The Atkin operator on spaces of overconvergent modular forms and arithmetic applications

Thesis

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.

Authors: Vonk, J; Jan Vonk

• Publication date: 2015
• DOI: 10.5287/ora-xkzdpnqbz
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: Oxford University, UK
• Language: English
• Keywords: Modular forms; computational number theory; p-adic geometry; Hecke operators
• Subjects: Algebraic geometry; Number theory