1968a Extensions of abelian varieties defined over a
finite field.
Invent. Math. 5 (1968), 63-84.
For two abelian varieties A and B over a finite field, prove a
formula relating the order of Ext1(A,B)
to the zeta functions of A and B.
pdf (Scanned
document; file is 1500KB.)
1968b The Tate-\v Safarevi\v c group of a constant abelian variety
Invent. Math. 6 (1968), 91-105.
Prove the full conjecture of Birch and Swinnerton-Dyer in the case of a
constant abelian variety over a global field of prime characteristic; in
particular, give the first examples of nonzero abelian varieties whose
Tate-Shafarevich groups are known to be finite.
pdf (Scanned document; file is 1300KB.)
1970a The homological dimension of commutative group schemes over a
perfect field.
J. of Algebra 16 (1970), 436--441.
Prove a Hochschild-Serre type spectral sequence for Exts of commutative group schemes
over a perfect field. Give a short proof that Ext2(A,B)=0 for abelian varieties
over an algebraically closed field.
pdf
(Scanned document; file is 970KB.)
1970b Elements of order p in the Tate-\v Safarevi\v c group,
Bull.
Prove that the Tate-Shafarevich group of an abelian variety over a global field
has only finitely many elements of order an integer m, even when m is
divisible by the characteristic. This statement becomes false when finitely
many primes are omitted, or the field constants of the global field is
algebraically closed.
pdf (Scanned
document; file is 832KB.)
1970c Weil-Châtelet groups over local fields,
Ann. Sci. Ecole Norm. Sup. 4 series, T3 (1970), 273-284; Addendum T5
(1972), 261-264.
For an abelian variety A and its dual B over a local field of
prime characteristic, prove that A(K) is dual to the
Weil-Chatelet group of B.
pdf (Scanned
document; file is 2510KB.)
1970d The Brauer group of a rational surface,
Invent. Math. 11 (1970), 304-307.
Proves the conjecture of Artin and Tate (relating the order of the Brauer group
of a surface over a finite field to its zeta function) for rational surfaces.
pdf (Scanned
document; file is 612KB.)
1971a Abelian varieties over finite fields (with W. C. Waterhouse),
Proc. Symp. Pure Math. 20 (1971), 53-64.
An introduction to Honda-Tate theory. Includes proofs of two results of Tate
announced in his 1966 paper but not proved there. Also states results of the two
authors.
pdf (Scanned
document; file is 2563KB.)
1972a On the arithmetic of abelian varieties,
Invent. Math. 17 (1972), 177-190.
Relates the arithmetic invariants of an abelian variety to those of a Weil
restriction of scalars of it. Deduces that the conjecture of Birch and
Swinnerton-Dyer holds for the former if and only if it holds for the latter.
Identifies the zeta functions of some abelian varieties
that acquire complex multiplication over an extension field.
pdf (Scanned
document; file is 2633KB.)
1972b Congruence subgroups of abelian varieties,
Bull. Sci. Math. 96 (1972), 333-338.
Proves that the congruence subgroup problem has a positive solution for abelian
varieties over global fields.
pdf (Scanned document; file is 1110KB.)
1972c Abelian varieties defined over their fields of moduli, I,
Bull. London Math. Soc. 4 (1972), 370-372; Correction 6 (1974), 145-146.
pdf (Scanned at 600dpi; 430KB.)
1973a On a theorem of Mazur and Roberts,
Amer. J. Math. 95 (1973), 80-86.
Gives a short proof of the theorem, which is a duality for the cohomology groups
of finite flat group schemes over complete discrete valuation rings.
pdf (Scanned document; file is
484KB.),
jstor
1975a On a conjecture of Artin and Tate,
Annals of Math. 102 (1975), 517-533.
Proves that, for a surface over a finite field, the Tate conjecture implies the
Artin-Tate conjecture (relating the order of the Brauer group of a surface over
a finite field to its zeta function).
pdf (Scanned document; file is 1277KB.),
jstor
1976a Duality in the flat cohomology of surfaces,
Ann. Sci. Ecole Norm. Sup. 9 (1976), 171-202.
Introduces the sheaves \nu (now called the logarithmic de Rham-Witt sheaves),
proves the flat duality theorem for surfaces (conjectured by M. Artin), and, for
sheaves killed by p, generalizes it to all smooth projective varieties.
pdf (Scanned
document; file is 5138KB.)
