NAG

Abstract

We show the following result supporting the uniform boundedness conjecture for torsion of abelian varieties: Let k be a field finitely generated over the rationals, X a smooth curve over k, and A an abelian scheme over X. Let l be a prime number and d a positive integer. Then there exists a non-negative integer N, such that, for any closed point x of X with [k(x):k] \leq d and any k(x)-rational, l-primary torsion point v of A_x, the order of v is \leq l^N. (Here, A_x stands for the fiber of the abelian scheme A at x.) As a corollary of this result, we settle the one-dimensional case of the so-called modular tower conjecture, posed by Fried in the context of the (regular) inverse Galois problem. The above result is obtained by combining geometric results (estimation of genus/gonality) and Diophantine results (Mordell/Mordell-Lang conjecture, proved by Faltings) for certain ``moduli spaces''. If we have time, we will also explain our recent progress on a variant of these geometric results, where the set of powers of the fixed prime l is replaced by the set of all primes. For an extension of the above results to more general l-adic representations, see Cadoret's talk on Friday.