The unbounded denominators conjecture

Abstract: (Joint work with Vesselin Dimitrov and Yunqing Tang). The arithmetic theory of modular forms usually considers functions on the upper half plane which transform nicely under a “congruence subgroup" of SL_2(Z), that is, a subgroup of SL_2(Z) containing all matrices congruent to 1 mod N for some integer N. But as was already known to Klein, SL_2(Z) has many finite index subgroups which are *not* congruence subgroups. It turns out that the modular forms for these non-congruence subgroups behave quite differently. One longstanding open problem which characterizes whether a modular form comes from a congruence subgroup or not is the so-called “unbounded denominators conjecture”. In this talk, we give an overview of the proof of this conjecture, starting with an introduction to the conjecture itself. Organisateur : quebecvermontnumbertheory@gmail.com

number theory

Audience: researchers in the topic

CRM-CICMA Québec Vermont Seminar Series

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