Title: Abelian surfaces with fixed 3-torsion
Speaker: Shiva Chidambaram - University of Chicago
Abstract: The Siegel modular variety A_2(3) which parametrizes abelian surfaces with full level 3 structure is birational to the Burkhardt quartic threefold. This was shown to be rational over 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)? It is geometrically rational and known to be unirational over Q by a map of degree at most 6. But it is not rational in general. The degree 6 cover corresponds to a choice of an odd theta characteristic. An explicit description of the universal genus 2 curve over this cover is obtained using invariant theoretic ideas. An application of this result is to render the transfer of modularity explicit, thereby producing infinitely many modular abelian surfaces. This is joint work with Frank Calegari and David Roberts.
