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The Masser-Wüstholz isogeny theorem

Posted by Martin Orr on Wednesday, 25 April 2012 at 14:09

Let A and B be two isogenous abelian varieties over a number field K. Can we be sure that there is an isogeny between them of small degree, where "small" is an explicit function of A and K? In particular, our bound should not depend on B; this means that the bound will imply Finiteness Theorem I, and hence the Shafarevich, Tate and Mordell conjectures.

The Masser-Wüstholz isogeny theorem answers this question, at least subject to a minor condition on polarisations (I think that this was removed in a later paper of Masser and Wüstholz but it is not too important anyway -- when deducing Finiteness Theorem I you can remove the polarisation issue with Zarhin's Trick).

Theorem. (Masser, Wüstholz 1993) Let A and B be principally polarised abelian varieties over a number field K. Suppose that there exists some isogeny A \to B. Then there is an isogeny A \to B of degree at most c \max([K:\mathbb{Q}], h(A))^\kappa where c and \kappa are constants depending only on the dimension of A.

We will prove this using the Masser-Wüstholz period theorem which I discussed last time.

Proof of the isogeny theorem

Recall from last time that in order to obtain an isogeny of bounded degree between a subvariety of A and a subvariety of B, it suffices to have a non-split subvariety of bounded degree in A \times B. The central idea of the proof is that we will use the period theorem, along with the assumption that A and B are isogenous, to construct such a non-split subvariety.

Actually we will construct a non-split subvariety C of A \times B^{2g}, of degree N say. Once we have done this, it is easy to finish the proof: by the lemma from last time, there are non-zero subvarieties A' \subset A and B' \subset B^{2g} and an isogeny f : A' \to B' of degree at most N^{12g^2}.

If we suppose that A is simple, then we must have A' = A and at least one of the projections p_i : B' \to B is an isogeny. Because the degree of B' is bounded, so is the degree of p_i and hence p_i \circ f is the desired isogeny A \to B of bounded degree.

If A is not simple, then we can do something similar to get an isogeny A'' \to B'' where A'' \subset A and B'' \subset B of bounded degree; then we quotient out by A'' and B'' and induct.

Finding a non-split subvariety

It remains to show that for any isogenous abelian varieties A and B, there is a non-split subvariety C \subset A \times B^{2g} with degree bounded polynomially by \delta, [K:\mathbb{Q}] and h(A).

Choose a non-zero period \omega \in H_1(A, \mathbb{Z}) and a basis \omega_1, \dotsc, \omega_{2g} for H_1(B, \mathbb{Z}). Let \tilde{\omega} be the period (\omega, \omega_1, \dotsc, \omega_{2g}) of A \times B^{2g}.

Let C be the smallest abelian subvariety of A \times B^{2g} whose tangent space contains \tilde{\omega}. Now the period theorem gives a bound for the degree of C, and the assumption that A is isogenous to B implies that C is non-split.

To prove the latter, let f be any isogeny A \to B. There are integers m_1, \dotsc, m_{2g} such that f_* \omega = m_1 \omega_1 + \dotsb + m_{2g} \omega_{2g}. Then C must be contained in C_f = \{ (x, y_1, \dotsc, y_{2g}) \in A \times B^{2g} \mid f(x) = m_1 y_1 + \dotsb + m_{2g} y_{2g} \}. Any subvariety of C_f with a non-trivial projection to A is non-split.

(Observe that the degree of f may be really big, and the degree of C_f is related to the degree of f, so the degree of C may be big. But this is OK because we are not using C_f to say anything about the degree of C - that comes from the period theorem.)

Bounding the degree of the subvariety

The period theorem tells us that \deg_{\lambda \times \mu^{2g}} C \leq c \max([k:\mathbb{Q}], h_F(A \times B^{2g}), H(\tilde{\omega}, \tilde{\omega}))^\kappa where \lambda and \mu are principal polarisations of A and B, and H is the Hermitian form associated with the polarisation \lambda \times \mu^{2g} of A \times B^{2g}.

