These are my live-TeXed notes for the course *Math 232br: Abelian Varieties* taught by Xinwen Zhu at Harvard, Spring 2012. The main reference book is [1]. See also [2] and [3]. Please let me know if you find any typos or mistakes!

2012/02/06

##
Introduction

Euler discovered an addition formula for elliptic integrals where and is a certain algebraic function. In modern language, the affine equation defines an elliptic curve and the group structure on it gives the addition formula. More generally, let be an algebraic curve with genus , then integration gives a map and an isomorphism is called the *Jacobian* of and has a natural group structure.

- is compact (hence is a complex torus).
- has a natural (unique) structure as a projective variety.

(Sketch)
- We need to show that the image of is a lattice. This follows from the isomorphism using Hodge theory.
- The second part follows from the following theorem and lemma. We only need to construct a symplectic pairing such that the corresponding Hermitian form on is positive definite. This pairing can given by the intersection pairing on .
¡õ

Let

be a complex vector space and

be the underlying real vector space. Then there exists an bijection between Hermitian forms on

and skew-symmetric forms

on

satisfying

given by sending

to

.

Notice that the group law on is compatible with its algebraic variety structure. This motivates us to make the following definition.

An

*abelian variety* over

is a projective variety with a group law, i.e., the multiplication and inversion are morphisms of algebraic varieties.

So we can associate an abelian variety to each algebraic curve with .

Another example of abelian varieties comes from number theory. Let be a totally real extension of degree and be am imaginary quadratic extension, i.e., is a *CM field*. Then has elements and the complex conjugation acts on it.

A

*CM-type* is a choice of

such that

has

elements and

.

Thus a CM-type gives an isomorphism by evaluation.

Let

be the ring of integers of

. Then

is an abelian variety.

(Sketch)
By weak approximation, one can choose an

totally imaginary and

for each

, then

is positive definite and restricts to an integral pairing

.

¡õ
For any lattice in a 1-dimensional complex vector space, is an abelian variety. Suppose , where . We define Then is positive definite and restricts to an integral pairing: and .

##
Basic questions

Is it true that for any lattice

in an

-dimensional complex vector space,

is an abelian variety? Conversely, is every abelian variety a complex torus?

The answer to the first question is false: even for , for *almost all* lattices , the complex torus is not an abelian variety. However, the converse is true: every abelian variety must be a complex torus.

2012/02/08

Let

be a connected complex compact Lie group, then

, where

is a complex vector space and

is a lattice.

Let

. The adjoint representation

is

*holomorphic*. But

is compact and connected and

is an open subset of a complex vector space, hence

must be constant. In particular,

and

is commutative. When

is commutative,

is a group homomorphism and is in fact a covering map. Hence

for some discrete subgroup

. Moreover,

must be a lattice as

is compact.

¡õ
It follows that abelian varieties are complex tori. The following holds for any complex torus, hence any abelian variety.

Now we introduce the general notion of abelian varieties over an arbitrary field. By a *variety* over , we mean a geometrically integral, separated and finite type -scheme.

Let

be a field. An

*abelian variety* over

is a smooth complete variety together with a point

and morphisms of algebraic varieties

,

such that

forms a group with multiplication

and inversion

.

Let

,

be two abelian varieties. A

*homomorphism* is a morphism of algebraic varieties compatible with the group structures. The set of homomorphisms from

to

is denoted by

. The category of abelian varieties is denoted by

.

The structure of

and

as an abstract group.

Let

be an abelian variety over

. Then

is commutative and divisible.

is a surjective homomorphism with kernel

for

, or

for

, where

can be any value between 1 and

.

(Mordell-Weil)
Suppose

is a number field. Then

is a finitely generated abelian group.

The structure of

.

Since is surjective, we know that is torsion-free. Over complex numbers, is a free abelian group of finite rank, so we know that is a finite generated abelian group. More generally, over an arbitrary field,

is a finite generally free abelian groups.

Let be a prime different from , then . It is a -module as is defined over .

The

*-adic Tate module* . This is a free

module of rank

with a continuous action of

.

The Tate module can be viewed as an analog of the homology group . The following result is analogous to the complex case but is harder to prove.

Let

be a homomorphism of abelian varieties. The induced map

is an injection.

Notice that the image of the induced map lies in . We have the following famous Tate conjecture concerning the image.

(Tate)
If

is a field finitely generated over its prime field. Then

is a bijection.

The Tate conjecture is proved by Tate for finite fields and by Faltings for number fields. Faltings theorem is much beyond the scope of this course, however, we may have a chance to talk about Tate's proof.

Line bundles on abelian varieties.

There is a short exact sequence

Here has a natural structure of abelian variety, usually denoted by , is called the *dual abelian variety*, and is a finitely generated free abelian group, is called the *Neron-Severi group*.

The cohomology of line bundles, Riemann-Roch problems.

Let

be an ample line bundle on

. Then

is very ample.

This is useful in constructing moduli spaces of abelian varieties, which we may or may not cover.

##
Group Structures

Let

be an abelian variety. Then

is commutative.

We can mimic the complex case and consider the adjoint representation. The issue is that does not hold true in general. Here we take another approach.

(Rigidity Lemma)
Let

be a complete variety and

,

be arbitrary varieties. Let

be a morphism of algebraic varieties such that

. Then there exists

such that

.

Let

,

be two abelian varieties and

be a morphism of algebraic varieties. Then

, where

and

.

We may assume

. We need to show that

is homomorphism of abelian varieties. Define

Then

. Then the Rigidity Lemma

2implies that

.

¡õ
(Proof of Theorem 10)
Apply Corollary

2 to the inversion

. It follows that

is commutative since

is a group homomorphism.

¡õ
2012/02/10

(Proof of Proposition 2)
Without loss of generality, we may assume

. Let

, where

is affine open. Then

is open in

. Let

, then

is closed. Because

is complete (hence universally closed), we know that

, the projection of

onto

, is closed in

. By construction,

. Hence

is nonempty and open. For any

, we have

. Because

is complete and

is affine, we know that

is a point. So we have proved that

, where

. But everything is separated, hence

.

¡õ
##
The Theorem of the Cube

Suppose

, then

is surjective.

(Sketch)
Consider the differential

. It is given by multiplication-by-

. Because

,

is an isomorphism. So

is smooth, hence surjective.

¡õ
However, the above argument fails when (the differential is zero). We need to develop some techniques of line bundles on abelian varieties to prove the surjectivity of in general.

(Theorem of the Cube)
Let

,

be complete varieties and

be an arbitrary variety. Let

,

,

. Let

be a line bundle on

. If

,

,

are trivial, then

is trivial.

Before giving the proof, we shall interpret the theorem of the cube in a more conceptual way and draw several consequences of it.

.

The functor

is called

*of order * (

*linear* when

,

*quadratic* when

) if

is injective (equivalently,

is surjective).

The Theorem of Cube implies that the functor

is quadratic: the map

is injective.

Let

be an abelian variety. Then

is linear. In fact,

is a bijection. For the injectivity, suppose

, then by the Rigidity Lemma

2we know that

, hence

. The surjectivity is obvious.

Suppose

, then

is quadratic.

Here come several corollaries of the Theorem of Cube.

Let

,

,

be complete varieties. Then every line bundle on

is of the form

.

Since

is equivalent to

.

¡õ
Let

be an abelian variety and

be an arbitrary variety. Let

be three morphisms. Then for any line bundle

on

,

Consider the universal case

and

are the projections. This follows from the Theorem of Cube by restricting to

,

and

. Other cases are pullbacks through

.

¡õ
Let

be a line bundle on

. Then

.

Applying Corollary

4 to the case

,

and

, we obtain that

Let

, then

, thus

as

. Hence

This completes the proof.

¡õ
If

is symmetric, i.e.,

, then

.

(Theorem of the Square)
For any

and

a line bundle on

, we have

where

is the translation-by-

map.

Apply Corollary

4 to

,

and

.

¡õ
2012/02/13

##
Review of cohomology theory on schemes

In order to prove the Theorem of the Cube, we need to digress to review some result on the cohomology of vector bundles over a flat family of varieties in this section.

