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 *SL _{2}( Z)*. The majority of these are
noncongruence subgroups. For example, the degree 3 Fermat curve

f(z) = q^{1/2}+... +70q^{5/2}+...
+(23000/3^{2})q^{7/2}+... +(6850312202/3^{5})
q^{13/2}+...

= ∑_{n ≥ 1} a(n)q^{n/2}.

Observe that the
Fourier coefficients of *f* are rational numbers with unbounded
denominators which indicates that *Φ _{3}* 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)q ^{n}*. Atkin and Swinnerton-Dyer
discovered remarkable congruence relations satisfied by the Fourier
coefficients of noncongruence form

a(n p) - b(p)a(n) + p a(n/p) ≡ 0 mod p^{1+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 *S _{2}(Φ_{3})*
isomorphic to the

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.

- When will 2-dimensional representations of the Galois group of a totally real field attached to noncongruence cuspforms arise from Hilbert modular forms?
- To what extent will the bounded denominator property on Fourier coefficients characterize a congruence modular form?
- 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
to a totally real number field?**Q** - 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.
- 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.

For more information email *workshops@aimath.org*

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