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.

Theorem 1
1. is compact (hence is a complex torus).
2. has a natural (unique) structure as a projective variety.
Proof (Sketch)
1. We need to show that the image of is a lattice. This follows from the isomorphism using Hodge theory.
2. 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
Theorem 2 Let , where is a lattice. Then the followings are equivalent:
1. can be embedded into a projective space.
2. There exists an algebraic variety such that .
3. There exists algebraically independent meromorphic functions on .
4. There exists a positive definite Hermitian form such that is integral.
Lemma 1 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.

Definition 1 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.

Definition 2 A CM-type is a choice of such that has elements and .

Thus a CM-type gives an isomorphism by evaluation.

Theorem 3 Let be the ring of integers of . Then is an abelian variety.
Proof (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

Question 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

Proposition 1 Let be a connected complex compact Lie group, then , where is a complex vector space and is a lattice.
Proof 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.

Corollary 1 Let be an abelian variety over of dimension . Then
1. is commutative and divisible. In particular, the multiplication-by- map is a surjective group homomorphism and has kernel .
2. .
3. Let , be two abelian varieties. Let be the group of homomorphisms as complex Lie groups from to . Then is an injection. (But it is not surjective in general since additional complex structure is needed.)

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.

Definition 3 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 .
Definition 4 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 .
Remark 1 It turns out every abelian variety is projective (cf. Corollary 12). We could replace "complete" by "projective" in our definition, but completeness is more convenient in constructing abelian varieties, since we do not know they are projective a priori.
Question The structure of and as an abstract group.
Theorem 4 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 .
Theorem 5 (Mordell-Weil) Suppose is a number field. Then is a finitely generated abelian group.
Question 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,

Theorem 6 is a finite generally free abelian groups.

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

Definition 5 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.

Theorem 7 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.

Conjecture 1 (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.

Question Line bundles on abelian varieties.
Theorem 8 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.

Question The cohomology of line bundles, Riemann-Roch problems.
Theorem 9 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

Theorem 10 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.

Proposition 2 (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 .
Corollary 2 Let , be two abelian varieties and be a morphism of algebraic varieties. Then , where and .
Proof We may assume . We need to show that is homomorphism of abelian varieties. Define Then . Then the Rigidity Lemma 2implies that . ¡õ
Proof (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 (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
Remark 2 Since the group law on an abelian variety is commutative, we shall use instead of to denote the multiplication and to denote the inverse map.

## The Theorem of the Cube

Lemma 2 Suppose , then is surjective.
Proof (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 11 (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.

Remark 3 Let be the category of pointed complete varieties i.e. a pairing over . Let be a contravariant functor, where is the category of abelian groups. Let be the projection and be the -the inclusion . We define and . Under this general setting, the following is always true:
Lemma 3 .
Definition 6 The functor is called of order (linear when , quadratic when ) if is injective (equivalently, is surjective).
Example 1 The Theorem of Cube implies that the functor is quadratic: the map is injective.
Example 2 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.
Example 3 Suppose , then is quadratic.

Here come several corollaries of the Theorem of Cube.

Corollary 3 Let , , be complete varieties. Then every line bundle on is of the form .
Proof Since is equivalent to . ¡õ
Corollary 4 Let be an abelian variety and be an arbitrary variety. Let be three morphisms. Then for any line bundle on ,
Proof 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
Corollary 5 Let be a line bundle on . Then .
Proof Applying Corollary 4 to the case , and , we obtain that Let , then , thus as . Hence This completes the proof. ¡õ
Corollary 6 If is symmetric, i.e., , then .
Corollary 7 (Theorem of the Square) For any and a line bundle on , we have where is the translation-by- map.
Proof Apply Corollary 4 to , and . ¡õ
Remark 4 If follows the Theorem of the Square that the map is a group homomorphism for any line bundle on , where is any field extension.

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:

1. .
2. Any short exact sequence gives a long exact sequence
3. 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

Theorem 12 (Comparison theorem) Suppose is separated, then for any cover of affine opens.
Corollary 8 (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.

