Frank Calegari

How many automorphic forms are there over imaginary quadratic fields?

Let G be a semisimple real group that does _not_ admit discrete series, and let Gv denote the unitary dual. What can one say about the multiplicities for which representations π ∈ Gv of cohomological type appear in certain spaces of cusp forms for G? We give new upper bounds; we explain why naive non-trivial lower bounds do not in general exist; and we finally describe a conjectural framework where there is ``enough'' cohomology to account for all Galois representations. This is joint work with Matthew Emerton. Of particular interest will be the special case of modular forms over imaginary quadratic fields.