Basic Theory of Abelian Varieties 1. Definitions

113 Pages·2000·1.09 MB·English

ABELIAN VARIETIES
J.S. MILNE
Abstract. ThesearethenotesforMath731,taughtattheUniversityofMichigan
in Fall1991,somewhat revised from those handed out during the course. They are
availableat www.math.lsa.umich.edu/∼jmilne/.
Please send comments and corrections to me at [email protected].
v1.1(July27,1998). Firstversiononthe web. These notes areinmoreprimitive
formthanmyothernotes —the reader shouldviewthem asanunfinished painting
in which some areas are only sketched in.
Contents
Introduction 1
Part I: Basic Theory of Abelian Varieties
1. Definitions; Basic Properties. 7
2. Abelian Varieties over the Complex Numbers. 9
3. Rational Maps Into Abelian Varieties 15
4. The Theorem of the Cube. 19
5. Abelian Varieties are Projective 25
6. Isogenies 29
7. The Dual Abelian Variety. 32
8. The Dual Exact Sequence. 38
9. Endomorphisms 40
10. Polarizations and Invertible Sheaves 50
11. The Etale Cohomology of an Abelian Variety 51
12. Weil Pairings 52
13. The Rosati Involution 53
14. The Zeta Function of an Abelian Variety 54
15. Families of Abelian Varieties 57
16. Abelian Varieties over Finite Fields 60
17. Jacobian Varieties 64
18. Abel and Jacobi 67
Part II: Finiteness Theorems
19. Introduction 70
20. N´eron models; Semistable Reduction 76
21. The Tate Conjecture; Semisimplicity. 77
(cid:2)c1998 J.S. Milne. You may make one copy of these notes for your own personal use.
1
0 J.S. MILNE
22. Geometric Finiteness Theorems 82
23. Finiteness I implies Finiteness II. 87
24. Finiteness II implies the Shafarevich Conjecture. 92
25. Shafarevich’s Conjecture implies Mordell’s Conjecture. 94
26. The Faltings Height. 98
27. The Modular Height. 102
28. The Completion of the Proof of Finiteness I. 106
Appendix: Review of Faltings 1983 (MR 85g:11026) 107
Index 110
ABELIAN VARIETIES 1
Introduction
The easiest way to understand abelian varieties is as higher-dimensional analogues
of elliptic curves. Thus we first look at the various definitions of an elliptic curve.
Fix a ground field k which, for simplicity, we take to be algebraically closed.
0.1. An elliptic curve is the projective curve given by an equation of the form
Y2Z = X3 +aXZ +bZ3, ∆ =df 4a3 +27b2 (cid:3)= 0. (*)
(char (cid:3)= 2,3).
0.2. An ellipticcurve is a nonsingular projectivecurve of genus one together with
a distinguished point.
0.3. An elliptic curve is a nonsingular projective curve together with a group
structure defined by regular maps.
0.4. (k = C) An elliptic curve is complex manifold of the form C/Λ where Λ is a
lattice in C.
We briefly sketch the equivalence of these definitions (see also my notes on Elliptic
Curves, especially §§5,10).
(0.1) =⇒(0.2). The condition ∆ (cid:3)= 0 implies that the curve is nonsingular; take
the distinguished point to be (0 : 1 : 0).
(0.2) =⇒(0.1). Let ∞ be the distinguished point on the curve E of genus 1. The
Riemann-Roch theorem says that
dim L(D) = deg(D)+1−g = deg(D)
where
L(D) = {f ∈ k(E) | div(f)+D ≥ 0}.
On taking D = 2∞ and D = 3∞ successively,wefind that thereis a rational function
x on E with a pole of exact order 2 at ∞ and no other poles, and a rational function
y on E with a pole of exact order 3 at ∞ and no other poles. The map
P (cid:13)→ (x(P) : y(P) : 1), ∞ (cid:13)→ (0 : 1 : 0)
defines an embedding
E (cid:18)→ P2.
On applying the Riemann-Roch theorem to 6∞, we find that there is relation (*)
between x and y, and therefore the image is a curve defined by an equation (*).
(0.1,2) =⇒(0.3): Let Div0(E) be the group of divisors of degree zero on E, and let
Pic0(E) be its quotient by the group of principal divisors; thus Pic0(E) is the group
of divisor classes of degree zero on E. The Riemann-Roch theorem shows that the
map
P (cid:13)→ [P]−[∞]: E(k) → Pic0(E)
is a bijection, from which E(k) acquires a canonical group structure. It agrees with
the structure defined by chords and tangents, and hence is defined by polynomials,
i.e., it is defined by regular maps.
