In algebraic geometry, the Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism on its cohomology groups.
One application of the Grothendieck trace formula is to express the zeta function of a variety over a finite field, or more generally the L-series of a sheaf, as a sum over traces of Frobenius on cohomology groups.
Let k be a finite field, l a prime number invertible in k, X a smooth k-scheme of dimension n, and
holds: where F is everywhere a geometric Frobenius action on l-adic cohomology with compact supports of the sheaf
Taking logarithmic derivatives of both formal power series produces a statement on sums of traces for each finite field extension E of the base field k: For a constant sheaf
to qualify as an l-adic sheaf) the left hand side of this formula is the number of E-points of X.