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Main steps of the proof of the period theorem

Posted by Martin Orr on Friday, 03 August 2012 at 11:06

Today I will explain how to prove the Masser-Wüstholz Period Theorem starting from the Key Proposition, an weaker existence result for abelian subvarieties of bounded degrees. The Key Proposition and the Tangent Space Lemma, which I mention briefly, are the main results proved by Masser and Wüstholz by the techniques of transcendental number theory on their way to the Period Theorem.

Recall that the Period Theorem is a bound for the degree of the smallest abelian subvariety of A whose tangent space contains a given period \omega. In the Key Proposition we find a subvariety of bounded degree whose tangent space satisfies the very weak condition of being inside a chosen hyperplane in T_0 A. In particular \omega need not be in the tangent space of the subvariety. However we use the Key Proposition and induction to prove the Strong Proposition, which gives a subvariety whose tangent space does contain \omega. Several applications of the Strong Proposition with different hyperplanes in T_0 A prove the Period Theorem.

Heights of vector subspaces

Let V be a vector space of dimension n over a number field k. We can identify the set of d-dimensional subspaces of V with the points of a projective variety, called the Grassmannian \operatorname{Gr}(d, V).

The simplest example of this is d = 1: by definition, points in \mathbb{P}^{n-1} correspond to lines in V. Hence \operatorname{Gr}(d, V) = \mathbb{P}^{n-1}. For a recipe for how to do this in general, see Wikipedia.

We now define the height h(W) of a subspace W \subset V to be the logarithmic Weil height of the corresponding point in the Grassmannian.

The Tangent Space Lemma

Let A be an abelian variety of dimension g defined over a number field k. The tangent space T_0 A to A at the origin is then a k-vector space of dimension g. (This is the tangent space in the algebraic geometry sense. The complex analytic tangent space, which contains the period lattice, is T_0 A \otimes_k \mathbb{C}.)

If B is an abelian subvariety of A also defined over k, then T_0 B is a k-vector subspace of T_0 A, and so has a height as defined above. Note that h(T_0 B) has nothing to do with h(B), the Faltings height of B. Indeed h(B) depends only on the isomorphism class of B, while h(T_0 B) depends on how B is embedded as a subvariety of A.

One of the two main results proved by Masser and Wüstholz on the way to the Period Theorem is the Tangent Space Lemma, which bounds h(T_0 B) in terms of the degree of B.

Tangent Space Lemma. Let A be a principally polarised abelian variety defined over a number field and B \subset A an abelian subvariety. There exists a constant c depending only on \dim A such that  h(T_0 B) \leq c \max(1, h(A), \log \Delta(B)).

The Strong Proposition

The other main ingredient in the proof of the Period Theorem is the Key Proposition. This is an existence result for subvarieties of A of bounded degree, although the subvariety H satisfies only a very weak condition on its tangent space -- in particular T_0 H_\mathbb{C} might not contain the period \omega.

Key Proposition. Let (A, \lambda) be a polarised abelian variety and B \subset A an abelian subvariety, defined over a number field k. Let W be a hyperplane in T_0 B defined over k, and suppose that W \otimes \mathbb{C} contains a non-zero period \omega of B.

Then there exists a non-zero abelian subvariety H of B such that T_0 H \subset W and  \Delta(H) \leq c \max(\Delta(B), rd, rdh(A), rdh(W))^m where m = \dim B, c is a constant depending only on \dim A, r = H_\lambda(\omega, \omega) and d = [k:\mathbb{Q}].

Note that this is essentially a proposition about B, with A playing little role in the conclusion. The only benefit of allowing A \neq B is that we are going to apply the proposition to various different subvarieties B of a fixed A, and we want the bound to always depend on the height of A rather than on the varying height of B.

We will use the Key Proposition to prove the Strong Proposition. This is the same statement as the Key Proposition, with the extra condition that \omega \in T_0 H_\mathbb{C} and with the exponent m in the bound replaced by some function of m. The implication Key Proposition => Strong Proposition is proved in the next section.