2 J.S. MILNE
(0.3) =⇒(0.2): We have to show that the existence of the group structure implies
that the genus is 1. Our first argument applies only in the case k = C. The Lefschetz
trace formula states that, for a compact oriented manifold X and a continuous map
α : X → X with only finitely many fixed points, each of multiplicity 1,
number of fixed points = Tr(α|H0(X,Z))−Tr(α|H1(X,Z))+···
If X has a group structure, then, for any nonzero point a ∈ X, the translation map
t : x (cid:13)→ x+a has no fixed points, and so
a
Tr(t ) =df Σ(−1)iTr(t |Hi(X,Q)) = 0.
a a
The map a (cid:13)→ Tr(t ): X → Z is continuous, and so Tr(t ) = 0 also for a = 0. But t
a a 0
is the identity map, and
(cid:1)
Tr(id) = (−1)i dimHi(X,Q) = χ(X) (Euler-Poincar´e characteristic).
Since the Euler-Poincar´e characteristic of a complete nonsingular curve of genus g is
2−2g, we see that if X has a group structure then g = 1. [The same argument works
over any field if one replaces singular cohomology with ´etale cohomology.]
We now give an argument that works over any field. If V is an algebraic variety
with a group structure, then the sheaf of differentials is free. For a curve, this means
that the canonical divisor class has degree zero. But this class has degree 2g−2, and
so again we see that g = 1.
(0.4) =⇒(0.2). The Weierstrass ℘-function and its derivative define an embedding
z (cid:13)→ (℘(z) : ℘(cid:1)(z) : 1) : C/Λ (cid:18)→ P2,
whose image is a nonsingular projective curve of genus 1 (in fact, with equation of
the form (*)).
(0.2) =⇒(0.4). This follows from topology.
Abelian varieties. Definition(0.1) simplydoesn’t generalize— thereis no simple
description of the equations defining an abelian variety of dimension1 g > 1. In
general, it is not possible to write down explicit equations for an abelian variety of
dimension > 1, and if one could, they would be too complicated to be of use.
I don’t know whether (0.2) generalizes. Abelian surfaces are the only minimal
surfaces with the Betti numbers
1,4,6,4,1
1Thecaseg =2issomethingofanexceptiontothisstatement. Everyabelianvarietyofdimension
2istheJacobianvariety(seebelow)ofacurveofgenus2,andeverycurveofgenus2hasanequation
of the form
Y2Z4 =f0X6+f1X5Z +···+f6Z6.
Flynn (Math. Proc. Camb. Phil. Soc. 107, 425–441) has found the equations of the Jacobian
variety of such a curve in characteristic (cid:3)= 2,3,5 — they form a set 72 homogeneous equations of
degree 2 in 16 variables (they take 6 pages to write out). See: Cassels, J.W.S., and Flynn, E.V.,
Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2, Cambridge, 1996.
ABELIAN VARIETIES 3
and canonical class linearly equivalent to zero. In general an abelian variety of di-
mension g has Betti numbers
(cid:2) (cid:3) (cid:2) (cid:3)
2g 2g
1, ,... , ,... ,1.
1 r
Definition(0.3)does generalize: wecandefineanabelianvarietytobeanonsingular
connected projective2 variety with a group structure defined by regular maps.
Definition (0.4) does generalize, but with a caution. If A is an abelian variety over
C, then
A(C) ≈ Cg/Λ
for some lattice Λ in Cg (isomorphism simultaneously of complex manifolds and of
groups). However, when g > 1, for not all lattices Λ does Cg/Λ arise from an abelian
variety. Infact,ingeneralthetranscendencedegreeoverCofthefieldofmeromorphic
functions Cg/Λ is < g, with equalityholding ifand only ifCg/Λ is an algebraic (hence
abelian) variety. There is a very pleasant criterion on Λ for when Cg/Λ is algebraic
(§2).
Abelian varieties and elliptic curves. As we noted, if E is an elliptic curve over
an algebraically closed field k, then there is a canonical isomorphism
P (cid:13)→ [P]−[0]: E(k) → Pic0(E).
This statement has two generalizations.
(A).Let C be a curveand choose a point Q ∈ C(k); then there isan abelian variety
J, called the Jacobian variety of C, canonically attached to C, and a regular map
ϕ : C → J such that ϕ(Q) = 0 and
Σ n P (cid:13)→ Σ n ϕ(P ): Div0(C) → J(k)
i i i i
induces an isomorphism Pic0(C) → J(k). The dimension of J is the genus of C.