1976b Flat homology,
Bull. Amer. Math. Soc. 82 (1976), 118-120.
For a scheme X over a field k, proves the existence of a flat
homology complex that universally computes the flat cohomology of any constant
commutative algebraic group over X (partially confirms, and partially
contradicts, a conjecture of Grothendieck).
pdf (Scanned
document; file is 494KB.)
1976c Duality in the flat cohomology of curves (with M. Artin),
Invent. Math. 35 (1976), 111-129.
Proves a duality theorem for the flat cohomology "groups" of a finite group
scheme over a smooth complete curve.
pdf (Scanned document; file is 2990KB.)
1979a Points on Shimura varieties mod p,
Proc. Symp. Pure Math. 33 (1979), part 2, 165-184.
For a Shimura variety defined by a totally indefinite quaternion algebra over a
totally real field, this article and its sequel 1979b prove a conjecture of
Langlands concerning the points on the good reduction of the variety.
pdf (Scanned document; file is
2036KB.)
1979b Etude d'une class d'isogénie.
In Variétiés de Shimura et Fonctions L, Publications Mathématiques de
l'Université Paris 7 (1979), 73-81.
Completes the proof of the theorem in 1979a.
Scan of preprint: pdf (file is 709KB.)
08.12.02 TeXed and translated. dvi ps pdf
1980a Etale Cohomology.
Princeton University Press (1980), 323pp.
A fairly comprehensive treatment of the subject.
[No online file, but see my course notes Lectures on Etale Cohomology. The book
is still in print.]
1981a The action of complex conjugation on a Shimura variety (with
K-y. Shih),
Annals of Math. 113 (1981), 569-599.
For a Shimura variety with a real canonical model, complex conjugation defines
an involution of the complex points, which it is necessary to know in order to
be able to compute the zeta function. Langlands conjectured a description of the
involution, and we proved it for all Shimura varieties of abelian type.
pdf (Scanned document; file is
2585KB.),
jstor
1981b Automorphism groups of Shimura varieties and reciprocity laws
(with K-y. Shih),
Amer. J. Math. 103 (1981), 1159-1175.
We deduce the existence of canonical models in the sense of Shimura from knowing
the existence of canonical models in the sense of Deligne. Contrary to expert
opinion at the time, this was a serious exercise.
pdf (Scanned document; file is
1661KB.),
jstor
1981c Some estimates from étale cohomology,
J. Reine Angew. Math. 328 (1981), 208-220.
Proves estimates for exponential sums that enabled to Hooley to solve a problem
that had stumped him for 20 years. See his plenary talk at the ICM 1983.
(Actually, don't, because he forgot to mention his debt to étale cohomology.)
pdf (Scanned at 600dpi; 928KB.),
1982a Hodge cycles and abelian varieties (notes of a seminar of P. Deligne),
in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in
Math. 900 (1982), Springer, Heidelberg, 9-100.
This is the seminar in which Deligne proved his famous theorem that all Hodge
cycles on abelian varieties are absolutely Hodge.
html
See also Books
1982b Tannakian categories (with P. Deligne),
in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math. 900
(1982), Springer, Heidelberg, 101-228.
An introduction to the theory of Tannakian categories, with some improvements to
the theory.
See Books
1982c Langlands's construction of the Taniyama group (with K-y. Shih),
in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math. 900
(1982), Springer, Heidelberg, 229-260.
The Taniyama "group" controls how automorphisms of the field of complex numbers
act on CM abelian varieties, their points of finite order, Shimura varieties,
automorphic functions, etc.. This is a more detailed account of its construction
than Langlands's original.
See Books
1982d Conjugates of Shimura varieties (with K-y. Shih),
in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math. 900
(1982), Springer, Heidelberg, 280-356.
Proves Langlands's conjugation conjecture for Shimura varieties of abelian type.
See Books
1982e Zero cycles on algebraic varieties in nonzero characteristic: Rojtman's theorem,
Compositio Math. 47 (1982), 271-287.
