Gross's proof of the ChowlaSelberg formula
Posted by Martin Orr on Friday, 04 March 2016 at 11:40
Today I am going to write about Gross's proof of the ChowlaSelberg formula (up to algebraic numbers).
As I discussed last time, the ChowlaSelberg formula is a formula for the periods of a CM elliptic curve in terms of values of the gamma function.
The idea of Gross's proof is to construct a family of abelian varieties equipped with a section of the de Rham cohomology which has constant periods, such that there is one abelian variety in the family where the period is easy to calculate, and another abelian variety in the family is a power of
and so knowing a period of this variety allows us to calculate the periods of
.
Which family of abelian varieties do we use?
We consider a moduli space of abelian varieties of dimension equipped with a homomorphism from
to their endomorphism ring, where
is a given imaginary quadratic field.
Moduli spaces of CM elliptic curves are zerodimensional, so in order to get a useful family of abelian varieties we have to look at
.
We can still get out information about an elliptic curve
because there will be a fibre of the family isogenous to
.
In the usual way, in order to get a moduli space, we have to work with polarised abelian varieties and impose a condition relating the polarisation to the action of .
I shall omit the details of this condition, but it does lead to a constraint on the abelian varieties in the family which I need to mention.
The field
acts on the complex tangent space
for each abelian variety in our family.
There are two eigenspaces for this action, corresponding to the two embeddings
.
The polarisation condition implies that the dimensions of these two eigenspaces, say
and
respectively, must be constant in the family.
The moduli space is a Shimura variety of PEL type associated with the unitary group of an Hermitian form of signature
.
It turns out that for Gross's proof we need .
On the other hand, the family of abelian varieties of split Weil type considered in the proof of Deligne's theorem on absolute Hodge classes is the same kind of family described above but with
.
What do we mean by the period of a de Rham cohomology class?
Let be an abelian variety over
.
Recall that we have isomorphisms
Hence any de Rham class
induces a linear map
.
We define the period lattice of
to be the image of
under this map.
The name period lattice is something of a misnomer as this is usually not a lattice, just a finitedimensional sub
vector space of
.
This definition is compatible with the definition given last time for the period lattice of an element of for an elliptic curve
, via the injection
which defines the Hodge filtration.
What do we mean by a de Rham section with constant periods?
Let be an abelian scheme, with
being an algebraic variety over
.
Given a section
of the de Rham cohomology bundle
, the above construction defines the period lattice of
for each
.
Saying that
has constant periods simply means that the period lattice of
is the same for all
.
Another equivalent fact is that has constant periods if and only if its image under the comparison isomorphism
is a section of the local system
.
This can be seen by looking at what happens over open subsets of
which are small enough for
to be trivialised.
How do we construct sections of the de Rham bundle with constant periods?
Pick a point .
One can prove that there are canonical bijections between the following sets:

invariant elements of

Global sections of

Global sections of the holomorphic vector bundle
with constant periods

Global sections of the algebraic vector bundle
with constant periods
If is a moduli space of abelian varieties and
is the associated universal abelian scheme, then it is easy to describe the action of
on
and thus we can use the above sequence of bijections to get global sections of
with constant periods.
Getting the right section of the de Rham bundle
Gross's proof requires us to take a global section of with constant periods, and work out which element of
it specialises to for two different points
.
In order to control how our section specialises, we shall impose a condition on how
acts on the section, and hence also how it acts on each specialisation of the section.
Define to be the subbundle of
on which
acts via the character
(recall that
denotes one of the embeddings
).
We shall need to use a couple of properties of
to complete the proof.
First we need to show that possesses a global section with constant periods.
The key fact here is that the condition "
acts via the character
" also defines a sublocal system
.
Clearly the comparison isomorphism matches up
and
.
Some linear algebra shows that the subspace on which
acts via
is nonzero and is
invariant.
Therefore we can take an element of this subspace and apply the recipe above to get a global section of
with constant periods.
Second we note that is defined over
.
One can also use the fact that "the GaussManin connection is defined over
" to show that the vector space of "global sections of
with constant periods" is defined over
.
Since this vector space is nonzero over
, it is also nonzero over
.
We conclude that has a nonzero section
with constant periods defined over
, and that the specialisation of this section at any point
lies in the subspace of
on which
acts via
.
Linear algebra shows that this subspace has dimension
, so
is a
multiple of any other element of this subspace.
How do we use relate this de Rham section to periods of an elliptic curve?
Let be an elliptic curve with complex multiplication by
.
Let
and
be a period and quasiperiod of
respectively.
The ring acts in several different ways on
.
On each factor, we can choose the action of
on
to be either
or
.
In order for
to be one of the fibres
of our family of abelian varieties, we have to choose an action of
which gives
times and
times.
This means that if we have a section of
(i.e. a section of
on which the action of
is given by
), then the periods of
are
multiples of
.
Using the Legendre period relation, we deduce that the periods of
are in
Thus if
, then knowing the periods of
tells us the periods of
(up to multiplication by
).
How do we actually calculate the periods of this de Rham section?
Because our section onf
has constant periods, the periods of
are the same as the periods of
for any other
.
Thus it will suffice to calculate the periods of
for any single point
.
We can do this by considering the Fermat curve
where
is the discriminant of the imaginary quadratic field
.
Let
denote the Jacobian of
.
Then
acts on
.
If we choose a suitable quotient of
, then we get a fibre
of our family
.
We can then calculate periods of as integrals of certain differential forms on
.
The calculations required to do this take up a large part of Gross's paper, plus an appendix by Rohrlich, but I am going to skip that here.