Joint IAS/Princeton University Number Theory Seminar

The classical conjectures of Ramanujan-Petersson and Sato-Tate on the Fourier coefficients of modular forms, or more generally on the Satake parameters of automorphic representations, are highly sensitive to questions of functoriality. For example, the coefficients of CM modular forms are equidistributed according to a very different law from that of non-CM forms, and the first historical counter examples to the naive generalization of the Ramanujan conjecture were found amongst the theta lifts on the group Sp4. A more recent analogue of these conjectures looks at the L^p norms of automorphic forms (with p=\infty corresponding to Ramanujan). Their concentration properties, at points or along certain cycles, are of general interest from both an analytic and arithmetic viewpoint. I will describe in this talk a few new results on the subject, joint with Simon Marshall, which attempt to clarify the structure of the problem: the L^p norms of an automorphic form are closely related to the asymptotic size of certain of its periods which in turn reflect the form's functorial origin. In particular, in a work in progress, we show the existence of Maass forms, defined on hyperbolic manifolds and in the image of the theta correspondence from Sp4, which concentrate to some degree along closed geodesics.