In 1980s, Gross–Zagier [GZ86] established a formula that relates the Neron–Tate height of Heegner points on modular curves to the central derivative of certain L-functions associated to modular forms. Around the same time, Waldspurger proved a formula, relating toric periods of modular forms to the central value of certain L-functions. Gross put both of these formula in the framework of represe...

We construct liftings of cuspidal automorphic forms from the metaplectic group S̃L2 to GSpin(1, 4) using the Maaß Converse Theorem. In order to prove the non-vanishing of the lift we derive Waldspurger’s formula for Fourier coefficients of half integer weight Maaß forms. We analyze the automorphic representation of the adelic spin group obtained from the lift and show that it is CAP to the Saito...

We find defining equations for the Shimura curve of discriminant 15 over Z[1/15]. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy groups of a spectrum of “topological automorphic forms” associated to this curve, as well as one associated to a quotient by an Atkin-Lehner involution.

We here formulate and prove several new results concerning the Eichler cohomology of automorphic forms on finitely generated Fuchsian groups of the first kind and, in particular, on //-groups. The Eichler cohomology we introduce is connected with automorphic forms of arbitrary real degree (as opposed to integral degree), with a suitably chosen underlying space of functions analytic in the upper...

By using spectral expansions in global automorphic Levi–Sobolev spaces, we estimate an average of the first Fourier coefficients of Maass cusp forms for SL2(Z), producing a soft estimate on the first numerical Fourier coefficients of Maass cusp forms, which is an example of a general technique for estimates on compact periods via application of a pre–trace formula. Incidentally, this shows that...