MumfordTate groups
Posted by Martin Orr on Monday, 04 October 2010 at 12:37
In this post I will define the MumfordTate group of an abelian variety.
This is a algebraic group, such that the Hodge structure is a representation of this group.
The MumfordTate 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 MumfordTate 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 nonzero determinant, which is an affine variety (over any field you like):
it is a Zariskiclosed 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 MumfordTate group
Now we are ready to define the MumfordTate group.
Definition. The MumfordTate 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 MumfordTate group of a generic abelian variety of dimension
has dimension
.
By construction, the MumfordTate group is a subgroup of , so
is a representation of
.
It enjoys several representationtheoretic special properties, and we can use these together with the classification of linear algebraic groups and their representations to calculate the MumfordTate group in many cases.
Uses of the MumfordTate group
The MumfordTate 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 MumfordTate 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 MumfordTate 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 MumfordTate group is a
algebraicgroup 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:

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

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 MumfordTate 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 MumfordTate 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.
Questions, in order of importance:
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
?
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
?)
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..."?
Why do isogenous abelian varieties have the "same"
Hodge structures?
Where does the
formula come from?
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
.
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.
The fact that
is usually not a basis of
is what makes the MumfordTate group interesting. This is simply because
is not an element of
, which is the set of rational multiples of elements of the lattice.
You are right on both counts.
I'll answer the others later.
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. 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 finiteindex 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 MumfordTate 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 nonblog things to write.