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Symmetric Power L-functions of the hyper-Kloosterman Family

C. Douglas Haessig, Steven Sperber

Abstract

The symmetric power L-function of the hyper-Kloosterman family is a rational function over the integers. Its degree and complex absolute values of its zeros and poles are now known through the work of Fu and Wan. The purpose of this paper is to study the p-adic absolute value of these zeros and poles. In particular, we give a uniform lower bound, independent of the symmetric power, of the q-adic Newton polygon of this $L$-function under suitable conditions. We also give similar results for any other linear algebra operation of the hyper-Kloosterman family, such as tensor, exterior, symmetric powers, or combinations thereof.

Symmetric Power L-functions of the hyper-Kloosterman Family

Abstract

The symmetric power L-function of the hyper-Kloosterman family is a rational function over the integers. Its degree and complex absolute values of its zeros and poles are now known through the work of Fu and Wan. The purpose of this paper is to study the p-adic absolute value of these zeros and poles. In particular, we give a uniform lower bound, independent of the symmetric power, of the q-adic Newton polygon of this -function under suitable conditions. We also give similar results for any other linear algebra operation of the hyper-Kloosterman family, such as tensor, exterior, symmetric powers, or combinations thereof.
Paper Structure (11 sections, 25 theorems, 143 equations)

This paper contains 11 sections, 25 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. The Trivial Factor $P(n, k, T)$
  3. Cohomological formula
  4. Cohomology
  5. Relative cohomology
  6. Connection maps on relative cohomology and reduced relative cohomology
  7. Reduced symmetric power cohomology
  8. Symmetric power cohomology
  9. Frobenius
  10. Estimates on the relative Frobenius
  11. Estimates on the symmetric power Frobenius

Key Result

Theorem 1.1

Suppose $d_k(n, p) = 0$. Then $L(\mathop{\mathrm{Sym}}\nolimits^k \mathop{\mathrm{Kl}}\nolimits_n / {\mathbb F} _q, T)$ is a polynomial in $1 + T {\mathbb Z} [T]$ of degree $\frac{1}{n+1} \binom{n+k}{n}$ whose $q$-adic Newton polygon lies on or above the $q$-adic Newton polygon of $\prod (1 - q^i

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • Proposition 3.2
  • proof : Proof of Theorem \ref{['T: cohom formula']}
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • ...and 35 more