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 ${\sigma }_1,\dots ,{\sigma }_r$ denote its distinct embeddings. Given $s\in F,$ we give an explicit formula for $\Vert \sigma (s)\Vert$ and ${\sum }_{i<j}{\sigma }_i(s){\sigma }_j(s),$ where $\Vert \sigma (s)\Vert =\sqrt{{\sum }_{i=1}^r({\sigma }_i(s){)}^2}.$ Let $\mathfrak{M}$ be a fractional ideal in $F$ and $\min \left(\mathfrak{M}\right):=\min \{\Vert \sigma (s)\Vert |s\in \mathfrak{M},s\ne 0\}.$ The set of shortest nonzero lattice points for $\mathfrak{M}$ is given by $\{s\in \mathfrak{M}:\Vert \sigma (s)\Vert =\min (\mathfrak{M})\}.$ 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.

12 pages. Comments are welcome

References in corpus (1)

• A lower bound for the discrepancy in a Sato-Tate type measure