Theorem 13 Let be a proper morphism of noetherian schemes. Suppose , then . In particular, when , is a finite dimensional -vector space.
Theorem 14 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
Example 4 Consider the situation . Then for any line bundle on , is flat over . This the case we will use later (cf. Theorem 15).
Definition 7 Let be a proper scheme over and . We define the Euler characteristic .
Corollary 9 Let and be as in Theorem 14. Then is a locally constant function on , where and .
Proof 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
Corollary 10 Let and be as in Theorem 14. Then is an upper semicontinuous function, i.e., is closed for any .
Proof 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
Corollary 11 Let and be as in Theorem 14. In addition, assume that is connected and reduced. Then then followings are equivalent:
1. is locally constant.
2. is locally free of finite rank and the natural map is an isomorphism.
Proof 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 .

Theorem 15 (Seesaw Theorem) Suppose is algebraically closed, is a complete variety over and is an arbitrary variety over . Let be a line bundle on . Then
1. is closed.
2. There exists some line bundle on such that .

We need the following easy lemma.

Lemma 4 A line bundle on is trivial if and only and .
Proof Suppose and . Choose two sections and , we obtain a morphism , which is an isomorphism since ( is complete). ¡õ
Proof (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
Remark 5 When is not necessarily algebraically closed, is the algebraic closure of in . Similarly one can show that there exists on such that , where .

## 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.

Lemma 5 For any , on , there exists an irreducible curve containing , .
Proof 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 (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

Remark 6 The proof of the Theorem of Cube is a bit tricky. If you do not like it, here is an easier proof for . By exponential sequence , we have an exact sequence Then we know that is quadratic since both and are quadratic by Kunneth formulas (cf. Remark 3).

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 .

Definition 8 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).

Lemma 6 Let be a line bundle on . Then if and only if .
Proof "": Pulling back through , we know for any .

"": Let . Then and are trivial since . So is trivial the Seesaw Theorem 15. ¡õ

Definition 9 For , we define . The it is clear that for , we have .
Lemma 7 is closed in . (So has a natural structure of an algebraic group.)
Proof Let . Then is closed by the Seesaw Theorem 15. ¡õ

Now we can state the main theorem of this section:

Theorem 16 Let ,where is an effective divisor on . Then the followings are equivalent:
1. is ample.
2. is finite.
3. is finite.
4. The linear system is base-point-free and defines a finite morphism .

This theorem has the following important consequence.

Corollary 12 Every abelian variety is projective.
Proof 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
Remark 7 In the process of proving the Riemann hypothesis for algebraic curves, Weil constructed the Jacobian of a curve as a (complete) abelian variety. But it was not known that abelian varieties are projective. Weil thus reestablished the foundation of algebraic geometry and introduced the notion of abstract varieties. Unfortunately, the foundation was rewritten by Grothendieck for the second time and Weil's language was mostly abandoned nowadays. The fact that any abelian variety is projective is really deep: it took more than 10 years before it was proved in 1950s.
Proof (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 .

Lemma 8 Let be an irreducible curve and be a divisor such that . Then for any , .
Proof (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

Corollary 13 is surjective.
Proof 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 .

Theorem 17 Suppose has dimension , then
1. .
2. is separable if and only if , where .
3. The inseparable degree of is at least .

2012/02/22

## Isogenies of abelian varieties

Definition 10 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.
Definition 11 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 .
Proposition 3
1. Let be a coherent sheaf on with generic rank . Then .
2. 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.

Theorem 18 .
Proof 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 (Proof of Proposition 3) For simplicity, we prove the case when is smooth and is finite.
1. 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.
2. By adjunction, , thus . Since is a coherent sheaf of generic rank , we also know that by part (a). ¡õ
Theorem 19
1. is separable if and only if , where .
2. The inseparable degree of is at least .
Proof
1. 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.)
2. 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
Corollary 14 Let be the -torsion points of an abelian variety . Then where .
Proof 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.

Definition 12 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:

Definition 13 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.

