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Gross's proof of the Chowla-Selberg 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 Chowla-Selberg formula (up to algebraic numbers). As I discussed last time, the Chowla-Selberg formula is a formula for the periods of a CM elliptic curve E 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 E and so knowing a period of this variety allows us to calculate the periods of E.

Which family of abelian varieties do we use?

We consider a moduli space of abelian varieties of dimension n equipped with a homomorphism from \mathcal{O}_F to their endomorphism ring, where F is a given imaginary quadratic field. Moduli spaces of CM elliptic curves are zero-dimensional, so in order to get a useful family of abelian varieties we have to look at n > 1. We can still get out information about an elliptic curve E because there will be a fibre of the family isogenous to E^n.

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 \mathcal{O}_F. 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 F acts on the complex tangent space T_0 A_{\mathbb{C}} for each abelian variety in our family. There are two eigenspaces for this action, corresponding to the two embeddings \tau, \bar\tau : F \to \mathbb{C}. The polarisation condition implies that the dimensions of these two eigenspaces, say p and q respectively, must be constant in the family.

The moduli space is a Shimura variety of PEL type associated with the unitary group of an F/\mathbb{Q}-Hermitian form of signature (p,q).

It turns out that for Gross's proof we need p \neq q. 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 p = q.

What do we mean by the period of a de Rham cohomology class?

Let A be an abelian variety over \mathbb{C}. Recall that we have isomorphisms  H^n_{dR}(A/\mathbb{C}) \cong H^n(A(\mathbb{C}), \mathbb{C}) \cong H_n(A(\mathbb{C}), \mathbb{C})^\vee. Hence any de Rham class \omega \in H^n_{dR}(A/\mathbb{C}) induces a linear map H_n(A(\mathbb{C}), \mathbb{C}) \to \mathbb{C}. We define the period lattice of \omega to be the image of H_n(A(\mathbb{C}), \mathbb{Q}) under this map. The name period lattice is something of a misnomer as this is usually not a lattice, just a finite-dimensional sub-\mathbb{Q}-vector space of \mathbb{C}.

This definition is compatible with the definition given last time for the period lattice of an element of (T_0 E_{\mathbb{C}})^\vee for an elliptic curve E, via the injection (T_0 E_{\mathbb{C}})^\vee \to H^1_{dR}(E/\mathbb{C}) which defines the Hodge filtration.

What do we mean by a de Rham section with constant periods?

Let \pi \colon \mathcal{A} \to S be an abelian scheme, with S being an algebraic variety over \mathbb{C}. Given a section \omega of the de Rham cohomology bundle \mathcal{H}^n_{dR}(\mathcal{A}/S), the above construction defines the period lattice of \omega_s for each s \in S(\mathbb{C}). Saying that \omega has constant periods simply means that the period lattice of \omega_s is the same for all s \in S(\mathbb{C}).

Another equivalent fact is that \omega has constant periods if and only if its image under the comparison isomorphism  \mathcal{H}^n_{dR}(\mathcal{A}/S) \to R^n \pi_* \mathbb{C} \otimes_{\mathbb{C}} \mathcal{O}_S is a section of the local system R^n \pi_* \mathbb{C}. This can be seen by looking at what happens over open subsets of S(\mathbb{C}) which are small enough for R^n \pi_* \mathbb{C} to be trivialised.

How do we construct sections of the de Rham bundle with constant periods?

Pick a point s \in S(\mathbb{C}). One can prove that there are canonical bijections between the following sets:

  1. \pi_1(S(\mathbb{C}), s)-invariant elements of H^n(\mathcal{A}_s(\mathbb{C}), \mathbb{C})

  2. Global sections of R^n \pi_* \mathbb{C}

  3. Global sections of the holomorphic vector bundle \mathcal{H}^n_{dR}(\mathcal{A}(\mathbb{C})/S(\mathbb{C})) with constant periods

  4. Global sections of the algebraic vector bundle \mathcal{H}^n_{dR}(\mathcal{A}/S) with constant periods

If S is a moduli space of abelian varieties and \mathcal{A} is the associated universal abelian scheme, then it is easy to describe the action of \pi_1(S(\mathbb{C}), s) on H^n(\mathcal{A}_s(\mathbb{C}), \mathbb{C}) and thus we can use the above sequence of bijections to get global sections of \mathcal{H}^n_{dR}(\mathcal{A}/S) with constant periods.