Let be a scheme. The category of quasicoherent sheaves on is an abelian category. If is further noetherian, we also consider the category of coherent sheaves on . Let be a morphism of schemes, then the pullback functor has a right adjoint . The *derived functor* of consists of a collection of functors together with natural transformations for each short exact sequence satisfying the following:

- .
- Any short exact sequence gives a long exact sequence
- For any commutative diagram of short exact sequences the following diagram commutes

When , we also write , the *-th cohomology* of .

Instead of giving the precise definition of , let us review how to compute them (using Cech complexes). Assume and is separated (hence the intersection of two affine opens is still affine). Let be a cover of by affine opens. Fixing an order on , we form the Cech complex of -modules by In particular, The differential is given by

(Comparison theorem)
Suppose

is separated, then

for any cover

of affine opens.

(Kunneth formula)
Let

,

be two separated schemes over a field

. Suppose

and

. Then

where

.

Now let us state two important properties of sheaf cohomology we shall use later without proof.

Let

be a proper morphism of noetherian schemes. Suppose

, then

. In particular, when

,

is a finite dimensional

-vector space.

Let

be a proper morphism of noetherian schemes. Let

, flat over

. Then there exists a finite complex

of locally free

-modules of finite rank such that for every morphism

,

where

and

fit in the pullback diagram

Consider the situation

. Then for any line bundle

on

,

is flat over

. This the case we will use later (cf. Theorem

15).

Let

be a proper scheme over

and

. We define the

*Euler characteristic* .

Let

and

be as in Theorem

14. Then

is a locally constant function on

, where

and

.

Let

be the finite complex in Theorem

14. Then

. Hence

. The result follows from the additivity of the Euler characteristic and the fact that

's are locally free.

¡õ
Let

and

be as in Theorem

14. Then

is an upper semicontinuous function, i.e.,

is closed for any

.

By Theorem

14,

, or

Since

is locally constant, it is enough to show that

is closed. Since

is locally given by a matrix, this set is locally cut out by vanishing conditions on the

minors, hence closed.

¡õ
Let

and

be as in Theorem

14. In addition, assume that

is connected and reduced. Then then followings are equivalent:

- is locally constant.
- is locally free of finite rank and the natural map is an isomorphism.

It is clear that (b) implies (a) . Conversely, suppose (a) is true, then

is locally constant by the previous proof. So

is locally free by the assumption on

. So the local splitting of the complex ensures that

.

¡õ
2012/02/15

Now we apply the above results to the situation and a line bundle on .

(Seesaw Theorem)
Suppose

is algebraically closed,

is a complete variety over

and

is an arbitrary variety over

. Let

be a line bundle on

. Then

- is closed.
- There exists some line bundle on such that .

We need the following easy lemma.

A line bundle

on

is trivial if and only

and

.

Suppose

and

. Choose two sections

and

, we obtain a morphism

, which is an isomorphism since

(

is complete).

¡õ
(Proof of the Seesaw Theorem 15)
The first part is clear using the above lemma together with the upper semicontinuity (Corollary

10). For the second part, since for any

,

, we know that

is locally free of rank 1 by Corollary

11. Then the adjunction map

is an isomorphism as it is an isomorphism on each fiber.

¡õ
##
Proof of the Theorem of the Cube

Now let us return to finish the proof of the Theorem of the Cube. We need the following lemma.

For any

,

on

, there exists an irreducible curve

containing

,

.

The case

is clear. We now assume

. Since

is complete, by Chow's Lemma (for any complete variety there is a surjective birational map from a projective variety to it), we may assume that

is projective. Let

be the blowup of

at

,

, then

is also projective. Fixing an embedding

, by Bertini's theorem, we can find a general hyperplane

such that

is irreducible of codimension 1. By construction,

, so

. Now the lemma follows from induction on

.

¡õ
(Proof of the Theorem of the Cube 11)
We may assume that

is a smooth projective curve by the above lemma. In fact, it is enough to show that

is trivial for all

from the Seesaw Theorem

15(applying to

). To show this, we can replace

by a curve containing two given points. In addition, we can replace it by its normalization and further assume

is smooth.

Suppose has genus . Pick a divisor on of degree such that (exercise: we can always do this). Let . By Serre duality, hence the support of does not intersect by the upper semicontinuity (Corollary 10). Thus the projection of onto is a closed subset not containing . In other words, there is open containing such that . So we can replace by by the Seesaw theorem 15.

In sum, now we can assume . Then Since the Euler characteristic does not vary when we move (in a flat family) and , we know that It follows that is locally free of rank 1 on by Corollary 11.

Let be an open cover of such that is trivial. We choose a trivialization . Then . Let be the set of zeros of . These 's can be glued into a codimension 1 closed subset such that . So by definition is the set of zeros of the nonzero section of . To show that is trivial is equivalent to showing that , or equivalently, .

Let such that , we would like to show that is empty. This intersection does not meet or as we choose . The projection of onto is a closed subset of codimension 1 not containing . So as is of codimension 1. On the other hand, does not intersect , so is empty. Hence is empty.
¡õ

2012/02/17

##
Abelian varieties are projective

In this section, we will use the Theorem of Cube to deduce some deep results of abelian varieties, including the fact that all abelian varieties are projective.

Recall the group homomorphism defined in Remark 4. Since we obtain a homomorphism .

We define

, i.e.,

if

for any

.

Thus we have an exact sequence We will see that admits a natural structure of an algebraic variety, hence is an abelian variety (the *dual abelian variety*, cf 22).

Let

be a line bundle on

. Then

if and only if

.

For

, we define

. The it is clear that for

, we have

.

is closed in

. (So

has a natural structure of an algebraic group.)

Let

. Then

is closed by the Seesaw Theorem

15.

¡õ
Now we can state the main theorem of this section:

This theorem has the following important consequence.

Every abelian variety is projective.

Pick

an affine open containing 0 and

a divisor. Then

is closed as it is the projection of the preimage of

under the map

. So

is complete. But

since we choose

. We conclude that

is a complete variety inside an affine variety, thus is finite. Now the the result follows from Theorem

16.

¡õ
(Proof of Theorem 16)
(a) implies (b): If not, then the identity connected component is an abelian variety of positive dimension. By Lemma 6, . Now pulling back through we know that . Since is ample we know that is ample. Since is an automorphism of , we know that is also ample. Hence is ample, a contradiction.

(b) implies (c): It is clear since by definition.

(c) implies (d): By the Theorem of the Square 7, we know that . In particular, . To prove the base-point-freeness, for any , we want to find some such that and , or equivalently, . This can be done since and are both divisors. So is base-point-free (for this part we have not used (c)).

The base-point-free linear system defines a map , which is proper (since and are complete). In order to show that it is finite, we only need to show that each fiber is finite. If not, then contracts a curve . Let , then either or . Moreover, for a generic , . We know that and does not meet for a generic . Hence does not meet for a generic . Using the finiteness of , it remains to prove the following lemma (applying to .

Let

be an irreducible curve and

be a divisor such that

. Then for any

,

.

(Proof of the lemma)
Let

. Then

. The multiplication

gives a line bundle

on

. So for any

,

since the Euler characteristic stays the same in a flat family. Hence

by Riemann-Roch. So either

or

. For any

and

, we have

, hence

,

.

¡õ
(d) implies (a): We may replace by . We want to show is surjective for each coherent sheaf and sufficiently large . Let be the finite morphism in the assumption, then . Applying we obtain a commutative diagram The lower map is surjective since is ample, so the above map is also surjective. (This is the general fact that the pullback of an ample line bundle through a finite morphism is ample).
¡õ

is surjective.

By the dimension reason and the homogeneity, we know that

is surjective if and only if

is finite. Let

be an ample line bundle (existence ensured by the projectivity). We know that

is ample by Corollary

5. Since

is trivial, we know that

. It follows that

is finite.

¡õ
In the next section we shall show the following properties of .

2012/02/22

##
Isogenies of abelian varieties

Let

,

be two abelian varieties. A homomorphism

is called an

*isogeny* if

is surjective and has finite kernel. So by Corollary

13,

is an isogeny.

Let

be a complete variety of dimension

and

be a line bundle on

. Let

be a coherent sheaf on

. Then

is a polynomial of degree

(this is the usual Hilbert polynomial when

is smooth). Let

be the leading coefficient

. We call

the

*degree* of

with respect to

. We also write

for short. Note that when the support of

has dimension

, the degree

.