Remark 8
1. For any morphism , the base change of a group scheme over has a natural structure of group scheme over .
2. We can define the right multiplication of by an element in . More precisely, for and , we have a right multiplication map . At the level of -points, this induces the usual right multiplication on .
3. Let be a group scheme over . Let be a open (resp. closed) subscheme of . We say is an open (resp. closed) group subscheme of if the group structure on is compatible with that of .
4. Let , be two group schemes over . A morphism is called a homomorphism if the following diagram commutes: Moreover, we define to be the pullback thus is a group scheme over . If, in addition, is a closed embedding, then is a closed group subscheme of .
Example 5
1. An abelian variety over is a group scheme over .
2. 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 .
3. 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 .
4. The multiplication-by- map is defined to be , . The kernel is a closed group subscheme and for any -scheme .
5. 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 .

Definition 14 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 .

Lemma 9 If , then . If , then .

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

Definition 15 A restricted Lie algebra over a field of characteristic is a Lie algebra together with a map such that
1. ,
2. ,
3. for some universal non-commutative polynomial depending only on .
Remark 9 The exact expression of is not important for us, but we will need the fact that , i.e., has no constant term.
Remark 10 If is abelian (we will see this is the case for the Lie algebra of an abelian variety), then and is -linear.

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.

Proposition 4 The map is an isomorphism, where .

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

Remark 11 Let be a scheme. To give a derivation is the same to giving an automorphism as -algebras of the form . In other words, let , then we can view as a morphism over such that its restriction to over is the identity. Under this correspondence, for be a group scheme, a derivation on is left invariant if and only if is a commutative diagram.
Proof (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 .

Lemma 10 Let and . Then .
Remark 12 We omit the proof, which is easy to check by reducing to the affine case. This is an analogy to the fact in differential geometry that .
Corollary 15 If is commutative, then is abelian.
Proof 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 .

Lemma 11
1. is open, closed and is a group subscheme of .
2. is geometrically irreducible.
3. is of finite type.
Proof (Sketch)
1. 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.
2. 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.
3. For any affine open, is surjective, hence is quasicompact. It is quasicompact and locally of finite type, hence is of finite type. ¡õ
Remark 13 We do not claim that itself is smooth. In fact there are many examples of reduced group schemes in characteristic .
Example 6 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.

Theorem 20 If , then is smooth (hence is smooth).
Proof (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).

Theorem 21
1. is represented by a scheme (hence a group scheme), which is locally of finite type over .
2. The connected component is quasiprojective, and is projective if is smooth.
Example 7 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.
Definition 16 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 .

Definition 17 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
1. ,
2. , .
3. .
Lemma 12 Let be a line bundle on and be the corresponding point. Then if and only if and are algebraically equivalent.
Proof Let . Then composition gives an algebraical equivalence between and via on .

Conversely, suppose and are algebraically equivalent. Then by definition we have a chain of schemes . Shrinking if necessary, we can equip each a trivialization on . Then these pairs gives rise to morphisms . By connectedness, all 's map into . In particular, lie in the same component. ¡õ

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

Definition 18 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 .

Theorem 22 .
Proof 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

Theorem 23 Let be an ample line bundle. Then the map is surjective.

To prove this theorem, we need the following lemma.

Lemma 13 Let be a nontrivial line bundle. Then for any .
Proof First let us show that , Otherwise, for some effective divisor . Since , we know that . Hence . Therefore we know that , hence .

In general, let be the smallest integer such that . The identity map gives us an identity map This implies that . But Kunneth formula and the minimality of , we have know , a contradiction. ¡õ

Proof (Proof of Theroem 23) We may assume . Assume is not of the form . We let be a line bundle on . We calculate in two ways via the Leray spectral sequences and .

To calculate , we look at the fibers . By construction is nontrivial by assumption and belongs to . It follows from the previous lemma that . So by Theorem 11. Hence is zero by the first Leray spectral sequence.

On the other hand, we can loot at the fibers . By construction , which is trivial if and only if . Since is ample, is finite by Theorem 16. Again by the previous lemma, We know that and is a coherent sheaf supported on a finite set . By the second Leray spectral sequences, we know . Because it is supported on a finite set, we conclude that is actually zero for any . Thus . This is a contradiction because . That completes the proof that muse be of the form . ¡õ

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 .

