Weil-Étale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes

Abstract

Communicated by Stephen

Compatibility of special value conjectures with the functional equation of zeta functions

“…This article is a continuation of our previous article [8] in which we formulated a conjecture describing the leading Taylor coefficient of the Zeta function ζ(X , s) of a proper regular arithmetic scheme X at integer arguments n ∈ Z. Our conjecture involved a rather inexplicit correction factor C(X , n) ∈ Q × , defined in terms of p-adic Hodge theory at all primes p, which we could only compute for X = Spec(O F ) where F is a number field all of whose completions F v are absolutely abelian.…”

“…where LΩ <n X /Z is the derived de Rham complex [14] modulo the n-th step in the Hodge filtration and RΓ W,c (X , Z(n)) is a perfect complex of abelian groups whose definition is dependent on assumptions (finite generation of étale motivic cohomology, Artin-Verdier duality for torsion motivic cohomology) denoted by L(X et , n), L(X et , d−n), AV(X et , n) in [8][Conj. 3.2,Conj.…”

“…3.1]. Also assuming the Beilinson conjectures in the form of conjecture B(X , n) of [8][Conj. 2.5] one can construct a natural trivialization…”

“…was defined in [8][Def. 5.6] under yet another assumption D p (X , n) [8][Conj. 5.5] as well as assumption R(F p , dim(X Fp )) (resolution of singularities) borrowed from [12].…”

“…Let ζ(X , s) be the Zeta function of X and ζ * (X , n) ∈ R × its leading Taylor coefficient at s = n. Our special value conjecture [8][Conj. 5.12] is the assertion…”