Bi-ordinary modular forms

In this talk, I will discuss some introductory ideas about modular forms and Galois representations, while also giving a sense of a recent result (joint work with Francesc Castella). Hida theory provides a p-adic interpolation of modular forms that have a property known as ordinary. The corresponding 2-dimensional p-adic Galois representations are known to be reducible when restricted to a decomposition group at p. Which ordinary modular forms satisfy the stricter property that the restriction to the decomposition group at p is both reducible and decomposable? We propose a length 1 “bi-ordinary” complex, built out of overconvergent modular forms of critical slope, whose cohomology we show supplies a satisfactory answer to this question.