# Noncongruence modular forms and modularity

August 17 to August 21, 2009

at the

American Institute of Mathematics, Palo Alto, California

organized by

Winnie Li, Tong Liu, Ling Long, and Ravi Ramakrishna

This workshop, sponsored by AIM and the NSF, will explore the arithmetic and analytic properties of noncongruence modular forms and their potential applications. A special focus will be on the connection between Scholl representations attached to noncongruence cuspforms and automorphic forms by applying modularity lifting theorems.

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 SL2( Z). The majority of these are noncongruence subgroups. For example, the degree 3 Fermat curve E: x3+y3=1 is the modular curve for the degree 3 Fermat group Φ3, contained in Γ(2). The space of weight 2 cuspforms for Φ3, denoted by S23), is 1-dimensional and generated by

f(z) = q1/2+... +70q5/2+... +(23000/32)q7/2+... +(6850312202/35) q13/2+...
= ∑n ≥ 1 a(n)qn/2.

Observe that the Fourier coefficients of f are rational numbers with unbounded denominators which indicates that Φ3 is noncongruence.

On the other hand, the celebrated Taniyama-Shimura modularity theorem established by Wiles et al. says that the l-adic representation attached to E comes from a weight 2 congruence normalized newform g(z) = ∑ b(n)qn. Atkin and Swinnerton-Dyer discovered remarkable congruence relations satisfied by the Fourier coefficients of noncongruence form f and congruence form g foralmost all primes p:

a(n p) - b(p)a(n) + p a(n/p) ≡ 0 mod p1+ordp n for all n≥ 1.

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 l-adic Galois representations attached to each d-dimensional space of noncongruence cuspforms of integral weight k ≥ 2 under general assumptions. The congruences above result from the Scholl representations attached to S23) isomorphic to the l-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?

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.