10,212 research outputs found
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 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
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 -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 -cohomology groups.Comment: October 2013: Improved exposition. Added note
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
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