- Let be a coherent sheaf on with generic rank . Then .
- Let be a dominant morphism of complete varieties of the same dimension and be a line bundle on , then .

Assuming Proposition 3, we can prove the following theorem promised before.

.

Let

be an ample line bundle on

. Replacing

with

, we may assume that

is symmetric. Then

(Corollary

6). By Proposition

3, we have

. On the other hand, since

by definition, we know that

. Hence

.

¡õ
Now let us come back to the proof of Proposition 3.

(Proof of Proposition 3)
For simplicity, we prove the case when

is smooth and

is finite.

- Let be open such that and be a divisor. Since is smooth, we can form a line bundle , where is the ideal sheaf of . We have a section such that the zero locus of is . Choosing a basis of the sections of , we know that extends to a section of on for any and large enough. We get an exact sequence The first map is injective since it is injective on and is torsion-free on . The quotient is torsion and has support in . Tensoring with , we have an exact sequence where is torsion and supported on a smaller dimension set (in particular, ). Using the additivity of the degrees, we know that Now the result follows from the fact that since they agree on an open subset and the quotient sheaf has lower dimensional support.
- By adjunction, , thus . Since is a coherent sheaf of generic rank , we also know that by part (a).
¡õ

- is separable if and only if , where .
- The inseparable degree of is at least .

- is separable if and only if is smooth at a generic point, if and only if is smooth at the origin by homogeneity. At the origin, the tangent map is multiplication by , thus is surjective if and only if . (Another way: the degree of an inseparable extension is always a power of , but we know that by the previous theorem.)
- Since the tangent map is zero, we know that is zero. Hence at the generic point, is zero. Therefore for any , , which implies that lies in the kernel of the differential map . Now the result follows from the fact that is purely inseparable of degree .
¡õ

Let

be the

-torsion points of an abelian variety

. Then

where

.

is equal to the cardinality of the generic fiber, thus is equal to the separable degree of

. From the previous theorem, for any prime

, we have

for

or

for

. Using the exact sequence

the result now follows from induction.

¡õ
2012/02/24

##
Group schemes

We have basically solved the first question in our introduction about the group structures of abelian varieties (cf. Theorem 4). In the sequel, we shall study the Tate modules and dual abelian varieties. We prepare some general notions of group schemes in this section.

A functor

is called a

*group functor*. A

*group object* in

is a triple

where

is a group functor and

is an isomorphism for some

. Namely, for every

, one assigns a group structure on the set

and for any

, we have a group homomorphism

.

Assume that finite products exist in (in particular, the final object exists). Then giving a group object is equivalent to giving a object in and morphisms , and satisfying the usual commutative diagrams:

Fix

a (locally) noetherian scheme. A

*group scheme* over

is a group object in the category of schemes over

.

Therefore a group scheme can be understood in the above two ways: as a representable group functor, or as a object with a group structure.

- An abelian variety over is a group scheme over .
- The
*additive group* is defined to be with the group structure given by , and . Alternatively, the group structure can be described as with usual addition for any -scheme .
- The
*multiplicative group* is defined to be with the group structure given by , , . Alternatively, the group structure can be described as with usual multiplication for any -scheme .
- The multiplication-by- map is defined to be , . The kernel is a closed group subscheme and for any -scheme .
- Let be an -scheme. The the
*Picard functor* , sending to the isomorphism classes of line bundles on modulo the isomorphism classes of line bundles on , is a group functor. Moreover, if is representable, then the corresponding scheme is called the *Picard scheme* of . We will study the Picard scheme of an abelian variety later.

##
Lie algebras and smoothness of group schemes

From now on, we assume that is a group scheme and the structural morphism is locally of finite type.

The sheaf of differentials is a coherent sheaf described as for any quasicoherent sheaf on . In particular, the elements in are called the *vector fields* on . For any base change we have and a natural pullback map . The image of a vector field on is a vector field on , which is also denoted by .

We say

is a

*right invariant vector field* if for any

,

and

,

holds. Similarly we can define left invariant vector fields on

. We denote the set of left (or right) invariant vector fields on

by

. Then

is a sheaf of

-modules on

.

2012/02/27

Now let us specify the base scheme . Let be a group scheme. We defined as the set of left (or right) invariant vector fields on , i.e., derivations such that .

In other words forms a *restricted Lie algebra* (or -Lie algebra) over :

Let be the tangent space of at the origin . From Grothendieck's point of view, where . The multiplication structure of as a group scheme coincides with its addition structure as a vector space. The tangent space can be canonically identified with the Lie algebra as follows.

The map

is an isomorphism, where

.

Before giving the proof, we shall make a remark on another point view of derivations.

(Sketch)
The inverse map

is given as follows. Let

, then the right translation

satisfies the above commutative diagram, hence by the previous remark gives a left invariant derivation

. (This is a general fact from Lie theory that vector fields generated by the right translation is left invariant.)

¡õ
Let . We have two projections and also a morphism given by .

If

is commutative, then

is abelian.

For any two derivations

and

, we can find

such that

and

, then

since

is commutative. Now by previous lemma we know that

.

¡õ
Now let be a group scheme of finite type and be the connected component containing .

- is open, closed and is a group subscheme of .
- is geometrically irreducible.
- is of finite type.

(Sketch)
- The connected component is always closed and it is open since topologically the is locally noetherian. The map factors through by connectedness, hence is a group scheme.
- It is a general fact that if a group scheme over is connected and contains a rational point, then it is geometrically connected (we do not prove it). Base change to and consider the induced reduced scheme , then is a reduced group scheme over , hence is smooth. It is connected and smooth, hence is geometrically irreducible.
- For any affine open, is surjective, hence is quasicompact. It is quasicompact and locally of finite type, hence is of finite type.
¡õ

In characteristic

, the group scheme

and

are not reduced, hence not smooth. Let

be the pullback of

via the

-fold Frobenius map

, then we have an induced map

via the following diagram

is a morphism of group schemes (notice that

is not), hence the kernel is a closed group subscheme, called the

*Frobenius kernel* of

. Thus

is the Frobenius kernel of

and

is the Frobenius kernel of

.

Nevertheless, any group scheme over a field of characteristic 0 is automatically smooth.

If

, then

is smooth (hence

is smooth).

(Sketch)
We may assume

. We only need to show that every completed local ring (by homogeneity, at the origin

) is isomorphic to the power series ring generated by

, where

form a basis for

. Let

be a dual basis, then they give left invariant vector fields

. We thus have a natural map into the completed local ring

. The inverse map is given by the Taylor expansion

.

¡õ
leap day!

##
Picard schemes and dual abelian varieties

Let be a projective variety and assume that we have a rational point . We introduced the Picard functor (cf. Example 5) where is the projection. In other words, consists of isomorphism classes of pairs Grothendieck proved the following representability theorem of Picard functors (which we will treat as a black box).

- is represented by a scheme (hence a group scheme), which is locally of finite type over .
- The connected component is quasiprojective, and is projective if is smooth.

The

-points

is equal to the groupoid

In other words, we fixed a

*rigidification* of a line bundle

, so that the pair

has no nontrivial automorphisms (which is important to the representability) and this groupoid is actually equivalent to a category of sets.

Consider

, then

it corresponds to a pair

on

. This pair is called the

*Poincare sheaf*.

By the functoriality, any line bundle is the pull back via the map corresponding to . For , is a line bundle on and corresponds to the line bundle represented by .

Let

,

be two line bundles on

. We say that

and

are

*algebraically equivalent* if there exists

connected schemes of finite type over

,

geometric points of

and

line bundles on

such that

- ,
- , .
- .

Let

be a line bundle on

and

be the corresponding point. Then

if and only if

and

are algebraically equivalent.

Now we apply the above general theory to the case of an abelian variety together with the rational point .

We define

. It is a connected projective group scheme over

. We will soon see that

is

*smooth* (even in positive characteristic), so

is a variety, called the

*dual abelian variety* of

.

Recall we defined (Definition 8) , i.e. if and only if for all . The notation suggests some connection between and .

.

