The canonical subgroup: a "subgroup-free" approach | Zendy

Eyal Z. Goren | Zendy; Payman L Kassaei | Zendy

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The canonical subgroup: a "subgroup-free" approach

Author(s) -

Eyal Z. Goren,

Payman L Kassaei

Publication year - 2006

Publication title -

commentarii mathematici helvetici

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.603

H-Index - 46

eISSN - 1420-8946

pISSN - 0010-2571

DOI - 10.4171/cmh/66

Subject(s) - mathematics , pure mathematics , continuation , unitary state , moduli , canonical form , algebra over a field , analytic continuation , group (periodic table) , realization (probability) , mathematical analysis , computer science , chemistry , physics , organic chemistry , quantum mechanics , political science , law , programming language , statistics

Beyond the crucial role they play in the foundations of the theory of overconver- gent modular forms, canonical subgroups have found new applications to analytic continuation of overconvergent modular forms. For such applications, it is essential to understand various "numerical" aspects of the canonical subgroup, and in particular, the precise extent of its over- convergence. In this paper, we develop a theory of canonical subgroups for a general class of curves (including the unitary and quaternionic Shimura curves), using formal and rigid ge- ometry. In our approach, we use the common geometric features of these curves rather than their (possible) specific moduli-theoretic description; it allows us to reproduce, for the classical cases, the optimal radii of definition for the canonical subgroup, usually derived by employing the theory of formal groups.

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