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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 principally polarised semiabelian varieties of relative dimension $g$ with proper generic fibre and 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 $\mathrm{End}_K(A) \otimes \mathbf{Q}_\ell$ 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

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

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 principally polarised semiabelian varieties of relative dimension $g$ with proper generic fibre and 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 $h((A \times A^\vee)^4) = 8h(A)$. By Zarhin's trick, $(A \times A^\vee)^4$ 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, 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

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 principally polarised semiabelian varieties of relative dimension $g$ with proper generic fibre and 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 $\mathrm{End}_K(A) \otimes \mathbf{Q}_\ell$ 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

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 principally polarised semiabelian varieties of relative dimension $g$ with proper generic fibre and 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 $h((A \times A^\vee)^4) = 8h(A)$. By Zarhin's trick, $(A \times A^\vee)^4$ 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, 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

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 principally polarised semiabelian varieties of relative dimension $g$ with proper generic fibre and 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 $h((A \times A^\vee)^4) = 8h(A)$. By Zarhin's trick, $(A \times A^\vee)^4$ 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, 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