Geodesic
Weil-Étale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes
Documenta mathematica, Tome 23 (2018), pp. 1425-1560
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We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme X at any integer n in terms of Weil-étale cohomology complexes. This extends work of S. Lichtenbaum [Compos. Math. 141, No. 3, 689–702 (2005; Zbl 1073.14024)] and T. Geisser [Math. Ann. 330, No. 4, 665–692 (2004; Zbl 1069.14021)] for X of characteristic p, of S. Lichtenbaum [Ann. Math. (2) 170, No. 2, 657–683 (2009; Zbl 1278.14029)] for X=Spec(OF​) and n=0 where F is a number field, and of the second author for arbitrary X and n=0 [B. Morin, Duke Math. J. 163, No. 7, 1263–1336 (2014; Zbl 06303878)]. We show that our conjecture is compatible with the Tamagawa number conjecture of S. Bloch and K. Kato [Prog. Math. 86, 333–400 (1990; Zbl 0768.14001)], and J.-M. Fontaine and B. Perrin-Riou [Proc. Symp. Pure Math. 55, 599–706 (1994; Zbl 0821.14013)] if X is smooth over Spec(OF​), and hence that it holds in cases where the Tamagawa number conjecture is known.
DOI : 10.4171/dm/651
Classification : 11G40, 14F20, 14F42
Mots-clés : Zeta functions, Arakelov theory, Zeta-values, Weil-étale cohomology 
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     author = {Baptiste Morin and Matthias Flach},
     title = {Weil-\'Etale {Cohomology} and {Zeta-Values} of {Proper} {Regular} {Arithmetic} {Schemes}},
     journal = {Documenta mathematica},
     pages = {1425--1560},
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     volume = {23},
     doi = {10.4171/dm/651},
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Baptiste Morin; Matthias Flach. Weil-Étale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes. Documenta mathematica, Tome 23 (2018), pp. 1425-1560. doi: 10.4171/dm/651

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