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Maths > Abelian varieties > Complex abelian varieties and the Mumford-Tate conjecture

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 \mathbb{Q}-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 \ell-adic representations attached to the abelian variety.

\mathbb{Q}-Hodge structures

In the last article I defined H_1 \mathbb{Z}-Hodge structures. Today I will work with H_1 \mathbb{Q}-Hodge structures, because it is nicer to work with vector spaces than \mathbb{Z}-modules. You just replace \mathbb{Z} by \mathbb{Q} everywhere in the definition:

Definition. An H_1 \mathbb{Q}-Hodge structure is a \mathbb{Q}-vector space V_{\mathbb{Q}} together with a complex structure on V_{\mathbb{Q}} \otimes_{\mathbb{Q}} \mathbb{R}.

I shall write V_\mathbb{Q} for the \mathbb{Q}-vector space of a Hodge structure, and V_k = V_{\mathbb{Q}} \otimes_{\mathbb{Q}} k for other fields k.

Given an H_1 \mathbb{Z}-Hodge structure, you get an H_1 \mathbb{Q}-Hodge structure by tensoring the \mathbb{Z}-module with \mathbb{Q}. 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 k be a field of characteristic 0. A linear algebraic group over k is a k-subvariety G of \mathrm{GL}_n, such that the set of \bar{k}-points G(\bar{k}) is a subgroup of \mathrm{GL}_n(\bar{k}).

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

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

Here \mathrm{GL}_n 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 (n^2 + 1)-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 \mathrm{GL}_n are morphisms of varieties.

Hodge structures as representations of \mathbb{C}^\times

Let V be a H_1 \mathbb{Q}-Hodge structure. Recall that this means that V comes with a homomorphism of \mathbb{R}-algebras h : \mathbb{C} \to \mathop{\mathrm{End}_{\mathbb{R}}} V_{\mathbb{R}}.

If we restrict h to \mathbb{C}^\times, then we get a group homomorphism \mathbb{C}^\times \to \mathrm{GL}(V_{\mathbb{R}}).

The restriction of h to \mathbb{C}^\times certainly contains enough information to recover the Hodge structure, because of course h(0) = 0. However not every multiplicative group homomorphism \mathbb{C}^\times \to \mathrm{GL}(V_{\mathbb{R}}) comes from a H_1 Hodge structure, because it might not behave well with respect to addition.

One benefit of considering the restriction of h to \mathbb{C}^\times is that there is a group homomorphism \mathbb{C}^\times \to \mathrm{GL}(V_{\mathbb{R}}) for every Hodge structure, whereas only H_1 and H^1 Hodge structures come with complex structures on V_{\mathbb{R}}.

The other benefit is that we have introducted group representations into the picture: a group homomorphism \mathbb{C}^\times \to \mathrm{GL}(V_{\mathbb{R}}) is the same thing as a representation of \mathbb{C}^\times on the real vector space V_{\mathbb{R}}.

However, since \mathbb{C}^\times is commutative, its representation theory is rather simple. All we can do with this representation is diagonalise it, recovering the eigenspaces V^{-1,0} and V^{0,-1} 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 M of \mathrm{GL}(V_{\mathbb{Q}}), defined over \mathbb{Q}, whose real points M(\mathbb{R}) contain h(\mathbb{C}^\times).

The words "defined over \mathbb{Q}" in this definition are essential: h(\mathbb{C}^\times) is itself an algebraic group over \mathbb{R}, but its defining equations usually have irrational, even transcendental, coefficients. The smallest \mathbb{Q}-algebraic group containing it may be much larger: h(\mathbb{C}^\times) has dimension 2 as a real variety (the same as the real dimension of \mathbb{C}^\times), but the Mumford-Tate group of a generic abelian variety of dimension g has dimension 2g^2 + g + 1.

By construction, the Mumford-Tate group is a subgroup of \mathrm{GL}(V_{\mathbb{Q}}), so V_{\mathbb{Q}} is a representation of M. 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 \ell-adic representations.

Let A be an abelian variety over a number field K, V its H_1 \mathbb{Q}-Hodge structure and M the Mumford-Tate group.

Then there is a Galois representation \rho_\ell on the \ell-adic Tate module T_\ell A and a natural isomorphism V \otimes_{\mathbb{Q}} \mathbb{Q}_\ell \cong T_\ell A \otimes_{\mathbb{Z}_\ell} \mathbb{Q}_\ell so we may view \rho_\ell as a homomorphism \mathrm{Gal}(\bar{K}/K) \to \mathrm{GL}(V_{\mathbb{Q}_\ell}).

Deligne proved that, after replacing K by a finite extension, the image of \rho_\ell is contained in M(\mathbb{Q}_\ell), and it is conjectured that M 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 \ell-adic representations. (It tends to be easier to prove things on the Hodge theory side.)

Example: elliptic curves

Let E be an elliptic curve over \mathbb{C}. We shall sketch a proof that the Mumford-Tate group is a \mathbb{Q}-algebraic-group version of F^\times if E has complex multiplication by F, and is \mathrm{GL}_2 if E does not have CM.

With respect to the basis \{ 1, i \} of V_{\mathbb{R}}, the complex structure is given by  h(x + yi) = \bigl( \begin{smallmatrix} x & -y \\ y & x \end{smallmatrix} \bigr).

