Lefschetz Fixed-point Theorem - Frobenius

Frobenius

Let be a variety defined over the finite field with elements and let be the lift of to the algebraic closure of . The Frobenius endomorphism (often just the Frobenius), notation, of maps a point with coordinates to the point with coordinates (i.e. is the geometric Frobenius). Thus the fixed points of are exactly the points of with coordinates in, notation for the set of these points: . The Lefschetz trace formula holds in this context and reads:

This formula involves the trace of the Frobenius on the étale cohomology, with compact supports, of with values in the field of -adic numbers, where is a prime coprime to .

If is smooth and equidimensional, this formula can be rewritten in terms of the arithmetic Frobenius, which acts as the inverse of on cohomology:

This formula involves usual cohomology, rather than cohomology with compact supports.

The Lefschetz trace formula can also be generalized to algebraic stacks over finite fields.