is one, so $y-x=z-y=x-z$ , all equal to some nonzero element $\lambda$ of $\mathbb{Q}$ . Adding these up shows that $y+z+x-x-y-z=0=3\lambda$ , a contradiction. ∎

Lemma 5.

Let $(\alpha,\beta,\gamma)\in\mathbb{Z}^{3}$ satisfy $\alpha+\beta+\gamma=1$ . Then $(\alpha,\beta,\gamma)$ is a normal element in the split $\mathbb{Q}$ -algebra $\mathbb{Q}^{3}$ .

Proof.

If $(\alpha,\beta,\gamma)$ is not normal, then

(114)

\det\begin{bmatrix}\alpha&\beta&\gamma\\ \beta&\gamma&\alpha\\ \gamma&\alpha&\beta\end{bmatrix}=3\alpha\beta\gamma-\alpha^{3}-\beta^{3}-\gamma^{3}=0.

Set $a=\alpha\beta+\beta\gamma+\gamma\alpha$ . First observe that $1=(\alpha+\beta+\gamma)^{2}=\alpha^{2}+\beta^{2}+\gamma^{2}+2a$ and so $\alpha^{2}+\beta^{2}+\gamma^{2}=1-2a$ . Next,

	$\displaystyle a=(\alpha\beta+\beta\gamma+\gamma\alpha)(\alpha+\beta+\gamma)$	$\displaystyle=3\alpha\beta\gamma+\alpha^{2}(\beta+\gamma)+\beta^{2}(\alpha+\gamma)+\gamma^{2}(\alpha+\beta)$
		$\displaystyle=3\alpha\beta\gamma+\alpha^{2}(1-\alpha)+\beta^{2}(1-\beta)+\gamma^{2}(1-\gamma)$
		$\displaystyle=3\alpha\beta\gamma+\alpha^{2}+\beta^{2}+\gamma^{2}-(\alpha^{3}+\beta^{3}+\gamma^{3}).$

Putting these together with (114) shows that

(115)

a=3\alpha\beta\gamma-\alpha^{3}-\beta^{3}-\gamma^{3}+(1-2a)=1-2a

which is impossible since $a\in\mathbb{Z}$ . ∎

A polynomial $f=t^{3}-t^{2}+at+b=(t-\alpha)(t-\beta)(t-\gamma)\in\mathbb{Z}[t]$ which splits into three linear factors over $\mathbb{Q}$ will be called normal if $x=(\alpha,\beta,\gamma)$ is a normal element of the split $C_{3}$ -algebra $K_{\mathrm{spl}}=\mathbb{Q}^{3}$ . Since $x$ is normal if and only if it has at most two identical coordinates, the split polynomial $f$ is normal if and only if it has at most two identical roots. From the above lemmas we see that the polynomials under consideration are all normal, which means they are all realized by rational points of $T$ .

Corollary 1.

Let $F$ denote the set of polynomials of the form $t^{3}-t^{2}+at+b\in\mathbb{Z}[t]$ which either have Galois group $C_{3}$ or split into three linear factors over $\mathbb{Q}$ . Then any $f\in F$ is normal.

Lemma 6.

We have

(116)

\#\{f\in F:\text{reducible, $\mathrm{disc}(f)\neq 0$},H(f)\leq H\}=\frac{\pi}{9\sqrt{3}}H^{2}-\tfrac{1}{6}H+O(H^{t})

for some $\frac{1}{2}<t<1$ and

(117)

\#\{f\in F:\text{reducible, $\mathrm{disc}(f)=0$},H(f)\leq H\}=\tfrac{1}{3}H+O(1).

In the error term one may take $t=\frac{131}{208}$ [7].

Refer to caption — Figure 5. The ellipse $E_{5}$ and the three lines of points with nontrivial stabilizer in $S_{3}$ .

Proof.

Consider the ellipse in $\mathbb{R}^{2}$ defined by

(118)

E_{H}:-a=x^{2}+y^{2}+xy-x-y=\tfrac{1}{3}(H^{2}-1).

The permutation action of the symmetric group $S_{3}$ stabilizes the affine hyperplane $x+y+z=1$ . If we identify $\mathbb{R}^{2}$ with this affine hyperplane via $(x,y)\mapsto(x,y,1-x-y)$ then the induced action of $S_{3}$ on $\mathbb{R}^{2}$ stabilizes the level sets of $x^{2}+y^{2}+xy-x-y$ , and therefore acts on the interior of $E_{H}$ . Let $E_{H}^{\circ}=E_{H}\cup\mathrm{int}(E_{H})$ . Then we have a canonical bijection

(119)

(E_{H}^{\circ}\cap\mathbb{Z}^{2})/S_{3}\xrightarrow{\sim}\{f\in F:\text{reducible},\,\,\,H(f)\leq H\}.