Corollary 16 If is ample, then is surjective with finite kernel. In particular, .
Proof It follows from Theorem 16 and 23. ¡õ
Theorem 24 is smooth (hence is an abelian variety).
Remark 14 is always smooth for abelian varieties and curves, but may not be smooth in general (e.g., for some surfaces).
Proof 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
Remark 15 This is not the best proof: we can really construct a line bundle in using a cocyle in .

## Hopf algebras

Now let be a complete variety over such that . Then

1. is a graded commutative -algebra with the product given by
2. 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.
Lemma 14 makes a finite dimensional positively graded commutative and cocommutative Hopf -algebra such that , .
Remark 16 We say a Hopf algebra (or bialgebra) is graded if is a graded vector space and all the morphisms preserve the gradings. We say a Hopf algebra (or bialgebra) is graded commutative if for any homogeneous elements , we have . Similarly we can define the notion of graded cocommutative Hopf algebras.

2012/03/05

Example 8 (Hopf algebras)
1. 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.
2. 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.
3. 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.
4. 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 .
Theorem 25 (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.

Corollary 17 Suppose is an abelian variety over of dimension , then and is an isomorphism. In particular, and .
Proof 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.

Definition 19 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, ).
Remark 17 Note that a polarization does not necessarily come from a line bundle over (there are counter examples, though hard to construct). The polarization really lives in the image of the map .
Definition 20 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.

Remark 18 By Lemma 12 and the fact in an algebraic family of line bundles the Euler characteristic does not change, we know that It is also true that by the same argument as in the proof of Theorem 24.
Definition 21 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 .
Proposition 5 is smooth (hence is an abelian variety).
Proof 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
Theorem 26 admits a canonical principal polarization.

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

Definition 22 We define the theta divisor to be the scheme-theoretic image . By definition,

2012/03/07

Example 9 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.
Proof (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.

Example 10 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

Lemma 15 Let be a line bundle on and . Then .
Proof 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 .

Definition 23 is a commutative finite group scheme over , called the Cartier dual of . The natural dual functor satisfies .
Definition 24 Let , be two commutative group schemes over . We define the functor
Proposition 6 .
Proof 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
Example 11 For , . Therefore . One also has (the dual pairing is given by the truncated exponent).

The following is the main result of this section.

Theorem 27 Let be an isogeny of abelian varieties. Then the induced morphism of dual abelian varieties is also an isogeny and .
Remark 19 Here is an informal reasoning which can be made rigorous. For a line bundle on , we let be the total space . Then is a scheme affine over and is a -torsor. Moreover, for and a line bundle on , we have . Applying to the case of abelian varieties, we obtain that the dual abelian variety can be identified as -torsors on satisfying the following commutative diagram These -torsors correspond to central extensions of commutative group schemes In other words, . Using this interpretation of dual abelian varieties, applying to the exact sequence gives the required exact sequence as 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:

1. there is a canonical isomorphism ;
2. restricting to the diagonal we get ;
3. 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 .

Theorem 28 (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.

Lemma 16 Let be an isogeny of abelian varieties. Then is fppf.
Proof We only need to check that is flat, which can be shown by generic flatness since is a morphism of group varieties. ¡õ
Proof (Proof of Theorem 27) equals to which is exactly the isomorphism classes of on such that . Now we apply the previous theorem to our case , and (it is fppf by the previous lemma), we obtain

Let . Then with two projections and (multiplication on ) to . Also with three projections , and to . By construction, and , we know that where the latter equality is because is an abelian (hence projective) variety. Now the condition can be translated into that for the counit since the diagonal map is given by . Similarly, the cocycle condition can be translated into that . So It follows that . ¡õ

Corollary 18 If is an isogeny, then .
Proof Because for . ¡õ
Definition 25 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 .
Example 12 Let be the Poincare line bundle on . Then by the definition of . Also we have . Denote .
Proposition 7 Let be a line bundle on . The we have the following commutative diagram
Proof 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
Corollary 19 is an isomorphism.
Proof 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 .

Definition 26 A morphism is called symmetric if .
Remark 20 Any polarization is be symmetric. So not every isogeny is a polarization because there exists non-symmetric isogenies.

