Abelian Varieties with given $p$-torsion representations

Abstract: The Siegel modular variety $\mathcal{A}_2(3)$, which parametrizes abelian surfaces with full level $3$ structure, was shown to be rational over $\Q$ by Bruin and Nasserden. What can we say about its twist $\mathcal{A}_2(\rho)$, parametrizing abelian surfaces $A$ with $\rho_{A,3} \simeq \rho$, for a given mod $3$ Galois representation $\rho : G_{\Q} \rightarrow \GSp(4, \F_3)$? While it is not rational in general, it is unirational over $\Q$ by a map of degree at most $6$, if $\rho$ satisfies the necessary condition of having cyclotomic similitude. In joint work with Frank Calegari and David Roberts, we obtain an explicit description of the universal object over a degree $6$ cover of $\mathcal{A}_2(\rho)$, using invariant theoretic ideas. One application of this result is towards an explicit transfer of modularity, yielding infinitely many examples of modular abelian surfaces with no extra endomorphisms. 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 cyclotomic similitude condition is also sufficient for a mod $p$ Galois representation to arise from the $p$-torsion of a $g$-dimensional abelian variety. When $(g,p)$ is not one of these six tuples, we will discuss a local obstruction for representations to arise as torsion.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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