## Algebra

•  Shiva Chidambaram, MIT
•  Abelian varieties with given p-torsion representation
•  10/06/2021
•  4:00 PM - 5:00 PM
•  Online (virtual meeting) (Virtual Meeting Link)
•  Preston Wake (wakepres@msu.edu)

The Siegel modular variety $\mathcal{A}_2(3)$, which parametrizes abelian surfaces with full level $3$ structure, was recently shown to be rational over $\mathbf{Q}$ by Bruin and Nasserden. What can we say about its twist $\mathcal{A}_2(\rho)$ that parametrizes abelian surfaces $A$ whose $3$-torsion representation is isomorphic to a given representation $\rho$? While it is not rational in general, it is always unirational over $\mathbf{Q}$ showing that $\rho$ arises as the $3$-torsion representation of infinitely many abelian surfaces. We will discuss how we can obtain an explicit description of the universal object over such a unirational cover of $\mathcal{A}_2(\rho)$ using invariant theoretic ideas, thus parametrizing families of abelian surfaces with fixed $3$-torsion representation. Similar ideas work in a few other cases, showing in particular that whenever $(g,p) = (1,2)$, $(1,3)$, $(1,5)$, $(2,2)$, $(2,3)$ and $(3,2)$, the necessary condition of cyclotomic similitude is also sufficient for a mod $p$ Galois representation to arise from the $p$-torsion of a $g$-dimensional abelian variety.

## Contact

Department of Mathematics
Michigan State University