[Paper Review] p-adic variation of L-functions of exponential sums, I

Hui June Zhu|arXiv (Cornell University)|Nov 18, 2001

Meromorphic and Entire Functions14 references3 citations

TL;DR

This paper establishes the existence of a generic Newton polygon for $L$-functions of one-variable exponential sums modulo $p$, proving that for large primes $p$, the Newton polygon of a generic polynomial $f(x)$ in $\mathbb{A}^d$ equals the generic Newton polygon $\mathrm{GNP}(\mathbb{A}^d;\mathbb{F}_p)$, and that as $p \to \infty$, this polygon converges to the Hodge polygon $\mathrm{HP}(\mathbb{A}^d)$. The result confirms a conjecture of Daqing Wan on $p$-adic variation of $L$-functions.

ABSTRACT

For a polynomial $f(x)$ in $(\mathbb{Z}_p\cap \mathbb{Q})[x]$ of degree $d>2$ let $L(f \bmod p;T)$ be the $L$-function of the exponential sum of $f \bmod p$. Let $\mathrm{NP}(f \bmod p)$ denote the Newton polygon of $L(f \bmod p;T)$. Let $\mathrm{HP}(f)$ denote the Hodge polygon of $f$, which is the lower convex hull in the real plane of the points $(n,n(n+1)/(2d))$ for $0\leq n\leq d-1$. We prove that there is a Zariski dense subset $\mathcal{U}$ defined over $\mathbb{Q}$ in the space $\mathbb{A}^d$ of degree-$d$ monic polynomials over $\mathbb{Q}$ such that for all $f$ in $\mathcal{U}(\mathbb{Q})$ we have $\lim_{p ightarrow\infty} \mathrm{NP}(f \bmod p) = \mathrm{HP}(f)$. Moreover, we determine the $p$-adic valuation of every coefficient of $L(f \bmod p;T)$ for $p$ large enough and $f$ in $\mathcal{U}(\mathbb{Q})$, and that of $L(x^d+a x \bmod p;T)$ for all $a eq 0$.

Motivation & Objective

• To resolve Daqing Wan's conjecture on the $p$-adic variation of $L$-functions of one-variable exponential sums.
• To prove the existence of a generic Newton polygon $\mathrm{GNP}(\mathbb{A}^d;\mathbb{F}_p)$ for degree-$d$ monic polynomials over $\mathbb{F}_p$ when $p$ is sufficiently large.
• To show that for a Zariski dense open subset $\mathcal{U} \subset \mathbb{A}^d$ defined over $\mathbb{Q}$, the Newton polygon of $f \otimes \mathbb{F}_p$ equals $\mathrm{GNP}(\mathbb{A}^d;\mathbb{F}_p)$ for large $p$.
• To establish the limit behavior of the Newton polygon as $p \to \infty$, showing convergence to the Hodge polygon $\mathrm{HP}(\mathbb{A}^d)$.
• To extend the results to the one-parameter family $f(x) = x^d + ax$ and prove that $\lim_{p \to \infty} \mathrm{NP}((x^d + ax) \otimes \mathbb{F}_p) = \mathrm{HP}(\mathbb{A}^d)$ for any nonzero $a \in \mathbb{Q}$.

Proposed method

• The paper uses $p$-adic analysis of $L$-functions associated with exponential sums $S_\ell(f \otimes \mathbb{F}_p)$, defined via the Artin-Hasse exponential function and roots of unity $\zeta_p$.
• It defines the Newton polygon $\mathrm{NP}(f \otimes \mathbb{F}_p)$ as the lower convex hull of points $(n, \mathrm{ord}_p b_n)$, where $b_n$ are coefficients of the $L$-function $L(f \otimes \mathbb{F}_p; T)$.
• The Hodge polygon $\mathrm{HP}(\mathbb{A}^d)$ is defined as the lower convex hull of points $(n, \frac{n(n+1)}{2d})$ for $0 \leq n \leq d-1$.
• The authors construct a Zariski dense open subset $\mathcal{U} \subset \mathbb{A}^d$ over $\mathbb{Q}$ such that for $f \in \mathcal{U}(\mathbb{Q})$, the Newton polygon stabilizes to $\mathrm{GNP}(\mathbb{A}^d; \mathbb{F}_p)$ for large $p$.
• For the one-parameter family $f(x) = x^d + ax$, the paper proves that $\mathrm{NP}((x^d + ax) \otimes \mathbb{F}_p)$ converges to $\mathrm{HP}(\mathbb{A}^d)$ as $p \to \infty$.
• Key technical tools include $p$-adic valuation estimates on products of factorials and binomial coefficients, and the use of Vandermonde determinants to control the leading terms in the $L$-function coefficients.

Experimental results

Research questions

• RQ1 Does the generic Newton polygon $\mathrm{GNP}(\mathbb{A}^d; \mathbb{F}_p)$ exist for sufficiently large primes $p$?
• RQ2 For a generic polynomial $f \in \mathbb{A}^d(\mathbb{Q})$, does $\mathrm{NP}(f \otimes \mathbb{F}_p)$ equal $\mathrm{GNP}(\mathbb{A}^d; \mathbb{F}_p)$ when $p$ is large?
• RQ3 What is the limit of $\mathrm{NP}(f \otimes \mathbb{F}_p)$ as $p \to \infty$ for a fixed $f \in \mathbb{A}^d(\mathbb{Q})$?
• RQ4 Does the limit of $\mathrm{NP}((x^d + ax) \otimes \mathbb{F}_p)$ as $p \to \infty$ equal the Hodge polygon $\mathrm{HP}(\mathbb{A}^d)$ for any nonzero $a \in \mathbb{Q}$?
• RQ5 Can the $p$-adic valuation of the coefficients of the $L$-function be controlled uniformly to establish convergence to the Hodge polygon?

Key findings

• For $p$ sufficiently large, the generic Newton polygon $\mathrm{GNP}(\mathbb{A}^d; \mathbb{F}_p)$ exists and is explicitly computable.
• There exists a Zariski dense open subset $\mathcal{U} \subset \mathbb{A}^d$ defined over $\mathbb{Q}$ such that for all $f \in \mathcal{U}(\mathbb{Q})$, $\mathrm{NP}(f \otimes \mathbb{F}_p) = \mathrm{GNP}(\mathbb{A}^d; \mathbb{F}_p)$ for large $p$.
• As $p \to \infty$, the Newton polygon $\mathrm{NP}(f \otimes \mathbb{F}_p)$ for $f \in \mathcal{U}(\mathbb{Q})$ converges to the Hodge polygon $\mathrm{HP}(\mathbb{A}^d)$.
• For the one-parameter family $f(x) = x^d + ax$ with $a \neq 0$, $\lim_{p \to \infty} \mathrm{NP}((x^d + ax) \otimes \mathbb{F}_p) = \mathrm{HP}(\mathbb{A}^d)$.
• The $p$-adic valuation of the $L$-function coefficients satisfies $\mathrm{ord}_p C_n \geq \frac{n(n+1)}{2d} + \epsilon_n'$, with equality if and only if $a \not\equiv 0 \pmod{p}$.
• The proof relies on controlling the leading terms in the $L$-function coefficients via $p$-adic valuation estimates and Vandermonde determinant identities to ensure the Newton polygon matches the Hodge polygon in the limit.