On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne. (English) Zbl 0769.14007

Pink, Richard

Ann. Math. (2) 135, No. 3, 483-525 (1992).

In B. Dwork’s proof of the rationality of the zeta function of a variety over a finite field [see Am. J. Math. 82, 631–648 (1960; Zbl 0173.48501)], the Lefschetz trace formula shown by A. Grothendieck [see Sém. Bourbaki 17 (1964/65), Exposé 279 (1966; Zbl 0199.24802)], plays an essential rôle. In order to obtain finer information one likes to split up the total cohomology using algebraic cycles and to describe the Galois representations on the factors. The splitting process is described in terms of correspondences and one wants to have a Lefschetz trace formula for the twist of a correspondence by the Frobenius, or more special, a Lefschetz trace formula for the twist by all sufficiently large powers of a fixed Frobenius. In this context P. Deligne has conjectured a certain Lefschetz trace formula in terms of correspondences over a finite field. This is shown in the present paper assuming the resolution of singularities holds. It generalizes L. Illusie’s proof in the one-dimensional case [see A. Grothendieck and L. Illusie in Sémin. Géométrie algébrique 1965-1966, SGA 5, Lect. Notes Math. 589, Exposé III, 73–137 (1977; Zbl 0355.14004)].
For the smooth case of an arbitrary dimension there is an independent proof by E. Shpiz [“Deligne’s conjecture in the constant coefficient case”, Ph. D. Thesis (Harvard Univ. 1990)].

Reviewer: P.Schenzel (Halle)

MSC:

14F20	Étale and other Grothendieck topologies and (co)homologies
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Keywords:

rationality of the zeta function; Lefschetz trace formula; resolution of singularities

Citations:

Zbl 0173.48501; Zbl 0199.24802; Zbl 0355.14004