Arithmetic properties of similitude theta lifts from orthogonal to symplectic groups

TL;DR

This paper develops a theta lift from orthogonal to symplectic groups, analyzing its arithmetic properties, Fourier coefficients, and Hecke eigenvalues, with applications to Siegel modular forms and p-adic integrality.

Contribution

It adapts Kudla and Millson's work to construct a theta lift for orthogonal groups, proving new results like Thom's Lemma and p-integrality of the lift.

Findings

Proved Thom's Lemma for hyperbolic 3-space.

Established p-integrality of the theta lift for almost all primes.

Calculated Hecke eigenvalues at various places for specific quadratic spaces.

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 GSp_n(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).