# Martin Orr's Blog

## Mumford-Tate groups

Posted by Martin Orr on Monday, 04 October 2010 at 12:37

In this post I will define the Mumford-Tate group of an abelian variety. This is a -algebraic group, such that the Hodge structure is a representation of this group. The Mumford-Tate group is important in the study of Hodge theory, and surprisingly also tells us things about the -adic representations attached to the abelian variety.

### -Hodge structures

In the last article I defined -Hodge structures. Today I will work with -Hodge structures, because it is nicer to work with vector spaces than -modules. You just replace by everywhere in the definition:

Definition. An -Hodge structure is a -vector space together with a complex structure on .

I shall write for the -vector space of a Hodge structure, and for other fields .

Given an -Hodge structure, you get an -Hodge structure by tensoring the -module with . The information you lose by doing this is equivalent to the information you lose by considering abelian varieties up to isogeny instead of up to isomorphism.

### Algebraic groups

I will need to say a little about algebraic groups, because the Mumford-Tate group is defined to be an algebraic group. Here's a concrete definition which will suffice for this article.

Definition. Let be a field of characteristic 0. A linear algebraic group over is a -subvariety of , such that the set of -points is a subgroup of .

I shall usually be lazy and call this an algebraic group.

If you want to know exactly what I mean by a -variety here, it is a reduced, separated scheme of finite type over . People often require their varieties to be irreducible, but it is convenient to allow algebraic groups to be reducible.

Here is the group of matrices of non-zero determinant, which is an affine variety (over any field you like): it is a Zariski-closed subset of affine -space, with coordinate functions given by the entries of the matrix and the reciprocal of the determinant. Observe that the multiplication and inverse maps for are morphisms of varieties.

### Hodge structures as representations of

Let be a -Hodge structure. Recall that this means that comes with a homomorphism of -algebras .

If we restrict to , then we get a group homomorphism .

The restriction of to certainly contains enough information to recover the Hodge structure, because of course . However not every multiplicative group homomorphism comes from a Hodge structure, because it might not behave well with respect to addition.

One benefit of considering the restriction of to is that there is a group homomorphism for every Hodge structure, whereas only and Hodge structures come with complex structures on .

The other benefit is that we have introducted group representations into the picture: a group homomorphism is the same thing as a representation of on the real vector space .

However, since is commutative, its representation theory is rather simple. All we can do with this representation is diagonalise it, recovering the eigenspaces and as at the end of the last article.

### The Mumford-Tate group

Now we are ready to define the Mumford-Tate group.

Definition. The Mumford-Tate group is the smallest algebraic subgroup of , defined over , whose real points contain .

The words "defined over " in this definition are essential: is itself an algebraic group over , but its defining equations usually have irrational, even transcendental, coefficients. The smallest -algebraic group containing it may be much larger: has dimension 2 as a real variety (the same as the real dimension of ), but the Mumford-Tate group of a generic abelian variety of dimension has dimension .

By construction, the Mumford-Tate group is a subgroup of , so is a representation of . It enjoys several representation-theoretic special properties, and we can use these together with the classification of linear algebraic groups and their representations to calculate the Mumford-Tate group in many cases.

### Uses of the Mumford-Tate group

The Mumford-Tate group first came up purely in Hodge theory, but for me it is interesting because many facts about the Hodge theory of abelian varieties have analogues for their -adic representations.

Let be an abelian variety over a number field , its -Hodge structure and the Mumford-Tate group.

Then there is a Galois representation on the -adic Tate module and a natural isomorphism so we may view as a homomorphism .

Deligne proved that, after replacing by a finite extension, the image of is contained in , and it is conjectured that is the smallest algebraic group to have this property. Even without proving that conjecture, proving things about the Mumford-Tate group still leads to smaller conjectures or sometimes theorems about the -adic representations. (It tends to be easier to prove things on the Hodge theory side.)

### Example: elliptic curves

Let be an elliptic curve over . We shall sketch a proof that the Mumford-Tate group is a -algebraic-group version of if has complex multiplication by , and is if does not have CM.

With respect to the basis of , the complex structure is given by

But is usually not a basis of . Suppose that be a basis of , with . With respect to this basis, we get

There are two cases:

1. has complex multiplication, say by the field .

Then we can choose and get The image of is those matrices of the form , which is a group defined by polynomials in , so this is the Mumford-Tate group.

The -points of form a group isomorphic to , so you can think of this group as a -algebraic-group version of .

2. does not have complex multiplication.

In this case, the image of is not defined by polynomials with rational coefficients (see comments for proof), so the Mumford-Tate group must be larger. In particular, has dimension 3 or 4, and if it has dimension 4 then it is equal to .