First we show that

. Let

. Consider

and

. Since

and

are trivial, we know that

,

and

are trivial, therefore

is trivial by the Theorem of the Cube

11. Therefore

is translation invariant. By the universality of

, it follows that any line bundle in

is translation invariant, thus lies in

.

2012/03/02

Now pick an ample line bundle on , by the following theorem the map is surjective, hence , therefore .
¡õ

Let

be an ample line bundle. Then the map

is surjective.

To prove this theorem, we need the following lemma.

Let

be a nontrivial line bundle. Then

for any

.

We can extend the definition of at the level of schemes: for any -scheme and . We define on . Because is connected, we know that lands in .

If

is ample, then

is surjective with finite kernel. In particular,

.

It follows from Theorem

16 and

23.

¡õ
is smooth (hence is an abelian variety).

To prove

is smooth, it is enough to show

(because

). By definition,

, which is equal to isomorphism classes of triples

Recall that isomorphism classes of line bundles on

is identified with

. By the exact sequence

where the first map is given by

. This exact sequence splits, so we get a split exact sequence

namely

So we can identify the tangent space

with

, which has dimension

as we shall show in the next section using algebraic facts of bialgebras (cf. Corollary

17).

¡õ
##
Hopf algebras

Now let be a complete variety over such that . Then

- is a graded commutative -algebra with the product given by
- Let be an abelian variety. Then we have a further structure: is a cocommutative coalgebra given by Moreover, for , and . The follow lemma is straightforward.

makes

a finite dimensional positively graded commutative and cocommutative Hopf

-algebra such that

,

.

2012/03/05

(Hopf algebras)
- Let be affine group scheme over . Then is a commutative Hopf -algebra. In fact, the category commutative Hopf algebras is equivalent to the category of affine group -schemes. If is commutative, then is cocommutative.
- The additive group gives a Hopf algebra structure on . If we put in an
*even* degree, then is graded commutative and *cocommutative*. Similarly for with put in an even degree.
- The group scheme gives with a Hopf algebra structure. If we put in
*odd* degree (reason: we need ), then it is graded commutative and *cocommutative*.
- If , are two graded commutative and cocommutative bialgebra, then is also a graded commutative and cocommutative bialgebra. For example, is equal to the exterior algebra since we require that .

(Borel)
Let

be a perfect field and

be a positively graded commutative and cocommutative

bialgebra. If

and

. Then

, where

's are of the form

,

or

in the previous examples.

This is a purely algebraic statement and we will not prove it here. We apply this theorem to , which is a graded commutative and cocommutative bialgebra.

Suppose

is an abelian variety over

of dimension

, then

and

is an isomorphism. In particular,

and

.

Because

for

. By Theorem

25 we know that

where

is a graded vector space in odd degrees. Let

be the degree 1 piece of

. Because

, we know that

. But

and

, which implies that

and

. Similarly, if

(for

) or

is not in

, then the element

has degree greater than

, a contradiction. Hence

.

¡õ
##
Polarizations and Jacobian varieties

Note that for any . So in some sense the isogeny is more fundamental than the ample line bundle itself.

A

*polarization* of an abelian variety

is an isogeny

such that

for some ample line bundle

on

.

is called a

*principal polarization* if

is an isomorphism (equivalently,

).

Let

be a complete curve over

with a rational point

. Then

is representable. We denote

by

, called the

*Jacobian variety* of

.

Our next goal is to show that is indeed an abelian variety and admits a canonical principal polarization.

For any

, the

*Abel-Jacobi map* is defined to be

At the level of

-points, this map can be defined as

, where

is the graph of

, which is a Cartier divisor inside

.

is smooth (hence is an abelian variety).

Since

, by Riemann-Roch,

is surjective and the fibers of

are projective space of dimension

as long as

, where

is the genus of

. In particular,

. But we already know that

, thus

is smooth.

¡õ
admits a canonical principal polarization.

We shall construct a canonical ample divisor on and show that it gives rise to a principal polarization.

We define the

*theta divisor* to be the scheme-theoretic image

. By definition,

2012/03/07

Let us first consider a special case:

is an elliptic curve. Then

and

It is well known that

is an ample divisor and

is an isomorphism. So every elliptic curve is canonically principally polarized. From this point of view, the correct generalization of elliptic curves should be

*polarized* abelian varieties rather than abelian varieties themselves.

(Theorem 26)
Let

. We obtain the pulling back of line bundles

. It is then enough to show

. To do this, we now give a different construction of

.

Let be a noetherian scheme. Let be a line bundle on and be the projection. We would like to construct a line bundle on such that for each , the fiber of it is the determinant of the cohomology By Theorem 14 there exists a complex of locally free sheaves on of finite rank such that . Let be a line bundle on , called the *determinant line bundle* of . This is independent of the choice of the complex and the formulation commutes with any base change.

Suppose

and

is the diagonal

. For

,

. For

, we obtain the canonical sheaf

(exercise).

Now apply to the case . We have a universal line bundle on . Let . For any , has degree . Hence by Riemann-Roch, . Let be the complex on representing via Theorem 14. We know that . Therefore . So the induced map is a map between line bundles, hence is either injective or zero. Consider This is the locus where , i.e, , which is equal to . This is exactly the divisor and thus is injective.

Using this construction of , we know that is an ideal sheaf. The quotient is supported on , hence is for some . So for some . We claim that . This is enough because implies that and .

Showing is equivalent to showing that for any , or, . Note that , this is equivalent to showing that . Since . By the base change using , it is equal to . Since and we know that

Let

be a line bundle on

and

. Then

.

By induction, it is enough to prove that

. By the exact sequence on

,

where

(the poles only occurs at

when it intersects

). All these are flat over the second factor, the result follows from taking the determinant line bundles.

¡õ
From this lemma, we know that . Since the left hand side is and the right hand side is , taking the inverses finishes the proof of Theorem 26.
¡õ

2012/03/09

##
Duality of abelian varieties

Let be a commutative finite (hence affine) group scheme over . Then is a finite dimensional commutative and cocommutative Hopf algebra over . Let be the dual of , it has a natural structure of commutative and cocommutative Hopf algebra over induced by that of .

is a commutative finite group scheme over

, called the

*Cartier dual* of

. The natural

*dual functor* satisfies

.

Let

,

be two commutative group schemes over

. We define the functor

.

Let

be a

-algebra. We want to show that

. By definition,

. Since

, we may regard an element of

as an element of

. For

, one can check that

if and only if

and

is invertible, hence corresponds exactly to the elements of

.

¡õ
For

,

. Therefore

. One also has

(the dual pairing is given by the truncated exponent).

The following is the main result of this section.

Let

be an isogeny of abelian varieties. Then the induced morphism of dual abelian varieties

is also an isogeny and

.

2012/03/19

To prove Theorem 27, we need a theorem of Grothendieck on fppf descent. Let be a morphism of schemes. Then we have the following morphisms (projections) Suppose , the pullback sheaf has some additional properties:

- there is a canonical isomorphism ;
- restricting to the diagonal we get ;
- if we further pullback to , we obtain a cocycle relation .

Motivated by this, we define the *category of descent data* We then have a functor extending the original functor .

(Grothendieck)
If

is fppf (faithfully flat, locally of finite presentation), then

is an equivalence of categories.

Applying to our case, we need the following lemma.

Let

be an isogeny of abelian varieties. Then

is fppf.

We only need to check that

is flat, which can be shown by generic flatness since

is a morphism of group varieties.

¡õ
If

is an isogeny, then

.

Because

for

.

¡õ
Let

,

be two abelian varieties over

of the same dimension. A line bundle

on

is called a

*divisorial correspondence* if

and

. A divisorial correspondence

induces a morphism

of abelian varieties (in general, a line bundle

on

gives a morphism

). The morphism

is actually a homomorphism since it sends 0 to 0. Let

be the flipping isomorphism. Then we have a homomorphism

.

Let

be the Poincare line bundle on

. Then

by the definition of

. Also we have

. Denote

.

Let

be a line bundle on

. The we have the following commutative diagram

Consider

. We claim that

To prove it, we use Seesaw Theorem

15. The restriction to

of

is

. So we only need to show that the restriction to

of

is the same as

. This is true because both are trivial and the claim follows. So we have

. Since the left hand side is

(by definition of

) and the right hand side is

, the required result then follows.