We want to remove H(\tilde{\omega}, \tilde{\omega}) from this bound, by bounding it in terms of the other quantities. Let us write \lVert v \rVert = \sqrt{H(v, v)}. This is a norm on the real vector space H_1(A \times B^{2g}, \mathbb{R}).

We have \lVert \tilde{\omega} \rVert^2 = \lVert \omega \rVert^2 + \lVert \omega_1 \rVert^2 + \dotsb + \lVert \omega_{2g} \rVert^2) so it will suffice to show that we can choose \omega and \omega_1, \dotsc, \omega_{2g} with bounded lengths.

The only condition that we have put on \omega is that it is a period i.e. in the lattice H_1(A, \mathbb{Z}). A fundamental domain for this lattice has volume 1 with respect to our chosen metric (this is equivalent to the polarisation \lambda being principal) and so Minkowski's theorem gives an upper bound for the length of the smallest period, depending only on the dimension g: \lVert \omega \rVert \leq c_1.

With regard to \omega_1, \dotsc, \omega_{2g}, they must be a basis for H_1(B, \mathbb{Z}). Again this lattice has covolume 1 and so by Minkowski's theorem, there is a basis such that \prod_{i=1}^{2g} \lVert \omega_i \rVert \leq c_2 for a constant c_2 depending only on g.

An upper bound for the product \prod \lVert \omega_i \rVert does not imply an upper bound for the individual lengths (or for \sum \lVert \omega_i \rVert^2) but we can deduce such a bound if we also have a lower bound for the \lVert \omega_i \rVert. The following such bound can be deduced from a refinement of Masser's Matrix Lemma.

Lemma. Let (B, \mu) be a principally polarised abelian variety defined over a number field K. There is a constant c_3 depending only on \dim B such that every non-zero period \omega of B satisfies \lVert \omega \rVert_\mu \geq c_3 ([K:\mathbb{Q}] h(B))^{-1/2}.

Finishing the proof

Combining the above, we get that the isogeny f : A \to B of least degree satisfies \deg f \leq c \max([K:\mathbb{Q}], h_F(A \times B^{2g}))^\kappa. (The constants c and \kappa are different in different inequalities in this post.)

However this bound depends on h_F(B), which we wanted to avoid. We use a basic fact about the Faltings height: if there is an isogeny f : A \to B, then h(B) \leq h(A) + \log \deg f. So we get that \deg f \leq c \max([K:\mathbb{Q}], h(A) + \log \deg f)^\kappa. It might seem silly that we want to bound \deg f and we have just introduced it on the right hand side, but it is only a polynomial in \log \deg f so with a bit of rearrangement we can absorb it into the constant.

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

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  1. Matrix lemma for elliptic curves From Martin's Blog

    Let be a principally polarised abelian variety of dimension over . We can associate with a complex matrix called the period matrix which roughly speaking describes a basis for the image of in (actually it is not really the period matrix as it is o...

Comments

  1. Barinder Banwait said on Friday, 27 April 2012 at 18:55 :

    1. "Any subvariety of C_f with a non-trivial projection to A is non-split". Why is this? Are you using that A is simple?

    2. Right at the end, "So we get that...": Are you claiming that h(A \times B^{2g}) \leq h(B)?

  2. Martin Orr said on Wednesday, 02 May 2012 at 08:53 :

    1. No. I use that C_f \cap (A \times \{(0, \dotsc, 0\}) = \ker f \times \{(0, \dotsc, 0)\} so any split subvariety contained in C_f is in fact contained in \ker f \times B^{2g}. Since \ker f is finite, the subvariety is contained in \{0\} \times B^{2g} i.e. has trivial projection to A.

    2. Again no. I confusingly left out some steps. We use the fact that h(A \times B) = h(A) + h(B) so h(A \times B^{2g}) = h(A) + 2g\,h(B) \leq (2g+1)h(A) + 2g \log \deg f then we can hide the constants in the big constant c.

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