Colloquium - Vesselin Dimitrov - The unbounded denominators conjecture in the theory of vector-valued modular forms
Title: The unbounded denominators conjecture in the theory of vector-valued modular forms
Abstract: Traditionally, the theory of modular forms has focused primarily on the most symmetrical case of the congruence subgroups of SL(2,Z). It is largely due to the Hecke operators that the congruence modular forms are a pillar of contemporary mathematics, taking central roles in the Langlands program, conformal field theory, and modern analytic as well as algebraic number theory. One basic outcome of the Hecke theory is that, due to a simultaneous diagonalization with algebraic integers for eigenvalues, all modular forms on the congruence subgroups of SL(2,Z) have Fourier expansions with bounded denominators at all cusps.
And yet it turns out that most finite index subgroups of SL(2,Z) are not described by congruence conditions on the matrix entries. It was observed empirically by Atkin and Swinnerton-Dyer in the late 1960s that the modular forms on noncongruence subgroups behave very differently than in the congruence case, and in particular they tend to have q-expansions with unbounded denominators. Correspondingly, it was also conjectured by Atkin that the usual Hecke operators act trivially on the noncongruence modular forms. This latter conjecture was proved by Serre and Berger in the early 1990s. The former surmise is the conjecture referred to in my title. I will explain our recent joint paper with Frank Calegari and Yunqing Tang which proves the unbounded denominators conjecture in the full generality of vector-valued modular forms for SL(2,Z), using arithmetic algebraization ideas alongside Serre and Berger's solution of Atkin's conjecture.
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