Mathematical Articles (with abstracts) - J.S. Milne, Top
1960s  1970s  1980s  1990s  2000s  2010s
The pdf files on my site are sometimes of a higher quality and more recent than those on the arXive.
Added mathjax to the page October 2015.

### 1960sTop

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 restriction later removed). In particular, it proved for the first time that the Tate-Shafarevich groups of some abelian varieties over global fields are 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 $\mathrm{Ext}^i(A,B)$ is finite, and expresses its order in terms of the zeta functions of $A$ and $B$ and the discriminant of the pairing $\mathrm{Hom}(A,B)\times\mathrm{Hom}(B,A)\to \mathbb{Z}$ sending two homomorphisms to the trace of their composite.

1968b The Tate- S(h)afarevic(h) 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 (an abelian variety over a global function field is constant if it comes from an abelian variety over the field of constants).

### 1970sTop

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 $\mathrm{Ext}^2(A,B)=0$ for abelian varieties $A$ and $B$ over an algebraically closed field.

1970b Elements of order p in the Tate S(h)afarevic(h) group, Bull. London Math. Soc. 2 (1970), 293-296.
Proves that Tate-Shafarevich group of an abelian variety over a global field of characteristic $p$ has only finitely many elements killed by a fixed integer $m$, even when $p$ divides $m$. In contrast to the number field case, this fails when finitely many primes are omitted from the definition of the Tate-Shafarevich group. The group of elements killed by $p$ may be infinite when the field of constants is taken to be 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.
Proves that, for an abelian variety $A$ and its dual $B$ over a local field $K$ of prime characteristic, the group $A(K)$ is canonically dual to the Weil-Chatelet group $H^1(K,B)$ of $B$. When $K$ has characterisic zero this was proved by Tate.

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.

1971a Abelian varieties over finite fields (with W. C. Waterhouse), Proc. Symp. Pure Math. 20 (1971), 53-64.
An introduction to the Honda-Tate-Weil theory of abelian varieties over finite fields. It includes proofs of two theorems of Tate announced in his 1966 Inventiones Math. paper but not proved there. It 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 its Weil restriction. Deduces that the conjecture of Birch and Swinnerton-Dyer holds for the former if and only if it holds for the latter. Therefore it suffices to prove the conjecture of Birch and Swinnerton-Dyer for abelian varieties over $\mathbb{Q}$ and $\mathbb{F}_p(X)$. The article identifies the zeta function of an abelian variety that acquires complex multiplication over an extension of the base number field as a Hecke-Weil $L$-series.

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). Provides an explict 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.

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 statement for the cohomology groups of finite flat group schemes over complete discrete valuation rings.

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.

1976a Duality in the flat cohomology of a surface, Ann. Sci. Ecole Norm. Sup. 9 (1976), 171-202.
Introduces the sheaves $\nu_n(r)$ (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 $p$. This is the long sought $p$-counterpart of the étale duality theorem for the sheaves $\mu_{l^n}^{\otimes r}$.

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 group schemes of a finite group scheme over a smooth complete curve.

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. (Originally, this was intended to be a report on work of Langlands, but that turned out to be incorrect, and my approach is quite different.)

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).

### 1980sTop

1980 Etale Cohomology, Princeton Mathematical Series 33, Princeton UP, 323+xiii pages (see Books).
As Grothendieck once scornfully put it, this was written for the sole purpose of allowing mathematicians to learn étale cohomology without reading the orginal sources (SGA 4, 1615 pages; SGA 5, 496 pages).

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 of the variety. In order to compute the factors at infinity of the zeta function of the variety, it is necessary to know this involution. Langlands conjectured a description of the involution, and we prove his conjecture for all Shimura varieties of abelian type (i.e., for all except those defined by groups of type $E_6$, $E_7$, or certain mixed types $D$)

1981b Automorphism groups of Shimura varieties and reciprocity laws (with K-y. Shih), Amer. J. Math. 103 (1981), 1159-1175.
We study the automorphisms of Shimura varieties, and 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 involving results not known to the afore-mentioned experts.

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.)

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.)

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).
Collects six original articles, four of which I describe below (the remaining two were by Deligne and by Ogus).

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 famous seminar of Deligne in which he proved that all Hodge cycles on abelian varieties are absolutely Hodge.
1982aT TeXed, somewhat revised and updated, with endnotes added (04.07.03).