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

Proposition 8 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

Definition 27 Let be a finite group scheme over . We say
• is local if (connected).
• is etale if is an etale -algebra. Recall that a -algebra of finite type is called etale if , equivalently, , where 's are finite separable field extensions of .
Example 13 is etale if and only if . It is local if and only if is a -power.
Example 14 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.

Lemma 17 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 .
Proof (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
Corollary 20 The category of etale -group schemes is equivalent to the category of finite groups with -action.
Example 15 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.
Proposition 9 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 .
Proof (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
Corollary 21 If is a -group scheme of finite type, then is an etale -group scheme and is a homomorphism.
Corollary 22 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.
Proof (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
Definition 28 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.
Corollary 23 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.
Proof Apply the previous decomposition twice. ¡õ
Remark 21 Suppose is algebraically closed. All etale-etale -group schemes must be a product of etale -group schemes of the form . Furthermore, we have a non-canonical isomorphism depending on a choice of primitive root of unity. Similarly, all etale-local -group schemes must be a product of and all local-etale -group schemes must be a product of . However, there are a lot of local-local -group schemes even in this case.
Remark 22 Suppose . Then since there are no nontrivial local -group schemes.

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

Definition 29 A local -group scheme is called of height one if for any , where is the maximal ideal at .
Lemma 18 Suppose is of height one, then the coordinate ring . In particular, is a -power.
Proof 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).

Lemma 19 is a group homomorphism.
Proof 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

Theorem 29 The functor is an equivalence of categories between height one group schemes and -Lie algebra.

2012/03/23

Proof (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
Remark 23 When , there is an analogous equivalence between Lie algebras and formal groups.
Corollary 24 If is commutative of height one, then is zero.
Proof The morphism gives which is multiplication by , hence is zero in characteristic . Now by the previous theorem, we know that . ¡õ
Corollary 25 If is local and commutative, then there exists some such that is zero.
Proof 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
Corollary 26 If is finite and commutative, then there exists some such that is zero.
Proof Use the decomposition in Corollary 22 and the previous corollary. ¡õ
Corollary 27 Let be an isogeny of abelian varieties. Then there exists some and an isogeny such that .
Proof We need another fact from Grothendieck's descent theory.
Theorem 30 Let be faithfully flat, of finite presentation and be a scheme. Then is an equalizer diagram.

Apply this theorem to and . We only need to show that the two compositions are the same. This can be done by choosing killing (which can be chosen as from the previous discussion). So factors through . ¡õ

Remark 24 From the proof one sees that can be chosen as .
Example 16 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.
Lemma 20 The Cartier dual isomorphic to .
Proof 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 .

Proposition 10 The natural morphism is an isomorphism.
Proof 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 .

Proposition 11 is an invariant under isogeny (called the -rank of ).
Proof 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
Corollary 28 .
Proof 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 .

Definition 30 Suppose is a free -module of finite rank with an action of . We define the Tate twists of by for and for .
Proposition 12 .
Proof From , we know that Passing to the limit we obtain that . ¡õ
Proposition 13 Let be an isogeny and be its kernel. Then we have a short exact sequence of -modules where is the -Sylow subgroup of .
Proof 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 .

Definition 31 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.

Definition 32 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.
Remark 25 For historical reason the -divisible group is defined to be an inductive system as opposed to the projective system in the -adic Tate module. Nevertheless, we can also define it as a projective system using .
Example 17 Let be an abelian variety. Define . This -divisible group is the right replacement of the -adic Tate module . In particular, one can recover from .
Definition 33 Since kills , we know that for some . We call the the height of . By induction, one sees that .
Example 18 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.

Definition 34 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 .
Lemma 21 Let be an isogeny. Then is invertible in .
Proof By Corollary 27, there exists isogenies and such that and , both of which are invertible in . Hence is invertible. ¡õ
Theorem 31 (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 .
Proof 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
Definition 35 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.

Corollary 29 Every abelian variety is isogenous to , where 's are pairwise non-isogenous simple abelian varieties. Moreover, this decomposition is unique up to permutation.
Remark 26 In fact more is true: is a semisimple abelian category. (Note that this is not true for since the kernel of an isogeny is not necessarily an abelian variety.)
Corollary 30 Let be an abelian variety, then is a semisimple algebra over , hence can be written as a product of matrix algebra of division algebras .
Remark 27 We will see that these semisimple algebras are actually finite dimensional.
Definition 36 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 , .
Definition 37 Define to be if is an isogeny and 0 otherwise.
Theorem 32 There is a unique way to extend to a (homogeneous) polynomial of degree .
Remark 28 This puts strong restriction on : must be discrete in the -vector space (hence cannot be too divisible).

