nt.number theory - lower bound for torsion of abelian varieties

Now let me address your last question. For the sake of simplicity, let us assume that $K$ is a global field of characteristic $p>2$ and the ring $End(A)$ of all endomorphisms of $A$ (over an algebraic closure of $K$) is the ring $Z$ of integers. Then my old results (Math. Notes: 21 (1978), 415--419 and 22 (1978), 493--498) imply that for all but finitely many primes $ell$ the Galois module $A[ell]$ is absolutely simple and the Galois group $Gal(K(A[ell])/K)$ is noncommutative.

I claim that for all but finitely many primes $ell$ the order of
$Gal(K(A[ell])/K)$ is divisible by $ell$ and therefore $[K(A[ell]):K] > ell$.

Indeed, suppose that for a given $ell$ the order of
$Gal(K(A[ell])/K)$ is not divisible by $ell$. Let us call such $ell$ exceptional. Then the natural representation of $Gal(K(A[ell])/K)$ in $A[ell]$ can be lifted to characteristic zero, i.e.,
$Gal(K(A[ell])/K)$ is isomorphic to a (finite) subgroup of $GL(2dim(A),C)$ where $C$ is the field of complex numbers. Now, by a theorem of Jordan, there exists a positive integer $N$ that depends only on $dim(A)$ anf such that $Gal(K(A[ell])/K)$ contains a normal commutative subgroup, whose index does not exceed $N$. This means that $K(A[ell])/K$ contains a Galois subextension $K_{0,ell}/K$ such that $[K_{0,ell}:K] le N$ and the field extension $K(A[ell])/K_{0,ell}$ is abelian.

Let $S$ be the (finite) set of places of bad reduction for $A$. The field extension $K(A[ell])/K$ is unramified outside $S$; the same is true for the subextension $K_{0,ell}/K$. (In the number field case one should add the divisors of $ell$ but we live in characteristic $p$!). Recall that the Galois extensions of global $K$ with degree $le N$ and ramification only at $S$ constitute a finite set. Let $L/K$ be the compositum of all such extensions. Clearly, $L$ is also a global field while $L/K$ is a finite Galois extension that contains $K_{0,ell}$. In addition, the Galois group $Gal(L(A[ell])/L)$ is commutative. Now considering $A$ as an abelian variety over $L$ and applying previously mentioned results, we obtain that the set of exceptional primes $ell$ is finite.

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