Deformation in p-adic families of Bianchi modular forms

Lead Research Organisation: University of Warwick
Department Name: Mathematics


The theory of modular forms and their generalisation, automorphic forms, is central to modern number theory. One way to study these theories is via the eigencurve construction, where geometric properties of these curve correspond to arithmetic results. I will focus on smoothness of eigencurves, at non-regular weights and for groups other than GL_2 over the rationals. In the case of GL_2 over quadratic imaginary fields, work by Loeffler and Zerbes, "On the Birch-Swinnerton-Dyer conjecture for modular abelian surfaces", shows smoothness would imply (under some additional hypotheses) special cases of the Birch-Swinnerton-Dyer conjecture.

For GL_2 / Q, smoothness is well known for points of weight greater than one. Recent work by Bellaiche and Dimitrov, "On the Eigencurve at classical weight 1 points", establishes this at weight one as well. Calegari and Mazur have explored these questions for GL_2 over number fields other than Q in "Nearly Ordinary Galois Deformations over Arbitrary Number Fields", but much remains unanswered. Moreover, these works approach the problem via deformations of the Galois representations attached to modular forms, whereas I aim to work more directly with modular forms and modular symbols. An additional benefit of this method is the possibility for computational results - I have developed an algorithm to numerically test for smoothness of cusp forms over Euclidean domains, and I hope to extend this to more general quadratic imaginary fields. I also hope to extend these ideas to provide a general criterion for smoothness.


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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/W523793/1 30/09/2021 29/09/2025
2593437 Studentship EP/W523793/1 03/10/2021 29/09/2025 James Rawson