Rational Tate classes

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 Hodge and Tate conjectures were known. v3. Submitted version

Descent for Shimura Varieties

We verify that the descent maps provided by Langlands's Conjugacy Conjecture do satisfy the continuity condition necessary for them to be effective. Thus Langlands's conjecture does imply the existence of canonical models. This replaces an earlier version of the paper --- the proof in this version is simpler, and the exposition more detailed.Comment: 6 page

Gerbes and abelian motives

Assuming the Hodge conjecture for abelian varieties of CM-type, one obtains a good category of abelian motives over the algebraic closure of a finite field and a reduction functor to it from the category of CM-motives. Consequentely, one obtains a morphism of gerbes of fibre functors with certain properties. We prove unconditionally that there exists a morphism of gerbes with these properties, and we classify them

Kazhdan's Theorem on Arithmetic Varieties

Define an arithmetic variety to be the quotient of a bounded symmetric domain by an arithmetic group. An arithmetic variety is algebraic, and the theorem in question states that when one applies an automorphism of the field of complex numbers to the coefficients of an arithmetic variety the resulting variety is again arithmetic. This article simplifies Kazhdan's proof. In particular, it avoids recourse to the classification theorems. It was originally completed on March 28, 1984, and distributed in handwritten form. July 23, 2001: Fixed about 30 misprints

The Tate Conjecture for Certain Abelian Varieties over Finite Fields

Tate's theorem (Invent. Math. 1966)implies that the Tate conjecture holds for any abelian variety over a finite field whose Q_l-algebra of Tate classes is generated by those of degree 1. We construct families of abelian varieties over finite fields for which this condition fails, but for which we are nevertheless able to prove the Tate conjecture.Comment: 28 page

Towards a proof of the conjecture of Langlands and Rapoport

A conference talk discussing the conjecture of Langlands and Rapoport concerning the structure of the points on a Shimura variety modulo a prime of good reduction.Comment: Text for a talk April 28, 2000, at the Conference on Galois Representations, Automorphic Representations and Shimura Varieties, Institut Henri Poincare, Paris, April 24-29, 200

Addendum to: Milne, Values of zeta functions of varieties over finite fields, Amer. J. Math. 108, (1986), 297-360

The original article expressed the special values of the zeta function of a variety over a finite field in terms of the $\hat{Z}$-cohomology of the variety. As the article was being completed, Lichtenbaum conjectured the existence of certain motivic cohomology groups. Progress on his conjecture allows one to give a beautiful restatement of the main theorem of the article in terms of $Z$-cohomology groups.Comment: October 2013: Improved exposition. Added note

The fundamental theorem of complex multiplication

The goal of this expository article is to present a proof that is as direct and elementary as possible of the fundamental theorem of complex multiplication (Shimura, Taniyama, Langlands, Tate, Deligne et al.). The article is a revision of part of my manuscript, Complex Multiplication, April 7, 2006.Comment: 33 page

Points on Shimura varieties over finite fields: the conjecture of Langlands and Rapoport

We state an improved version 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 PEL-type

The Tate conjecture over finite fields (AIM talk)

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. v2: Revised expanded (24 pages).Comment: A pdf file with better fonts, style file, etc. is available at http://www.jmilne.org/math/ v2: Revised expanded (24 pages