For a smooth projective variety X over a field of prime characteristic
p, the canonical map CH0(X)(p) -->
Alb(X)(p) is an isomorphism. Here CH0(X)
is the group of zero cycles modulo rational equivalence, and CH0(X)(p)
is its p-primary component.
pdf (Scanned at 600dpi; 808KB.),
1982f Comparison of the Brauer group with the Tate-\v Safarevi\v c group,
J. Fac. Sci. Univ. Tokyo (Shintani Memorial Volume) IA 28 (1982), 735-743.
For a surface fibred over a curve over global field, relates the order of the Brauer group of the surface to that of the Tate-Shafarevich group of the Jacobian of the generic fibre in a general situation where the two are not equal. Deduce that, for many Jacobians A over global fields of prime characteristic, the first part of the Birch/Swinnerton-Dyer conjecture (order of the zero) implies the whole conjecture (formula for the order of the Tate-Shafarevich group).
pdf (Scanned document; file is
1956KB.)
1983a The action of an automorphism of C on a Shimura variety and its
special points
In: Arithmetic and Geometry, Papers dedicated to I.R. Shafarevich on the
occasion of his sixtieth birthday, Progress in Math. 35 (1983), Birkhauser
Verlag, 239-265.
Proves Langlands's conjugation conjecture, and
hence the existence of canonical models, for all Shimura varieties.
pdf (Scanned document;
file is 4,188KB.)
1986a Values of zeta functions of varieties over finite fields,
Amer. J. Math. 108, (1986), 297-360.
States the conjecture (improved version of a conjecture of Lichtenbaum) relating
special values of zeta functions to motivic cohomology, and proves a Z^
version of it (including the p-part).
pdf (Scanned document; file is
3,221KB.),
jstor
1986b Abelian varieties,
in Arithmetic Geometry (Proc. Conference on Arithmetic Geometry, Storrs, August
1984) Springer, 1986, 103-150.
An introductory survey of abelian varieties.
pdf (Scan at 600dpi; 3.7MB.)
1986c Jacobian varieties,
in Arithmetic Geometry (Proc. Conference on Arithmetic Geometry, Storrs,
August 1984) Springer, 1986, 167-212.
An introductory survey of Jacobian varieties.
pdf (Scan at 600dpi; 3.8MB.)
1986d Arithmetic Duality Theorems
Perspectives in Mathematics, No. 1, Academic Press, 1986, 432pp.
Includes proofs of all the main duality theorems in algebraic number theory
and arithmetic geometry, some of which were previously unavailable. The chapters
are: I, Galois Cohomology; II, Etale Cohomology; III, Flat Cohomology.
A pdf files for the first and second editions are available at
Books
1988a Automorphic vector bundles on connected Shimura varieties,
Inventiones math., 92 (1988), 91-128.
Proves the analogue for automorphic vector bundles of Langlands's conjugacy
conjecture (over connected Shimura varieties).
pdf(Scanned document; file is 4100KB; offsite)
1988b Motivic cohomology and values of zeta functions,
Compos. math. 68 (1988), 59-102.
Adds the p-Kummer axiom to those of Beilinson and Lichtenbaum for the
motivic complex, and explains how the conjecture (Lichtenbaum+M) relating the
special values of zeta functions to motivic cohomology will follow from the main
result in Milne 1986a once a complex has been shown to exist satisfying certain
of the axioms.
pdf (Scan at 600dpi; 2.6MB.)
1990a Canonical Models of (Mixed) Shimura Varieties and Automorphic Vector
Bundles.
In: Automorphic Forms, Shimura Varieties, and L-functions, (Proceedings of a Conference
held at the University of Michigan, Ann Arbor, July 6-16,
1988), pp283--414.
Surveys what was known, or conjectured, about canonical models of Shimura
varieties and related objects at the time it was written (1988).
16.09.97. Put the original dvi-file on the web.
22.06.01. New files (LaTeX 2e); fixed some minor misprints; added some footnotes dvi;
17.08.02. Fixed a problem with the numbering in Chapter III. pdf.
1992a The points on a Shimura variety modulo a prime of good reduction.
In: The Zeta Function of Picard Modular Surfaces, Publ. Centre de Rech. Math.,
Examines the conjecture of Langlands and Rapoport and its consequences;
introduces the notion of a canonical integral model.
14.06.01. Fixed a problem with the dvi-file (because of an errant input, it
repeated pages 57--68); added two pages of notes; added a pdf file. dvi, pdf.
1994a Motives over finite fields.
