Critical Hida theory, bi-ordinary complexes, and weight 1 coherent cohomology

Abstract: Coleman made observations about overconvergent modular forms of weight at least 2 and critical slope which imply that they are almost spanned by two subspaces corresponding to two different kinds of twist of ordinary overconvergent modular forms. He also showed that the “almost” is accounted for by a square-nilpotent action of Hecke operators. Motivated by questions about Galois representations associated to these forms, we intersect these two twists to define “bi-ordinary” forms. But we do this in a derived way: the sum operation from the two twisted ordinary subspaces to the space of critical forms defines a length 1 “bi-ordinary complex," making the bi-ordinary forms the 0th degree of bi-ordinary cohomology and realizing the square-nilpotent Hecke action as a degree-shifting action. Relying on classical Hida theory as well as the higher Hida theory of Boxer-Pilloni, we interpolate this complex over weights. We can deduce “R=T” theorems in the critical and bi-ordinary cases from R=T theorems in the ordinary case. And specializing to weight 1 under a supplemental assumption, we show that the bi-ordinary complex with its square-nilpotent Hecke action specializes to weight 1 coherent cohomology of the modular curve with a degree-shifting action of a Stark unit group. The action is a candidate for a p-adic realization of conjectures about motivic actions of Venkatesh, Harris, and Prasanna. This is joint work with Francesc Castella.

number theory

Audience: researchers in the topic

London number theory seminar

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