The Stark conjecture in modular* settings

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 algebraic invariants of K by the class number formula. The Artin L-function is a further generalization to any Galois representation ρ; its leading coefficient at s=0 is predicted to be given by "Stark units" and algebraic invariants of ρ by 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 analogues (modulo* p, p-adic, and function field) being resolved in the last decade. I'll report on new developments from the theory of derived Hecke operators (degree-shifting maps on cohomology) on modular* forms that describe Stark units modulo p and p-adically.