An Explicit Theta Lift to Siegel Paramodular Forms

TL;DR

This paper constructs an explicit theta lift from automorphic representations on GL(2) over a real quadratic extension to GSp(4) over a number field, linking Hilbert modular forms to Siegel paramodular forms.

Contribution

It provides an explicit construction of the theta lift and describes local data for paramodular invariance at finite places, except in wild ramification cases.

Findings

Explicit local data for theta lift at finite places.

Construction of a map from Hilbert modular forms to Siegel paramodular forms.

Identification of cases with wild ramification where the lift is not explicitly described.

Abstract

Let $E/L$ be a real quadratic extension of number fields. We construct an explicit map from an irreducible cuspidal automorphic representation of $\mathrm{GL}(2,E)$ which contains a Hilbert modular form with $\Gamma_0$ level to an irreducible automorphic representation of $\mathrm{GSp}(4,L)$ which contains a Siegel paramodular form and exhibit local data which produces a paramodular invariant vector for the local theta lift at every finite place, except when the local extension has wild ramification.