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Motives over finite fields. (English) Zbl 0811.14018
Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 1, 401-459 (1994).
In 1949 André Weil made his famous conjectures on solutions of equations over finite fields. These conjectures explained how a “good” cohomology theory (with values in a field of characteristic 0) would allow one to give amazingly precise information about the zeta function of a (smooth projective) variety over the finite field \(\mathbb{F}_ q\). These conjectures gave birth to a huge explosion of the most profound algebra at the hands of Serre, Grothendieck, M. Artin, and others, which led to the final proof of these conjectures by P. Deligne in 1973. But it is safe to say that these conjectures, and the algebra they generated, have been basic in much of the great successes of modern number theory and algebraic geometry.
As is well known, cohomology comes in many different “flavors”; one has Betti cohomology, de Rham cohomology, \(\ell\)-adic cohomology, \(p\)-adic cohomology, etc. All these cohomology theories tend to have certain things in common, such as Poincaré duality, cohomology classes of cycles and so on. One of Grothendieck’s deepest – and as yet not completely fulfilled – programs was to use these properties to “sculpt” the category of varieties (over an arbitrary field) into a new category (called the category “motives”) which, in some sense, is the “universal” cohomology theory. For instance, in a linear space, all projectors split and so one wants to impose the same property on motives and so on. In order to actually carry out these constructions geometrically one needs lots of cycles and so Grothendieck made his famous “standard conjectures” which are still unknown.
In the case where we are back to working over the finite field \(\mathbb{F}_ q\), one can give a construction of the category of motives via Deligne’s proof of the Weil conjectures (one uses the Frobenius morphism and its various characteristic polynomials to construct the needed projectors). Upon assuming the Tate conjecture (relating cycles to the order of pole of the zeta function and to \(\ell\)-adic representations) one can give an “almost entirely satisfactory” description of the category of motives over finite fields. The paper being reviewed gives a very readable complete treatment of these motives and describes a reduction functor from the category of CM-motives (over the algebraic closure of \(\mathbb{Q}\)) to the category of motives over finite fields.
For the entire collection see [Zbl 0788.00053].

14F99 (Co)homology theory in algebraic geometry
14A20 Generalizations (algebraic spaces, stacks)
14G15 Finite ground fields in algebraic geometry
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18G60 Other (co)homology theories (MSC2010)
11G09 Drinfel’d modules; higher-dimensional motives, etc.