1967 The conjectures of Birch and Swinnerton-Dyer for constant abelian varieties over function fields (thesis)
Proves the full conjecture of Birch and Swinnerton-Dyer for constant abelian varieties over global fields of nonzero characteristic (with a small proviso). In particular, it contains the first examples of abelian varieties whose Tate-Shafarevich groups are known to be finite. The results were improved and published in Milne 1968a, 1968b.
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, proves that the group Ext1(A,B) is finite, and expresses its order in terms of the zeta functions of A and B and the discriminant of a pairing on the Hom groups.
1968b The Tate-\v Safarevi\v c group of a constant abelian variety, Invent. Math. 6 (1968), 91-105.
Proves the full conjecture of Birch and Swinnerton-Dyer for constant abelian varieties over global fields of nonzero characteristic.
1970b Elements of order p in the Tate-\v Safarevi\v c group, Bull. London Math. Soc. 2 (1970), 293-296.
Proves 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. When the characteristic divides m, the statement becomes false when finitely many primes are omitted, or the finite field of constants is replaced by an algebraically closed field.
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, the article proves that A(K) is dual to the Weil-Chatelet group of B.
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 InvM paper but not proved there. Also states results of the two authors.
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 only over an extension field.
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 (when stated correctly). This provides a description of one of the terms in the Cassels-Tate exact sequence.
1972c Abelian varieties defined over their fields of moduli, I, Bull. London Math. Soc. 4 (1972), 370-372; Correction 6 (1974), 145-146.
1981d Abelian Varieties with Complex Multiplication (for Pedestrians) Handwritten notes (19.09.81), widely distributed.
The notes give a simplified proof of Deligne's extension of the Main Theorem of Complex Multiplication to all automorphisms of the complex numbers. (Shortly afterwards, Deligne further simplified the proof in a letter to Tate.)
1981dT TeXed the article, updated the references, corrected a few misprints, and added a table of contents, some footnotes, and an addendum (07.06.1998). arXiv:math/9806172
1982a Hodge cycles on abelian varieties (notes of a seminar of P. Deligne), in Hodge Cycles, Motives, and Shimura Varieties, LNM 900, Springer, 9-100.
This is the seminar in which Deligne proved his famous theorem that all Hodge cycles on abelian varieties are absolutely Hodge.
1982aT TeXed, somewhat revised and updated, with endnotes added (04.07.03).
1982c Langlands's construction of the Taniyama group (with K-y. Shih), in Hodge Cycles, Motives, and Shimura Varieties, LNM 900, Springer, 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.
1986b Abelian varieties, in Arithmetic Geometry (Proc. Conference on Arithmetic Geometry, Storrs, August 1984) Springer, 1986, 103-150.
An introductory guide.
1986c Jacobian varieties, in Arithmetic Geometry (Proc. Conference on Arithmetic Geometry, Storrs, August 1984) Springer, 1986, 167-212.
An introductory guide.
1999a Lefschetz Classes on Abelian Varieties Duke Math. J. 96:3, 1999, pp. 639-675.
The Lefschetz classes are those in the Q-algebra generated by divisor classes. The article shows that for abelian varieties, they are exactly the classes fixed by an algebraic group. Deduce that the classes on abelian varieties known to be algebraic (Kunneth components of the diagonal, inverse to the Lefschetz map L etc.), are, in fact, Lefschetz, and that the Lefschetz classes on abelian varieties define a category of correspondences. Deciding whether the algebra of Hodge classes (or the algebra of Tate classes) on a given abelian variety is generated by divisor classes becomes a matter of deciding whether two reductive groups are equal.
1997aP Preprint.
1999b Lefschetz Motives and the Tate Conjecture Compositio Math. 117 (1999), pp. 47-81.
Studies the categories of motives (Lefschetz and neq) generated by abelian varieties over finite fields and CM abelian varieties in characteristic zero and the functors between them. Proves 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.
1997bP Preprint.
2001a The Tate conjecture for certain abelian varieties over finite fields. Acta Arith. 100 (2001), no. 2, pp.135--166; arXiv:math/9911218
Proves the Tate conjecture for a family of abelian varieties whose Ql -algebras of Tate classes are 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.
2001aP Preprint.
2002a Polarizations and Grothendieck's Standard Conjectures, Ann. of Math. 155 (2002), pp. 599--610; arXiv:math/0103175
Proves that Grothendieck's Hodge standard conjecture holds for abelian varieties in all characteristics if the Hodge conjecture holds for complex abelian varieties of CM-type. For abelian varieties with no exotic algebraic classes, the article proves the Hodge standard conjecture unconditionally.
