Emerton-Gee Stacks, Serre Weights, and Breuil-Mézard Conjectures for GSp_4

We construct a moduli stack of rank 4 symplectic projective étale (phi,Gamma)-modules and prove its geometric properties for any prime p>2 and finite extension K/Q_p. When K/Q_p is unramified, we adapt the theory of local models recently developed by Le-Le Hung-Levin-Morra to study the geometry of potentially crystalline substacks in this stack. In particular, we prove the unibranch property at torus fixed points of local models and deduce that tamely potentially crystalline deformation rings are domain under genericity conditions. As applications, we prove, under appropriate genericity conditions, an GSp_4-analogue of the Breuil-Mézard conjecture for tamely potentially crystalline deformation rings, the weight part of Serre's conjecture formulated by Gee-Herzig-Savitt for global Galois representations valued in GSp_4 satisfying Taylor--Wiles conditions, and a modularity lifting result for tamely potentially crystalline representations.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2023

Author dc:creator

• Lee, Heejong

Advisor dc:contributor.advisor

• Herzig, Florian

• automorphic representation
• Galois representation
• Generalized Serre weight conjectures
• Langlands Program

Statement dc:rights

• Attribution-NonCommercial-NoDerivatives 4.0 International

Licence dc:rights.uri

• http://creativecommons.org/licenses/by-nc-nd/4.0/

Handle dc:identifier.uri

http://hdl.handle.net/1807/130442

OAI identifier oai:identifier

oai:utoronto.scholaris.ca:1807/130442