Congruences Between Derivatives of Artin L-Series at s=0

Abstract We conjecture a family of integral congruence relations between the values at zero of different order derivatives of Artin $L$-series over general number fields. We show that this prediction specialises to recover the “refined class number formula for $\mathbb{G}_{m}$” independently conjectured for abelian extensions by Mazur and Rubin and by the third author, prove it for Artin $L$-series of characters that factor through a natural class of Frobenius extensions of $\mathbb{Q,}$ and provide concrete supporting evidence in other special cases. We also use techniques of non-commutative Iwasawa theory to prove, modulo only a standard $\mu $-vanishing hypothesis, the analogous family of congruence relations for $p$-adic Artin $L$-series over arbitrary totally real number fields.