Bounds for theta sums in higher rank I

TL;DR

This paper develops new upper bounds for multi-variable theta sums with smooth and box truncations, extending classical results and improving previous estimates by leveraging automorphic representations and growth analysis in homogeneous spaces.

Contribution

It generalizes a 1977 classical result to higher dimensions and enhances prior bounds using automorphic methods and cusp growth analysis.

Findings

Established new upper bounds for multi-variable theta sums.

Extended classical one-variable bounds to higher dimensions.

Improved previous estimates by Cosentino and Flaminio.

Abstract

Theta sums are finite exponential sums with a quadratic form in the oscillatory phase. This paper establishes new upper bounds for theta sums in the case of smooth and box truncations. This generalises a classic 1977 result of Fiedler, Jurkat and K\"orner for one-variable theta sums and, in the multi-variable case, improves previous estimates obtained by Cosentino and Flaminio in 2015. Key steps in our approach are the automorphic representation of theta functions and their growth in the cusps of the underlying homogeneous space.