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 on the universal abelian variety $mathcal{X}_g$ implies, by pulling back along a collection of Abel--Jacobi maps, the vanishing results in the tautological ring of $mathcal{M}_{g,n}$ of Looijenga, Ionel, Graber--Vakil, and Faber--Pandharipande. We also show that Pixtons double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem~$star$ of Graber and Vakil, and we provide an explicit algorithm for expressing any tautological class on $overline{mathcal{M}}_{g,n}$ of sufficiently high codimension as a boundary class.