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The Chowla-Selberg formula

Posted by Martin Orr on Monday, 28 December 2015 at 20:00

The Chowla-Selberg formula is an equation which expresses the periods of CM elliptic curves in terms of values of the gamma function at rational arguments. Colmez conjectured a generalisation of the Chowla-Selberg formula to higher-dimensional CM abelian varieties, and an averaged version of Colmez's conjecture was recently proved by Andreatta, Goren, Howard and Madapusi Pera and independently by Yuan and Zhang. This has been much talked about in the world of abelian varieties, because Tsimerman used the averaged Colmez conjecture to complete the proof of the André-Oort conjecture for \mathcal{A}_g.

I thought that rather than looking directly at the Colmez conjecture, it would be good to start with the simpler Chowla-Selberg formula i.e. the elliptic curves case. In this post I will talk about a couple of ways of stating the formula, in terms of periods of CM elliptic curves or in terms of Faltings heights.

Periods of CM elliptic curves

Recall that we defined the period lattice of an elliptic curve E over \mathbb{C} to be the kernel of the exponential map \exp_E \colon T_0 E_{\mathbb{C}} \to E(\mathbb{C}). If we choose an element of the cotangent space \omega \in (T_0 E_{\mathbb{C}})^\vee, then the image under \omega of the period lattice of E is a lattice in \mathbb{C}, which we call the period lattice of \omega and denote \Lambda(\omega).

Suppose that E is defined over \bar{\mathbb{Q}}. We can then insist that \omega be in (T_0 E_{\bar{\mathbb{Q}}})^\vee. Since (T_0 E_{\bar{\mathbb{Q}}})^\vee is a one-dimensional \bar{\mathbb{Q}}-vector space, the period lattices \Lambda(\omega) associated with different choices of \omega \in (T_0 E_{\bar{\mathbb{Q}}})^\vee differ by scalar factors in \bar{\mathbb{Q}}^\times. Hence the \bar{\mathbb{Q}}-vector space generated by \Lambda(\omega) is a \bar{\mathbb{Q}}-vector subspace of \mathbb{C} (of dimension at most 2) which is canonically associated with E. We call the elements of \bar{\mathbb{Q}}.\Lambda(\omega) periods of E.

Suppose that E has complex multiplication by an order \mathcal{O} in an imaginary quadratic field F. The action of \mathcal{O} on E induces an action on T_0 E_{\mathbb{C}}. This action agrees with the multiplication action induced by one of the two embeddings F \to \mathbb{C}, and also stabilises \ker \exp_E. It follows that the period lattice of any \omega \in (T_0 E_{\mathb{C}})^\vee is a rank-1 \mathcal{O}-submodule of \mathbb{C}. Since \mathcal{O} \subset \bar{\mathbb{Q}}, the \bar{\mathbb{Q}}-vector space generated by \Lambda(\omega) has dimension 1.

The fact that E has complex multiplication implies that it is defined over \bar{\mathbb{Q}}. As remarked above, \bar{\mathbb{Q}}.\Lambda(\omega) is a \bar{\mathbb{Q}}-vector space which is independent of the choice of \omega \in (T_0 E_{\bar{\mathbb{Q}}})^\vee. We have just shown that this space has dimension 1. It therefore makes sense to talk about "the period of E up to multiplication by \bar{\mathbb{Q}}^\times." (On the other hand, Schneider's theorem implies that if E is a non-CM elliptic curve over \bar{\mathbb{Q}}, then its space of \bar{\mathbb{Q}}-periods has dimension 2.)

In Gross's version, the Chowla-Selberg formula tells us the value of the periods of the CM elliptic curve E up to multiplication by an element of \bar{\mathbb{Q}}^\times: \lambda_E \in \sqrt{\pi} \prod_{a=1}^{d-1} \Gamma(a/d)^{w \varepsilon(a)/4h} \cdot \bar{\mathbb{Q}}^\times. On the right hand side of the formula, d is the fundamental discriminant of F, w is the number of units in \mathcal{O}_F, h is the class number of F and \varepsilon is the Dirichlet character associated with the quadratic extension F/\mathbb{Q}.

Note that according to this formula, the value of \lambda_E (up to multiplication by \bar{\mathbb{Q}}^\times) is the same for all elliptic curves with CM by the same field. This is expected, because all elliptic curves with CM by a given field are isogenous to each other.

A versions with the Faltings height

The original Chowla-Selberg formula was an exact equation, not just an equation "up to multiplication by \bar{\mathbb{Q}}^\times." In the original formula, the ambiguity in the choice of period was removed by using the modular discriminant \Delta. Colmez later used the Faltings height instead.

