Number Theory Seminar

In this work, we study Fontaine-Laffaille, essentially self-dual deformations of a mod p non-semisimple Galois representation of dimension n with its Jordan-Holder factors being three mutually non-isomorphic absolutely irreducible representations. We show that under some conditions on certain Selmer groups, the universal deformation ring is a discrete valuation ring. Given enough information on the Hecke side, we also prove an R=T theorem. We then apply our results to abelian surfaces with cyclic rational isogenies and certain 6-dimensional representations arising from automorphic forms congruent to Ikeda lifts. In particular, our result identifies the special L-value conditions for the uniqueness of the abelian surface isogeny class, and assuming the Bloch-Kato conjecture, an R=T theorem for the 6-dimensional representations.