Torsion of abelian schemes and rational points on moduli spaces (joint work with Anna Cadoret)
Seminar Room 1, Newton Institute
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.