分野別セミナー

Let $k$ be a field finitely generated over $\mathbb{Q}$ and denote by $\Gamma_k$ its absolute Galois group. For an abelian variety $A$ over $k$, a prime $p$ and a character $\chi:\Gamma_{k}\rightarrow
\mathbb{Z}_{p}^{\ast}$, define $A[p^{\infty}](\chi)$ to be the module of $p$-primary torsion of $A(\overline{k})$ on which $\Gamma_k$ acts as $\chi$-multiplication. Assume that $\chi$ does not appear as a
subrepresentation of the $p$-adic representation associated with an abelian variety over $k$. Then $A[p^{\infty}](\chi)$ is always finite, but the exponent of $A[p^{\infty}](\chi)$ may depend on $A$, a priori. Our main result is about the uniform boundedness of $A[p^{\infty}](\chi)$ when $A$ varies in
a $1$-dimensional family. More precisely, we have:Thm. 1: If $S$ is a curve over $k$ and $A$ is an abelian scheme over $S$, then there exists an integer $N:=N(A,S,k,p,\chi)$, such that $A_{s}[p^{\infty}](\chi)\subset A_{s}[p^N]$ holds for any $s\in S(k)$. Such a result has been widely open, even when
$\chi$ is trivial, except for the case of elliptic curves.
This arithmetic result is obtained as a corollary of the following geometric result on the $p$-primary torsion of abelian varieties over function fields of curves, combined with Mordell's conjecture
(Faltings' theorem).
Thm. 2: Let $K$ be the function field of a curve over an algebraically closed field of characteristic $0$
and $A$ an abelian variety over $K$. Assume for simplicity that $A$ contains no nontrivial isotrivial subvariety. Then, for any $c\geq 0$, there exists an integer $N:= N(c,A,S,k,p)\geq 0$ such that
$A[p^{\infty}](K')\subset A[p^N]$ for all finite extension $K'/K$ with $K'$ of genus $\leq c$.

Thm. 1 and 2 actually remain true over fields of characteristic different from $p$.After sketching the proofs of theorems 1 and 2 in characteristic 0, I will mention some of their conjectural generalizations (higher dimension, gonality, dependence on $p$).