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].