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Title: The Atkin operator on spaces of overconvergent modular forms and arithmetic applications
Author: Vonk, Jan Bert
ISNI:       0000 0004 5362 5645
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2015
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We investigate the action of the Atkin operator on spaces of overconvergent p-adic modular forms. Our contributions are both computational and geometric. We present several algorithms to compute the spectrum of the Atkin operator, as well as its p-adic variation as a function of the weight. As an application, we explicitly construct Heegner-type points on elliptic curves. We then make a geometric study of the Atkin operator, and prove a potential semi-stability theorem for correspondences. We explicitly determine the stable models of various Hecke operators on quaternionic Shimura curves, and make a purely geometric study of canonical subgroups.
Supervisor: Kim, Minhyong; Lauder, Alan G. B. Sponsor: Engineering and Physical Sciences Research Council ; Andrew Mullins Scholarship ; Leatherseller's Company ; St. Catherine's College ; European Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Algebraic geometry ; Number theory ; Modular forms ; Hecke operators ; p-adic geometry ; computational number theory