However, rather than talking about the periods of a single CM elliptic curve, the exact Chowla-Selberg formula talks about the produc tof the periods of all elliptic curves in the set \operatorname{Ell}(\mathcal{O}_F) = \{ \text{isom classes of elliptic curves } E \text{ s.t.\ } \operatorname{End}(E) \cong \mathcal{O}_F \} for some imaginary quadratic field F. As we remarked above, the curves in \operatorname{Ell}(\mathcal{O}_F) are all isogenous to each other and so their periods are the same up to multiplication by \bar{\mathbb{Q}}^\times. Furthermore, we know that the size of \operatorname{Ell}(\mathcal{O}_F) is equal to the class number of F. Hence, up to \bar{\mathbb{Q}}, there is exactly the same information in a period of a single CM elliptic curve or in the product of the periods of all curves in \operatorname{Ell}(\mathcal{O}_F).

If E is defined over \mathbb{Q}, then we can use a Néron model for E to get a canonical choice of element \omega \in (T_0 E_\mathbb{Q})^\vee (up to sign anyway). In this case there is a very simple definition of the Faltings height: h_{\mathrm{Fal}}(E) = -\frac{1}{2} \log \frac{i}{2} \int_{E(\mathbb{C})} \omega \wedge \bar\omega.

In the CM case, we can relate the Faltings height to the periods. Observe that \frac{i}{2} \int_{E(\mathbb{C})} \omega \wedge \bar\omega is the area of a fundamental domain of the period lattice \Lambda(\omega). If \lambda_1 and \lambda_2 form a basis for \Lambda(\omega), then this area is given by \operatorname{Im}(\lambda_1 \bar{\lambda}_2). Suppose that E has complex multiplication by F. Then \lambda_2 = s\lambda_1 for some s \in F^\times, and so \operatorname{Im}(\lambda_1 \bar{\lambda}_2) = \operatorname{Im}(s) \lvert \lambda_1 \rvert^2. Since \operatorname{Im}(s) is an algebraic number, we conclude that h_{\mathrm{Fal}}(E) \in -\log \lvert \lambda_1 \rvert + \log \bar{\mathbb{Q}}^\times.

If E is not defined over \mathbb{Q} but over some number field K, then there is no canonical choice of \omega \in (T_0 E_K)^\vee. The Faltings height is defined as a sum over all places of K, in a way which removes the dependence on choice of \omega. The terms for finite places are logarithms of algebraic numbers, while the terms for archimedean places look like -\frac{1}{2[K:\mathbb{Q}]} \sum_{\sigma \colon K \to \mathbb{C}} \log \int_{E^\sigma(\mathbb{C})} \omega \wedge \bar\omega.

As in the \mathbb{Q} case, in the case of a CM elliptic curve, up to an element of \log \bar{\mathbb{Q}}^\times, this comes out as -\frac{1}{[K:\mathbb{Q}]} \sum_{\sigma \colon K \to \bar{\mathbb{Q}}} \log \lvert \lambda_{E^{\sigma}} \rvert. If \operatorname{End}(E) = \mathcal{O}_F, then the set of Galois conjugates of E is exactly \operatorname{Ell}(\mathcal{O}_F). We conclude that h_{\mathrm{Fal}}(E) \in -\frac{1}{h} \sum_{E' \in \operatorname{Ell}(\mathcal{O}_K)} \log \lvert \lambda_{E'} \rvert + \log \bar{\mathbb{Q}}^\times where h = [K:\mathbb{Q}] = \# \operatorname{Ell}(\mathcal{O}_F) is the class number of F.

Hence the version of the Chowla-Selberg formula stated above is equivalent to h_{\mathrm{Fal}}(E) \in -\frac{w}{4h} \sum_{a=1}^{d-1} \varepsilon(a) \log \Gamma(a/d) - \frac{1}{2} \log \pi + \log \bar{\mathbb{Q}}^\times. It turns out that we can make this into the following exact equality: h_{\mathrm{Fal}}(E) = -\frac{w}{4h} \sum_{a=1}^{d-1} \varepsilon(a) \log \Gamma(a/d) - \frac{1}{2} \log (2\pi) + \frac{3}{4} \log d. (I am not confident that I have got all the constants correct in the above equation.)

Colmez observed that we can rewrite the right hand side of the above in terms of logarithmic derivatives of L-functions: h_{\mathrm{Fal}}(E) = -\frac{1}{2} \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \frac{L'(\varepsilon, 0)}{L(\varepsilon, 0)} + \frac{1}{4} \log d.

Proofs

Chowla and Selberg proved their formula analytically, using the Kronecker limit formula. It seems that they announced the formula in 1949 but did not publish their proof until 1967, after other people had already published a proof (which I think was also analytic).

Gross proved his weakened ("up to \bar{\mathbb{Q}}^\times") version in 1978 by looking at a family of abelian varieties (with base a unitary Shimura variety) in which there is a constant period. Gross's proof is similar to the proof of Deligne's theorem on absolute Hodge classes for Weil classes, and indeed inspired Deligne's proof. I might talk about this proof in my next post.

Colmez conjectured a generalisation of the formula to CM abelian varieties of higher dimensions. He extended Gross's method to prove the exact formula in the elliptic curve case.

Tags abelian-varieties, faltings, maths, number-theory

Trackbacks

  1. Gross's proof of the Chowla-Selberg formula From Martin's Blog

    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...

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