Compatibility of Special value conjectures with the functional equation of Zeta functions

Abstract

We prove that the special value conjecture for the Zeta function ζ(X , s) of a proper, regular arithmetic scheme X that we formulated in [8][Conj. 5.12] is compatible with the functional equation of ζ(X , s) provided that the factor C(X , n) we were not able to compute in loc. cit. has the simple explicit form suggested in [9].We now explain the significance of this result. Let ζ(X , s)Conjecture 1.3. (Functional Equation) Let X be a regular scheme of dimension d, proper and flat over Spec(Z). Then ζ(X , s) has a meromorphic continuation to all s ∈ C and By [8][Prop. 5.29], the exact triangle (3) and Weil-étale duality [8][Thm. 3.22] induce a canonical isomorphismcompatible with ξ ∞ and the trivializations (2) of ∆(X /Z, n) and ∆(X /Z, d − n). We obtain Theorem 1.4. Assume that ζ(X , s) satisfies Conjecture 1.3. Then Conjecture 1.1 for (X , n) is equivalent to Conjecture 1.1 for (X , d − n).