Modular curves and the double ramification cycle.

The Double Ramification Cycle (DRC) is a closed substack of the stack of smooth curves of genus g, of interest in enumerative geometry. We will explain how the DRC may be viewed as a kind of generalisation of modular curves to abelian varieties of arbitrary dimension. In particular, we will show how the Strong Torsion Conjecture (on rational torsion points on abelian varieties) is equivalent to a conjecture on the rational points on the DRC. We will describe recent progress in constructing good compactifications and integral models for the DRC.