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

Abstract

We construct a well-behaved Weil-étale 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-étale 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 $X$.