An Explicit Theta Lift to Siegel Paramodular Forms

Abstract Let E / L be a real quadratic extension of number fields. We construct an explicit map from an irreducible cuspidal automorphic representation of $$\mathop \textrm{GL}(2,E)$$ GL ( 2 , E ) which contains a Hilbert modular form with $$\Gamma _0$$ Γ 0 level to an irreducible automorphic representation of $$\mathop \textrm{GSp}(4,L)$$ 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.