To prove this theorem, we need the following lemma.

Lemma 22 Pick an ample line bundle on such that (this is always true though we did not prove it). Then .
Proof (Proof of Theorem 32) The function is of degree in the sense that by Theorem 18. There is a unique way to extend to such that it is homogeneous function of degree : for any , there exists some such that ; then . We need to show that this extended function is actually a polynomial.

It is enough to show that for any , is a polynomial of . The previous lemma gives . So it suffices to show that is a polynomial of . Applying Corollary 4to the case , and , we know that where . So Therefore where is independent of . Summing up implies that is of the form for some and independent of . Hence is a polynomial of . ¡õ

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.

Theorem 33 (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.

Proof (Proof of Lemma 22) Suppose is an isogeny. By Proposition 3, . This is equal to using the previous theorem. (Or apply directly the weaker Theorem 34below).

Now suppose is not an isogeny. We need to show that . From the previous theorem. . By the projection formula, this is also equal to. Because is not an isogeny, the image of is a proper subvariety of . So is actually a polynomial of degree , thus . ¡õ

Remark 29 Note that if is very ample, then is the degree of corresponding embedding , which is the self-intersection number . Hence when is very ample, . In general, for an arbitrary , we can attach the first Chern class in the first Chow group (when is defined over , we can take ). Then and there is a natural degree map (when is defined over , ). The general Riemann-Roch formula says that This general formula follows (tautologically) from the very ample case.

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.

Definition 38 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).
Example 19 If is an isogeny of abelian varieties. Then is a -torsor over .
Theorem 34 Suppose is finite, is a -torsor (hence is finite and ) and is proper. Then for any coherent sheaf on , we have
Proof 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
Lemma 23 Any line bundle on an abelian variety over can be written as , where is symmetric and .
Proof 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 (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

Theorem 35 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).
Proof 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

Corollary 31 The Neron-Severi group is a finitely generated free abelian group (of rank ).
Proof By definition, there is an injection given by . The latter one has finite rank by the previous theorem. ¡õ
Corollary 32 is a finite dimensional semisimple -algebra.

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

Definition 39 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 .
Proposition 14 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 .
Definition 40 is called the reduced norm and is called the reduced trace.
Proof (Sketch) When , the norm and descents to . Similarly, the trace (as it factors through the 1-dimensional quotient and also descents to .

In general, . We have and . Thus we find that . Since comes from , we know that 's are all the same, hence is of the desired form. The argument is similar for trace forms. ¡õ

2012/04/04

Theorem 36
1. The degree function agrees with , where . In other words, for , .
2. Let and (equal to the characteristic polynomial of ). Then .
Proof 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

Definition 41 Write the characteristic polynomial of . We call the norm of and the trace of .
Remark 30 If we decompose and . Then . In fact, is the second leading coefficient 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.

Proposition 15 If is simple, then .

This result can be refined as follows.

Proposition 16 If and is simple, then .
Proof 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
Remark 31 This refinement fails when (e.g. for supersingular elliptic curves).
Definition 42 An abelian variety is called of CM-type if there exists a commutative subalgebra of degree .
Remark 32 When simple, is a division algebra and its commutative subalgebras are actually subfields. Since the maximal subfield of a division algebra has degree , by Proposition 15, actually equals to if is simple and of CM-type.
Proposition 17 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 .
Proof 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.

Proposition 18 Let and . Then
Proof Choose an isomorphism , then , which is equal to . ¡õ
Remark 33 This proposition implies that . Hence prove the desired commutativity of the previous diagram.
Remark 34 By the same argument, in general, for an isogeny, we have for any and .

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 .

