Hyperbolicity and torsion in families of abelian varieties

Abstract: The hyperbolicity of the moduli space Ag of abelian varieties implies that nontrivial families of abelian varieties exist over a quasiprojective curve B only when the Euler number χ(B) is negative.  Conjecturally the existence of torsion sections of higher order requires the genus (or even the gonality) of B to be large.  We investigate a general picture for understanding such phenomena---and prove the conjecture in a special case---by bounding Seshadri constants of the Hodge bundle along special subvarieties of the moduli space.  Time permitting we will discuss a related result:  that a family of elliptic curves over a base of fixed gonality is determined up to isogeny by its p-torsion local system.  This is joint work with J. Tsimerman.