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Title: Perfectoid geometry of p-adic modular forms
Author: Heuer, Ben
ISNI:       0000 0004 8500 5627
Awarding Body: King's College London
Current Institution: King's College London (University of London)
Date of Award: 2019
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We prove a perfectoid tilting isomorphism that describes the Hecke module of over- convergent t-adic modular forms of Andreatta-Iovita-Pilloni at the boundary of weight space in terms of p-th power sequences of overconvergent p-adic modular forms of weights converging to the boundary. This isomorphism relies on a theory of perfectoid modular forms, which we define using Scholze's modular curves of infinite level. For the proof, we study p-adic families of perfectoid modular forms over a perfected p-adic weight space, as well as equicharacteristic families of t-adic modular forms. Our results give a close analogy between perfectoid modular forms and perfectoid algebras: We prove that integral sheaves of perfectoid modular forms are almost acyclic, and construct an analogue of the canonical lift and the #-map, which canonically extend a t-adic modular form into a family over a large weight space annulus. Finally, we give some conjectural relations to Coleman's Spectral Halo conjecture.
Supervisor: Kassaei, Payman L. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available