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The Masser-Wüstholz period theorem

Posted by Martin Orr on Friday, 30 March 2012 at 12:20

I wanted to write a post about the Masser-Wüstholz isogeny theorem, which gives a quantitative version of Finiteness Theorem I. But it turned out to be too long so for today I will focus on the main ingredient in the proof of the isogeny theorem: the Masser-Wüstholz period theorem.

The period theorem gives a bound for the degree of the smallest abelian subvariety of a fixed abelian variety A having a given period of A in its tangent space. In this post I will explain the statement of the period theorem, in particular defining the degree of a (polarised) abelian variety, and give some properties of the degree which will be used in the proof of the isogeny theorem.

Period Theorem. (Masser, Wüstholz 1993) Let A be an abelian variety defined over a number field k with a principal polarisation \lambda. For any non-zero period \omega of A, the smallest abelian subvariety A_\omega of A whose tangent space contains \omega satisfies  \deg_\lambda A_\omega \leq C \max([k:\mathbb{Q}], h_F(A), H_\lambda(\omega, \omega))^\kappa where C and \kappa are constants depending only on \dim A.

Masser and Wüstholz gave a value for \kappa of (g-1) 4^g g! where g = \dim A. For myself, I am only interested in the existence of such a bound, but work has been done on improving it. If I correctly understand a recent preprint of Gaudron and Rémond, they show that \kappa = 3g + \epsilon suffices.

Periods

Let A be an abelian variety of dimension g.

A period of A is an element of the kernel of the exponential map T_0(A_\mathbb{C}) \to A(\mathbb{C}). Let \Lambda denote this kernel; recall that \Lambda is a lattice in the complex vector space T_0(A_\mathbb{C}).

One of the ways of interpreting a polarisation is as a positive definite Hermitian form H_\lambda on T_0(A_\mathbb{C}). This gives us the quantity H_\lambda(\omega, \omega) on the right hand side of the period theorem.

The degree of a polarised abelian variety

Let A be an abelian variety of dimension g and \lambda a polarisation of A. There are two notions of degree of (A, \lambda), which differ by a factor of g!.

First we define what Masser and Wüstholz call the normalised degree \Delta_\lambda(A). This is the volume of a fundamental domain for \Lambda in T_0(A_\mathbb{C}), measured using the norm \mathop{\mathrm{Re}} H.

An equivalent way of defining the normalised degree is as the square root of the degree of \lambda, viewed as an isogeny A \to A^\vee.

The unnormalised degree \deg_\lambda A comes from interpreting the polarisation as an ample line bundle on A_{\bar{k}}. Intersection theory gives a general notion of the degree of any line bundle on a projective variety.

The two degrees are related by the Riemann-Roch theorem for abelian varieties:  \deg_\lambda A = g! \Delta_\lambda(A).

For our purposes it will be convenient to work always with the normalised degree. I stated the period theorem with the unnormalised degree because that is the standard statement, but of course we can replace it by the normalised degree if we want.

The degree of the graph of an isogeny

Let (A, \lambda) and (B, \mu) be two polarised abelian varieties. For the rest of this post we will consider degrees of subvarieties C of A \times B. I shall write \Delta(C) with no subscript to mean the normalised degree of C polarised by the restriction of \lambda \times \mu.

Note that \Delta(A) = \Delta_\lambda(A), \Delta(B) = \Delta_\mu(B) and \Delta(A \times B) = \Delta(A) \Delta(B) (the last relation would not be true with unnormalised degrees - there is a factor depending on the dimensions).

Suppose that there is an isogeny f : A \to B of degree n. Then the graph of f is an abelian subvariety \Gamma_f of A \times B. We can bound its degree from below by the degree of the isogeny f.

Lemma. (\deg f)\Delta(B) \leq \Delta(\Gamma_f).

Proof. Let p, q be the projections \Gamma_f \to A and \Gamma_f \to B. Note that p is an isomorphism and \deg f = \deg q.

We get polarisations p^* \lambda and q^* \mu on \Gamma_f. Let U, V be the real parts of the associated Hermitian forms, which we may interpret as positive-definite symmetric matrices.

We have  \Delta_{q^* \mu}(\Gamma_f) = \deg(q) \Delta(B) so it will suffice to prove that  \Delta(\Gamma_f) \geq \Delta_{q^* \mu}(\Gamma_f).

Recall that \Delta(\Gamma_f) means the degree of \Gamma_f with respect to the restriction of \lambda \times \mu, which is given by p^* \lambda + q^* \mu. So we have  \det(U + V) = \Delta(\Gamma_f) \quad \text{and} \quad \det(V) = \Delta_{q^* \mu}(\Gamma_f).

Thus the lemma follows from the fact that  \det(U + V) \geq \det(V) for any positive-definite symmetric real matrices U and V. This inequality is left as an exercise for the reader.

From subvarieties to isogenies

When can we reverse the process of going from an isogeny to its graph? That is, when does an abelian subvariety of A \times B tell us something about an isogeny? Some choices of subvariety such as A \times \{ 0 \} are clearly of no use, so we introduce the following definition.

Definition. We say that a subvariety of A \times B is split if it has the form U \times V for some U \subset A and V \subset B.

Suppose that there is a non-split abelian subvariety C \subset A \times B. First if A and B are simple, then the projections p : C \to A and q : C \to B must be isogenies so A and B are isogenous.

We would like to show that there is an isogeny A \to B whose degree is bounded by a function of \Delta(C). There is an isogeny p^* : A \to C such that p^* \circ p = [\deg p]. Then \deg p^* = (\deg p)^{2g-1}, where g = \dim A.

By the above lemma, \deg p and \deg q are bounded above by \Delta(C) so there is an isogeny q \circ p^* : A \to B of degree at most  \Delta(C)^{2g}.

Dropping the assumption that A and B are simple we can prove the following:

Lemma. If A \times B has a non-split abelian subvariety C then there are non-zero abelian subvarieties A' \subset A and B' \subset B and an isogeny f : A' \to B' such that  \Delta(A'), \Delta(B') \leq \Delta(C)^2 \quad \text{and} \quad \deg f \leq \Delta(C)^{6n} where n = \max(\dim A, \dim B).

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

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