Absolute Hodge classes in ladic cohomology
Posted by Martin Orr on Friday, 26 June 2015 at 11:30
We can define absolute Hodge classes in adic cohomology in the same way as absolute Hodge classes in de Rham cohomology. We can then prove Deligne's theorem, that Hodge classes on an abelian variety are absolute Hodge, for adic cohomology. Because it is easy to prove that absolute Hodge classes in adic cohomology are potentially Tate classes, this implies half of the MumfordTate conjecture.
In particular, it implies that if is an abelian variety over a number field, then a finite index subgroup of the image of the adic Galois representation on is contained in the points of the MumfordTate group of . This is the goal I have been working towards for some time on this blog.
Deligne's definition of absolute Hodge classes considered adic cohomology (for all ) and de Rham cohomology simultaneously.
The accounts I read of this theory focussed on the de Rham side, leading me to believe that the de Rham part was essential and the adic part an optional extra.
This is why I wrote the past few posts about de Rham cohomology and am now adding the adic version on at the end, even though I am more interested in the adic version.
Now that I understand what is going on, I realise that I could have used only adic cohomology from the beginning.
One day I might write up a neater account which uses
adic cohomology only.
Before the main part of this post, talking about absolute Hodge classes in adic cohomology, I need to talk about Tate twists in adic cohomology. These are more significant than Tate twists in singular cohomology because they change the Galois representations involved. This resulted in some mistakes in my previous posts on Tate classes, which I think I have now fixed.
Tate twists in adic cohomology
In singular cohomology, using Tate twists is largely a matter of convenience as you can avoid them by putting in factors of . On the other hand, using Tate twists in adic cohomology (for example, when defining the algebraic cycle class map) is essential. There are two reasons for this:

The Tate twist object
in the category of Hodge structures has a canonical generator and so we get canonical isomorphisms of the underlying lattices
for any Hodge structure (at least, canonical up to sign). On the other hand, the Tate twist object
in the adic world is a rank 1
module but has no canonical isomorphism to
, so
and
are only noncanonically isomorphic.

If we are working over a nonalgebraically closed base field , the Tate twist object has a nontrivial action of so
and
are not isomorphic as Galois representations.
The adic Tate twist object is defined to be the inverse limit
where
is the group of th roots of unity in .
Thus
is a rank 1
module and
acts on
as multiplication by the cyclotomic character
(this is essentially the definition of
).
We define notations ,
and
for any adic Galois representation
in the same way as we defined analogous notations for the Tate twist of Hodge structures.
The adic cycle class map
has image in
.
The Tate twist is necessary here to get a Galoisequivariant cycle class map, because it has to map subvarieties defined over to a invariant classes, but there are no Galoisinvariant classes in the untwisted
.
Over , there is a standard isomorphism between the Betti and adic Tate twist objects, which is defined as the inverse limit of the system of maps
Hence if is an embedding, we get comparison isomorphisms
Definition of absolute Hodge classes in adic cohomology
Because there is a comparison isomorphism between adic cohomology and singular cohomology for complex projective varieties, we can define absolute Hodge classes in adic cohomology with a definition which looks exactly the same as the definition for de Rham cohomology.
Let be an algebraically closed field which can be embedded in and a smooth projective variety over . Deligne also defines absolute Hodge classes over nonalgebraically closed base fields, but this definition does not seem particularly useful to me.
For each embedding , we say that an adic cohomology class
is a Hodge class relative to
if
where
is the comparison isomorphism between
adic étale cohomology and singular cohomology.
We say that is an absolute Hodge class in adic cohomology if is a Hodge class relative to every embedding
.
Deligne's theorem on absolute Hodge classes in adic cohomology
One can and prove state Deligne's theorem on absolute Hodge classes for adic cohomology just as for de Rham cohomology.
Theorem. Let
be an algebraically closed field
embeddable in . If
is an abelian variety over
and
is a Hodge class relative to one embedding
, then
is an absolute Hodge class.
The proof uses Principle B for adic cohomology.
Theorem (Principle B). Let
be a family of abelian varieties over
, with connected base
. Let
be a global section of
.
If there is a point
such that
is an absolute Hodge class on
, then for every
,
is an absolute Hodge class on
.
Here the notation means that we evaluate the section of at to get
, then take the preimage of
under the comparison isomorphism
The proofs we sketched before for Principle B and for Deligne's theorem work for adic cohomology with essentially no changes. This is because we used Blasius' statement and proof of Principle B, using the theorem of the fixed part. In Deligne's original paper, he had to use separate methods to prove Principle B for de Rham and for adic cohomology.
Application to the MumfordTate conjecture
Let be an abelian variety over a number field .
Recall that the MumfordTate conjecture is equivalent to the claim that potentially Tate classes are precisely the span of Hodge classes. We can use Deligne's theorem to prove one half of this, namely that all Hodge classes are potentially Tate.
In order to deduce this from Deligne's theorem, we just have to prove:
Lemma. Every absolute Hodge class on is a potentially Tate class on .
The key point in the proof of the lemma is that the Galois action on preserves the absolute Hodge classes.
This is simply because absolute Hodge classes are defined as classes satisfying a certain property for all embeddings
, and
permutes these embeddings.
Hence the action of on
restricts to an action on the absolute Hodge classes.
But the set of absolute Hodge classes is a finitedimensional
vector space because it is contained in
for some .
Thus we get a representation
.
Furthermore, this representation is continuous for the
adic topology on
and its image is therefore a profinite group.
But any countable profinite group is finite.
We deduce that the action of on absolute Hodge classes factors through a finite quotient, and hence that all absolute Hodge classes are potentially Tate.
Using Deligne's Principle A, we conclude that
Theorem. For any abelian variety over a number field, the identity component of the
adic algebraic monodromy group of is contained in the extension of scalars of the MumfordTate group
.