¡õ
is an isomorphism.

Pick

an ample line bundle, then

and

are isogenies. Because

, we know

is also an isogeny. Now

implies that

.

¡õ
Thus we can identify and via . Under this identification we have .

A morphism

is called

*symmetric* if

.

Finally, it is easy to check the following proposition by definition.

Let

be a morphism and

a line bundle on

. Then

, namely we have the following commutative diagram

2012/03/21

##
Finite group schemes and torsion

is etale if and only if

. It is local if and only if

is a

-power.

is local.

We will soon see that local group schemes and etale group schemes are building blocks of finite group schemes: the connected component of is local -group and the quotient is etale. Let us study etale group schemes first.

Fix

a separable closure of

. Then the category of finite etale

-algebras (= category of finite etale

-schemes) is equivalent to the category of finite

-sets. The equivalence is given by sending a

-scheme

to

.

(Sketch)
This is basically the main theorem of Galois theory. Let

be a finite etale

-scheme. Then

admits a

action. Conversely, if

is a

-set, we form the

-algebra

, where the action of

is the diagonal action.

¡õ
The category of etale

-group schemes is equivalent to the category of finite groups with

-action.

The etale

-group scheme

,

, corresponds to the finite group

(

-th roots of unity in

) with the natural

-action. This action defines

by sending

, where

is the cyclotomic character so that

for

a primitive

-th root of unity.

Let

be a

-scheme of finite type. Then there exists a finite etale

-scheme

together with a morphism

which is universal in the following sense: if

with

finite etale, then there exists a unique

such that

. In addition,

is faithfully flat and the fibers of

are connected components of

.

(Sketch)
We define

by

together with the natural

-action. Over

,

is a product of copies of

and the map

is simply the structure map (clearly flat). This is equivariant with respect to the

-action, therefore descents to

.

¡õ
If

is a

-group scheme of finite type, then

is an etale

-group scheme and

is a homomorphism.

Every finite

-group scheme

fits into the following exact sequence (meaning the map

is faithfully flat with kernel

):

Moreover, if

is perfect, this exact sequence splits canonically.

(Sketch)
For the first part, we take

and

. When

is perfect,

is a

-group scheme (since when

is perfect, the fiber product of two reduced schemes is still reduced), so we obtain a morphism

. One can check the composition

is an isomorphism by checking this over

.

¡õ
Let

be a commutative finite

-group scheme. We say

is

*etale-etale* if

is etale and

is etale. We define similarly the notion of

*etale-local*,

*local-etale* and

*local-local*.

Suppose

is perfect. Let

be commutative finite

-group scheme. Then

can be decomposed into a product of these four types of groups

Moreover, this decomposition is unique.

Apply the previous decomposition twice.
¡õ

Local groups are more complicated than etale groups. Fortunately, they can be built by more basic blocks.

A local

-group scheme

is called

*of height one* if

for any

, where

is the maximal ideal at

.

Suppose

is of height one, then the coordinate ring

. In particular,

is a

-power.

Let

such that

form a basis of

. Since

is local, we know that

is a local ring, thus there is a surjection

. We need to show that there is no relation between

for

. Taking the dual basis of

, we can produce left invariant vector fields

on

such that

. Suppose

is a relation with the smallest degree, then

gives a relation with a lower degree, a contradiction.

¡õ
Let be any -group scheme of finite type. Let be the Frobenius morphism and be the relative Frobenius (cf. Example 6).

is a group homomorphism.

We want to check that for any scheme

,

is a group homomorphism. This is because

commutes with any reasonable base change (in particular, we can base change the multiplication diagram).

¡õ
We denote . The morphism factors through the local group and we have a cartesian diagram of schemes Thus we have a cartesian diagram and where . In particular, is a local group (topologically one point) of height one and . We cannot recover from its Lie algebra, however, we have the following

The functor

is an equivalence of categories between height one group schemes and

-Lie algebra.

2012/03/23

(Sketch)
Let us construct the inverse functor. From a Lie algebra

, we can construct an associative algebra, its universal enveloping algebra

. In fact

is also a cocommutative Hopf algebra, where the comultiplication is given by

for any

(and extends uniquely to

). Futhermore,

is exactly

. Now using the

operation from the

-Lie algebra structure, we define

. One can check

is a

*finite dimensional* cocommutative Hopf algebra. The inverse functor then sends

to

.

¡õ
If

is commutative of height one, then

is zero.

The morphism

gives

which is multiplication by

, hence is zero in characteristic

. Now by the previous theorem, we know that

.

¡õ
If

is local and commutative, then there exists some

such that

is zero.

By iterating we obatin morphisms

Since

is local, this gives inclusions

for some

(which can be chosen as the dimension of the coordinate ring

). Using the previous corollary,

is killed by

. By induction we can show that

is killed by

since the image of

lies in

.

¡õ
If

is finite and commutative, then there exists some

such that

is zero.

Use the decomposition in Corollary

22 and the previous corollary.

¡õ
Let

be an isogeny of abelian varieties. Then there exists some

and an isogeny

such that

.

The morphism

factors through the relative Frobenius

. In other words, there exists

such that

. Then

(since Frobenius commutes with any morphism) which implies that

. The morphism

is called the

*Verschiebung*.

The Cartier dual

isomorphic to

.

By Theorem

27, we only need to show

. By definition, the morphism

sends

to

, which is equal to

by Corollary

5 and

for

.

¡õ
Now write , where . Then and are both closed subgroups of .

The natural morphism

is an isomorphism.

By the previous lemma,

is etale-etale since

. We also know

These two parts together give the decomposition in Corollary

22 for

. The result then follows from the uniqueness of such decomposition.

¡õ
From Remark 21, we know that for some integers and .

is an invariant under isogeny (called the

*-rank* of

).

Let

be an isogeny and

. Then

induces a morphism

and comparing the orders gives

for any

, hence

. By Corollary

27, we have another isogeny

. By the same reason one knows that

.

¡õ
.

Apply the previous proposition to the isogeny

and use Lemma

20.

¡õ
2012/03/26

##
Tate modules and p-divisible groups

Recall that (Definition 5) for we defined the *-adic Tate module* (equal to since is etale). This is a free -module of rank with a continuous action of . An isogeny induces an continuous map . This notion is valid for any commutative group schemes other than abelian varieties. For example, the -adic Tate module of the multiplicative group is a free module of rank 1 where acts via the cyclotomic character .

Suppose

is a free

-module of finite rank with an action of

. We define the

*Tate twists of * by

for

and

for

.

.

From

, we know that

Passing to the limit we obtain that

.

¡õ
Let

be an isogeny and

be its kernel. Then we have a short exact sequence of

-modules

where

is the

-Sylow subgroup of

.

Observe that

From the exact sequence

we obtain a long exact sequence

The first term is zero since

is finite and the last term is 0 since

is divisible. So we only need to understand the group

. Using the isomorphism

, we know that

since

is killed by something prime

which is an isomorphism on

. Now apply

to the exact sequence

gives a long exact sequence

The first and last terms are zero since

kills

. Therefore

Since

is etale, we have

. The result then follows.

¡õ
Now let us consider the case .

We define

(equal to

. It is a free

-module of rank

, where

is the

-rank of

.

This -adic Tate module is not as good as the -adic Tate module because it only sees the etale quotient part and loses other information.

Let

be a base scheme. A

*-divisible group* (or

*Barsotti-Tate group*)

is an inductive system

, where

,

is commutative and finite flat over

and

is a closed embedding such that

factors as

, where

is faithfully flat.

Let

be an abelian variety. Define

. This

-divisible group is the right replacement of the

-adic Tate module

. In particular, one can recover

from

.

Since

kills

, we know that

for some

. We call the

the

*height* of

. By induction, one sees that

.

Let

be an abelian variety of dimension

, then the height of

is

.

2012/03/28

##
The Poincare complete reducibility and the degree polynomial

Denote by the category of abelian varieties over . This is a quite complicated category to study. We introduce a slightly simpler category.

We define

to be the category of abelian varieties

*up to isogeny*: the objects are abelian varieties over

and the morphisms between

and

are elements of

.