(B). Let A be an abelian variety. Then there is a “dual abelian variety” A∨ such
that Pic0(A) = A∨(k) and Pic0(A∨) = A(k) (we shall define Pic0 in this context
later). In the case of an elliptic curve, E∨ = E. In general, A and A∨ are isogenous,
but they are not equal (and usually not even isomorphic).
Appropriately interpreted, most of the statements in Silverman’s books on elliptic
curves hold for abelian varieties, but because we don’t have equations, the proofs are
more abstract. Infact, every(reasonable) statementabout ellipticcurvesshould have
a generalization that applies to all abelian varieties. However, for some, for example,
theTaniyama conjecture,the correctgeneralizationis difficultto state3. To pass from
a statement about elliptic curves to one about abelian varieties, replace 1 by g (the
∨
dimension of A), and half the copies of E by A and half by A . I give some examples.
2For historical reasons, we define them to be complete varieties rather than projective varieties,
but they turn out to be projective anyway.
3Blasius has pointed out that, by lookingat infinity types, one can see that the obvious general-
ization of the Taniyama conjecture, that every abelian variety over Q is a quotient of an Albanese
variety of a Shimura variety, can’t be true.
4 J.S. MILNE
Let E be an elliptic curve over an algebraically closed field k. For any integer n
not divisible by the characteristic the set of n-torsion points on E
E(k) ≈ (Z/nZ)2,
n
and there is a canonical nondegenerate pairing
E(k) ×E(k) → µ (k) (Weil pairing).
n n n
Let A be an abelian variety of dimension g over an algebraically closed field k. For
any integer n not divisible by the characteristic,
A(k) ≈ (Z/nZ)2g,
n
and there is a canonical nondegenerate pairing
A(k) ×A∨(k) → µ (k) (Weil pairing).
n n n
Let E be an elliptic curve over a number field k. Then E(k) is finitely generated
(Mordell-Weil theorem), and there is a canonical height pairing
E(k)×E(k) → Z
which becomes nondegenerate when tensored with Q. Let A be an abelian variety
over a number field k. Then A(k) is finitely generated (Mordell-Weil theorem), and
there is a canonical height pairing
A(k)×A∨(k) → Z
which becomes nondegenerate when tensored with Q.
For an elliptic curve E over a number field k, the conjecture of Birch and
Swinnerton-Dyer states that
[TS(E)][Disc]
L(E,s) ∼ ∗ (s−1)r as s → 1,
[E(k) ]2
tors
where ∗ is a minor term, TS(E) is the Tate-Shafarevich group of E, Disc is the
discriminant of the height pairing, and r is the rank of E(k). For an abelian variety
A, the Tate generalized the conjecture to the statement
[TS(A)][Disc]
L(A,s) ∼ ∗ (s−1)r as r → 1.
[A(k) ][A∨(k) ]
tors tors
We have L(A,s) = L(A∨,s), and Tate proved that [TS(A)] = [TS(A∨)] (in fact the
two groups, if finite, are canonically dual), and so the formula is invariant under the
interchange of A and A∨.
Remark 0.5. We noted above that the Betti number of an abelian variety of
dimensiong are1,(2g),(2g),...,(2g),...,1. ThereforetheLefschetztraceformulaimplies
1 2 r
that Σ(−1)r+1(2g) = 0. Of course, this can also be proved by using the binomial
r
theorem to expand (1−1)2g.
Exercise 0.6. Assume A(k) and A∨(k) are finitely generated, of rank r say, and
that the height pairing
(cid:19)·,·(cid:20): A(k)×A∨(k) → Z
ABELIAN VARIETIES 5
is nondegenerate when tensored with Q. Let e ,...,e be elements of A(k) that are
1 r
linearly independent over Z, and let f ,...,f be similar elementsof A∨(k); show that
1 r
|det((cid:19)e ,f (cid:20))|
i j
(A(k) : ΣZe )(A∨(k) : ΣZf )
i j
is independent of the choice of the e and f . [This is an exercise in linear algebra.]
i j
The first part of these notes covers the basic theory of abelian varieties over arbi-
trary fields, and the second part is an introduction to Faltings’s proof of Mordell’s
Conjecture.
