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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:655130 |
| Date | January 2015 |
| Creators | Vonk, Jan Bert |
| Contributors | Kim, Minhyong; Lauder, Alan G. B. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:081e4e46-80c1-41e7-9154-3181ccb36313 |
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