Special values of $L$-functions on regular arithmetic schemes of dimension $1$

DOI10.1016/J.JNT.2022.07.002arXiv2108.00811WikidataQ114156436 ScholiaQ114156436MaRDI QIDQ6374311

Adrien Morin

Publication date: 2 August 2021

Abstract: We construct a well-behaved Weil-'etale complex for a large class of $mathbbZ$-constructible sheaves on a regular irreducible scheme $U$ of finite type over $mathbbZ$ and of dimension $1$. We then give a formula for the special value at $s=0$ of the $L$-function associated to any $mathbbZ$-constructible sheaf on $U$ in terms of Euler characteristics of Weil-'etale cohomology; for smooth proper curves, we obtain the formula of arXiv:2009.14504. We deduce a special value formula for Artin $L$-functions twisted by a singular irreducible scheme $X$ of finite type over $mathbbZ$ and of dimension $1$. This generalizes and improves all results in arXiv:1611.01720; as a special case, we obtain a special value formula for the arithmetic zeta function of $X$.

Mathematics Subject Classification ID

Étale and other Grothendieck topologies and (co)homologies (14F20) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) (14G10) (L)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture (11G40) Zeta functions and (L)-functions of number fields (11R42)

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