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.

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.

1982d Conjugates of Shimura varieties (with K-y. Shih),
in Hodge Cycles, Motives, and Shimura Varieties, LNM 900, Springer, 280-356.
Langlands's conjugation conjecture describes the action of an automorphism of $\mathbb{C}$ on the collection of Shimura varieties and their special points. In particular, it identifies the conjugate of a Shimura variety under an automorphism of $\mathbb{C}$ as another specific Shimura variety. We prove the conjecture for all Shimura varieties of abelian type (i.e., for all except those defined by groups of type $E_6$, $E_7$, or certain mixed types $D$).

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 $k$ of prime characteristic $p$, proves that the canonical map $\mathrm{CH}_0(X)(p)\to \mathrm{Alb}(X)(p)$ is an isomorphism. Here $\mathrm{CH}_0(X)$ is the group of zero cycles modulo rational equivalence, and $\mathrm{CH}_0(X)(p)$ is its $p$-primary component. When $k$ has characteristic zero, this is a theorem of Rojtman.

1982f Comparison of the Brauer group with the Tate-S(h)afarevic(h) 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 who assumed that there was a section). Using (1975a), one can 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).

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 (see 1982d), and hence the existence of canonical models (Shimura's conjecture), for all Shimura varieties.

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

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 $\hat{\mathbb{Z}}$-version of it (including the p-part). See also (2015b) below for an update.

1986b Abelian varieties, in Arithmetic Geometry (Proc. Conference on Arithmetic Geometry, Storrs, August 1984) Springer, 1986, 103-150.
An introductory guide. See AVs for a corrected version.

1986c Jacobian varieties, in Arithmetic Geometry (Proc. Conference on Arithmetic Geometry, Storrs, August 1984) Springer, 1986, 167-212.
An introductory guide. See JVs for a corrected version.

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).

1988a Automorphic vector bundles on connected Shimura varieties, Inventiones math., 92 (1988), 91-128.
Extends the proof of Langlands's conjugation conjecture (see 1983a) to the standard principal bundle, and hence to automorphic vector bundles (over connected Shimura varieties).

1988b Motivic cohomology and values of zeta functions, Compos. math. 68 (1988), 59-102.
Introduces a new p-Kummer axiom for the motivic complex of Beilinson and Lichtenbaum. 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. See (2015b) below.

### 1990sTop

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
Corrects a sign error in the theory of Shimura varieties.

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., Montreal (Eds. R. Langlands and D. Ramakrishnan), 1992, pp151--253.
Introduces the notion of a canonical integral model of a Shimura variety. Specifically, Langlands (1976) suggested that Shimura varieties should have smooth models at the primes where $K_p$ is hyperspecial, but said that "I do not know how they should be characterized". The article characterizes them. It also examines the conjecture of Langlands and Rapoport, and proves that it implies the conjectural formula of Langlands and Kottwitz.
1992aP Preprint (with two pages of notes added 14.06.01).

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.

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 $\mathbb{C}$, 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.)

1999a Lefschetz Classes on Abelian Varieties Duke Math. J. 96:3, 1999, pp. 639-675.
The Lefschetz classes are those in the $\mathbb{Q}$-algebra generated by divisor classes. The article shows that for abelian varieties, they are exactly the classes fixed by an algebraic group (necessarily not connected). Thus, 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. It is shown 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. The Lefschetz classes on abelian varieties can be used to define a category of motives.
The article was submitted in January 1997, and the results had been announced (to K. Murty and others) in 1995.
1999a 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.
The article was submitted in January 1997.
1997bP Preprint.

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 stated by Langlands in his Corvallis article). Hence the conjecture does imply the existence of canonical models.
1997cP Preprint.

### 2000sTop

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. Introduces and explains the importance of the "rationality conjecture".

2001a The Tate conjecture for certain abelian varieties over finite fields. Acta Arith. 100 (2001), no. 2, pp.135--166; arXiv:math/9911218
Tate's theorem (1966) proves the Tate conjecture for any abelian variety over a finite field whose algebra of Tate classes is generated by divisor classes. The article proves the Tate conjecture for a class of abelian varieties that fail this condition. As far as I know, these are the first examples of such varieties.
2001aP Preprint.

