BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Chengyang Bao (Imperial College London) DTSTART:20251015T150000Z DTEND:20251015T160000Z DTSTAMP:20260418T065522Z UID:LNTS/171 DESCRIPTION:Title: A pplications of patching the coherent cohomology of modular curves\nby Chengyang Bao (Imperial College London) as part of London number theory se minar\n\nLecture held in Imperial College London\, Room 139.\n\nAbstract\n We apply the Taylor--Wiles--Kisin patching method to study certain partial normalizations of crystalline deformation rings associated with two-dimen sional representations \\bar{r} : G_{\\Q_p} \\to \\GL_2(\\F)\, where $\\F$ is a finite field of characteristic $p \\ge 5$. Using the $q$-expansion p rinciple\, we obtain a multiplicity-one result\, which implies that the pa rtial normalization of the crystalline deformation ring is Cohen--Macaulay . As applications\, we give a simple criterion for when a crystalline defo rmation ring coincides with its partial normalization\, thereby establishi ng new cases where these rings are Cohen--Macaulay. We also prove a Zarisk i-density result for crystalline points in characteristic $p$\, and we app ly our method to deduce a multiplicity-one result for Serre's mod-$p$ quat ernionic modular forms. \n \nMost of these results originated from attempt s to explain computational data from my thesis on computing crystalline de formation rings via the Taylor--Wiles--Kisin patching method. I will concl ude with some expected properties of crystalline deformation rings suggest ed by the data that remain open.\n LOCATION:https://researchseminars.org/talk/LNTS/171/ END:VEVENT END:VCALENDAR