Theta functions and symplectic groups, II

Theta functions on covers of symplectic groups

We study the automorphic theta representation $Theta_{2n}^{(r)}$ on the $r$-fold cover of the symplectic group $Sp_{2n}$‎. ‎This representation is obtained from the residues of Eisenstein series on this group‎. ‎If $r$ is odd‎, ‎$nle r

Symplectic Microgeometry Ii: Generating Functions

We adapt the notion of generating functions for lagrangian submanifolds to symplectic microgeometry. We show that a symplectic micromorphism always admits a global generating function. As an application, we describe hamiltonian flows as special symplectic micromorphisms whose local generating functions are the solutions of Hamilton-Jacobi equations. We obtain a purely categorical formulation of...

Arithmetic theta lifting and L-derivatives for unitary groups, II

p-ADIC L-FUNCTIONS FOR ORDINARY FAMILIES ON SYMPLECTIC GROUPS

We construct the p-adic standard L-functions for ordinary families of Hecke eigensystems of the symplectic group Sp(2n)/Q using the doubling method. We explain a clear and simple strategy of choosing the local sections for the Siegel Eisenstein series on the doubling group Sp(4n)/Q, which guarantees the nonvanishing of local zeta integrals and allows us to p-adically interpolate the restriction...

Arithmetic properties of similitude theta lifts from orthogonal to symplectic groups

Building on 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 3space, which together with results of Kudla and Millson imply an interpretation of the Fourier coeff...