Expresses the number of points of a variety over a finite field
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. There are several generalizations: the Frobenius endomorphism can be replaced by a more general endomorphism, in which case the points over a finite field are replaced by its fixed points, and there is also a more general version for a
sheaf over the variety, where the cohomology groups are replaced by cohomology with coefficients in the sheaf.
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. This is one of the steps used in the proof of the
Weil conjectures.
Freitag, Eberhard;
Kiehl, Reinhardt (1988), Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 13, Berlin, New York:
Springer-Verlag,
ISBN978-3-540-12175-6,
MR0926276