Some Notations. Our conventions concerning varieties are the same as those in my
notes on Algebraic Geometry, which is the basic reference for these notes. For exam-
ple,anaffinealgebraoverafieldk isafinitelygeneratedk-algebra AsuchthatA⊗ kal
k
al
has no nonzero nilpotents for one (hence every) algebraic closure k of k. With such
a k-algebra, we associate a ring space Specm(A) (topological space endowed with a
sheaf of k-algebras) and an affine variety over k is a ringed space isomorphic to one of
this form. A variety over k is a ringed space (V,O ) admitting a finite open covering
V
V = ∪U such that (U ,O |U ) is an affine variety for each i and which satisfies the
i i V i
separation axiom. If V is a variety over k and K ⊃ k, then V(K) is the set of points
of V with coordinates in K and V or V is the variety over K obtained from V by
K /K
extension of scalars. Occasionally, we also use schemes.
We often describe regular maps by their actions on points. Recall that a regular
map φ: V → W of k-varieties is determined by the map of points V(kal) → W(kal)
that it defines. Moreover, to give a regular map V → W is the same as to give maps
V(R) → W(R), for R running over the affine k-algebras, that are functorial in R (AG
3.29 and AG p135).
Throughout, k is an arbitrary field. The symbol ksep denotes a separable closure
of k, i.e., a field algebraic over k such that every separable polynomial in k[X] has a
root in ksep. For a vector space N over a field k, N∨ denotes the dual vector space
Hom (N,k).
k
We use the following notations:
X ≈ Y X and Y are isomorphic;
∼
X = Y X and Y are canonically isomorphic (or there is a given or unique isomorphism);
df
X = Y X is defined to be Y, or equals Y by definition;
X ⊂ Y X is a subset of Y (not necessarily proper).
References. Lang, S., Abelian Varieties, Interscience, 1959.
Lange, H., and Birkenhake, Ch., Complex Abelian Varieties, Springer, 1992.
Milne, J.S., Abelian varieties, in Arithmetic Geometry (ed. Cornell, G., and Sil-
verman) pp103-150 (cited as AV).
Milne, J.S., Jacobian varieties, ibid., pp167-212 (cited as JV).
Mumford, D., Abelian Varieties, Oxford, 1970.
Murty, V. Kumar, Introduction to Abelian Varieties, CRM, 1993.
Serre: Lectures on the Mordell-Weil theorem, Vieweg, 1989.
6 J.S. MILNE
Silverman, J., The Arithmetic of Elliptic Curves, Springer, 1986.
Silverman, J., Advanced Topics in the Arithmetic of Elliptic Curves, Springer,
1994.
Weil, A., Sur les Courbes Alg´ebriques et les Vari´et´es qui s’en D´eduisent, Hermann,
1948.
Weil, A., Vari´et´es Ab´eliennes et Courbes Alg´ebriques, Hermann, 1948.
Mumford’s book is the only modern account of the subject, but as an introduction
it is rather difficult. It treats only abelian varieties over algebraically closed fields;
in particular, it does not cover the arithmetic of abelian varieties. Weil’s books
contain the original account of abelian varietiesover fieldsother thanC. Serre’s notes
give an excellent treatment of some of the arithmetic of abelian varieties (heights,
Mordell-Weil theorem, work on Mordell’s conjecture before Faltings — the original
title “Autour du th´eor`eme de Mordell-Weil is more accurate than the English title.).
Murty’s notes concentrate on the analytic theory of abelian varieties over C except
for the final 18 pages. The book by Lange and Birkenhake is a very thorough and
complete treatment of the theory of abelian varieties over C.
Wheneverpossible, use my other course notes as references(because theyare freely
available to everyone).
GT: Group Theory (Math 594).
FT: Field and Galois Theory (Math 594).
AG: Algebraic Geometry (Math 631).
ANT: Algebraic Number Theory (Math 676).
MF: Modular Functions and Modular Forms (Math 678).
EC: Elliptic Curves (Math 679).
LEC: Lectures on Etale Cohomology (Math 732).
CFT: Class Field Theory (Math 776).
ABELIAN VARIETIES 7
Part I: Basic Theory of Abelian Varieties
1. Definitions; Basic Properties.
A group variety over k is a variety V together with regular maps
m: V × V → V (multiplication)
k
inv: V → V (inverse)
and an element e ∈ V(k) such that the structure on V(kal) defined by m and inv is
a group with identity element e.