Let

be an isogeny. Then

is invertible in

.

By Corollary

27, there exists isogenies

and

such that

and

, both of which are invertible in

. Hence

is invertible.

¡õ
(Poincare complete reducibility)
Let

be an abelian subvariety. Then there exists

such that the multiplication morphism

is an isogeny. In other words, abelian varieties are completely reducible in

.

Pick an ample line bundle

on

and write the closed immersion as

. Proposition

8gives

and we know that

has finite kernel since

does. Let

. So

is finite. Since

is surjective, counting dimensions we know that

is an isogeny.

¡õ
An abelian variety

is called

*simple* if it does not contain any abelian subvariety other than 0 and

.

Write and for short. The previous lemma implies that if simple, then is a division algebra over .

Applying the completely reducibility successively we obtain the following corollaries.

Every abelian variety

is isogenous to

, where

's are pairwise non-isogenous simple abelian varieties. Moreover, this decomposition is unique up to permutation.

Let

be an abelian variety, then

is a semisimple algebra over

, hence can be written as a product of matrix algebra of division algebras

.

Let

be a field and

be a vector space (not necessarily finite dimensional). A function

is called

*homogeneous polynomial of degree * if the restriction of

to any finite dimensional subspace is a polynomial function of degree

, or equivalently, for any

,

is a homogeneous polynomial of degree

in

,

.

Define

to be

if

is an isogeny and 0 otherwise.

There is a unique way to extend

to a (homogeneous) polynomial

of degree

.

To prove this theorem, we need the following lemma.

Pick an ample line bundle

on

such that

(this is always true though we did not prove it). Then

.

2012/03/30

##
The Riemanm-Roch Theorem for abelian varieties

In this section we will prove the following version of Riemann-Roch theorem for abelian varieties.

(Riemann-Roch)
Let

be a line bundle on

. Then

is a homogeneous polynomial of deg

. In particular,

Assuming this theorem, we can prove Lemma 22and thus finish the proof of Theorem 32.

Now let us turn to the proof of Theorem 33. Let be a group scheme over of finite type and be a scheme over of finite type equipped with the trivial -action.

A

*-torsor* (or,

*principal -bundle*)

over

is a scheme

with a right

-action together with a

-equivariant morphism

such that the natural morphism

is an isomorphism (the action

factors through

since

acts on

trivially).

If

is an isogeny of abelian varieties. Then

is a

-torsor over

.

Suppose

is finite,

is a

-torsor (hence is finite and

) and

is proper. Then for any coherent sheaf

on

, we have

Using additivity of the Euler characteristic in short exact sequences and noetherian induction, we can assume the theorem holds for

and

is integral. Let

be the generic rank of

. Then there exists

open such that

. Extend

to a coherent sheaf

. The projections

and

are isomorphisms on

, hence their kernels and cokernels are supported on lower dimensions. The additivity of the Euler characteristic and induction hypothesis thus allow us reduce to showing the theorem for

, or for any

*one* coherent sheaf on

. Let us prove it for

, namely to show

By the flat base change

we know that

. Since

(

is finite flat), the result follows.

¡õ
Any line bundle

on an abelian variety

over

can be written as

, where

is symmetric and

.

We want to find some

such that

is symmetric, i.e.,

. If

, then

works (

is divisible over

). It remains to prove that

, which is equivalent to showing that

since

. The right-hand-side is equal to

because

is in

. Now it remains to show that

This follows from Theorem of the Square.

¡õ
Finally,

(Proof of Theorem 33)
It is enough to show that

. This is true the previous theorem if

is symmetric (in this case

). In general, Lemma

23 tells us we can write

here

is symmetric and

. In particular,

is algebraically equivalent to

. Using the invariance of Euler characteristic in algebraic families, we find that

This completes the proof.

¡õ
2012/04/02

##
Endomorphisms of abelian varieties

Let

,

be two abelian varieties over

. Then the natural map

is injective. In particular,

is a free

-module of finite rank (as

is torsion-free).

Suppose we have isogenies

and

with

,

simple. Using the injectivity of

, we reduce to the case where

and

are simple. If

and

are not isogenous, then we are done. Otherwise, choosing an isogeny

gives a bijection between

and

, thus we are reduced to the case

. To show this case, it is enough to show that for any

finitely generated, there is an injection

. By Theorem

34 ,

can not be too divisible. More precisely, let

, then

is also finitely generated. In fact,

inside

and

is a finite dimensional

-vector space. Moreover,

is a polynomial such that

for any

. Therefore

is discrete inside

, hence

is finitely generated.

Now we may assume that is finitely generated and . We need to show that is injective. Let be a -basis of . Suppose for . If not all , we can assume there exists some . Choose such that . Then . Let , then , hence by the definition of the Tate module. Thus we can write by Theorem 30, where lies in . This implies , a contradiction.
¡õ

The Neron-Severi group

is a finitely generated free abelian group (of rank

).

By definition, there is an injection

given by

. The latter one has finite rank by the previous theorem.

¡õ
is a finite dimensional semisimple

-algebra.

Our next goal is to classify the possibilities of the endomorphism algebra .

Let

be a finite dimensional simple

-algebra. A function

is called a

*norm form* if

is a polynomial and

. A function

is called a

*trace form* if

is linear and

.

Let

be a finite dimensional simple

-algebra and

be the center of

(which must be a field). Then there exists a norm form and a trace form

,

(with

) such that any norm form

is of the form

for some

and any trace form

is of the form

for some linear map

.

is called the

*reduced norm* and

is called the

*reduced trace*.

2012/04/04

- The degree function agrees with , where . In other words, for , .
- Let and (equal to the characteristic polynomial of ). Then .

First observe that both

and

are norm forms on

. We claim that any

, the

-adic absolute values

. This claim implies the first part of the theorem: write

, then

and

by the previous proposition; also

is dense in

.

Now let us show the claim. Since is dense in and both sides are homogeneous polynomial of degree , we may only check on . If is not an isogeny, then is not an isomorphism (the image has lower dimension), hence both sides are equal to 0. If is an isogeny. By the exact sequence in Proposition 13, is injective and has cokernel . Therefore , which is equal to .

To see the polynomial in the second part of the theorem has integer coefficients, we observe that by definition is an integer for any , hence . Since is finite over , then there exists such that . So all the roots of , hence all roots of are algebraic integers. But , we know that .
¡õ

Write

the characteristic polynomial of

. We call

the

*norm* of

and

the

*trace* of

.

Now let us further analyze the struction of . Suppose is simple and is the center of the division algebra of . Then and for some integers . Thus is a polynomial of degree . Because, is a polynomial of degree , we obtain the following result.

If

is simple, then

.

This result can be refined as follows.

By Lefschetz principle, we may assume

. Write

as a

-vector space quotient by a lattice

, then

acts on the

-vector space

(linear algebra over division algebras still makes sense). So

.

¡õ
An abelian variety

is called of

*CM-type* if there exists a commutative subalgebra

of degree

.

If

is of CM-type by a field

, then

is isogenous to

for some

simple and of CM-type. If

and

is simple and of CM-type, then

is a commutative field of degree

over

.

Since

embeds into

, it must embed into some

. Since the maximal subfield of

has degree

over

, we know that

since

is of CM-type. On the other hand,

by the previous corollary. Therefore all equality must hold: there is only one simple factor

and

. The rest follows from the previous proposition and the previous remark:

implies

.

¡õ
2012/04/06

##
Weil pairings

To further classify the endomorphism algebras, we need the knowledge of the Weil pairing.

Recall that from Proposition 12. The proof of Proposition 12 is indeed not quite complete: in order to take the limit, we need the compatibility of the identifications when varies. More precisely, we need the following commutative diagram In other words, we need to understand the *Weil pairing* . By definition, this pairing is induced by the duality .

Let be an isogeny. The duality is given as follows at the level of -points. Suppose and . By definition of , we can pick an isomorphism , then . Since (notice as ), we know that is another isomorphism . Hence is actually a number and the pairing takes this value.

Choose an isomorphism

, then

, which is equal to

.

¡õ
Let us slightly rewrite the Weil pairing in a more explicit form. Let for some Cartier divisor . gives an embedding , where is the sheaf of rational functions on . Composing with , we obtain a rational function . In other words, the divisor . Therefore we have (a bit more explicitly) for any .

