BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Shiva Chidambaram (MIT) DTSTART:20210921T203000Z DTEND:20210921T213000Z DTSTAMP:20260423T130251Z UID:MITNT/31 DESCRIPTION:Title: A belian Varieties with given $p$-torsion representations\nby Shiva Chid ambaram (MIT) as part of MIT number theory seminar\n\nLecture held in Room 2-143 in the Simons building (building 2).\n\nAbstract\nThe Siegel modula r variety $\\mathcal{A}_2(3)$\, which parametrizes abelian surfaces with f ull level $3$ structure\, was shown to be rational over $\\Q$ by Bruin and Nasserden. What can we say about its twist $\\mathcal{A}_2(\\rho)$\, para metrizing abelian surfaces $A$ with $\\rho_{A\,3} \\simeq \\rho$\, for a g iven 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 con dition of having cyclotomic similitude. In joint work with Frank Calegari and David Roberts\, we obtain an explicit description of the universal obj ect over a degree $6$ cover of $\\mathcal{A}_2(\\rho)$\, using invariant t heoretic ideas. One application of this result is towards an explicit tran sfer of modularity\, yielding infinitely many examples of modular abelian surfaces with no extra endomorphisms. Similar ideas work in a few other ca ses\, showing in particular that whenever $(g\,p) = (1\,2)$\, $(1\,3)$\, $ (1\,5)$\, $(2\,2)$\, $(2\,3)$ and $(3\,2)$\, the cyclotomic similitude con dition is also sufficient for a mod $p$ Galois representation to arise fro m the $p$-torsion of a $g$-dimensional abelian variety. When $(g\,p)$ is n ot one of these six tuples\, we will discuss a local obstruction for repre sentations to arise as torsion.\n LOCATION:https://researchseminars.org/talk/MITNT/31/ END:VEVENT END:VCALENDAR