Special values of \(L\)-functions on regular arithmetic schemes of dimension 1. (English) Zbl 1536.14018
Summary: 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 [T. H. Geisser and T. Suzuki, J. Reine Angew. Math. 793, 281–304 (2022; Zbl 1515.11064)]. We deduce a special value formula for Artin \(L\)-functions of integral representations twisted by a singular irreducible scheme \(X\) of finite type over \(\mathbb{Z}\) and of dimension 1. This generalizes and improves all results in [M. H. Tran, “Weil-étale cohomology and special values of \(L\)-functions”, Preprint, arXiv:1611.01720]; as a special case, we obtain a special value formula for the arithmetic zeta function of \(X\).
MSC:
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 14F20 | Étale and other Grothendieck topologies and (co)homologies |
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 11R42 | Zeta functions and \(L\)-functions of number fields |
Citations:
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