To prove the Period Theorem from the Strong Proposition: Let m = \dim A_\omega. We can choose g-m hyperplanes W_1, \dotsc, W_{g-m} in T_0 A such that  T_0 A_\omega = \bigcap W_i. Furthermore we can choose the W_i such that h(W_i) \leq h(T_0 A_\omega) for all i.

Applying the Strong Proposition to each W_i, with B = A, we get abelian subvarieties H_i \subset A of bounded degree such that \omega \in T_0 H_i \otimes \mathbb{C} and T_0 H_i \subset W_i. The first condition implies that A_\omega \subset \bigcap H_i and the second that \bigcap T_0 H_i \subset \bigcap W_i = T_0 A_\omega. Hence A_\omega is the identity connected component of \bigcap H_i, which implies that  \Delta(A_\omega) \leq \prod \Delta(H_i).

The Strong Proposition gives us bounds for each \Delta(H_i). We just have to get rid of the h(W_i) term from these bounds. But by the Tangent Space Lemma,  h(W_i) \leq h(T_0 A_\omega) \leq c \max(1, h(A), \log \Delta(A_\omega)). This gives  \Delta(A_\omega) \leq c \max(d, r, h(A), \log \Delta(A_\omega))^k. and we can get rid of the \log \Delta(A_\omega) by increasing the constant, completing the proof of the Period Theorem.

From Key Proposition to Strong Proposition

The Strong Proposition is deduced from the Key Proposition by induction on m = \dim B. We begin by applying the Key Proposition directly to the given B and W, getting H \subset B such that T_0 H \subset W.

If \omega \in T_0 H_\mathbb{C} then we can stop. In particular this always happens if m = 2 since then \dim H = \dim W = 1 so that T_0 H = W. This gives the base case of the induction.

Otherwise, let H^\perp be the orthogonal complement of H in B. This is the abelian subvariety of B such that T_0 H_\mathbb{C} and T_0 H^\perp_\mathbb{C} are orthogonal with respect to H_\lambda.

The tangent space of B is the direct sum of the tangent spaces of H and H^\perp. This need not be true for the period lattices: the sum of the period lattices of H and H^\perp has finite index in the period lattice of B. Let this index be b; we can prove that b \leq \Delta(H)^2.

Now W_1 = W \cap T_0 H^\perp has codimension 1 in T_0 H^\perp so we can apply the Strong Proposition inductively to H^\perp and W_1. To do this we need a period of H^\perp contained in W_1. We can write b\omega = \omega_0 + \omega_1 with \omega_0 and \omega_1 periods of H and H^\perp respectively, and use \omega_1.

So applying the Strong Proposition inductively gives us an abelian subvariety H_1 \subset H^\perp such that  \Delta(H_1) \leq c \max(\Delta(H^\perp), r_1 d, r_1 dh(A), r_1 dh(W_1))^k where  r_1 = H_\lambda(\omega_1, \omega_1) \leq b^2 H_\lambda(\omega, \omega) \leq \Delta(H)^4 r.

By a Theorem of Schmidt, Struppeck and Vaaler,  h(W_1) = h(W \cap T_0 H^\perp) \leq h(W) + h(T_0 H^\perp) + c and by the Tangent Space Lemma,  h(T_0 H^\perp) \leq c \max(1, h(A), \log \Delta(H^\perp)). We can also show that \Delta(H^\perp) \leq \Delta(H)\Delta(B) so the above bound simplifies to  \Delta(H_1) \leq c \max(\Delta(B), \Delta(H), rd, rdh(A), rdh(W))^k.

Finally the abelian subvariety H + H_1 \subset B satisfies the conditions of the Strong Proposition (its tangent space is contained in W and contains \omega), and its degree satisfies  \Delta(H + H_1) \leq \Delta(H) \Delta(H_1). Using the above bounds for \Delta(H_1) and the bound for \Delta(H) from the Key Proposition completes the proof of the Strong Proposition.

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

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