Arithmetic properties of similitude theta lifts from orthogonal to symplectic groups
Abstract
By adapting the work of Kudla and Millson we obtain a lifting of cuspidal cohomology classes for the symmetric space associated to GO(V) for an indefinite rational quadratic space V of even dimension to holomorphic Siegel modular forms on GSpn(A). For n=2 we prove Thom's Lemma for hyperbolic 3-space, which together with results of Kudla and Millson imply an interpretation of the Fourier coefficients of the theta lift as period integrals of the cohomology class over certain cycles. This allows us to prove the p-integrality of the lift for a particular choice of Schwartz function for almost all primes p. We further calculate the Hecke eigenvalues (including for some "bad" places) for this choice in the case of V of signature (3,1).
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