2002aS As orginally submitted (14.08.01, 16 pages).
2002aP After being shortened, following the suggestions of the referee (19.09.01, 11 pages).
2004c Periods of abelian varieties, Compositio Math. 140 (2004), 1149--1175; arXiv:math/0209076
Proves various characterizations of the period torsor of abelian varieties, and corrects some errors in the literature. Beyond its intrinsic interest, the period torsor controls the arithmetic of holomorphic automorphic forms.
2004cP Preprint.
2007c The fundamental theorem of complex multiplication, arXiv:0705.3446.
Presents 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).
2007d Rational Tate classes on abelian varieties, arXive:0707.3167
Investigates whether there exists a theory of "rational Tate classes" for abelian varieties having the properties that the algebraic classes would have if the Tate conjecture were known (July 20, 2007).
2007e The Tate conjecture over finite fields (AIM talk),
My notes for a talk at The Tate Conjecture workshop at AIM, July 2007, somewhat revised and expanded (19.09.07 19 pages).
1970d The Brauer group of a rational surface, Invent. Math. 11 (1970), 304-307.
Proves that the Brauer group of a rational surface over a finite field is finite and has the order predicted by the conjecture of Artin and Tate.
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 that the Brauer group of the surface is finite and has the order predicted by the Artin-Tate conjecture.
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 (generalization of a theorem of Artin and Tate). Deduce that, for many Jacobians over global fields of nonzero 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).
1986a Values of zeta functions of varieties over finite fields, Amer. J. Math. 108, (1986), 297-360.
States a conjecture (generalization of the Artin-Tate conjecture; stronger form of a conjecture of Lichtenbaum) relating special values of zeta functions of smooth projective varieties over finite fields to motivic cohomology, and proves a Z^ version of it (including the p-part).
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, Milne) 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.
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.
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 extends it to all smooth projective varieties for sheaves killed by p
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).
1976c Duality in the flat cohomology of curves (with M. Artin), Invent. Math. 35 (1976), 111-129 (Serre volume).
Proves a duality theorem for the flat cohomology "groups" of a finite group scheme over a smooth complete curve.
2005c The de Rham-Witt and Zp-cohomologies of an algebraic variety (with Niranjan Ramachandran), Advances in Mathematics (Artin volume), 198 (2005), 36--42.
Proves that the Zp-cohomology groups of a complete smooth algebraic variety, originally defined in Milne 1976a 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 the variety.
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.
1994aP Preprint, uncorrupted by the AMS copy editors.
2003b Gerbes and abelian motives arXiv.math/0301304
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. I prove unconditionally that there exists a morphism of gerbes with these properties, and I classify them (critical re-examination of work of Langlands and Rapoport).
2004a Integral Motives and Special Values of Zeta Functions (with N. Ramachandran), J. Amer. Math. Soc. 17 (2004), 499-555; arXiv:math/0204065
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.
2004aP Final preprint, 58 pages (28.03.2004, submitted 22.05.2002.).
2006a Motives over Fp, arXiv:math/0607569.
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).
Motivic complexes over finite fields and the ring of correspondences at the generic point (with Niranjan Ramachandran) arXiv.math/0607483.
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).
1970a The homological dimension of commutative group schemes over a perfect field. J. of Algebra 16 (1970), 436--441.
Proves a Hochschild-Serre type spectral sequence for Exts of commutative group schemes over a perfect field, and completes the computation of the Ext groups of abelian varieties over finite fields. Gives a short proof that Ext2(A,B)=0 for abelian varieties over an algebraically closed field.
1980 Etale Cohomology, Princeton Mathematical Series 33, Princeton UP, 323+xiii pages (see Books).
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.)
1982b Tannakian categories (with P. Deligne), in Hodge Cycles, Motives, and Shimura Varieties, LNM 900, Springer, 101-228.
An introduction to the theory of Tannakian categories, with some improvements to the theory.
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 an algebraically closed field of prime characteristic p, proves that 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.
1984 Kazhdan's Theorem on Arithmetic Varieties. Handwritten notes, 42 pages, 28.03.84.
Let V be a quotient of a bounded symmetric domain by an arithmetic group. The Baily-Borel theorem says that V is an algebraic variety, and Kazhdan's theorem says that when you apply an automorphism of the complex numbers to the coefficients of the polynomials defining V, the resulting variety is a quotient of the same form. This article simplifies Kazhdan's proof. In particular, it avoids recourse to the classification theorems.
1984T TeXed the article and added a few footnotes (22.06.01/12.07.01). arXiv:math/0106197
1987 The (failure of the) Hasse principle for centres of semisimple groups, manuscript.
Proves that the Hasse principle holds for the centres of some semisimple groups over number fields, and fails for others.
