L-functions of symmetric powers of the generalized Airy family of exponential sums: ell-adic and p-adic methods

TL;DR

This paper investigates the L-functions of symmetric powers of generalized Airy exponential sums over finite fields, employing both ell-adic and p-adic methods to analyze their roots and degrees.

Contribution

It provides explicit formulas for the degree of these L-functions and characterizes the absolute values of their roots using two different mathematical frameworks.

Findings

Explicit formula for the degree of the L-function

Determination of complex absolute values of roots

Analysis of p-adic absolute values of roots

Abstract

For \psi a nontrivial additive character on the finite field F_q, the map t \mapsto \sum_{x \in F_q} \psi(f(x)+tx) is the Fourier transform of the map t \mapsto \psi(f(t))$. As is well-known, this has a cohomological interpretation, producing a continuous ell-adic Galois representation. This paper studies the L-function attached to the k-th symmetric power of this representation using both ell-adic and p-adic methods. Using ell-adic techniques, we give an explicit formula for the degree of this L-function and determine the complex absolute values of its roots. Using p-adic techniques, we study the p-adic absolute values of the roots.