Analytic continuation of Hilbert modular forms and applications to modularity

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• Payman Kassaei (King's College London)
• Tuesday 07 May 2013, 16:15-17:15
• MR13.

If you have a question about this talk, please contact Teruyoshi Yoshida.

In his foundational work on the theory of p-adic modular forms, N. Katz observed that there is a positive lower bound for the “growth condition” of an overconvegent p-adic modular eigenform with nonzero Up-eigenvalue. In more modern language, this states that any such form can be analytically continued from its initial domain of definition to a not ”too small” region of the rigid analytic modular curve. Years later, K. Buzzard, by considering these forms in their true level (i.e., level divisible by p) proved that such forms can be further extended to a certain “large” region of the modular curve. These results were used by Buzzard and Taylor to prove modularity lifting results which led to a proof of certain cases of the Strong Artin conjecture.

It has been known for a while how to extend these results to the Hilbert case when p is split in the totally real field of degree g > 1, as the problem looks formally like a product of g copies of the modular curve case. In the inert case, however, a mixing happens that fundamentally changes the nature of the problem. In this talk, I will explain new results on domains of automatic analytic continuation for overconvergent Hilbert modular forms in the case p is unramified in the totally real field. These results can be used to prove many cases of the strong Artin conjecture for Hilbert modular forms. Some of the work that will be presented is joint with Sasaki and Tian.

This talk is part of the Number Theory Seminar series.

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