Getting the right section of the de Rham bundle

Gross's proof requires us to take a global section of \mathcal{H}^n_{dR}(\mathcal{A}/S) with constant periods, and work out which element of H^n(\mathcal{A}_s(\mathbb{C}), \mathbb{C}) it specialises to for two different points s \in S(\mathbb{C}). In order to control how our section specialises, we shall impose a condition on how F acts on the section, and hence also how it acts on each specialisation of the section.

Define \mathcal{V}_{dR} to be the subbundle of \mathcal{H}^n_{dR}(\mathcal{A}/S) on which F acts via the character \tau^n (recall that \tau denotes one of the embeddings F \to \mathbb{C}). We shall need to use a couple of properties of \mathcal{V}_{dR} to complete the proof.

First we need to show that \mathcal{V}_{dR} possesses a global section with constant periods. The key fact here is that the condition "F acts via the character \tau^n" also defines a sub-local system \mathcal{V}_{top} \subset R^n \pi_* \mathbb{C}. Clearly the comparison isomorphism matches up \mathcal{V}_{dR} and \mathcal{V}_{top}.

Some linear algebra shows that the subspace \mathcal{V}_{top,s} \subset H^n(\mathcal{A}_s(\mathbb{C}), \mathbb{C}) on which F acts via \tau^n is non-zero and is \pi_1(S(\mathbb{C}), s)-invariant. Therefore we can take an element of this subspace and apply the recipe above to get a global section of \mathcal{V}_{dR} with constant periods.

Second we note that \mathcal{V}_{dR} is defined over \bar{\mathbb{Q}}. One can also use the fact that "the Gauss-Manin connection is defined over \bar{\mathbb{Q}}" to show that the vector space of "global sections of \mathcal{V}_{dR} with constant periods" is defined over \bar{\mathbb{Q}}. Since this vector space is non-zero over \mathbb{C}, it is also non-zero over \bar{\mathbb{Q}}.

We conclude that \mathcal{V}_{dR} has a non-zero section \omega with constant periods defined over \bar{\mathbb{Q}}, and that the specialisation of this section at any point t \in S(\mathbb{C}) lies in the subspace of H^n_{dR}(\mathcal{A}_t/\bar{\mathbb{Q}}) on which F acts via \tau^n. Linear algebra shows that this subspace has dimension 1, so \omega_t is a \bar{\mathbb{Q}}-multiple of any other element of this subspace.

How do we use relate this de Rham section to periods of an elliptic curve?

Let E be an elliptic curve with complex multiplication by \mathcal{O}_F. Let \lambda_E and \eta_E be a period and quasi-period of \omega_E respectively.

The ring \mathcal{O}_F acts in several different ways on E^n. On each factor, we can choose the action of F on T_0 E_{\mathbb{C}} to be either \tau or \bar\tau. In order for E^n to be one of the fibres \mathcal{A}_s of our family of abelian varieties, we have to choose an action of F which gives \tau p times and \bar\tau q times.

This means that if we have a section v of \mathcal{V}_{dR} (i.e. a section of \mathcal{H}^n_{dR}(\mathcal{A}/S) on which the action of F is given by \tau^n), then the periods of v_s are \bar{\mathbb{Q}}-multiples of \lambda_E^p \eta_E^q. Using the Legendre period relation, we deduce that the periods of v_s are in  (2 \pi i)^q \, \lambda_E^{p-q} \, \bar{\mathbb{Q}}^\times. Thus if p \neq q, then knowing the periods of v_s tells us the periods of E (up to multiplication by \bar{\mathbb{Q}}^\times).

How do we actually calculate the periods of this de Rham section?

Because our section v onf \mathcal{V}_{dR} has constant periods, the periods of v_s are the same as the periods of v_t for any other t \in S(\mathbb{C}). Thus it will suffice to calculate the periods of v_t for any single point t.

We can do this by considering the Fermat curve  C(d) : x^d + y^d = 1 where -d is the discriminant of the imaginary quadratic field F. Let J(d) denote the Jacobian of C(d). Then F acts on J(d). If we choose a suitable quotient of J(d), then we get a fibre \mathcal{A}_t of our family \mathcal{A}/S.

We can then calculate periods of \mathcal{A}_t as integrals of certain differential forms on C(d). 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.

Tags abelian-varieties, hodge, maths, number-theory, shimura-varieties

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