The only connected algebraic subgroups of of dimension 3 are and conjugates of the upper triangular subgroup. certainly contains elements whose determinant is not 1, so we only have to rule out conjugates of the triangular group. This could probably be done by direct calculation, but it is much easier to use a little more theory to say that must be reductive, which the triangular group is not.

So the Mumford-Tate group of an elliptic curve without CM is always .

For higher dimensions, such explicit calculations would be cumbersome, and instead we use the classification of algebraic groups.

1. Tate modules From Martin's Blog

I said after my last post that I would write something about -adic representations coming from abelian varieties. I have finally got around to doing so: here I will tell the story of how these representations are defined, and show that the Tate mo...

2. Images of Galois representations From Martin's Blog

In this post, I will continue to talk about the -adic representations attached to abelian varieties, and in particular the images of these representations. I will define algebraic groups approximating , which are often more convenient to work with...

3. Hodge classes on abelian varieties From Martin's Blog

In this post I will define Hodge classes and state the Hodge conjecture. I will restrict my attention to the case of abelian varieties and say the minimum amount necessary to be able to discuss the relationships between the Hodge, Tate and ...

4. Deligne's theorem on absolute Hodge classes From Martin's Blog

Today I will outline the proof of Deligne's theorem that Hodge classes on an abelian variety are absolute Hodge. The proof goes through three steps of reducing to increasingly special types of abelian varieties, until finally one reaches a case ...

1. Barinder Banwait said on Friday, 08 October 2010 at 17:09 :

Questions, in order of importance:

1. Regarding the example, why is it that, when E does not have CM, the image of h is not rational? You say that is irrational, but why? Is a primitive element for the number field ?

2. Why is {1,i} NOT a basis of , but a basis for ? (Is it because, for real vector spaces, you can always choose a basis so that it is standard, but you can't do this over ?)

3. In your case 2 of the example (which is amusingly labelled case 1), you have a paragraph starting "The only connected algebraic subgroups of GL2 of dimension 4 are...". Should that read "... of dimension 3..."?

4. Why do isogenous abelian varieties have the "same" -Hodge structures?

5. Where does the formula come from?

6. Will you write a post elaborating more on the connection with -adic representations? I would like to understand why , as well as consequences of Im .

2. Martin Orr said on Sunday, 10 October 2010 at 10:43 :
1. Yes, I was wrong that must be irrational whenever E does not have CM. Maybe it is harder than I thought to show that the MT group must be by bare hands. I shall think about this.

2. The fact that is usually not a basis of is what makes the Mumford-Tate group interesting. This is simply because is not an element of , which is the set of rational multiples of elements of the lattice.

3. You are right on both counts.

3. Martin Orr said on Monday, 11 October 2010 at 20:15 :

Here is a correct argument that the image of is not defined over for an elliptic curve without CM:

If is in an imaginary quadratic field, then must be equal to that field (since both the field and are -vector spaces of dimension 2) and so the curve has CM.

If is not in an imaginary quadratic field, then at least one of and is irrational (exercise for reader).

Let be the ideal of complex polynomials which vanish on the image of . The proof relies on the observation that if were defined over , then would be closed under all automorphisms of fixing (i.e. all automorphisms of ).

Now if is irrational, then we can pick an automorphism of such that . The ideal contains , so also , and hence contains . With the assumption that , this implies that is zero on all of , which is false.

If is irrational, then do the same thing with the polynomial .

4. Martin Orr said on Wednesday, 13 October 2010 at 22:13 :

4. Here is a formal proof that if and are isogenous abelian varieties, then their -Hodge structures are isomorphic: let be an isogeny. Then there is an isogeny such that and are multiplication by some integer on and respectively.

Let and be the induced morphisms of -Hodge structures. Now is also a morphism of -Hodge structures, inverse to . (We need to work with -HS rather than -HS here to be able to divide by .)

Less formally, if you think about a -HS as being a full lattice embedded in , then the associated -HS is the -vector space generated by together with its embedding in . If are isogenous abelian varieties then we can scale their lattices so that is a finite-index subgroup of , and I hope it is then clear that and generate the same -vector space.

5. is the dimension of the group of transformations of a -dimensional vector space which take a symplectic form on that space to a scalar multiple of itself. The of an abelian variety has a symplectic form, called a Riemann form or polarisation, and the Mumford-Tate group preserves it up to scalars, so is contained in .

In the case of elliptic curves, the polarisation doesn't really do anything (and ) so I didn't need to talk about it in the example.

6. Yes, but it might not be for a couple of weeks as I have some non-blog things to write.