Such a quadruple (V,m,inv,e) is a group in the category of varieties over k. This
means that
G −(−id−,e→) G× G −−m−→ G, G −(−e,−id→) G× G −−m−→ G
k k
are both the identity map (so e is the identity element), the maps
id×inv
G ∆>G× G > G× G m >G
k > k
inv×id
are both equal to the composite
G → Specmk →e G
(so inv is the map taking an element to its inverse) and the following diagram com-
mutes (associativity)
G× G× G 1→×m G× G
k k k
↓ m×1 ↓ m
G× G →m G.
k
To prove that the diagrams commute, recall that the set where two morphisms of
varieties disagree is open (because the target variety is separated, AG 3.8), and if
it is nonempty the Nullenstellensatz (AG 1.6) shows that it will have a point with
coordinates in kal.
It follows that for every k-algebra R, V(R) acquires a group structure, and these
group structures depend functorially on R (AG p76).
Let V be a group variety over k. For a point a of V with coordinates in k, we
define t : V → V (right translation by a) to be the composite
a
V → V ×V →m V.
x (cid:13)→ (x,a) (cid:13)→ xa
Thus, on points t is x (cid:13)→ xa. It is an isomorphism V → V with inverse t .
a inv(a)
A group variety is automatically nonsingular: as does any variety, it contains a
nonempty nonsingular open subvariety U (AG 4.21), and the translates of U cover
V.
By definition, only one irreducible component of a variety can pass through a non-
singular point of the variety(AG p63). Thus a connected group schemeis irreducible.
Aconnected group varietyis geometricallyconnected, i.e.,remains connectedwhen
we extend scalars to the algebraic closure. To see this, we have to show that k is
8 J.S. MILNE
algebraically closed in k(V) (AG 9.2). Let U be any open affine neighbourhood of
e, and let A = Γ(U,O ). Then A is a k-algebra with field of fractions k(V), and e
V
is a homomorphism A → k. If k were not algebraically closed in k(V), then there
would be a field k(cid:1) ⊃ k, k(cid:1) (cid:3)= k, contained in A. But for such a field, there is no
homomorphism k(cid:1) → k, and a fortiori, no homomorphism A → k.
A complete connected group variety is called an abelian variety. As we shall see,
they are projective,and (fortunately) commutative. Their group laws will be written
additively. Thus t is now x (cid:13)→ x+a and e is usually denoted 0.
a
Rigidity. The paucity of maps between projective varieties has some interesting
consequences.
Theorem 1.1 (Rigidity Theorem). Consider a regular map α: V ×W → U, and
assume that V is complete and that V ×W is geometrically irreducible. If there are
points u ∈ U(k), v ∈ V(k), and w ∈ W(k) such that
0 0 0
α(V ×{w }) = {u } = α({v }×W)
0 0 0
then α(V ×W) = {u }.
0
In other words, if the two “coordinate axes” collapse to a point, then this forces
the whole space to collapse to the point.
Proof. Sincethehypotheses continuetoholdafterextendingscalarsfromk to kal,
we can assume k is algebraically closed. Note that V is connected, because otherwise
V × W wouldn’t be connected, much less irreducible.
k
We need to use the following facts:
(i) If V is complete, then the projection map q: V × W → W is closed (this is
k
the definition of being complete AG 5.25).
(ii) If V is complete and connected, and ϕ: V → U is a regular map from V into
an affine variety, then ϕ(V) = {point} (AG 5.28).
Let U be an open affine neighbourhood of u . Because of (i), Z =df q(α−1(U −U ))
0 0 0
is closed in W. By definition, Z consists of the second coordinates of points of V ×W
notmapping intoU . Thusapoint w ofW liesoutsideZ ifand onlyα(V ×{w}) ⊂ U .
0 0
In particular w lies outside Z, and so W − Z is nonempty. As V × {w}(≈ V) is
0
complete and U is affine, α(V ×{w}) must be a point whenever w ∈ W −Z: in fact,
0
α(V ×{w}) = α(v ,w) = {u }. Thus α is constant on the subset V × (W −Z) of
0 0
V ×W. As V ×(W −Z) is nonempty and open in V ×W, and V ×W is irreducible,
V ×(W −Z) is dense V × W. As U is separated, α must agree with the constant
map on the whole of V ×W.
Corollary 1.2. Every regular map α: A → B of abelian varieties is the compos-
ite of a homomorphism with a translation.
Proof. The regular map α will send the k-rational point 0 of A to a k-rational
pointbofB. Aftercomposingαwithtranslationby−b,wemayassumethatα(0) = 0.
Consider the map
ϕ: A×A → B,
ϕ(a,a(cid:1)) = α(a+a(cid:1))−α(a)−α(a(cid:1)).