A lattice point $(\alpha,\beta)$ has a nontrivial stabilizer in $S_{3}$ if and only if either $\alpha=\beta$ or $1-\alpha-\beta\in\{\alpha,\beta\}$ , so the number of lattice points in $E_{H}^{\circ}$ with a nontrivial stabilizer is $H+O(1)$ . The area of $E_{H}^{\circ}$ is $A_{H}=\frac{2\pi}{3\sqrt{3}}(H^{2}-1)$ , so the number of lattice points in $E_{H}^{\circ}$ is $A_{H}+O(H^{t})$ for some $t<1$ (conjecturally $t=\tfrac{1}{2}+\varepsilon$ ). Thus

(120)

\#\{f\in F:\text{reducible, $\mathrm{disc}(f)\neq 0$},H(f)\leq H\}=\tfrac{1}{6}(A_{H}-H+O(H^{t}))

and

(121)

\#\{f\in F:\text{reducible, $\mathrm{disc}(f)=0$},H(f)\leq H\}=\tfrac{1}{3}H+O(1).\qed

Theorem 2 is now easily proven by subtracting off the count for reducible polynomials in Lemma 6 from the Dirichlet coefficients of $Z(s)$ as expressed in Theorem 5. We may now prove Theorem 1.

Proof of Theorem 1.

Let $d_{n}$ denote the $n$ th Dirichlet coefficient of $Z_{0}(2s)$ . By the modular interpretation for $\mathcal{G}/C_{3}$ (Theorem 4), $d_{n}$ is equal to the number of equivalence classes $(K,x)$ of Galois $C_{3}$ -algebras $K/\mathbb{Q}$ equipped with a trace one normal element $x\in K$ and toric height $\sqrt{n}$ . Let $K^{\prime}$ denote the twist of the $C_{3}$ -algebra $K$ by the outer automorphism of $C_{3}$ . Each rational point $(K,x)$ falls into one of the following cases (cf. examples from §2):

(1)

$K$ is an abelian cubic field,
(2)

$K$ is the split $C_{3}$ -algebra $K_{\text{spl}}=\mathbb{Q}^{3}$ and $x$ has exactly two identical coordinates, or
(3)

$K$ is the split $C_{3}$ -algebra $K_{\text{spl}}=\mathbb{Q}^{3}$ and $x$ has distinct coordinates.

(It cannot happen that $K=K_{\text{spl}}$ and $x$ has three identical coordinates since $x$ would not be normal.) In these cases, respectively, we have

(1)

$K\not\cong K^{\prime}$ and $(K,x)\neq(K^{\prime},x)$ ,
(2)

$K\cong K^{\prime}$ and $(K,x)=(K^{\prime},x)$ , or
(3)

$K\cong K^{\prime}$ and $(K,x)\neq(K^{\prime},x)$ .

The characteristic polynomial $f$ of $x$ nearly determines the rational point $(K/\mathbb{Q},x)$ — in these cases, respectively, $f$ arises as the characteristic polynomial for

(1)

precisely the two rational points $(K,x)$ and $(K^{\prime},x)$ ,
(2)

only the rational point $(K_{\text{spl}},x)$ , or
(3)

precisely the two rational points $(K_{\text{spl}},x)$ and $(K_{\text{spl}}^{\prime},x)$ .²²2Let $\sigma$ be a transposition in $S_{3}$ . Then $(K_{\text{spl}},\sigma x)$ has the same characteristic polynomial as $(K_{\text{spl}},x)$ but it does not give us another rational point since $(K_{\text{spl}}^{\prime},x)=(K_{\text{spl}},\sigma x)$ . Thus these two rational points account for $(K_{\text{spl}},\sigma x)$ for any $\sigma\in S_{3}$ .

