On the Newton polygons of twisted $L$-functions of binomials

Fix a rational prime $p$. For $q=p^{a}$ a power of $p$, denote by ${\mathbb{F}}_{q}$ the finite field with $q$ elements, ${\mathbb{Q}}_{q}$ the unramified extension of ${\mathbb{Q}}_{p}$ of degree $a$ and ${\mathbb{Z}}_{q}$ its ring of integers. Let $f(x)\in{\mathbb{F}}_{q}[x]$ be a polynomial of degree $d$ with Teichmüller lifting $\hat{f}(x)\in{\mathbb{Z}}_{q}[x]$. Let $\chi:{\mathbb{F}}_{q}^{\times}\to{\mathbb{C}}_{p}^{\times}$ be a multiplicative character and $\omega:{\mathbb{F}}_{q}^{\times}\to{\mathbb{Z}}_{q}^{\times}$ the Teichmüller lifting. Then we can write $\chi=\omega^{-u}$ for some $0\leq u\leq q-2$.

For a non-trivial additive character $\psi_{m}:{\mathbb{Z}}_{p}\to{\mathbb{C}}_{p}^{\times}$ of order $p^{m}$, define the twisted $L$-function

where $S_{k,u}(f,\psi_{m})$ is the twisted exponential sum

If $p\nmid d$, then $L_{u}(s,f,\psi_{m})$ is a polynomial of degree $p^{m-1}d$ by Adolphson-Sperber [AS87, AS91, AS93], Li [Li99], Liu-Wei [LW07] and Liu [Liu07].

We will use the twisted $T$-adic exponential sums developed by Liu-Wan [LW09] and Liu [Liu02, Liu09]. Define the twisted $T$-adic $L$-function

where $S_{k,u}(f,T)$ is the twisted $T$-adic exponential sum

Then $L_{u}(s,f,\psi_{m})=L_{u}(s,f,\pi_{m})$ where $\pi_{m}=\psi_{m}(1)-1$.

Denote by

the characteristic function, which is $T$-adic entire in $s$. Then

Since the $\pi_{m}^{a(p-1)}$-adic Newton polygon of $C_{u}(s,f,\pi_{m})$ does not depend on the choice of $\psi_{m}$, we denote it by ${\mathrm{NP}}_{u,m}(f)$. Denote by ${\mathrm{NP}}_{u,T}(f)$ the $T^{a(p-1)}$-adic Newton polygon of $C_{u}(s,f,T)$. As shown in [LW09] and [Liu07], ${\mathrm{NP}}_{u,m}(f)$ lies over the infinity $u$-twisted Hodge polygon $H_{[0,d],u}^{\infty}$, which has slopes

If we write $0\leq s_{0}\leq\cdots\leq s_{p^{m-1}d-1}\leq 1$ the $q$-adic slopes of $L_{u}(s,f,\pi_{m})$, then the $q$-adic slopes of $C_{u}(s,f,\pi_{m})$ are

That’s to say, the $\pi_{m}^{a(p-1)}$-adic Newton polygon of $L_{u}(s,f,\pi_{m})$ is the restriction of ${\mathrm{NP}}_{u,m}(f)$ on $[0,p^{m-1}d]$, and it determines ${\mathrm{NP}}_{u,m}(f)$.

The prime $p$ is required large enough in the following results. When $\chi=\omega^{-u}$ is trivial, in [Zhu14] and [LLN09], they gave a lower bound of the Newton polygons. They defined a polynomial on the coefficients of $f$, called Hasse polynomial. If the Hasse polynomial is nonzero, then the Newton polygons coincide this lower bound.

Assume that $f(x)=x^{d}+\lambda x^{e}$ is a binomial. Since the exponential sums can be transformed to the twisted case when $d$ and $e$ are not coprime, we assume $(d,e)=1$ in this paper. When $u=0$, we list the known cases here.

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  $p\equiv 1\bmod d$, it’s well-known that the Newton polygons coincides the Hodge polygon.

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  $e=1$, see [Yan03, §1, Theorem], [Zhu14, Theorem 1.1] and [OY16, Theorem 1.1].

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  $e=d-1,p\equiv-1\bmod d$, see [OZ16].

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  $e=2,p\equiv 2\bmod d$, see [ZN21].

For arbitrary $u$, Liu-Niu [LN11] obtained the Newton polygons when $e=1$. Zhang-Niu [ZN21] also give a conjectural description of the Newton polygons when $p\equiv e\bmod d$.

We list the notations we will use.

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  $i,j,v,w,k,\ell,n$ indices.

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  $f(x)=x^{d}+\lambda x^{e}\in{\mathbb{F}}_{q}[x]$ a binomial with $d>e\geq 1,(d,e)=1,\lambda\neq 0$.

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  $\omega^{-u}:{\mathbb{F}}_{q}^{\times}\to{\mathbb{C}}_{p}^{\times}$, where $\omega$ is the Teichml̈ler lifting and $0\leq u\leq q-2$.

