The local monodromy as a generalized algebraic correspondence (with an appendix by Spencer Bloch)

Abstract

For an algebraic, normal-crossings degeneration over a local field the local monodromy operator and its powers naturally define Galois equivariant classes in the $\ell$-adic (middle dimensional) cohomology groups of some precise strata of the special fiber of a normal-crossings model associated to the fiber product degeneration. The paper addresses the question whether these classes are algebraic. It is shown that the answer is positive for any degeneration whose special fiber has (locally) at worst triple points singularities. These algebraic cycles are responsible for and they explain geometrically the presence of poles of local Euler L-factors at integers on the left of the left-central point.