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SUMMARY:Carl Wang-Erickson (University of Pittsburgh)
DTSTART:20240605T133000Z
DTEND:20240605T143000Z
DTSTAMP:20260418T070029Z
UID:LNTS/137
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/137/">C
 ritical Hida theory\, bi-ordinary complexes\, and weight 1 coherent cohomo
 logy</a>\nby Carl Wang-Erickson (University of Pittsburgh) as part of Lond
 on number theory seminar\n\nLecture held in K0.18\, King's Building\, Stra
 nd Campus\, King's College London.\n\nAbstract\nColeman 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 t
 o two different kinds of twist of ordinary overconvergent modular forms. H
 e also showed that the “almost” is accounted for by a square-nilpotent
  action of Hecke operators. Motivated by questions about Galois representa
 tions associated to these forms\, we intersect these two twists to define 
 “bi-ordinary” forms. But we do this in a derived way: the sum operatio
 n from the two twisted ordinary subspaces to the space of critical forms d
 efines a length 1 “bi-ordinary complex\," making the bi-ordinary forms t
 he 0th degree of bi-ordinary cohomology and realizing the square-nilpotent
  Hecke action as a degree-shifting action. Relying on classical Hida theor
 y 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 an
 d bi-ordinary cases from R=T theorems in the ordinary case. And specializi
 ng to weight 1 under a supplemental assumption\, we show that the bi-ordin
 ary 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. T
 his is joint work with Francesc Castella.\n
LOCATION:https://researchseminars.org/talk/LNTS/137/
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