The Tate conjecture over finite fields (AIM talk) J.S. Milne Abstract These are my notes for a talk at the The Tate Conjecture workshop at the American Institute of Mathematics in Palo Alto, CA, July 23–July 27, 2007, somewhat revised and expanded. The intent of the talk was to review what is known and to suggest directions for research. v1 (September 19, 2007): first version on the web. v2 (October 10, 2007): revised and expanded. v2.1 (November 7, 2007) Minor fixes. v2.2 (May 7, 2008) Sections 1 and 5 rewritten; other minor changes. Contents 1 The conjecture, and some folklore 2 2 Divisors on abelian varieties 6 3 K3 surfaces 10 4 Algebraic classes have the Tannaka property 10 5 Groups acting on the cohomology 11 6 The Hodge conjecture and the Tate conjecture 13 7 Rational Tate classes 16 8 Reduction to the case of codimension 2 18 9 The Hodge standard conjecture 21 10 On the existence of a good theory of rational Tate classes 22 Conventions All varieties are smooth and projective. Complex conjugation on C is denoted by . The symbol F denotes an algebraic closure of Fp, and ` always denotes a prime p. On the other hand, l is ¤ allowed to equal p. For a variety X, H .X/ L H i .X/ and H 2 .X/. / L H 2i .X/.i/; D i D i both are graded algebras. The group of algebraic cycles of codimension r on a variety X (free Z- module generated by the closed irreducible subvarieties of codimension r) is denoted Zr .X/, and c denotes the cycle class map c Zr .X/ H 2r .X/.r/. I denote a canonical (or a specifically given) W ! isomorphism by . ' MSC2000: 14C25, 11G25 1 1 THE CONJECTURE, AND SOME FOLKLORE 2 1 The conjecture, and some folklore Statements Let X be a variety over F. A model X1 of X over a finite subfield k1 of F gives rise to a commutative diagram: r c 2r Z .X/ H .X; Q`.r// x ! x ? ? ? ? r c 2r Z .X1/ H .X1; Q`.r//: ! Every algebraic cycle X arises from an algebraic cycle on some X1, and so the image of the cycle class map is contained in r def [ 2r Gal.F=k1/ T `.X/ H .X; Q`.r// : D X1=k1 In his talk at the AMS Summer Institute at Woods Hole in July, 1964, Tate conjectured a converse:1 r r TATE CONJECTURE T .X; `/. The Q`-vector space T `.X/ is spanned by the classes of algebraic r r cycles on X, i.e., c.Z .X// Q` T .X/. D ` r r 2r Let A`.X/ be the Q-span of the image of the cycle class map c Z.X/ H .X; Q`.r//, and, r W ! r for a model X1 of X over a finite subfield k1 of F, let A`.X1/ be the subspace of A`.X/ spanned r r r Gal. =k / r r by the image of Z .X1/ A .X/. Then A .X/ F 1 A .X1/, and so Conjecture T .X; `/ ! ` ` D ` implies that r 2r Gal.F=k1/ c.Z .X1// Q` H .X; Q`.r// : (1) D On the other hand, if (1) holds for all models of X over (sufficiently large) finite subfields of F, then Conjecture T r .X; `/ is true. def L r 2 Let T .X/ T .X/ — it is a Q`-subalgebra of H .X; Q`. // whose elements ` D 0 r dim X ` are called the Tate classesÄ Ä on X. I write T r .X/ (resp. T .X; `/, resp. T .X/) for the statement that Conjecture T r .X; `/ is true for all ` (resp. all r, resp. all r and `), and call it the Tate conjecture. In the same talk, Tate mentioned the following “conjectural statement”: CONJECTURE Er .X; `/ (EQUALITY OF EQUIVALENCE RELATIONS). The kernel of the cycle class r r 2r map c Z .X/ H .X; Q`.r// consists exactly of the cycles numerically equivalent to zero. W ! Equivalently: the pairing r dim X r A .X/ A .X/ Q ` ` ! defined by cup-product is nondegenerate. Both conjectures are existence statements for algebraic classes. It is well known that Conjecture E1.X/ holds for all X (see Tate 1994, 5). Let X be a variety over F. The choice of a model of X over a finite subfield of F defines a n Frobenius map X X. For example, for a model X1 P over Fq, acts as W ! q q .a1 a2 : : :/ .a a : : :/ X1.F/ X1.F/; X1.F/ X.F/: W W 7! 