University of Chicago
Department of Mathematics

The Riemann zeta function and its siblings, Dirichlet's L-functions, were first introduced because of their connections with the distribution of prime numbers. "Special values" of these functions - essentially values at integers - were observed to be particulary nice. For example, the value of the Riemann zeta function at 1-2n is -B(2n)/2n where B(2n) is the 2nth Bernoulli number. Subsequently it was discovered that these special values are connected to the orders of class groups of various abelian number fields. Proofs of these connections (due to Ribet, Mazur, Wiles,...) have made use of modular forms and of Eisenstein series in particular; the special values often show up as constant terms of Eisenstein Series. In this talk I will review these connections and the role of Eisenstein series in their proofs and will discuss current efforts to make use of Eisenstein series and Eisenstein-like objects on groups of higher rank to connect special values of some of zeta's cousins, the L-functions of modular forms, to the orders of Galois cohomology groups.