Rigid geometry, Lefschetz-Verdier trace formula and Deligne’s conjecture

Abstract

In abstract algebraic geometry, the Lefschetz-Verdier trace formula expresses the global trace of an endomorphism induced on cohomology by a correspondence as a sum of local contributions, called local terms, near fixed points. Though the formula holds quite generally, the explicit calculation of local terms is very hard to carry out, and no good formula is known for general correspondences. Over the complex numbers a useful formula is known for a good class called weakly hyperbolic [G-M]. On the other hand, in positive characteristics, P. Deligne conjectured that the local terms would be much simplified if we composed a given correspondence with a high power of Frobenius. Take a proper variety X over an algebraic closure k of a finite field Fq, a correspondence a : Y ~ X xk X, and a smooth Qr K on an open set U of X fixed by a (denote the inclusion by j , and { is invertible in k). Then