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

Abstract

Let $F$ be a multi-quadratic totally real number field. Let $σ_1,\dots, σ_r$ denote its distinct embeddings. Given $s \in F,$ we give an explicit formula for $\| σ(s)\|$ and $\sum_{i<j} σ_i(s)σ_j(s),$ where $\| σ(s)\|=\sqrt{\sum_{i=1}^r(σ_i(s))^2}.$ Let $\mathfrak{M}$ be a fractional ideal in $F$ and $\min\left( \mathfrak{M}\right):=\min\{\|σ(s)\| \, | \, s \in \mathfrak{M}, s\neq 0 \}.$ The set of shortest nonzero lattice points for $\mathfrak{M}$ is given by $\{s\in \mathfrak{M} : \| σ(s)\|=\min(\mathfrak{M}) \}.$ 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.