Carl Lian
Washington U. - St Louis
Title
d-elliptic loci and quasi-modular forms
Abstract
Let $N_{g,d}$ be the subvariety of $M_g$ parametrizing genus $g$ curves admitting a degree $d$ cover of an elliptic curve. For fixed $g$, it is conjectured that the cohomology classes of $N_{g,d}$ on $M_g$ are the Fourier coefficients of a cycle-valued quasi-modular form in $d$. A key difficulty is that these classes are often non-tautological, so lie outside the reach of many known techniques. Via the Torelli map, the conjecture can be moved to one on certain Noether-Lefschetz loci on $A_g$, where there is access to different tools. I will explain some evidence for these conjectures, gathered from results of many people, some of which are joint with François Greer and Naomi Sweeting.