Uniform rationality of Poincar\'e series of p-adic equivalence relations and Igusa's conjecture on exponential sums

TL;DR

This thesis presents new results demonstrating the uniform rationality of Poincaré series associated with p-adic equivalence relations and advances understanding of Igusa's conjecture on exponential sums.

Contribution

It introduces novel proofs and insights into the uniform rationality of Poincaré series and provides progress on Igusa's conjecture regarding exponential sums.

Findings

Established uniform rationality of Poincaré series for certain p-adic equivalence relations

Provided new evidence supporting Igusa's conjecture on exponential sums

Developed methods applicable to broader classes of p-adic and exponential sum problems

Abstract

This thesis contains some new results on the uniform rationality of Poincar\'e series of p-adic equivalence relations and Igusa's conjecture on exponential sums