Suppose

is a line bundle on

. The bilinear pairing

is skew-symmetric.

We need to prove that

. Suppose

and

. Then

. Let

be a rational function such that

. As explained above, we want to show that

. Write

and

. Then

Therefore

and

So

is a constant. In particular

implies that

for any

.

¡õ
Let

be a polarization. Then the pairing

is symplectic (skew-symmetric and nondegenerate).

Note that

is nondegenerate, the corollary follows since

is an isogeny (hence has finite kernel).

¡õ
More generally, an isogeny defines a pairing in a similar fashion. So here is a natural question: does every *symplectic* pairing defined by an isogeny come from a polarization? It turns out (Theorem 38) this is true over algebraically closed fields (and in general, the twice of it comes from a polarization). Our next goal is to show this fact.

Let

be an isogeny and

be a line bundle on

. Then

It follows from definition and the fact that

(Proposition

8).

¡õ
2012/04/09

Sending

to

gives an isomorphism

. In other words,

is a linear functor.

Any line bundle in

is translation-invariant. The injectivity then follows from the Theorem of Square

7 . So it is an isomorphism since both sides are abelian varieties of the same dimension.

¡õ
Let

be the Poincare line bundle on

. Then

(Notice that

.)

By the skew-symmetry, it is enough to show that

and

. Denote

, then by Lemma

24 we know that

which shows the first part.

By Lemma 25, using duality induced by the Poincare line bundle (Corollary 19). Under this identification, we see hat is given by . It follows that which shows the second part.
¡õ

As a consequence, we can prove the following theorem characterizing skew-symmetric pairings: they are "almost" induced from a polarization.

The Neron-Severi group

can be identified with the symmetric homomorphisms from

to

.

Use the equivalence of (a) and (d) in the previous theorem.
¡õ

##
Rosati involutions

Now we shall move back to study the endomorphisms of abelian varieties. Pick a polarization . We define , where . The following can be checked directly.

Therefore, is an anti-involution of . Since the polarization is not necessarily principal, this anti-involution does not necessarily preserve the integral structure . Moreover, if are two polarizations, then , where . Hence the two Rosati involutions induced by 's are related by a inner automorphism . So only the conjugacy class of the Rosati involution is canonically defined. The following is almost a tautology.

. In other words,

gives a homomorphism of algebras with anti-involutions, where

is the canonical anti-involution on

induced by the skew-symmetric pairing

.

Now strong restriction can be put on the structure of .

The Rosati involution is

*positive*: for any nonzero

,

.

2012/04/11

Now it remains to prove the following theorem.

For any

nondegenerate (i.e.,

is finite),

.

We will not give a complete proof of this theorem. Instead, we will prove that where is a constant. This is enough to be applied in the proof of the previous theorem because we only care about the ratio of two Euler characteristics.

Let

(called the

*Mumford line bundle* of

) on

. The idea is to calculate the Euler characteristic of

in two ways. On the one hand, by Theorem

34 On the other hand, for all

,

is a nontrivial line bundle lies in

, hence by Lemma

13, all the cohomology vanishes. Hence

can only have nonvanishing cohomology on

, i.e.,

is supported on the zero dimensional set

. The Leray spectral degenerates and we conclude that

By the projection formula, we know that

, thus

Hence

But

is an isomorphism, using Kunneth's formula we know that

Therefore

as desired.

¡õ
##
Classification of endomorphism algebras of abelian varieties

As a consequence of the positivity of the Rosati involution (which is deep), we know that if is a simple abelian variety, then is a finite dimensional division algebra with an anti-involution such that is positive. The classification of such algebras is done by Albert and those algebras are called *Albert division algebras* for the obvious reason.

Let

be an Albert division algebra and

be the center of

(equivalently, let

be an Albert field). Let

(it is either the whole

or an index 2 subfield). Then

is totally real and either

or

is a totally imaginary quadratic extension of

.

The proof is easy and purely algebraic.

Let

and

be the real and (non-conjugate) complex embeddings of

(thus

). Then we have an isomorphism

Moreover, the trace is simply the sum of the factors. For

,

, we know that

is a positive semidefinite quadratic form on

by continuity. But

is nondegenerate, so

must be positive definite. It follows that there can not be any complex embeddings, i.e.,

is totally real.

If , there is nothing to prove. Otherwise, and is a quadratic extension. We want to show that each . Notice that is a product of (when ) or (when ) and the involution acts as the complex conjugation or the flip respectively. By the positivity of the involution, we know that there are no factors, hence for every embedding.
¡õ

2012/04/16

Now we give the full classification of Albert algebras.

Suppose for some simple abelian variety over . Let us list the numerical invariants of these four types. Write , and . Let and .

Most of the entries are easy to derive using Proposition 15 and 16. The remaining boxed ones follows from the following

Suppose

for some simple abelian variety

over

. If

is a subfield, then

.

We have

by Corollary

34. Picking

b e a polarization gives an isomorphism between

and

. The symmetric homomorphisms corresponds to the elements of

under the Rosati involution induced by

. Therefore we have an isomorphism

. Since the Euler characteristic extends to a homogeneous polynomial

of degree

, we know

is also a polynomial function homogeneous of degree

. Explicitly, if

, then

. By Theorem

40,

. Hence

is a norm. But we already know

is a polynomial, therefore

is also a norm. In this way we obtain a norm of degree

on the field

, hence

must divide

by Proposition

14.

¡õ
In the case of elliptic curves, we have

. Applying the above classification result, we know that the endomorphism algebra of an elliptic curve is either

(Type I), a quaternion algebra over

ramified at

(Type III, when

), or an imaginary quadratic field (Type IV).

2012/04/18

##
Abelian varieties over finite fields

Suppose the base field is a finite field. Recall (Example 6) that we have a (relative) Frobenius morphism , which is a homomorphism when is a group scheme. Write , since and , we obtain -fold (relative) Frobenius .

Let

be an abelian variety. We write

and

the characteristic polynomial of

. So

is a polynomial of degree

with the constant coefficient

by Theorem

36.

- is semisimple.
- (Riemann Hypothesis) Let be a the root of . Then the absolute value of is under any embedding .

Fix

a polarization. We claim that

. In fact, by definition the claim is equivalent to

. Since

commutes with any morphism, this is equivalent to

. Thus it is enough to show that

. At the level of

-points,

is a line bundle on

. Its image under

corresponds to the composition

Tracing through definition we find that the image of

under

is the line bundle

, and the image of

under

is the line bundle

, which is the same as

on

. The claim is proved.

Since is an isogeny, we know that is invertible and (since the constant term of is nonzero). Therefore by the claim. Let be an ideal, then is also an ideal. Let under the bilinear form . The positivity of implies that by the positivity. A dimension count then shows that . Hence is semisimple. In other words, is finite etale over (without nilpotents). Since is commutative, we can write as a product of fields. We can check that fixes by the positivity. So is either totally real or CM by Lemma 28and for any embedding , . Now if is a root of , then is some . The second part follows because .
¡õ

A

*Weil -number* is an algebraic integer

such that for every embedding

,

. Two Weil

-numbers

,

are called

*conjugate* if there exists an isomorphism

sending

to

. Denote the set of Weil

-numbers by

and the conjugacy classes by

.

If

is a simple abelian variety, then

is a Weil

-number.

is a field when

is simple. The result follows from the previous theorem.

¡õ
Let be the isomorphism classes of simple objects in . Then we have a well defined map Here comes the amazing theorem due to Honda-Tate.

(Honda-Tate)
The map

is a bijection.

If there is a real embedding

, then we must in the first two cases. Assume now

for any

. Write

, then

since

is a Weil

-number. We conclude that

is totally real. Moreover,

satisfies a quadratic equation

over

, hence

is CM.

¡õ
The following is the starting point of the famous Tate conjecture.

(Tate)
The injective map (c.f. Theorem

35)

is a bijection, where

.

We omit Tate's beautiful proof but draw some important consequences.

2012/04/20

(a) implies (b) and (b) implies (c) are clear. (b) implies (a) follows from the previous theorem. (c) implies (b) follows from the fact that the Frobenius acts on

and

semisimply with characteristic polynomials

and

respectively.

