Algebra/Topology Seminar

Speaker: Adrien Morin

Title: Weil-étale cohomology and the ETNC for constructible sheaves

Abstract: Let X be a variety over a finite field. Given an order R in a semisimple algebra A over the rationals and a constructible étale sheaf F of R-modules over X, one can consider a natural equivariant L-function associated with F. We will formulate and prove a special value conjecture at negative integers for this L-function, expressed in terms of Weil-étale cohomology, provided that the latter is “well-behaved”; this is a geometric analogue of, and implies, the equivariant Tamagawa number conjecture for an Artin motive and its negative twists over a global function field. It also generalizes the results of Lichtenbaum and Geisser on special values at negative integers for zeta functions of varieties, and the work of Burns-Kakde in the case of the equivariant L-functions coming from a finite G-cover of varieties.