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

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 $\mathrm{\parallel }\sigma (s)\mathrm{\parallel }$ and ${\sum }_{i<j}{\sigma }_i(s){\sigma }_j(s),$ where $\mathrm{\parallel }\sigma (s)\mathrm{\parallel }=\sqrt{{\sum }_{i=1}^r({\sigma }_i(s){)}^2}\mathrm{.}$ Let $\mathfrak{M}$ be a fractional ideal in $F$ and $\min \left(\mathfrak{M}\right):=\min \{\mathrm{\parallel }\sigma (s)\mathrm{\parallel }\mathrm{\mid }s\in \mathfrak{M},s\mathrm{\ne }0\}\mathrm{.}$ The set of shortest nonzero lattice points for $\mathfrak{M}$ is given by $\{s\in \mathfrak{M}:\mathrm{\parallel }\sigma (s)\mathrm{\parallel }=\min (\mathfrak{M})\}\mathrm{.}$ 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.