¡õ
The followings are further results on the structure of for the case of the finite field .

- The center of is .
- Every abelian variety over is of CM-type.
- Assume that is simple. Let and . Then for a place of , we have

- By the previous theorem, , which means is the commutant of in . Using the double commutant theorem, we know that is the commutant of . Namely, is the center of .
- We may assume simple. Write and . Then . Write and . Then and , where each is a -vector spaces. Write , then we know that . Also, we know that . Thus Now Cauchy-Schwarz implies that the equality must hold. Hence and . In particular, is of CM-type.
- From the proof of (b) we know that splits at all finite places of above , hence . When has a real place, is one of Type I — III in Albert's classification. Type I is impossible by the restriction ( and ). For Type II and Type III, and . But the restriction shows that Type II is also impossible. Hence is of Type III and must ramify at all real places. The information at can be obtained similarly by the -divisible group version of Tate conjecture.
¡õ

2012/04/23

- . Then is even, and . Since is of CM-type by the previous theorem, must be a quaternion algebra over . So for and . Hence and is the unique quaternion algebra over ramified only at . We know that and , thus is an elliptic curve. The -rank of is zero, since the division quaternion algebra can not acts on . We say such an elliptic curve is
*supersingular*.
- is totally real and is odd. Then , is the quaternion algebra over ramified at two real places. We know that , thus is an abelian surfaces. When base change to , . So is isogenous to the product of two supersingular elliptic curves.
- is an imaginary quadratic extension of . Then is an irreducible quadratic polynomial, thus and is an elliptic curve. Because, , we find that and . There are two cases:
- does not split in , then there is only one place over . Looking at the action of on , we see that is a supersingular elliptic curve. We claim that there exists some such that , or equivalently, is a root of unity. Since , we know that for any . Since is a Weil -number, . By the product formula, we know that . So the claim is proved. By the first case , is a supersingular elliptic curve.
- splits in . Let be the two places over .

We claim this case is an *ordinary* elliptic curve (i.e., its -rank is 1). Otherwise the -rank is 0, there is only local-local part, hence the -divisible group corresponds to a formal group of dimension 1 and height 2, hence is a quaternion algebra. Because is an injection and , we obtain a contradiction.

We have proved the following results.

All supersingular elliptic curves over

are isogenous, with the endomorphism algebra

.

We have seen that every abelian variety over a finite field is of CM-type (Theorem 45). The converse is "almost" true.

(Grothendieck)
Let

be an abelian variety over

,

. If

is of CM-type, then

is isogenous to an abelian variety defined over a finite field.

We have the following stronger results for elliptic curves.

Let

be an elliptic curve over

,

. Then

if and only if

cannot be defined over a finite field.

Assume that

cannot be defined over

. Assume

for simplicity. We take the Legendre family

for

. Assume that

is transcendental over

. Let

. We can regard

as the function field of

. So we have a family of elliptic curves

and

is the generic fiber of

with

. Let

be a finite extension of

such that all endomorphisms of

are defined over

. Let

be the normalization of

in

. By base change we obtain a family

over

. Then

embeds into

for any closed point

by Proposition

22 below. By choosing two points

such that the two

's are supersingular with different endomorphism algebras, we conclude that

since it must embed into the two quaternion algebras simultaneously.

¡õ
2012/04/25

Let

. Assume

Then

is supersingular if and only if the polynomial

is zero, where

. This polynomial is called the

*Hasse polynomial*.

See Hartshorne.
¡õ

Over

, there are only finitely many supersingular elliptic curves up to isomorphism. All are defined over

.

Because the Hasse polynomial only has finitely many zeros.
¡õ

For

,

, so

. Over

, there is only one supersingular elliptic curve

, which is already defined over

. One can show

in this case. So

. The quadratic twist

for some

is isomorphic to

over

. Using the morphism

, one can check that

, hence

for a quadratic twist. We conclude that

. (In general, for

, define the quadratic twist

. Let

and

be the corresponding Galois representation arising from the Tate modules. Then

, where

is the quadratic character of

.)

##
Neron models

Let

be a DVR,

be an abelian scheme and

. Then

.

Let

be an abelian variety over

,

a discrete valuation of

and

. Consider the functor

given by

. If

is representable, then we say

is the Neron model of

. So if the Neron model exists, then it is unique.

If

is an abelian scheme. Then

is the Neron model of

. In this case, we say

has

*good reduction*.

For a general abelian variety, the existence of the Neron model is a highly nontrivial result.

(Neron)
The Neron model exists.

##
Abelian varieties of CM-type

Let us come back to the case .

Let be a CM field of degree over and be a CM-type. By Theorem 3 is an abelian variety with CM by , where is the embedding obtained by evaluating the elements of .

If is a complex abelian variety of dimension with CM by . Then acts on . Since both and are -dimension over , we know that is a 1-dimensional -vector space. Hence acts on .

There is a Hodge filtration Taking dual gives that This is an exact sequence of -modules. Moreover, acts on via , where is some subset of cardinality .

Hodge theory tells us that . So if acts on , then acts on . Hence is actually a CM-type. When we identify with , then under the embedding . Hence is isogenous to .

2012/04/30

We have proved:

Let

be an abelian variety over

with CM-type

. Then the abelian variety

is isogenous to

. In other words, a CM-type determines an isogeny class of abelian varieties.

Let

be an abelian variety over

with CM by

. Then

is defined over

. In fact, there is a unique model of

over

.

- Uniqueness. Suppose are two algebraically closed fields of characteristic 0. Then is an isomorphism for any two abelian varieties (actually we can replace by any separably closed field). In fact, the Hom set is represented by an etale finite group scheme. Observe that as over any algebraically closed field. Suppose and . Then we know that as the torsion subgroups are Zariski dense and they coincide on the torsion subgroups.
- Existence. Let be an abelian variety over with CM-type . Let finitely generated over such that and are both defined over . Then we obtain an abelian scheme . For any closed point , acts on . We claim is an abelian variety over of CM-type . The action of is clear. To see it has CM-type , we look at the action of on . This action factors through on the generic fiber, hence itself factors through . So and are isogenous by the previous theorem. Let be the kernel of . Since and are defined over , we know that itself is defined over .
¡õ

Let

be an abelian of CM-type over a number field

. Let

be a prime of

over

. Then after a possible finite base change of

,

has a good reduction at

.

Suppose is an abelian variety with CM-type and good reduction at , then the reduction is an abelian variety over the finite field , hence gives a Weil -number . Suppose is another abelian variety with CM-type and good reduction at , then for some since and will be isogenous after a finite extension by the previous proposition. In this way the CM-type determines the Weil -number up to roots of unity. Moreover, can be viewed as an element of by the following lemma.

The Weil

-numbers corresponding to a CM-type

lie inside

.

Since

is the maximal subfield in

(of degree

), we know that the commutant of

inside

is

. The lemma then follows because

commutes with the action of

.

¡õ
(Shimura-Taniyama Formula)
Assume that

contains

and

is an abelian variety with CM-type

and good reduction at a place

of

. Let

be a place of

. Then

where

.

Finally, as an application of the Shimura-Taniyama formula, let us sketch the proof of the Honda-Tate theorem 43.

(Sketch)
The injectivity follows from Corollary

36, so we only need to check the surjectivity. Assume

is a CM field (the real cases are easy). Let

be the division algebra over

given by Theorem

45, we know that

. There exists a CM subfield

containing

of degree

over

(we omit the details). Fix an algebraic closure

and write

. Then

, where

. We claim that there exists a CM-type

such that for any place

of

,

This claim allows us to construct an abelian variety possibly defined over a finite extension (due to the problem of roots of unity) with the required Weil

-number using reduction of complex abelian varieties at a prime. Finally we apply the Weil restriction functor to obtain the required abelian variety with Weil

-number

.

¡õ
#### References

[1]Mumford, D, Abelian varieties, Oxford Univ Press, 1970.

[2]Milne, James S., Abelian Varieties (v2.00), Available at www.jmilne.org/math/.

[3]Gerard van der Geer and Ben Moonen, Abelian varieties, http://staff.science.uva.nl/~bmoonen/boek/BookAV.html.