1W 2W W ! ' 1Tate’s talk is included in the mimeographed proceedings of the conference, which were distributed by the AMS to only a select few. Despite their great historical importance — for example, they contain the only written statement by Artin and Verdier of their duality theorem, and the only written account by Serre and Tate of their lifting theorem — the AMS has ignored all requests to make the proceedings more widely available. Fortunately, Tate’s talk was reprinted in the proceedings of an earlier conference (Arithmetical algebraic geometry. Proceedings of a Conference held at Purdue University, December 5–7, 1963. Edited by O. F. G. Schilling, Harper & Row, Publishers, New York 1965). 1 THE CONJECTURE, AND SOME FOLKLORE 3 Any such map will be called a Frobenius map of X (or a q-Frobenius map if it is defined by a model n n n2N n1N over Fq). If 1 and 2 are p 1 - and p 2 -Frobenius maps of X, then for some 1 D 2 N > 1.2 For a Frobenius map of X, we use a subscript a to denote the generalized eigenspace with eigenvalue a, i.e., S Ker . a/N : N CONJECTURE S r .X; `/ (PARTIAL SEMISIMPLICITY). Every Frobenius map of X acts semisim- 2r ply on H .X; Q`.r//1 (i.e., it acts as 1). 0 Weil proved (1948, 70, Theor´ eme` 38) that, for an abelian variety A over F, End .A/ is a 1 semisimple Q-algebra, from which it follows that the Frobenius maps act semisimply on H .A; Q`/ i Vi 1 and on all the cohomology groups H .A; Q`/ H .A; Q`/. In particular, Conjecture S.X/ ' holds when X is an abelian variety over F. The Frobenius endomorphism 1 of X defined by a model X1=Fq acts on the etale´ cohomology q r of X as the inverse of the canonical generator x x of Gal.F=Fq/, and so T `.X/ consists of the 2r 7! elements of H .X; Q`.r// fixed by some Frobenius map of X. Folklore THEOREM 1.4. Let X be a variety over F of dimension d, and let r N. The following statements 2 are equivalent: (a) T r .X; `/ and Er .X; `/ are true for a single `. r r d r (b) T .X; `/, S .X; `/, and T .X; `/ are true for a single `. r r r d r d r (c) T .X; `/, E .X; `/, S .X; `/, E .X; `/, and T .X; `/ are true for all `, and the Q- r r r subspace A`.X/ of T `.X/ generated by the algebraic classes is a Q-structure on T` .X/, i.e., r r A .X/ Q` T .X/. ` ˝Q ' ` (d) the order of the pole of the zeta function Z.X; t/ at t q r is equal to the rank of the group D of numerical equivalence classes of algebraic cycles of codimension r. Of course, the key point is that, for a model X1=Fq of X, the characteristic polynomial of acting on H .X; Q`/ has integer coefficients independent of ` p, which was not proved until nine years ¤ after Tate made his conjecture (by Deligne). That (d) follows from the remaining conditions and Deligne’s result is stated already by Tate in his Woods Hole talk (p101 of Tate 1965). Its proof involves only linear algebra — see Tate 1994, 2. I call statement (d) of the Theorem the full Tate conjecture. 2d THEOREM 1.5. Let X be a variety over F of dimension d. If S .X X; `/ is true, then every Frobenius map acts semisimply on H .X; Q`/. r PROOF. If a occurs as an eigenvalue of on H .X; Q`/, then 1=a occurs as an eigenvalue of on 2d r H .X; Q`.d// (by Poincare´ duality), and r 2d r 2d H .X; Q`/a H .X; Q`.d//1=a H .X X; Q`.d//1 ˝ (Kunneth¨ formula), from which the claim follows. 2 Some authors use “Tate conjecture” to mean the full Tate conjecture rather than Conjecture T , often without making this explicit. 2 Because any two models of X become isomorphic over a finite subfield of F, and the Frobenius map is replaced ŒK k1 by W when the model X1=k1 is replaced by X1K =K. 1 THE CONJECTURE, AND SOME FOLKLORE 4 Does the Tate conjecture hold integrally? Let r r 2r T .X; Z`/ T .X/ H .X; Z`.r//: ` D ` \ r 2r Thus T` .X; Z`/ consists of the elements of H .X; Z`.r// fixed by some Frobenius element. One can then ask whether r r c.Z .X// Z` T .X; Z`/: (2) D ` r 2r Since T `.X; Z`/ contains the torsion subgroup of H .X; Z`.r//, and Atiyah and Hirzebruch (1962) had recently shown that not all torsion classes in Betti cohomology are algebraic, Tate was not tempted to conjecture (2).