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A motivic conjecture of Milne
, 2007
"... Let k be an algebraically closed field of characteristic p> 0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates the étale cohomology with Zp coefficients to the crystalline cohomology with integral coefficients, in the wider context ..."
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Cited by 6 (5 self)
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Let k be an algebraically closed field of characteristic p> 0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates the étale cohomology with Zp coefficients to the crystalline cohomology with integral coefficients, in the wider context of pdivisible groups endowed with families of crystalline tensors over a finite, discrete valuation ring extension of W(k). The result implicitly extends work of Faltings. As a main new tool we construct global deformations of pdivisible groups endowed with crystalline tensors over certain regular, formally smooth schemes over W(k).
Rational Tate classes
, 2007
"... In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conject ..."
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Cited by 4 (3 self)
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In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjecture were known. In this article I investigate whether there exists a theory of “rational Tate classes ” on varieties over finite fields having the properties that the algebraic classes would have if the Tate conjecture were known. In particular, I prove that there exists at most one “good ” such theory.
Rational Tate classes on abelian varieties
 In preparation
"... In despair, as Deligne (2006) put it, of proving the Hodge and Tate conjectures, one can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjec ..."
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Cited by 2 (2 self)
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In despair, as Deligne (2006) put it, of proving the Hodge and Tate conjectures, one can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjecture were known. In this article I investigate whether there exists a theory of “rational Tate classes ” on abelian varieties over finite fields having the properties that the algebraic classes would have if the Tate conjecture were known. Contents 1 Numerical and homological equivalence 3
Points on Shimura varieties over finite fields: the . . .
, 2008
"... We state a strengthened form of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of PELtype. ..."
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Cited by 2 (0 self)
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We state a strengthened form of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of PELtype.
On the Conjecture of Langlands and Rapoport
, 2007
"... FORENOTE (2007): The remarkable conjecture of Langlands and Rapoport (1987) gives a purely grouptheoretic description of the points on a Shimura variety modulo a prime of good reduction. In an article in the proceedings of the 1991 Motives conference (Milne 1994, §4), I gave a heuristic derivation ..."
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FORENOTE (2007): The remarkable conjecture of Langlands and Rapoport (1987) gives a purely grouptheoretic description of the points on a Shimura variety modulo a prime of good reduction. In an article in the proceedings of the 1991 Motives conference (Milne 1994, §4), I gave a heuristic derivation of the conjecture assuming a sufficiently good theory of motives in mixed characteristic. I wrote the present article in order to examine what was needed to turn the heuristic argument into a proof, and I distributed it to a few mathematicians (including Vasiu). Briefly, for Shimura varieties of Hodge type (i.e., those embeddable into Siegel modular varieties) I showed that the conjecture is a consquence of three statements: (a) a good theory of rational Tate classes (see statements (a,b,c,d) in §3 below); (b) existence of an isomorphism between integral étale and de Rham cohomology for an abelian scheme over the Witt vectors (see 0.1, 5.4 below); (c) every point in Shp(F) lifts to a special point in Sh(Q al). At the time I wrote the article, I erroneously believed that my work on Lefschetz classes etc. (Milne 1999a,b, 2002, 2005) implied (a). This work does show that (a) (and much more) follows