In: Motives (Eds. Jannsen, Kleiman,Serre), AMS, Proc. Symp Pure Math. 55, 1994,
Part 1, pp. 401--459.
Studies the Tannakian category of motives over a finite field assuming the Tate
conjecture. It computes the associated groupoid, the polarizations, studies the
reduction functor from the category of CM-motives in characteristic zero. It is
partly expository, because many of the results were known to Grothendieck,
Langlands, Rapoport, and Deligne, but often not published.
dvi, pdf.
1994b Shimura varieties and motives
In: Motives (Eds. U. Jannsen, S. Kleiman, J.-P. Serre), Proc. Symp. Pure Math.,
AMS, 55, 1994, Part 2, pp447--523.
Proves that all Shimura varieties of abelian type with rational weight can be
realized as the moduli schemes of abelian motives with additional structure,
and draws some consequences. The paper also computes the affine group scheme
attached to the category of abelian motives over C, and contains a
heuristic derivation of the conjecture of Langlands and Rapoport.
dvi, pdf.
1997a Shimura Variety.
Encyclopedia of Mathematics, Supplement Vol I, Kluwer Acad, Publ., 1997,
pp448-449.
dvi, pdf
.
1999a Lefschetz Classes on Abelian Varieties
Duke Math. J. 96:3, 1999,
pp. 639-675.
A Lefschetz class on a smooth projective variety is an element of
the Q-algebra generated by divisor classes. Show that for any abelian variety
and any Weil cohomology, there is a reductive group (necessarily nonconnected) such
that the Lefschetz cohomology classes are precisely those fixed by the group. As
a consequence, we obtain for abelian varieties over algebraically closed fields:
---that the various classes
predicted to be algebraic by Grothendieck's standard conjectures are not merely
algebraic, but are even Lefschetz;
---if c is a nonzero Lefschetz
cohomology class, then there exists a Lefschetz cohomology class d such that
c.d
is nonzero;
---the question of whether the
algebra of Hodge classes (or the algebra of Tate classes) on a given abelian
variety is generated by divisor classes becomes a question of whether two
reductive groups are equal.
30.12.96; 01.09.97 (minor changes) dvi,
pdf, jnl
1999b Lefschetz Motives and the Tate Conjecture
Compositio Math. 117 (1999), pp. 47-81.
We prove that it is possible to define Q-linear Tannakian categories of
abelian motives using the Lefschetz classes as correspondences. As an
application, we prove that the Hodge conjecture for complex abelian varieties
of CM-type implies the Tate conjecture for all abelian varieties over finite
fields, thereby reducing the latter to a problem in complex analysis.
31.12.96; 28.08.97 (minor changes). dvi, pdf,
jnl
1999c Descent for Shimura Varieties
Michigan Math. J., 46 (1999), pp.
203--208.
This note checks that the descent maps provided by Langlands's conjugacy conjecture (see his Corvallis
article) do satisfy the continuity condition necessary for them to be effective.
Hence the conjecture does imply the existence of canonical models.
22.09.98 dvi, pdf,
jnl
2001a The Tate conjecture for certain abelian varieties over finite
fields.
Acta Arith. 100 (2001), no. 2, pp.135--166.
Proves the Tate conjecture for a family of abelian varieties whose Ql
-algebra of Tate classes is not generated by those of degree one
(and so the conjecture doesn't follow from Tate's theorem (1966)) --- as far as
I know, these are the first such varieties.
01.08.99; 11.02.01 (very minor changes). dvi, pdf
2002a Polarizations and Grothendieck's Standard Conjectures
Ann. of Math. 155 (2002), pp. 599--610.
We prove that Grothendieck's Hodge standard conjecture holds for abelian
varieties in arbitrary characteristic if the Hodge conjecture holds for complex
abelian varieties of CM-type. For abelian varieties with no exotic algebraic
classes, we prove the Hodge standard conjecture unconditionally.
14.08.01. 16 pages. dvi, pdf
19.09.01. Shortened following suggestions by the referee. 11 pages. dvi, pdf, jnl
2003 Gerbes and abelian motives
Not submitted for publication.
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. Consequently, 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.
