Automorphic Forms and Arithmetic

I will talk about my recent rationality results on the ratios of critical values for Rankin-Selberg L-functions for GL(n) x GL(m) over a totally imaginary base field. In contrast to a totally real base field, when the base field is totally imaginary, some delicate signatures enter the reciprocity laws for these special values. These signatures depend on whether or not the totally imaginary base field contains a CM subfield. This is a generalization of my work with Günter Harder on rank-one Eisenstein cohomology for GL(N), where N = n + m. The rationality result comes from interpreting Langlands's constant term theorem in terms of an arithmetically defined intertwining operator between Hecke summands in the cohomology of the Borel-Serre boundary of a locally symmetric space for GL(N). The signatures arise from Galois action on certain local systems that intervene in boundary cohomology.

Elliptic curves E over the rational numbers are modular: this means there is a nonconstant map from a modular curve to E. When instead the coefficients of E belong to a function field, it still makes sense to talk about the modularity of E (and this is known), but one can also extend the idea further and ask whether E is 'r-modular' for r=2,3.... To define this generalization, the modular curve gets replaced with Drinfeld's concept of a 'shtuka space'. The r-modularity of E is predicted by Tate's conjecture. In joint work with Adam Logan, we give some classes of elliptic curves E which are 2- and 3-modular.