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  $H_{[0,d],u}^{\infty}$, the infinity $u$-twisted Hodge polygon with slopes in (1.1).

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  $c=\frac{q-1}{(q-1,u)}$ the order of $\omega^{-u}$, then $u=\frac{(q-1)\mu}{c}$ for some $(\mu,c)=1$.

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  $P_{u,e,d}$ a polygon with slopes $w(i)$, defined in (1.2).

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  $b$ the least positive integer such that $p^{b}u\equiv u\bmod{(q-1)}$ (equivalently, $p^{b}\equiv 1\bmod c$).

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  $0\leq u_{i}\leq p-1$ such that $u=u_{0}+u_{1}p+\cdots+u_{a-1}p^{a-1}$, $u_{i}=u_{b+i}$.

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  $\overline{x}$ the minimal non-negative residue of $x$ modulo $d$.

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  $\delta_{P}$ takes value $1$ if $P$ happens; $0$ if $P$ does not happen.

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  $I_{n}=\left\{1,\dots,n\right\},I_{n}^{*}=\left\{0,1,\dots,n\right\}$.

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  $S_{n}$ (resp. $S_{n}^{*}$) the set of permutations of $I_{n}$ (resp. $I_{n}^{*}$).

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  $C_{t,n}$ the minimum of $\sum_{i=0}^{n}\overline{e^{-1}(pi-\tau(i)+t)}$ for $\tau\in S_{n}^{*}$ and $S_{t,n}^{\circ}$ the set of $\tau\in S_{n}^{*}$ such that the summation reaches minimal. Set $C_{t,-1}=0$ for convention.

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  $R_{i,\alpha}=\overline{e^{-1}(pi+\alpha)},\ r_{i,\alpha}=\overline{e^{-1}(t-\alpha-i)}$, see Proposition 2.1. We will the subscript $\alpha$ if there is no confusion.

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  ${\mathbf{C}}_{t,n,\alpha}$ the maximal size of $\left\{i\in I_{n}^{*}\mid R_{i,\alpha}+r_{\tau(i),\alpha}\geq d\right\}$ for $\tau\in S_{n}^{*}$. We will the subscript $\alpha$ if there is no confusion.

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  $y_{t,i}^{\tau}=\overline{e^{-1}(pi-\tau(i)+t)},\ x_{t,i}^{\tau}=d^{-1}(pi-\tau(i)+t-ey_{t,i}^{\tau})$ the unique solution of $dx+ey=pi-\tau(i)+t$ with $0\leq y\leq d-1$.

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  $h_{n,k},h_{u,e,d}$ the Hasse numbers defined in (1.3).

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  ${\mathbf{p}}$ the minimal non-negative residue of $p$ modulo $cd$.

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  $H_{\mu,c,{\mathbf{p}},e,d}\in{\mathbb{Z}}$ a constant defined in (3.1).

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  $E(X)$ the $p$-adic Artin-Hasse series, see (2.1).

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  $\pi$ a $T$-adic uniformizer of ${\mathbb{Q}}_{p}\llbracket T\rrbracket$ given by $E(\pi)=1+T$, with a fixed $d(q-1)$-th root $\pi^{\frac{1}{d(q-1)}}$.

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  $E_{f}(X)$, see (2.2).

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  $M_{u}=\frac{u}{q-1}+{\mathbb{N}}$.

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  ${\mathcal{L}}_{u}$ a Banach space, see (2.3).

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  ${\mathcal{B}}_{u}$ a subspace of ${\mathcal{L}}_{u}$, see (2.4).

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  ${\mathcal{B}}={\mathcal{B}}_{u}\oplus{\mathcal{B}}_{pu}\oplus\cdots\oplus{\mathcal{B}}_{p^{b-1}u}$.

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  $\psi:{\mathcal{L}}_{u}\rightarrow{\mathcal{L}}_{p^{-1}u}$ defined as $\psi\left(\sum_{v\in M_{u}}b_{v}X^{v}\right)=\sum_{v\in M_{p^{-1}u}}b_{pv}X^{v}$.

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  $\sigma\in{\mathrm{Gal}}({\mathbb{Q}}_{q}/{\mathbb{Q}}_{p})$ the Frobenius, which acts on ${\mathcal{L}}_{u}$ via the coefficients.

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  $\Psi=\sigma^{-1}\circ\psi\circ E_{f}:{\mathcal{B}}_{u}\to{\mathcal{B}}_{p^{-1}u}$ the Dwork’s $T$-adic semi-linear operator.

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  $c_{n}$ the coefficients of $\det(1-\Psi s\mid{\mathcal{B}})$, see (2.6).

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  $s_{k}\equiv p^{k}u\bmod{q-1}$ with $0\leq s_{k}\leq q-2$.

