Rationality of trace and norm L-functions

TL;DR

This paper introduces the r-th local norm L-function for l-adic sheaves on algebraic groups over finite fields, proves its rationality, and demonstrates its applications in estimating rational points and exponential sums.

Contribution

It defines a new class of local norm L-functions for algebraic groups and proves their rationality, extending the understanding of Frobenius trace sums over finite field extensions.

Findings

Proves the rationality of the r-th local norm L-function.

Shows applications in estimating rational points on curves.

Demonstrates bounds on exponential sums invariant under group actions.

Abstract

For a given l-adic sheaf F on a commutative algebraic group over a finite field k and an integer r we define the r-th local norm L-function of F at a point t in G(k) and prove its rationality. This function gives information on the sum of the local Frobenius traces of F over the points of G(k_r) (where k_r is the extension of degree r of k) with norm t. For G the one-dimensional affine line or the torus, these sums can in turn be used to estimate the number of rational points on curves or the absolute value of exponential sums which are invariant under a large group of translations or homotheties.