Congruences between derivatives of abelian L-functions at s=0

Let K/k be a finite abelian extension of global fields. We prove that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s=0 of derivatives of the Dirichlet L-functions associated to K/k. We also show that these congruences provide a universal approach to the 'refined abelian Stark conjectures' formulated by, inter alia, Stark, Gross, Rubin, Popescu and Tate. We thereby obtain the first proofs of, amongst other things, the Rubin-Stark conjecture and the 'refined class number formulas' of both Gross and Tate for all extensions K/k in which K is either an abelian extension of Q or is a function field.