# question regarding Faltings' proof of the Tate conjecture for Abelian varieties over number fields

Why does Faltings in his "Endlichkeitssätze für Abelsche Varietäten über Zahlkörpern" in the proof of Theorem 3/4 assume that $W$ is a maximal isotropic $\pi$-invariant subspace? Tate also assumes this in his proof of the Tate conjecture for Abelian varieties over finite fields, but in http://www.jmilne.org/math/CourseNotes/AV.pdf p. 136, Lemma 2.4 there is no such assumption.

Tate needed isotropic in order to be able to apply Weil's theorem to get a polarization of the same degree on the quotient variety, but Faltings uses Zarhin's trick.

• Faltings uses a weaker finiteness theorem than Milne does and then uses Zarhin's trick to prove the statement for any invariant isotropic subspace. If one wants to use Zarhin's trick directly then one must first prove that any abelian variety has only finitely many direct factors (up to isomorphism). – naf Aug 12 '17 at 10:59
• @ulrich: Why don't you make your comment into an answer? – user19475 Aug 12 '17 at 16:25

There are the following Theorems 1 and 2 in Faltings' Finiteness Theorems for Abelian Varieties over Number Fields:

Theorem 1. There are only finitely many isomorphism classes of pairs of semiabelian varieties of relative dimension $g$ with proper generic fibre and principal polarisation of bounded height.

One can prove that this holds without the assumption principally polarised'', which simplifies the proof of the Key Lemma (that for $W \subseteq V_\ell A$ a $\mathbf{Q}_\ell[\pi]$-submodule there exists $u \in \mathrm{End}_K(A) \otimes \mathbf{Q}_\ell$ with $u(V_\ell A) = W$):

Theorem 1'. There are only finitely many isomorphism classes of semiabelian varieties of relative dimension $g$ with proper generic fibre and bounded height.

Proof of Theorem 1'. One can prove that $h(A) = h(A^\vee)$ for an abelian variety $A$ with semistable reduction, so $h((A \times_K A^\vee)^4) = 8h(A)$. By Zarhin's trick, $(A \times_K A^\vee)^4$ is principally polarised, so by Theorem 1, there are only finitely many isomorphism classes. Since an abelian variety has only finitely many direct factors up to isomorphism (because the set of idempotents in $\mathrm{End}_K(A) \otimes \mathbf{Q}$ modulo units is finite since this ring is semisimple), there are only finitely many isomorphism classes for $A$.

Theorem 2. The sequence $h(A_n)$ becomes stationary. (Note the Erratum.)

Now the answer to my question is:

To apply Theorem 1, one has to assume $W \subseteq V_\ell A$ maximal isotropic in order to have the $A_n$ principally polarised. Using Zarhin's trick, one can use Theorem 1' to avoid this.

Here are my talk notes on Faltings' proof of the Tate conjecture for Abelian varieties over number fields: https://www.timokeller.name/Vortrag%20Endomorphismen.pdf