Journal of the Ramanujan Mathematical Society

Johnson-Leung Jennifer ¹, Brooks Roberts ¹

Affiliations
1 Department of Mathematics, University of Idaho, Moscow, ID 83844, US

Abstract

In this paper, we calculate the Fourier coefficients of the paramodular twist of a Siegel modular form of paramodular level N by a nontrivial quadratic Dirichlet character mod p for p a prime not dividing N. As an application, these formulas can be used to verify the nonvanishing of the twist for particular examples. We also deduce that the twists of Maass forms are identically zero.

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Brooks Roberts ¹

Affiliations
1 Department of Mathematics, University of Idaho, Moscow ID 83844, US

Abstract

Let X be an even dimensional symmetric bilinear space defined over a totally real number field F with adeles A, and let σ = ®_vσ_v be an irreducible cuspidal automorphic representation of 0(X, A) tempered at the finite places. We give a sufficient condition for the nonvanishing of the theta lift O_n(σ) of σ to the symplectic group Sp(n, A) (2n by 2n matrices) for 2n ≥ dim X for a large class of X. As a corollary, we show that if 2n = dim X and all the local theta lifts O_n(σ_v) are nonzero, then ©_n(σ) is nonzero if the standard L-function L ^S(s, σ) is nonzero at 1, and ☉_{n_1}(σ) is nonzero if L^S(s, σ) has a pole at 1. The proof uses only essential structural features of the theta correspondence, along with a new result in the theory of doubling zeta integrals.

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