Theta functions, L-functions, and modular forms

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The principal subjects of my research are arithmetic properties of theta functions, especially applications of Mumford's algebraic theta functions to basic questions in arithmetic. While Mumford's theory has had spectacular applications to the construction of arithmetic moduli spaces associated to abelian varieties, it seems that nobody has looked at the applications of the theory to concrete problems in arithmetic. We propose one such project, coming from quadratic forms, and propose a method of solution (joint work with Jonathan Hanke). A second project, more theoretical, project along these lines is the construction of geometric modular forms of weight 3/2, and to construct a geometric version of the theta correspondence in this context. We also propose two entirely different lines of research. The first deals with Iwasawa theory and deformation theory of Galois representations associated to modular forms of weight one. This is joint work with R. Greenberg, and is closely related to recent research of Bellaiche and Dimitrov. The second project seeks to construct modular symbols associated to Eisenstein series over totally real fields. This generalizes earlier work of Stevens, Heumann, and the present author.