Theorem 37 Suppose is a line bundle on . The bilinear pairing is skew-symmetric.
Remark 35 The dual abelian variety is not seen in the Weil pairing of elliptic curves: the reason is that for elliptic curves we have a canonical choice of a principal polarization given by the line bundle .
Proof 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
Corollary 33 Let be a polarization. Then the pairing is symplectic (skew-symmetric and nondegenerate).
Proof 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.

Lemma 24 Let be an isogeny and be a line bundle on . Then
Proof It follows from definition and the fact that (Proposition 8). ¡õ

2012/04/09

Lemma 25 Sending to gives an isomorphism . In other words, is a linear functor.
Proof 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
Proposition 19 Let be the Poincare line bundle on . Then (Notice that .)
Proof 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.

Theorem 38 Let be a homomorphism. Then the following are equivalent:
1. is symmetric (i.e., ).
2. is skew-symmetric.
3. for some line bundle .
4. Over , for some .
Proof (d) implies (a), (b): we have shown them in Proposition 7 and Theorem 37.

(c) implies (d): this uses the theory of theta groups, see Mumford Section 23, Theorem 3, p. 231.

(b) implies (c): Let . Then Lemma 24 implies that Using the previous proposition, this is equal to , hence is equal to by the skew-symmetry. It follows that .

(a) implies (c): Notice that . Therefore By the symmetry of , we know that . Therefore . ¡õ

Corollary 34 The Neron-Severi group can be identified with the symmetric homomorphisms from to .
Remark 36 Over , both side should be replaced by certain etale group schemes.
Proof 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.

Lemma 26 , and .

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.

Lemma 27 . 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 .

Theorem 39 The Rosati involution is positive: for any nonzero , .

2012/04/11

Remark 37 In complex geometry, a polarization on a complex manifold is simply a choice of an ample line bundle, i.e., a line bundle with a metric of positive curvature. This does not generalize to arbitrary fields. But for abelian varieties, the positivity of the Rosati involution somehow reflects the fact that polarizations are coming form ample line bundles.
Proof We may assume that for very ample and . We claim that The theorem easily follows from this claim since is ample and is effective.

To prove the claim, consider the homomorphism . Its degree is equal to , by the definition of the Rosati involution, this is equal to So we need to understand the ratio of the degrees of and in order to extract , the second leading coefficient of . From Theorem 40 below, we can compute the degrees and obtain By the Riemann-Roch Theorem 33, we know that therefore which finishes the proof. ¡õ

Now it remains to prove the following theorem.

Theorem 40 For any nondegenerate (i.e., is finite), .
Remark 38 By Theorem 16, a nondegenerate line bundle asscociated to an effective divisor is automatically ample.

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.

Proof 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
Remark 39 In fact, , though we omit the proof here.

## 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.

Lemma 28 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.

Proof 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.

Theorem 41 (Albert)An Albert algebra is one of the following types:
• Type I: is totally real.
• Type II: is totally real, is a quaternion algebra over such with the anti-involution corresponding to transpose of matrices.
• Type III: is totally real, is a quaternion algebra over with the standard anti-involution such that , where is the Hamilton quaternion algebra over .
• Type IV: ( is a CM-field), is a division algebra over of dimension . For every finite place of , , where is the local invariant of and is the automorphism induced by the anti-involution on . Moreover, if , then . In this case, we have with the anti-involution corresponding to conjugate transpose.
Remark 40 When , the anti-involution induces an isomorphism as central simple algebras over . But is always isomorphic to a matrix algebra . Therefore has order either 1 or 2 in . Every involution on a quaternion algebra differs from the standard one by an inner automorphism (by the Skolem-Noether theorem). One can then show there are only Type I — Type III. The argument for Type IV is similar. We omit the details of this beautiful proof here.

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

Proposition 20 Suppose for some simple abelian variety over . If is a subfield, then .
Proof 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
Remark 41 One might ask to what extent the above restrictions are complete. In characteristic zero, the answer is known due to Albert. On the other hand, not much seems to be known in positive characteristics.
Example 20 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 .

Definition 43 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.
Theorem 42
1. is semisimple.
2. (Riemann Hypothesis) Let be a the root of . Then the absolute value of is under any embedding .
Remark 42 This case of abelian varieties is the main motivation for Weil to formulate the general Weil conjecture.
Proof 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

Definition 44 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 .
Corollary 35 If is a simple abelian variety, then is a Weil -number.
Proof 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.

