A Coates-Sinnott-type Theorem for First Derivatives of Artin $L$-Functions

Abstract

Let $K/k$ be a finite abelian extension of number fields with Galois group $G$ and let $n\geq 2$. We prove, assuming the relevant $p$-part of the equivariant Tamagawa number conjecture, first-derivative analogues of the Deligne-Ribet integrality theorem and of the Coates-Sinnott conjecture. We construct a rank-one leading term from the first derivatives at $s=1-n$ of the $S$-truncated Artin $L$-functions and show that it satisfies an integral annihilation property. We then attach to this leading term a fractional ideal of $\mathbb Q_p[G]$ and prove that, up to the natural torsion factor coming from $K_{2n-1}(O_K)$, this ideal annihilates the even $K$-group $K_{2n-2}(O_{K,S})$. The proof uses determinant methods, $\Sigma$-modified \'etale complexes, and a cancellation argument which removes the auxiliary Euler factors.