Partial zeta functions, partial exponential sums, and p-adic estimates

TL;DR

This paper studies partial zeta functions of algebraic varieties over finite fields, revealing their rationality, computing examples, and establishing bounds for associated exponential sums and L-functions.

Contribution

It introduces the concept of partial zeta functions, analyzes their properties, and provides Chevalley-Warning type bounds for related exponential sums and L-functions.

Findings

Partial zeta functions can be rational despite their complexity.

Explicit description of the partial zeta function for an affine curve with varying unit poles.

Chevalley-Warning type bounds are established for partial exponential sums and L-functions.

Abstract

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation their rationality is surprising, and even simple examples are delicate to compute. For instance, we give a detailed description of the partial zeta function of an affine curve where the number of unit poles varies, a property different from classical zeta functions. On the other hand, they do retain some properties similar to the classical case. To this end, we give Chevalley-Warning type bounds for partial zeta functions and L-functions associated to partial exponential sums.