Abstract

The special value ζ(0) of Riemann zeta function was calculated to be -½ by Euler in 1740. The Dedekind zeta function is a generalization of the Riemann zeta function to any number field K; the leading coefficient of its Taylor series at s=0 is precisely described in terms of units and the class number of K in the class number formula. The Artin L-function is a further generalization to any Galois representation ρ of Gal(K/ℚ); its leading coefficient at s=0 is predicted to be given by units and algebraic invariants of ρ in the Stark conjecture. Very few cases of the Stark conjecture have been proved in the last 50 years, but a recent flurry of activity has led to many of the refinements & analogues (modulo p, p-adic, and function field) formulated by Brumer, Tate, Gross, Rubin, Popescu, Kurihara, and others being resolved in the last decade. I'll report on some of the newest developments, especially on the Harris–Venkatesh conjecture about the action of derived Hecke operators on weight 1 modular forms, and the questions that they raise for the Stark conjecture.