Double Complexes and Euler L-Factors

Abstract

In this paper we take into consideration a conjecture of Bloch equating in a suitable range and under some standard conjectures, the rank of the motivic cohomology of the special fiber of a semistable degeneration over the ring of integers of a number field with the order of zero of the local Euler factor of the L-function over the semistable prime corresponding to the special fiber. We then develop, following the philosophy that the fiber ‘at infinity’ of an arithmetic variety should be considered as ‘maximally degenerate’, a construction that goes in parallel to the one we use for the non-archimedean fiber. Namely, a definition of a double complex and a weighted operator N on it that plays the role of the logarithm of the (local) monodromy map at infinity. We like to see this archimedean theory as the analogue of the limiting mixed Hodge structure theory of a degeneration of projective varieties over a disc. In particular, this yields to a description of the (motivic) Deligne cohomology as homology of the mapping cone of N. The main result arising from this construction is the proof of a conjecture of Deninger. Namely we show that the archimedean Γ-factor of the Zeta-function of the archimedean fiber can be seen as the characteristic polynomial of an archimedean Frobenius acting on the subgroup of the invariants of N into the hypercohomology of our double complex.