EVENT DETAILS AND ABSTRACT

Title: Congruences for modular forms of half-integer weight
Speaker: Vinayak Vatsal
Speaker Info: University of British Columbia
Brief Description:
Special Note:
Abstract:

Suppose \(F\) and \(G\) are holomorphic cuspidal newsforms of even weight and trivial characters of levels M and N respectively, such that \(F\) and \(G\) are congruent modulo a prime \(P\) in the algebraic closure of \(\mathbf{Q}\). We can then pose the question of whether or not the modular forms associated to \(F\) and \(G\) by the Shimura-Waldspurger correspondence are also congruent modulo \(P\). In considering this question, one quickly realizes that the in the most naive form the answer to this question is negative, but the reason for the failure turns out to be quite subtle. One is faced with the obvious fact that there’s no evident way to single out a specific form on the metaplectic group that corresponds to \(F\) or \(G\), but a more subtle issue is that the usual Shimura-Waldspurger correspondence does not even yield a canonical bijection on the level of automorphic representations. In attempting to formulate a statement that might conceviably be true, one has to consider in some detail the structure of the Waldspurger packets on the metaplectic group, and the existence of a congruence on the metaplectic side is related to a hypothetical multiplicity one theorem for metaplectic modular forms in positive characteristic. Informal speculations along these lines were first made some years ago by K. Prasanna, and we will attempt to make some of his speculations more precise and state an actual conjecture.