$p$-adic estimates for multiplicative character sums

TL;DR

This paper discusses conjectures and results related to $p$-adic and archimedean estimates for multiplicative character sums over smooth projective varieties, including a proof for the case of projective space.

Contribution

It formulates conjectures on character sum estimates and proves one conjecture for projective space using recent results by Dollarhide.

Findings

Proposed conjectures on $p$-adic and archimedean estimates.

Verified one conjecture for ${ m P}^n$ using Dollarhide's results.

Reviewed foundational work on character sums over finite fields.

Abstract

This article is an expanded version of the talk given by the first author at the conference "Exponential sums over finite fields and applications" (ETH, Z\"urich, November, 2010). We state some conjectures on archimedian and $p$-adic estimates for multiplicative character sums over smooth projective varieties. We also review some of the results of J. Dollarhide, which formed the basis for these conjectures. Applying his results, we prove one of the conjectures when the smooth projective variety is ${\mathbb P}^n$ itself.