Bounds for p-adic exponential sums and log-canonical thresholds

TL;DR

This paper introduces a conjecture on p-adic exponential sums that generalizes existing conjectures, providing uniform bounds and new insights into log-canonical thresholds, with evidence supporting its validity.

Contribution

It proposes a new conjecture on exponential sums without homogeneity restrictions and establishes related bounds for log-canonical thresholds, advancing the theoretical understanding.

Findings

Uniform bounds for exponential sums in p, y, and some m values.

New bounds for log-canonical thresholds related to the conjecture.

Evidence supporting the conjecture's validity.

Abstract

We propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new predicted uniform behavior. The exponential sums have summation sets consisting of integers modulo $p^m$ lying $p$-adically close to $y$, and the proposed bounds are uniform in $p$, $y$, and $m$. We give evidence for the conjecture, by showing uniform bounds in $p$, $y$, and in some values for $m$. On the way, we prove new bounds for log-canonical thresholds which are closely related to the bounds predicted by the conjecture.