1987T Translated the article into TeX (11.12.01).
2005bv2 Quotients of Tannakian Categories arXiv:math/0508479 (11 pages). 24.08.05; 09.07.07 (minor expository improvements).
Classifies the "quotients" of a tannakian category in which the objects of a some tannakian subcategory become trivial; examines the properties of such quotient categories.
2007a Semisimple Lie algebras, algebraic groups, and tensor categories, (09.05.07, 37 pages)
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.
2007b Semisimple algebraic groups in characteristic zero, arXiv:0705.1348
Short version of Milne 2007a (09.05.07, 12 pages).
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.
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.
1979bT TeXed and translated the article into English (08.12.02).
1979c Shimura varieties: conjugates and the action of complex multiplication, 154pp, October 1979 (with K-y. Shih).
This manuscript was broken into three, and published as: 1981a (Annals); 1982c (LNM 900), 1982d (LNM 900).
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
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 some expert opinion at the time, this was a serious exercise
1982 Hodge Cycles, Motives, and Shimura Varieties (with Pierre Deligne, Arthur Ogus, Kuang-yen Shih), Lecture Notes in Math. 900, Springer-Verlag, 414 pages (see Books).
1982d Conjugates of Shimura varieties (with K-y. Shih),
in Hodge Cycles, Motives, and Shimura Varieties, LNM 900, Springer, 280-356.
Proves Langlands's conjecture, describing the action of an automorphism of C on the collection of Shimura varieties and their special points, for all Shimura varieties except those defined by groups of type E6, E7, or mixed type D.
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.
1988a Automorphic vector bundles on connected Shimura varieties, Inventiones math., 92 (1988), 91-128.
Extends the proof of Langlands's conjugation conjecture (Milne 1983) to the standard principal bundle, and hence to automorphic vector bundles (over connected Shimura varieties).
1990 Automorphic Forms, Shimura Varieties, and L-functions, Proceedings of a Conference held at the University of Michigan, Ann Arbor, July 6--16, 1988. (Editor with L. Clozel). See Books.
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).
1990aT Translated the old TeX files into LaTeX 2e; fixed some minor misprints; added some footnotes (22.06.01).
1990b Letter to Deligne 28.03.90
Concerning the signs in the theory of Shimura varieties.
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 (those in the category generated by abelian varieties) 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.
1994bP Preprint, uncorrupted by the AMS copy editors.
1995a Talks at IAS on Shimura varieties
The notes for 4 lectures I gave at IAS in early 1995 giving an introduction to Shimura varieties, and discussing the problems that arise in the attempt to understand their zeta functions.
1995b Shimura Variety. Encyclopedia of Mathematics, Supplement Vol I, Kluwer Acad, Publ., 1997, pp448-449.
One-page definition.
1995c On the conjecture of Langlands and Rapoport arXiv:0707.3173
This manuscript, which dates from 1995, examines what is needed to prove the conjecture of Langlands and Rapoport concerning the structure of the points on a Shimura variety modulo a prime of good reduction.
(Sept 1995, distributed to a few mathematicians; July 2007, added forenote and placed on the web.)
1999c Descent for Shimura Varieties, Michigan Math. J., 46 (1999), pp. 203--208; arXiv:alg-geom/9712031
This note checks that the descent maps provided by Langlands's conjugacy conjecture do satisfy the continuity condition necessary for them to be effective (as claimed by Langlands in his Corvallis article). Hence the conjecture does imply the existence of canonical models.
1997cP Preprint.
2000 Towards a proof of the conjecture of Langlands and Rapoport. 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, 2000.
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.
2002b MR review of: Harris and Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Math. Studies, Princeton UP, 2001.
With endnotes not part of review sent to MR.
2003a Canonical models of Shimura curves, Preliminary draft (04.04.03), 40 pages.
As an introduction to Shimura varieties, and, in particular, to Deligne's Bourbaki and Corvallis talks, I explain the main ideas and results of the general theory of Shimura varieties in the context of Shimura curves.
2004b MR review of Shimura, Collected Papers
With footnotes not part of the review sent to MR.
2005a Introduction to Shimura varieties, In Harmonic Analysis, the Trace Formula and Shimura Varieties (James Arthur, Robert Kottwitz, Editors) AMS, 2005, (Lectures at the Summer School held at the Fields Institute, June 2 -- June 27, 2003).
The article is an introduction the arithmetic theory of automorphic functions and holomorphic automorphic forms.
2005aX Expanded version, containing footnotes and endnotes not in the published version; 149 pages, 23.10.04.