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

Let Formula: see text be a multi-quadratic totally real number field. Let Formula: see text denote its distinct embeddings. Given Formula: see text we give an explicit formula for Formula: see text and Formula: see text where Formula: see text Let Formula: see text be a fractional ideal in Formula: see text and Formula: see text The set of shortest nonzero lattice points for Formula: see text is given by Formula: see text We provide shortest nonzero lattice points for Formula: see text 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 Theorem JS20, Theorem 1.6.