Moduli spaces of low dimensional abelian varieties with torsion

Abstract: The Siegel modular variety $A_2(3)$ which parametrizes abelian surfaces with split level 3 structure is birational to the Burkhardt quartic threefold. This was shown to be rational over $\mathbb{Q}$ by Bruin and Nasserden. What can we say about its twist $A_2(\rho)$ for a Galois representation $\rho$ valued in $GSp(4, F_3)$? While it is not rational in general, it is unirational over $\mathbb{Q}$ by a map of degree at most 6, showing that $\rho$ arises as the 3-torsion of infinitely many abelian surfaces. In joint work with Frank Calegari and David Roberts, we obtain an explicit description of the universal object over a degree 6 cover using invariant theoretic ideas. Similar ideas work for $(g,p) = (1,2), (1,3), (1,5), (2,2), (2,3)$ and $(3,2)$. When $(g,p)$ is not one of these six tuples, we discuss a local obstruction for representations to arise as torsion.

algebraic geometry

Audience: researchers in the discipline

American Graduate Student Algebraic Geometry Seminar

Series comments: The American Graduate Student Algebraic Geometry Seminar (AGSAGS) is a virtual seminar by and for algebraic geometry graduate students.

The goal of this seminar is for graduate students to share their research through online talks and to provide an algebraic geometry graduate networking system. Grad students, postdocs, and professors are welcome to attend.

Seminars will be held on Mondays at 4 p.m. Eastern on Zoom. We hope this time is convenient for graduate students in the Americas, hence the name AGSAGS. Prior registration is required and interested participants should register here: sites.google.com/view/agsags/registration. In addition to graduate talks, there will be occasional social events.