Powers of the theta divisor and relations in the tautological ring

We show that the vanishing of the (g + 1)-st power of the theta divisor in the cohomology and Chow rings of the universal abelian variety implies, by pulling back along a collection of Abel–Jacobi maps, the vanishing results in the tautological ring of M_g,n of Looijenga, Ionel, Graber–Vakil, and Faber–Pandharipande. We also show that Pixton’s double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem of Graber and Vakil. Moreover, our proof provides an algorithm for expressing any tautological class on M_g,n of sufficiently high codimension as a tautological class supported on the boundary.