Critical Lambda-adic modular forms and bi-ordinary complexes
Francesc Castella, Carl Wang-Erickson
TL;DR
This work develops a comprehensive $\Lambda$-adic theory for critical slope overconvergent modular forms, extending Hida theory to interpolate critical forms across $p$-adic weights and constructing a length-1 bi-ordinary complex whose cohomology encodes 2-dimensional Galois representations that split at $p$. It uses Coleman's de Rham presentation, Boxer–Pilloni’s higher Hida theory, and a $p$-adic dual operator $U'$ to realize a robust interpolation of de Rham and cohomological data, yielding $R= T$ theorems for critical and bi-ordinary Hecke algebras and a Λ-flat, degree-shifting action on bi-ordinary cohomology. The paper also develops a deformation-theoretic framework for ordinary, twist/anti-ordinary, bi-ordinary, and critical representations, establishing pushout structures and a precise link between Galois representations and Hecke algebras; it connects these to CM and Stark-unit phenomena, including a weight-1 action under specialized hypotheses. Collectively, these results provide a unified interpolation theory for critical and bi-ordinary phenomena, with implications for p-split representations and arithmetic applications via Stark units and CM forms. The framework opens avenues for further exploration of $p$-adic families beyond ordinary forms and deepens the interaction between modular cohomology, Galois representations, and Iwasawa-theoretic arithmetic.
Abstract
We produce a flat $Λ$-module of $Λ$-adic critical slope overconvergent modular forms, producing a Hida-type theory that interpolates such forms over $p$-adically varying integer weights. This provides a Hida-theoretic explanation for an observation of Coleman that the rank of such forms is locally constant in the weight. The key to the interpolation is to use Coleman's presentation of de Rham cohomology in terms of overconvergent forms to link critical slope overconvergent modular forms with the part of the first coherent cohomology of modular curves interpolated by Boxer-Pilloni's higher Hida theory. The novelty is that we interpolate a critical period in cohomology using modular forms, complementing the classical Hida-theoretic interpolation of an ordinary period. Using this interpolation, we also interpolate bi-ordinary complexes in various weights into a perfect and self-dual complex of length 1 over $Λ$. By design, the cohomology of the bi-ordinary complex supports 2-dimensional $p$-adic representations of ${\rm Gal}(\bar{\bf Q}/{\bf Q})$ that become reducible and decomposable upon restriction to a decomposition group at $p$. As applications and motivations for the above constructions, we prove "$R = T$" theorems for the critical and bi-ordinary Hecke algebras, produce a degree-shifting Hecke action on the co-homology of bi-ordinary complexes, and specialize this degree-shifting action to weight 1 to produce, under a supplemental assumption, an action of a Stark unit on the part of weight 1 coherent cohomology over ${\bf Z}_p$ that is isotypic for an ordinary eigenform with complex multiplication.