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

2026/08/02 by Saad El Boukhari
Mathematics · #math.NT #math.KT

paper · pdf

25 pages

arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

Let K/k be a finite abelian extension of number fields with Galois group G and let n≥ 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, Σ-modified étale complexes, and a cancellation argument which removes the auxiliary Euler factors.

Citations

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• On the theory of higher rank Euler, Kolyvagin and Stark systems, III: applications
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• An Equivariant Main Conjecture in Iwasawa Theory and Applications
• On motivic cohomology with Z/l coefficients