2002a Polarizations and Grothendieck's Standard Conjectures, Ann. of Math. 155 (2002), pp. 599--610; arXiv:math/0103175
Proves that the Hodge conjecture for complex abelian varieties of CM-type implies Grothendieck's Standard Conjecture of Hodge type for abelian varieties in nonzero characteristic. For abelian varieties with no exotic algebraic classes, the article proves the Standard Conjecture of Hodge type 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).

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.

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 $\mathbb{Z}$-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 $\mathrm{Ext}$ groups.
2004aP Final preprint, 58 pages (28.03.2004, submitted 22.05.2002.).

2004b MR review of Shimura, Collected Papers
With footnotes not part of the review sent to MR.

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.

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 to the theory of Shimura varieties or, in other words, 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.

2005c The de Rham-Witt and $\mathbb{Z}_p$-cohomologies of an algebraic variety (with Niranjan Ramachandran), Advances in Mathematics (Artin volume), 198 (2005), 36--42.
A major problem in the 1960s was to find a good p-counterpart for the étale cohomology groups $H^i(X,\mathbb{Z}_{\ell}(r)$. I found an ad hoc solution to this problem by defining some étale sheaves $\nu_n(r)$ in terms of the de Rham-Witt complex of Bloch, Deligne, and Illusie. In this article we show that the groups I defined arise naturally as the absolute cohomology groups given by the category $D_c^b(R)$ of bounded complexes of graded modules over the Raynaud ring (this category had been introduced and studied by Ekedahl, Illusie, and Raynaud). This result helps confirm $D_c^b(R)$ as the correct $p$-analogue of the category of bounded constructible $\mathbb{Z}_{\ell}$-complexes (Deligne-Ekedahl), and replaced my ad hoc definition of $H^i(X,\mathbb{Z}_p(r))$ with a natural definition.

2006a Motives over $\mathbb{F}_p$, arXiv:math/0607569.
In April, 2006, Kontsevich asked me whether the category of motives over $\mathbb{F}_p$ ($p$ 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 $\mathbb{Q}$ having certain properties (see Question 6.5).

2007a Semisimple Lie algebras, algebraic groups, and tensor categories, (09.05.07, 37 pages)
It is shown that, in characteristic zero, the classification theorems for semisimple algebraic groups and their representations can be derived quite simply and naturally from the corresponding theorems for Lie algebras by using a little of the theory of tensor categories. This has now been incorporated into my notes LAG

2007b Semisimple algebraic groups in characteristic zero, arXiv:0705.1348
Short version of Milne 2007a (09.05.07, 12 pages).

2007c The fundamental theorem of complex multiplication, arXiv:0705.3446.
Presents a proof, 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).

2007 Quotients of Tannakian Categories Theory Appl. Categ. 18 (2007), No. 21, 654--664. arXiv:math/0508479 (11 pages).
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.

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; the intent of the talk was to review what is known and suggest directions for research.
v1, 19.09.07, 19 pages; v2, 10.10.07 Revised and exanded (24 pages); v2.1, 07.11.07 Minor fixes; v2.2, 07.05.08 Rewrote sections 1 and 5 (27 pages).

2008b 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.

2009a Motivic complexes over finite fields and the ring of correspondences at the generic point (with Niranjan Ramachandran) Pure & App. Math. Quarterly (Tate issue), 5 (2009), 1219-1252; 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 $\mathbb{Q}$-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).

2009b Rational Tate classes, Moscow Math. J. (Deligne Issue) 9 (2009), 111--141; arXive:0707.3167
Investigates whether there exists a theory of "rational Tate classes" for abelian varieties over finite fields having the properties that the algebraic classes would have if the Tate conjecture were known (in particular, extending Deligne's theory of absolute Hodge classes on abelian varieties to mixed characteristic). It is proved that there exists at most one "good" such theory, and that Grothendieck's standard conjectures automatically hold for it.

2009c Motives---Grothendieck's dream (Chinese).
Mathematical Advances in Translation, Vol.28, No.3, 193-206, 2009 (Institute of Mathematics, Chinese Academy of Sciences) (translation by Xu Kejian, Qingdao University).

### 2010sTop

2010 Nonhomeomorphic conjugates of connected Shimura varieties (with Junecue Suh)
Amer. J. Math. 132 (2010), no. 3, 731--750; arXiv:0804.1953
We show that conjugation by an automorphism of the complex numbers may change the topological fundamental group of a locally symmetric variety over $\mathbb{C}$. As a consequence, we obtain a large class of algebraic varieties defined over number fields with the property that different embeddings of the number field into $\mathbb{C}$ give complex varieties with nonisomorphic topological fundamental groups.

