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

TL;DR

This paper develops a Weil-étale cohomology framework for regular arithmetic schemes of dimension one, providing new formulas for special values of associated $L$-functions, including Artin $L$-functions and zeta functions, generalizing prior results.

Contribution

It introduces a Weil-étale complex for $Z$-constructible sheaves on one-dimensional schemes and derives explicit formulas for their $L$-function special values, extending previous work.

Findings

Constructed Weil-étale complexes for schemes over $Z$

Derived formulas for $L$-function values at $s=0$

Generalized special value formulas to Artin $L$-functions and zeta functions

Abstract

We construct a well-behaved Weil-\'etale complex for a large class of $\mathbb{Z}$-constructible sheaves on a regular irreducible scheme $U$ of finite type over $\mathbb{Z}$ and of dimension $1$. We then give a formula for the special value at $s=0$ of the $L$-function associated to any $\mathbb{Z}$-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 $\mathbb{Z}$ 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 $X$.