Let $F$ denote the set of polynomials $t^{3}-t^{2}+at+b\in\mathbb{Z}[t]$ which either have Galois group $C_{3}$ or split into three linear factors over $\mathbb{Q}$ . Then any $f\in F$ is automatically normal (Corollary 1) so arises as the characteristic polynomial for some rational point in $T$ . The preceding analysis shows that the number $w_{f}$ of rational points of $T$ with characteristic polynomial equal to a given $f\in F$ is given by (10). Thus among $f\in F$ with $H(f)=\sqrt{n}$ we have that

\displaystyle 2\#\{\text{irreducible}\}=d_{n}-\#\{\text{reducible, }\mathrm{disc}(f)=0\}-2\#\{\text{reducible, }\mathrm{disc}(f)\neq 0\}.

Now we sum over $f$ with $H(f)\leq H$ . Then

(122)

2\sum_{F_{\mathrm{irr}},H(f)\leq H}1=\sum_{n\leq H^{2}}d_{n}-\sum_{\begin{subarray}{c}F_{\mathrm{red}},H(f)\leq H\\ \mathrm{disc}(f)=0\end{subarray}}1-2\sum_{\begin{subarray}{c}F_{\mathrm{red}},H(f)\leq H\\ \mathrm{disc}(f)\neq 0\end{subarray}}1.

By Lemma 6 this is

(123)

\sum_{n\leq H^{2}}d_{n}-\tfrac{1}{3}H-2\left(\frac{\pi}{9\sqrt{3}}H^{2}-\tfrac{1}{6}H+O(H^{t})\right)=\sum_{n\leq H^{2}}d_{n}-\frac{2\pi}{9\sqrt{3}}H^{2}+O(H^{t}).

Applying standard Tauberian theorems to $Z_{0}(s)$ and using the information about the poles and meromorphic continuation in Lemma 3 and Proposition 6 shows that

(124)

\sum_{n\leq H^{2}}d_{n}=\tfrac{1}{2}c_{2}H^{2}\log H+\tfrac{1}{2}c_{1}H^{2}+O_{\varepsilon}(H^{1+\varepsilon})

for any $\varepsilon>0$ . Putting this all together shows that

(125)

\sum_{F_{\mathrm{irr}},H(f)\leq H}1=\tfrac{1}{4}c_{2}H^{2}\log H+\tfrac{1}{4}c_{1}H^{2}-\frac{\pi}{9\sqrt{3}}H^{2}+O_{\varepsilon}(H^{1+\varepsilon}).\qed

This is the asymptotic count for polynomials of bounded toric height. By the comparison between toric height and root height (Remark 1), the asymptotic count for polynomials of bounded root height is obtained by replacing $H$ with $\sqrt{3}H$ .

Remark 3.

By the Riemann hypothesis one expects $\prod_{p\equiv 1\ (\mathrm{mod}\ 3)}\left(1-\frac{1}{p^{2z}}\right)^{4}$ to have analytic continuation to the region $\mathrm{Re}(z)>\frac{1}{4}$ , and also for $\zeta_{\mathbb{Q}(\sqrt{-3})}(z)$ to be nonvanishing at $z=1/3$ , in which case the $O_{\varepsilon}(H^{1+\varepsilon})$ in (125) should in fact be $aH^{2/3}\log H+bH^{2/3}+O(H^{t})$ for some computable nonzero constants $a,b$ where $t=\frac{131}{208}$ [7] is the best known exponent for the error term in the Gauss circle problem.

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On monic abelian trace-one cubic polynomials

Abstract.

1. Introduction

Theorem 1.

Theorem 2.

An integral Diophantine problem

Theorem 3.

Relation between the problems

Further remarks

Acknowledgements

2. The orbit parametrization

2.1. TT as a moduli space

Warning 1.

Theorem 4 ([12, §2]).

Example 1.

Example 2.

Example 3.

2.2. TT as a torus

Proposition 1.

Proof.

3. Toric heights

Definition 1.

Proposition 2.

Proof.

Definition 2 ([1]).

3.1. Computing the local toric height

Lemma 1.

Proof.

3.2. Completing the orbit parametrization

Proposition 3.

Proof.

Remark 1.

4. The Poisson summation formula

Theorem 5.

Proposition 4.

Remark 2.

Proof.

Proposition 5.

Proof.

Lemma 2.

Proof.

Proof of Theorem 5.

Lemma 3.

Proof.

Proposition 6.

Proof.

Proof of Theorem 3.

5. Proofs of Theorem 1 and Theorem 2

Lemma 4.

Proof.

Lemma 5.

Proof.

Corollary 1.

Lemma 6.

Proof.

Proof of Theorem 1.

Remark 3.

References

2.1. $T$ as a moduli space

2.2. $T$ as a torus