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The matrix lemma for elliptic curves

Posted by Martin Orr on Friday, 25 May 2012 at 14:00

Let A be a principally polarised abelian variety of dimension g over \mathbb{C}. We can associate with A a g \times g complex matrix called the period matrix which roughly speaking describes a basis for the image of H_1(A, \mathbb{Z}) in T_0 A(\mathbb{C}) (actually it is not really the period matrix as it is only defined up to the action of \mathop{\mathrm{Sp}}_{2g}(\mathbb{Z}) on the Siegel upper half space; we can make it nearly unique by forcing it to be in a particular fundamental domain).

The matrix lemma says that, if A is defined over a number field, then the entries of the imaginary part of the period matrix cannot be too large with respect to the height of A (Faltings height or modular height).

Matrix lemma. (Masser 1987) Let A be a principally polarised abelian variety of dimension g over a number field K. Let \tau be the period matrix for A in the standard fundamental domain of the Siegel upper half space. There is a constant c depending only on g such that all the entries of \tau satisfy  \lvert \mathop{\mathrm{Im}} \tau_{ij} \rvert \leq c [K:\mathbb{Q}] \max(1, h(A)).

Last time I used a lower bound for the lengths of non-zero periods in the proof of the isogeny theorem. This follows from the matrix lemma as we can easily relate lengths of periods and the period matrix.

Today I will prove the matrix lemma for elliptic curves. The general proof requires various facts about Siegel modular forms and also uses a funny choice of level structure due to Igusa (I do not understand why). But the basic structure of the proof is already visible in the elliptic curves case and we can be concrete about the modular forms involved, using only facts I learned in Part III.

A note about heights

The lemma is valid taking h(A) to be either the stable Faltings height of A or the Weil height of a point representing A in the moduli space of ppAVs of dimension g. This is because Faltings proved a bound for the difference between the two heights. The proof of the matrix lemma uses the height from the moduli space. But in applying it to the isogeny theorem, we need the Faltings height because we need the bound  h(B) \leq h(A) + \log \deg f for varieties related by an isogeny f : A \to B, and so far as I know this can only be proven using the Faltings height.

I was a bit disappointed when I realised this because I had thought that the Masser-W├╝stholz isogeny theorem gave a proof of Finiteness Theorem I independent of Faltings' work. Of course the comparison between heights is not all of Faltings' proof, but according to Milne, "Technically, this is by far the hardest part of the proof."

In order to define the height of a point in the moduli space we need to choose an embedding of the moduli space into projective space. I think we have to choose the right embedding for the lemma to work (and also for the comparison with the Faltings height to work) but I am not very clear about that -- I think this might be related to the Igusa level structure which I mentioned.

For today, I am sticking to elliptic curves, where there is nothing to worry about. We can simply use the Weil height of the j-invariant.

The proof

Let E be an elliptic curve over the number field K. Let \tau be a complex number in the upper half plane such that \{ 1, \tau \} generate the period lattice for E. We can choose \tau in the standard fundamental domain \mathcal{F} defined by  \lvert \tau \rvert \geq 1, \; \lvert \mathop{\mathrm{Re}} \tau \rvert \leq \frac{1}{2}.

The period lemma asserts that there is an absolute constant c such that  \mathop{\mathrm{Im}} \tau \leq c [K:\mathbb{Q}] \max(1, h(j(E))). We will prove this by bounding \min(\lvert j(\tau) \rvert^{-1}, \lvert 1 - j(\tau) \rvert^{-1}) in two ways, one involving \mathop{\mathrm{Im}} \tau and the other involving h(j(E)).

Recall that  j(E) = j(\tau) = \frac{f(\tau)}{\Delta(\tau)} where f and \Delta are modular forms of weight 12, f being a certain multiple of E_4^3 and \Delta being a cusp form.

In the compactified fundamental domain, f vanishes only at \omega^{\pm} = \frac{1}{2}(\pm 1 + \sqrt{-3}). We have \Delta(\omega^{\pm}) \neq 0; indeed \Delta has no zeroes except at the cusp. So \Delta-f is non-zero in some neighbourhoods of \omega^+ and \omega^-. Hence there is a constant c_1 > 0 such that  \max(\lvert f(\tau) \rvert, \lvert \Delta(\tau) - f(\tau) \rvert) \geq c_1 \text{ for all } \tau \in \mathcal{F}.

Cusp forms decay exponentially as we approach the cusp i.e. there are constants c_2, c_3 > 0 such that  \lvert \Delta(\tau) \rvert \leq c_2 \exp(- c_3 \mathop{\mathrm{Im}} \tau) \text{ for all } \tau \in \mathcal{F}. (This follows from the existence of q-expansions.)

Combining these two inequalities, we get that  \min(\lvert j(\tau) \rvert^{-1}, \lvert 1 - j(\tau) \rvert^{-1}) \leq c_1^{-1} c_2 \exp(- c_3 \mathop{\mathrm{Im}} \tau).

Since E is defined over the number field K, the same is true of j(\tau). Using the definition of the Weil height, it is not hard to prove Liouville's inequality:  \lvert j(\tau) \rvert \leq \exp([K:\mathbb{Q}] h(j(\tau))).

By basic properties of heights, h(1-j(\tau)) \leq h(j(\tau)) + \log 2 so  \lvert 1 - j(\tau) \rvert \leq \exp \left(c_5 [K:\mathbb{Q}] \max(1, h(j(\tau))) \right).

Hence  \min(\lvert j(\tau) \rvert^{-1}, \lvert 1 - j(\tau) \rvert^{-1}) \geq \exp \left(-c_5 [K:\mathbb{Q}] \max(1, h(j(\tau))) \right).

Combining the two bounds for \min(\lvert j(\tau) \rvert^{-1}, \lvert 1 - j(\tau) \rvert^{-1}), we get that  c_1^{-1} c_2 \exp(- c_3 \mathop{\mathrm{Im}} \tau) \geq \exp \left(-c_5 [K:\mathbb{Q}] \max(1, h(j(\tau))) \right) which simplifies to the desired  \mathop{\mathrm{Im}} \tau \leq c [K:\mathbb{Q}] \max(1, h(j(\tau))).

Tags abelian-varieties, alg-geom, maths, number-theory

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  1. Jacob said on Sunday, 30 July 2017 at 10:31 :

    This comment comes many years too late, but i found these notes quite helpful so wanted to contribute at least something: their proof is indeed independent of faltings proof. Milne was referring to another, much more subtle point about faltings heights within a K-isogeny class. The ineuqality, on the other hand, is a trivial consequence of looking at the definition in terms of the sum of volumes wrt a top form minus its index as a generator for top forms: Just pull back a neron form from B and use it for A. Note that both sides change by at most a constant multiple of logdeg f.

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