26.01.03. v1. 42 pages
21.02.03. v1.1, minor fixes, 42 pages dvi,
pdf
2004 Integral Motives and Special Values of Zeta Functions (with
J. Amer. Math. Soc. 17 (2004), 499-555.
For each field k, we define a category of rationally decomposed mixed motives
with Z-coefficients. When k is finite, we show that the category is Tannakian,
and we prove formulas relating the behaviour of zeta functions near integers to
the orders of Ext groups
22.05.2002. Submitted version. 53 pages.
02.08.2003. Submitted version. 56 pages. Many expository improvements.
28.03.2004. Final submitted version. 58 pages. Minor improvements.
dvi, pdf,
jnl(pdf)
2004 Periods of abelian varieties
Compositio Math. 140 (2004), 1149--1175.
We prove various characterizations of the period torsor of abelian
varieties, and we correct some errors in the literature. Beyond its intrinsic
interest, the period torsor controls the arithmetic of holomorphic automorphic
forms.
06.09.2002. E-version. 40 pages. dvi, pdf
08.11.2002. Submitted version. Revised and shortened from previous version. 35
pages. pdf
27.02.2003. Submitted version. Rewrote the introduction. 34 pages. pdf
19.04.2003. Submitted version. Fixed misstatement in the introduction 34 pages. pdf
18.05.2003. Submitted version. Minor changes. 34 pages. pdf
2005 Introduction to Shimura varieties
In Harmonic Analysis, the Trace Formula and Shimura Varieties
(James Arthur, Robert Kottwitz, Editors) AMS, 2005, ISBN 0-8218-3844-X.
Lectures at the Summer School held at the Fields Institute, June 2 -- June 27, 2003
My article is an introduction to the theory of Shimura varieties, or, in
other words, to the arithmetic theory of automorphic functions and holomorphic
automorphic forms.
The published version is essentially the same as the following, except that it omits most footnotes/endnotes.
12.01.04. Almost final version (131pp).
28.03.04. Final version (148pp).
23.10.04. Added an index of definitions; minor changes to TeXing (149pp) pdf.
2005 The de Rham-Witt and
Zp-cohomologies of an algebraic variety (with Niranjan Ramachandran)
Advances in Mathematics, 198 (2005), 36--42.
We prove that the Zp-cohomology groups of a complete
smooth algebraic variety, originally defined in Milne 1986a, also arise by
taking Hom (in the triangulated category of coherent complexes over the Raynaud ring) from an
identity object into the object defined by de Rham-Witt complex on X.
31.08.04. Submitted version. 6 pages.
20.09.04. Fixed misstatement in Introduction.
15.01.05. Final version (very minor changes). pdf
2005 Quotients of Tannakian Categories
Preprint 11 pages; eventually, it may be included in a revised version of Deligne and Milne 1982.
I classify the "quotients" of a tannakian category in which the objects of a tannakian
subcategory become trivial, and I examine the properties of such quotient categories.
24.08.05. pdf
2006 Motivic complexes over finite fields and
the ring of correspondences at the generic point
(with Niranjan Ramachandran)
Already in the 1960s Grothendieck understood that one could obtain an almost
entirely satisfactory theory of motives over a finite field when one assumes the
Tate conjecture. In this note we prove a similar result for motivic complexes.
In particular Beilinson's Q-algebra of "correspondences at the generic
point" is then defined for all connected varieties. We compute this for all
smooth projective varieties (hence also for varieties birational to such a
variety).
03.12.05. pdf First version on the web.
19.07.06. pdf Submitted version (plus an appendix).
2006 Motives over Fp
In April, 2006, Kontsevich asked me whether the category of motives over
Fp (p prime) has a fibre functor over a number field of
finite degree since he had a conjecture that more-or-less implied this. This
article is my response. Unfortunately, since the results are generally
negative or inconclusive, they are of little interest except perhaps for the
question they raise on the existence of a cyclic extension of Q
having certain properties (see Question 6.5).
22.07.06. pdf
2007 Semisimple Lie algebras, algebraic groups,
and tensor categories
It is shown that the classification theorems for semisimple algebraic groups
in characteristic zero can be derived quite simply and naturally from the
corresponding theorems for Lie algebras by using a little of the theory of
tensor categories.
09.05.07. pdf 37 pages.
2007 Semisimple algebraic groups in characteristic zero
Short version of the above article.
09.05.07. pdf 12 pages.
2007 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).
23.05.07. pdf 34 pages.