Noncongruence modular forms and modularity

Original Announcement

By a theorem of Belyi, any smooth projective curve defined over a number field is isomorphic to a modular curve for some finite index subgroup of $SL(2,\Z)$. The majority of these are noncongruence subgroups. For example, the degree 3 Fermat curve $E: x^3+y^3=1$ is the modular curve for the degree 3 Fermat group $\Phi_3$ , contained in $\Gamma(2)$. The space of weight 2 cuspforms for $\Phi_3$, denoted by $S_2(\Phi_3)$, is 1-dimensional and generated by \begin{align} f(z) =& q^{1/2}+...+70 q^{5/2}+...+23000/3^2 q^{7/2} +...+6850312202/3^3 q^{13/2} \cr =& \sum a(n) q^{n/2} . \end{align} Observe that the Fourier coefficients of $f$ are rational numbers with unbounded denominators which indicates that $\Phi_3$ is noncongruence.

On the other hand, the celebrated Taniyama-Shimura modularity theorem established by Wiles et al. says that the $\ell$-adic representation attached to $E$ comes from a weight 2 congruence normalized newform $g(z) = \sum b(n)q^n$. Atkin and Swinnerton-Dyer discovered remarkable congruence relations satisfied by the Fourier coefficients of noncongruence form $f$ and congruence form g for almost all primes $p$: $$ a(n p) - b(p)a(n) + p a(n/p) \equiv 0 \bmod p^{1+ord_p n}. $$ They further suggested that such three-term congruence relations on Fourier coefficients of noncongruence forms should hold in general for a basis depending on $p$ with suitably chosen algebraic integers replacing $b(p)$ and $p$.

Major breakthroughs in the study of noncongruence cuspforms were achieved by A. Scholl. In order to understand the Atkin and Swinnerton-Dyer congruence relations, Scholl constructed a compatible family of $2d$-dimensional $\ell$-adic Galois representations attached to each $d$-dimensional space of noncongruence cuspforms of integral weight $k \ge 2$ under general assumptions. The congruences above result from the Scholl representations attached to $S_2(\Phi_3)$ isomorphic to the $\ell$-adic Galois representations attached to $g(z)$.

Proposed below is a partial list of topics to be discussed during the workshop. The participants are welcomed to comment on it and suggest related topics of their interests.

1. When will 2-dimensional representations of the Galois group of a totally real field attached to noncongruence cuspforms arise from Hilbert modular forms?
2. To what extent will the bounded denominator property on Fourier coefficients characterize a congruence modular form?
3. Can the conductor of the Scholl representations be determined in terms of the data of the noncongruence cuspforms? If so, how to do it effectively? Can this be extended from $\Q$ to a totally real number field?
4. Search for fast algorithms to enumerate noncongruence subgroups. As an application, one can determine the noncongruence subgroup of least index in the modular group having exceptional eigenvalue of the Laplace operator.
5. What are the analytic properties of noncongruence Maass Waveforms? What are the distributions of their coefficients and the orders of their scattering matrices?

Material from the workshop

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop: