Daniel Vallières. The extended abelian Stark conjecture.
The word extended in the title refers to trying to remove the hypothesis on split primes in the classical statement of the abelian Stark conjecture. This problem was studied by Stark and his student Erickson in the rank one case. Popescu and his student Emmons extended the conjecture to the higher order of vanishing case very much in the spirit of the Rubin-Stark conjecture. In this talk, we will briefly explain these previous works of them and then talk about our recent work in trying to understand better the extended situation.