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  $\Gamma=\left(\gamma_{(v,\frac{s_{k}}{q-1}+i),(w,\frac{s_{\ell}}{q-1}+j)}\right)$ the matrix coefficient of $\Psi$ on ${\mathcal{B}}$, see (2.7).

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  $\Gamma^{(k)}$ the sub-matrix of $\Gamma$ defined in (2.7).

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  $A^{(k)}=A\cap\Gamma^{(k)}$ the sub-matrix of a principal minor $A$ of $\Gamma$.

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  ${\mathcal{A}}_{n}$ the set of all principal minor $A$ of order $bn$, such that every $A^{(k)}$ has order $n$.

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  $\phi(n)\in{\mathbb{N}}\cup\left\{+\infty\right\}$ the minimal $x+y$ where $dx+ey=n,x,y\in{\mathbb{N}}$.

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  $\gamma_{(\frac{s_{k}}{q-1}+i,\frac{s_{\ell}}{q-1}+j)}$, see (2.9).

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  $\left(x\right)_{\left[n\right]}:=x(x-1)\cdots(x-n+1),\left(x\right)_{\left[0\right]}:=1$ the falling factorial.

In this paper, we give an explicit lower bound of Newton polygons of twisted $L$-functions of binomial $f(x)=x^{d}+\lambda x^{e}$. We reduce the Hasse polynomial to a certain integer (3.1). Then $p>(d-e)(2d-1)$ does not divide this constant, if and only if this lower bound coincides the Newton polygons. Finally, we show that this condition holds for $e=d-1$.

Denote by $P_{u,e,d}$ the polygon such that

Denote by $w(n)=P_{u,e,d}(n+1)-P_{u,e,d}(n)$. Then

This polygon lies above the Hodge polygon $H_{[0,d],u}^{\infty}$ with same points at $d{\mathbb{Z}}$, and $w(n+d)=1+w(n)$. Moreover, we have $w(n)\leq w(n+1)$ if $p>(d-e)(2d-1)$. See Proposition 2.1.

Define

Here $H_{\mu,c,{\mathbf{p}},e,d}\in{\mathbb{Z}}$ is a constant defined in (3.1) and ${\mathbf{p}}$ is the minimal positive residue of $p$ modulo $cd$. Thus we have the following corollary.

The following result extends [OZ16], as they considered the untwisted case with an additional condition $p\equiv-1\bmod d$.

We give the following conjecture, which generalizes the conjecture in [ZN21]. Note that $h_{u,e,d}$ may be zero since $S_{u_{k},n}^{\circ}$ may be empty, so we require that $p$ is large with respect to $c$, as in Corollary 1.3 and Theorem 1.4.

For any integer $t$, we denote

We set $C_{t,-1}=0$ for convention. For any integer $\alpha$, we denote

and

This part is almost the same with [LN11, SS2,3]. Denote by

the $p$-adic Artin-Hasse series. Then $\lambda_{n}=1/n!$ if $n<p$. Denote by

Then

where $(x,y)$ runs through non-negative solutions of $dx+ey=k$.

Denote by $M_{u}=\frac{u}{q-1}+{\mathbb{N}}$. Define

and

Define a map

The power series $E_{f}$ defines a map on ${\mathcal{B}}_{u}$ via multiplication. Let $\sigma\in{\mathrm{Gal}}({\mathbb{Q}}_{q}/{\mathbb{Q}}_{p})$ be the Frobenius, which acts on ${\mathcal{L}}_{u}$ via the coefficients. Then the Dwork’s $T$-adic semi-linear operator $\Psi=\sigma^{-1}\circ\psi\circ E_{f}$ sends ${\mathcal{B}}_{u}$ to ${\mathcal{B}}_{p^{-1}u}$. Hence $\Psi$ acts on

We have a linear map

on ${\mathcal{B}}$ over ${\mathbb{Z}}_{q}\llbracket\pi^{\frac{1}{d(q-1)}}\rrbracket$. Since $\Psi$ is completely continuous in the sense of [Ser62], the following determinants are well-defined.

Write $s_{k}\equiv p^{k}u\bmod{q-1}$ with $0\leq s_{k}\leq q-2$. Then $s_{b-k}=s_{-k}=u_{k}+u_{k+1}p+\cdots+u_{k+a-1}p^{a-1}$. Let $\xi_{1},\dots,\xi_{a}$ be a normal basis of ${\mathbb{Q}}_{q}$ over ${\mathbb{Q}}_{p}$. The space ${\mathcal{B}}$ has a basis

over ${\mathbb{Z}}_{p}\llbracket\pi^{\frac{1}{d(q-1)}}\rrbracket$. Let $\Gamma=\left(\gamma_{(v,\frac{s_{k}}{q-1}+i),(w,\frac{s_{\ell}}{q-1}+j)}\right)_{{\mathbb{N}}\times I_{a}\times I_{b}}$ be the matrix of $\Psi$ on ${\mathcal{B}}$ with respect to this basis. Then

where

Hence we have