2012 What is a Shimura variety?
Notices Amer. Math. Soc. 59 (2012), no. 11, 1560--1561.
The theory of Shimura varieties grew out of the applications of modular functions and modular forms to number theory. Roughly speaking, Shimura varieties are the varieties on which modular functions live.

2013a Motives---Grothendieck's dream.
Open problems and surveys of contemporary mathematics. Edited by Lizhen Ji, Yat-Sun Poon and Shing-Tung Yau. Surveys of Modern Mathematics, 6. International Press, Somerville, MA; Higher Education Press, Beijing, 2013.
English version of 2009c

2013b Shimura varieties and moduli
Handbook of Moduli (Gavril Farkas, Ian Morrison, Editors), International Press of Boston, 2013, Vol II, 462--544.
Connected Shimura varieties are the quotients of hermitian symmetric domains by discrete groups defined by congruence conditions. We examine their relation with moduli varieties.

2013c A Proof of the Barsotti-Chevalley Theorem on Algebraic Groups,
A fundamental theorem of Barsotti and Chevalley states that every smooth algebraic group over a perfect field is an extension of an abelian variety by a smooth affine algebraic group. In 1956 Rosenlicht gave a short proof of the theorem. We explain his proof in the language of modern algebraic geometry.
First posted November 24, 2013; last revised October 18, 2015.

2014 The work of John Tate. Published in: The Abel Prize 2008-2012. Edited by Helge Holden and Ragni Piene. Springer, Heidelberg, 2014, pp.259-347.
This is my article on Tate's work for the second volume in the book series on the Abel Prize winners.

2013d Motivic complexes and special values of zeta functions (with Niranjan Ramachandran). arXiv:1311.3166 November 13, 2013.
Beginning with the conjecture of Artin and Tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function of an algebraic variety over a finite field in terms of other invariants of the variety. In this article, we present the ultimate such conjecture, and provide evidence for it. In particular, we enhance Voevodsky's $\mathbb{Z}[1/p]$-category of étale motivic complexes with a $p$-integral structure, and show that, for this category, our conjecture follows from the Tate and Beilinson conjectures. However, unlike other conjectures of this nature, it doesn't require the Tate conjecture to be true --- a "good" theory of rational Tate classes would suffice. As the conjecture is stated in terms of motivic complexes, it (potentially) applies also to algebraic stacks, log varieties, simplicial varieties, etc.
Note: (2013d) is a sequel to (2015a), which is the technical heart of the work, and has appeared in JIMJ. I wrote both manuscripts in 2013 based on joint research with NR and posted them on the arXiv. The manuscript (2013b) is basically complete and (I believe) correct, but I think too rough for formal publication. NR promises (2013) to polish it for publication. Questions concerning it should be directed to NR as I have completed my part of the work on this project and have moved on to other things.

2015a The p-cohomology of algebraic varieties and special values of zeta functions (with Niranjan Ramachandran). J. Inst. Math. Jussieu 14 (2015), no. 4, 801--835 ( arXiv:1310.4469 16 October 2013).
The $p$-cohomology of an algebraic variety in characteristic $p$ lies naturally in the category $D_c^b(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc. and the special values of their zeta functions. These results provide compelling evidence that $D_c^b(R)$ is the correct target for $p$-cohomology in characteristic $p$.

2015b Addendum to: Milne, Values of zeta functions of varieties over finite fields, Amer. J. Math. 108, (1986), 297-360. Amer. J. Math. 137 (2015), 1--10 (arXiv:0804.1953).
The original article expressed the special values of the zeta function of a variety over a finite field in terms of the $\hat{\mathbb{Z}}$-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 $\mathbb{Z}$-cohomology groups.

2015c The Riemann hypothesis over finite fields: from Weil to the present day. In: The Legacy of Bernhard Riemann after One Hundred and Fifty Years (Lizhen Ji, Frans Oort, Shing-Tung Yau Editors), ALM 35, 2015, pp.487-565.
The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves over finite fields led him to state his famous "Weil conjectures", which drove much of the progress in algebraic and arithmetic geometry in the following decades. The article describes Weil's work and some of the ensuing progress.