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 Lichtenbaum [65] and Geisser [36] for X of characteristic p, of Lichtenbaum [66] for X = Spec(OF ) and n = 0 where F is a number field, and of the second author for arbitrary X and n = 0 [72]. We show that our conjecture is compatible with the Tamagawa number conjecture of Bloch, Kato, Fontaine and Perrin-Riou [31] if X is smooth over Spec(OF ), and hence that it holds in cases where the Tamagawa number conjecture is known. Contents 1. Introduction 2. Motivic cohomology of proper regular schemes and the Beilinson conjectures 2.1. The Beilinson regulator on the level of complexes 2.2. The Beilinson conjectures and arithmetic duality with R-coefficients 2.3. Motivic cohomology of the Artin-Verdier compactification 3. Weil-étale cohomology of proper regular schemes 3.1. Notations 3.2. Assumptions 3.3. The complex RΓ W (X , Z(n)) 3.4. Rational coefficients 3.5. Torsion coefficients 3.6. Relationship with the Lichtenbaum-Geisser definition over finite fields 3.7. Weil-étale duality 3.8. The complex RΓ W,c (X , Z(n)) 4. Weil-Arakelov cohomology of proper regular schemes 4.1. Weil-Arakelov cohomology with R(n)-coefficients 4.2. Weil-Arakelov cohomology of X 4.3. Weil-Arakelov duality for X 4.4. Weil-Arakelov cohomology with compact support 5. Special values of zeta functions 5.1. De Rham cohomology 5.2. The fundamental line 5.3. The complex RΓ ′ eh (X Fp , Z p (n)) and Milne's correcting factor 5.4. The local factor c p (X , n) 5.5. The main conjecture 5.6. Compatibility with the Tamagawa Number Conjecture 2010 Mathematics Subject Classification. 14F20 (primary) 14F42 11G40 . 1 2 M. FLACH AND B. MORIN5.7. Relationship with the functional equation 5.8. Proven cases and examples 6. Appendix A: Artin-Verdier duality 6.1. Introduction 6.2. The motivic complex Z(n) X 6.3. The Artin-Verdier étale topos X et 6.4. Tate cohomology and the functor R π * 6.5. The motivic complex Z(n) X 6.6. Functoriality 6.7. Relationship with Milne's cohomology with compact support 6.8. Products 6.9. Artin-Verdier Duality 6.10. The conjecture AV(f, n) 6.11. The projective bundle formula 7. Appendix B: Motivic and syntomic cohomology 7.1. l-adic cohomology 7.2. p-adic cohomology ReferencesIn particular, the complex RΓ ar,c (X , R(n)) has vanishing Euler characteristic:b) The groups H i ar,c (X , R/Z(n)) are compact, commutative Lie groups for all i. Note here that the cohomology groups of a complex of locally compact abelian groups carry an induced topology which however need not be locally compact. Indeed, the groups H i ar,c (X , Z(n)) will not always be locally compact.The conjectural relation to the Zeta-function of X is given by the following two statements.c) The function ζ(X , s) has a meromorphic continuation to s = n and
Motivic cohomology of proper regular schemes and the Beilinson conjecturesThroughout this...