Theorem 43 (Honda-Tate) The map is a bijection.
Remark 43 This theorem is highly nontrivial, for example, it tells we can start with a Weil number (e.g. if is even) to produce an abelian variety!
Lemma 29 Let be a Weil -number. Then there are only three cases:
1. , and .
2. , , is totally real and .
3. For any embedding , , is CM.
Proof 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.

Theorem 44 (Tate) The injective map (c.f. Theorem 35) is a bijection, where .
Remark 44 The image of acts as on , where is the arithmetic Frobenius. In fact, is the absolute Frobenius of , hence is the identity.

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

Corollary 36 Let , be abelian varieties over . Then the followings are equivalent:
1. is isogenous to a subabelian variety of defined over .
2. There is a -equivariant injective map for some .
3. . In particular, when , are isogenous over , then we must have . This proves the injectivity of the map in Theorem 43.

2012/04/20

Proof (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 .

Theorem 45
1. The center of is .
2. Every abelian variety over is of CM-type.
3. Assume that is simple. Let and . Then for a place of , we have
Proof
1. 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 .
2. 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.
3. 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

Example 21
1. . 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.
2. 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.
3. 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:
1. 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.
2. 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.

Theorem 46 Let be an elliptic curve over . Then there are 3 possibilities:
1. , is supersingular, .
2. is an imaginary quadratic field, and does not splits, is supersingular,
3. is an imaginary quadratic field, and splits, is ordinary.
Theorem 47 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.

Theorem 48 (Grothendieck) Let be an abelian variety over , . If is of CM-type, then is isogenous to an abelian variety defined over a finite field.
Remark 45 The word "isogenous" cannot be replaced by "isomorphic" in this theorem.

We have the following stronger results for elliptic curves.

Theorem 49 Let be an elliptic curve over , . Then if and only if cannot be defined over a finite field.
Proof 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

Proposition 21 Let . Assume Then is supersingular if and only if the polynomial is zero, where . This polynomial is called the Hasse polynomial.
Proof See Hartshorne. ¡õ
Corollary 37 Over , there are only finitely many supersingular elliptic curves up to isomorphism. All are defined over .
Proof Because the Hasse polynomial only has finitely many zeros. ¡õ
Remark 46 We can also count explicitly the number of supersingular elliptic curves defined over .
Example 22 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

Proposition 22 Let be a DVR, be an abelian scheme and . Then .
Proof Let be an endomorphism. Let be its image. Take the closure of in (), then is a proper flat subgroup scheme of (flatness means torsion-freeness over a DVR). We claim that the connected component is smooth (hence is an abelian scheme). Let be the generic point of the special fiber and be the local ring of at .

Then is DVR (1-dimensional). By the properness and valuative criterion, the endomorphism of the generic fiber automatically extends to an open subset of the special fiber. But is the graph of the extension. So is generically reduced, hence reduced, which proves the claim. The first projection is an isomorphism , because it is a morphism between abelian schemes and is generically an isomorphism. The second projections to then gives a morphism extending . ¡õ

Definition 45 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.
Proposition 23 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.

Theorem 50 (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:

Theorem 51 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.
Proposition 24 Let be an abelian variety over with CM by . Then is defined over . In fact, there is a unique model of over .
Proof
1. 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.
2. 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
Proposition 25 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 .
Proof The proof is based on the following theorem.
Theorem 52 (Neron-Ogg-Shafarevich) For any abelian variety , has good reduction at if and only if the inertia subgroup acts trivially on .

Come back to our case, since has CM by , we know that , where the second isomorphism is due to the dimension reason. So maps to a compact subgroup in , hence maps to . Since is pro- up to finite index and, by local class field theory, the image of is pro- up to finite index, the Neron-Ogg-Shafarevich then tells us has good reduction after a finite extension. ¡õ

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.

Lemma 30 The Weil -numbers corresponding to a CM-type lie inside .
Proof 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
Theorem 53 (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 .
Remark 47 We omit the proof of this important result. Notice this formula makes sense because changing by a root of unity does not affect the result.

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

Proof (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.