Shortest nonzero lattice points in a totally real multi-quadratic number field and applications

TL;DR

This paper derives explicit formulas for shortest lattice points in multi-quadratic totally real number fields, relates them to Diophantine equations, and applies these results to refine asymptotics in the Petersson trace formula for Hilbert cusp forms.

Contribution

It provides explicit formulas for shortest lattice points in multi-quadratic fields and connects them to Diophantine equations, with applications to trace formulas in automorphic forms.

Findings

Explicit formulas for lattice point norms in multi-quadratic fields

Refined asymptotic for Petersson trace formula

Lower bound results analogous to Jung and Sardari

Abstract

Let $F$ be a multi-quadratic totally real number field. Let $\sigma_1,\dots, \sigma_r$ denote its distinct embeddings. Given $s \in F,$ we give an explicit formula for $\| \sigma(s)\|$ and $\sum_{i<j} \sigma_i(s)\sigma_j(s),$ where $\| \sigma(s)\|=\sqrt{\sum_{i=1}^r(\sigma_i(s))^2}.$ Let $\mathfrak{M}$ be a fractional ideal in $F$ and $\min\left( \mathfrak{M}\right):=\min\{\|\sigma(s)\| \, | \, s \in \mathfrak{M}, s\neq 0 \}.$ The set of shortest nonzero lattice points for $\mathfrak{M}$ is given by $\{s\in \mathfrak{M} : \| \sigma(s)\|=\min(\mathfrak{M}) \}.$ We provide shortest nonzero lattice points for $\mathfrak{M}$ in terms of rational solutions to a given Diophantine equation. As an application, we get a refined asymptotic for the Petersson trace formula for the space of Hilbert cusp forms. We then use the refined asymptotic to obtain a lower bound analogue to a theorem by Jung and Sardari.