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.### 1960s Top

### 1970s Top

### 1980s Top

### 1990s Top

### 2000s Top

### 2010s Top

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.

**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).

**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).

**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.)**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

**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.

**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.

**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).

The origin of this article is a "popular" lecture, *What is a Motive?*, that I gave at the
University of Michigan on February 3, 2009.

**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.