But \{ 1, i \} is usually not a basis of V_{\mathbb{Q}}. Suppose that \{ 1, \tau \} be a basis of V_{\mathbb{Q}}, with \tau = a + ib. With respect to this basis, we get  h(x + yi) = \frac{1}{b} \left( \begin{array}{cc} bx-ay & -(a^2+b^2)y \\ y & bx+ay \end{array} \right).

There are two cases:

  1. E has complex multiplication, say by the field F = \mathbb{Q}(\sqrt{-d}).

    Then we can choose \tau = \sqrt{-d} and get  h(x + yi) = \left( \begin{array}{cc} x & -\sqrt{d}y \\ y/\sqrt{d} & x \end{array} \right). The image of h is those matrices of the form \bigl( \begin{smallmatrix} u & -dv \\ v & u \end{smallmatrix} \bigr), which is a group defined by polynomials in \mathbb{Q}, so this is the Mumford-Tate group.

    The \mathbb{Q}-points of M form a group isomorphic to F^\times, so you can think of this group as a \mathbb{Q}-algebraic-group version of F^\times.

  2. E does not have complex multiplication.

    In this case, the image of h is not defined by polynomials with rational coefficients (see comments for proof), so the Mumford-Tate group must be larger. In particular, M has dimension 3 or 4, and if it has dimension 4 then it is equal to \mathrm{GL}_2.

    The only connected algebraic subgroups of \mathrm{GL}_2 of dimension 3 are \mathrm{SL}_2 and conjugates of the upper triangular subgroup. h(\mathbb{C}^\times) 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 M must be reductive, which the triangular group is not.

    So the Mumford-Tate group of an elliptic curve without CM is always \mathrm{GL}_2.

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

Tags abelian-varieties, alg-geom, alg-groups, hodge, maths


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  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 a^2 + b^2 is irrational, but why? Is a + ib a primitive element for the number field V_\mathbb{Q}?

    2. Why is {1,i} NOT a basis of V_\mathbb{Q}, but a basis for V_\mathbb{R}? (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 \mathbb{Q}?)

    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" H_1 \mathbb{Q}-Hodge structures?

    5. Where does the 2g^2 + g + 1 formula come from?

    6. Will you write a post elaborating more on the connection with l-adic representations? I would like to understand why V \otimes \mathbb{Q}_l \cong T_lA \otimes \mathbb{Q}_l, as well as consequences of Im \rho_l \subseteq M(\mathbb{Q}_l).

  2. Martin Orr said on Sunday, 10 October 2010 at 10:43 :
    1. Yes, I was wrong that a^2 + b^2 must be irrational whenever E does not have CM. Maybe it is harder than I thought to show that the MT group must be GL_2 by bare hands. I shall think about this.

    2. The fact that {1, i} is usually not a basis of V_{\mathbb{Q}} is what makes the Mumford-Tate group interesting. This is simply because i is not an element of V_{\mathbb{Q}}, which is the set of rational multiples of elements of the lattice.

    3. You are right on both counts.

    I'll answer the others later.

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

    Here is a correct argument that the image of h is not defined over \mathbb{Q} for an elliptic curve without CM:

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

    If \tau = a+ib is not in an imaginary quadratic field, then at least one of a and a^2 + b^2 is irrational (exercise for reader).

    Let I be the ideal of complex polynomials which vanish on the image of h. The proof relies on the observation that if \operatorname{Im} h were defined over \mathbb{Q}, then I would be closed under all automorphisms of \mathbb{C} fixing \mathbb{Q} (i.e. all automorphisms of \mathbb{C}).

    Now if a is irrational, then we can pick an automorphism \sigma of \mathbb{C} such that \sigma(a) \neq a. The ideal I contains  X_{22} - X_{11} - 2aX_{21} , so also  X_{22} - X_{11} - 2\sigma(a)X_{21} , and hence I contains 2(a - \sigma(a))X_{21}. With the assumption that a \neq \sigma(a), this implies that X_{21} is zero on all of \operatorname{Im} h, which is false.

    If a^2+b^2 is irrational, then do the same thing with the polynomial X_{12} + (a^2 + b^2)X_{21}.

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

    4. Here is a formal proof that if A and B are isogenous abelian varieties, then their H_1 \mathbb{Q}-Hodge structures are isomorphic: let f : A \to B be an isogeny. Then there is an isogeny g : B \to A such that f \circ g and g \circ f are multiplication by some integer n on B and A respectively.

    Let f_* and g_* be the induced morphisms of H_1 \mathbb{Q}-Hodge structures. Now \frac{1}{n} g_* is also a morphism of \mathbb{Q}-Hodge structures, inverse to f_*. (We need to work with \mathbb{Q}-HS rather than \mathbb{Z}-HS here to be able to divide by n.)

    Less formally, if you think about a H_1 \mathbb{Z}-HS as being a full lattice \Lambda embedded in \mathbb{C}^g, then the associated H_1 \mathbb{Q}-HS is the \mathbb{Q}-vector space generated by \Lambda together with its embedding in \mathbb{C}^g. If A, B are isogenous abelian varieties then we can scale their lattices so that \Lambda_A is a finite-index subgroup of \Lambda_B, and I hope it is then clear that \Lambda_A and \Lambda_B generate the same \mathbb{Q}-vector space.

    5. 2g^2 + g + 1 is the dimension of the group \operatorname{GSp}_{2g} of transformations of a 2g-dimensional vector space which take a symplectic form on that space to a scalar multiple of itself. The H_1 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 \operatorname{GSp}_{2g}.

    In the case of elliptic curves, the polarisation doesn't really do anything (and \operatorname{GSp}_2 = \operatorname{GL}_2) 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.

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