Finite Index Subgroups of the Modular Group and Their Modular Forms

1. Finite Index Subgroups of the Modular Group

        The special linear group SL2(Z) consists of all 2-by-2 integral matrices with determinant 1. It is one of the most well-known and important discrete groups. A finite index subgroup of SL2(Z)  is said to be congruence if it contains the kernel of a modulo N homomorphism from SL2(Z) to SL2(Z/NZ) for some positive integer N; otherwise, it is called a noncongruence subgroup. The existence of noncongruence subgroups of SL2(Z) was confirmed by Fricke and Pick. In contrast, any finite index subgroup of SLn(Z) with n 3 is congruence. A famous theorem of Belyi implies any compact complex smooth irreducible curve defined over the algebraic closure of Q, can be realized as a modular curve for a finite index subgroup of SL2(Z). Consequently, finite index subgroups arise naturally in many fields such as the theory of dessin d’enfant. Among them most of them are noncongruence subgroups.

2. Computational Package

KFarey: SAGE package for arithmetic subgroups (by Chris Kurth).

        KFarey is a computational package for arithmetic subgroups. In KFarey, a congruence subgroup can be specified classically. In general, an arithmetic subgroup is identified with a pair of permutations of order 2 and 3 respectively and subject to some conditions (Millington). Currently, KFarey can compute the basic invariants of the arithmetic subgroup such as index, genus, level, Farey symbols (Kulkarni), an independent set of generators, identify whether the group is congruence (Hsu), etc. KFarey can also output the special polygonal of the arithmetic subgroup. Please send your suggestions or comments on KFarey to Chris Kurth.

A sample output of KFarey

Expository paper: Computations with finite index subgroups of PSL2(Z) using Farey Symbols, Chris A. Kurth and Ling Long, 2007

3. Noncongruence Modular Forms

Relations between congruence and noncongruence modular forms at a first glance: Given a elliptic curve E defined over Q, by the Taniyama-Shimura-Weil conjecture, there exisits a weight 2 congruence newform g such that L(E,s)=L(g,s). On the other hand, by the Belyi's theorem, E is isomorphic to some modular curve a finite index subgroup G, which is mostly likely to be noncongruence. In such a setting, the space of weight 2 cuspform for G is one-dimensional. Let f be a basis of this space. Suppose f is a genuine noncongruence cuspform. It is natural to ask:
        What are the relations between f and g?
        What can we say about the properties of f?

Theorem (W.C. Li, Tong Liu, and L. Long 2008):  Suppose f=Σa(n)qn is a genuine noncongruence cuspform with rational Fourier coefficients such that it is associated with a compatible family of 2-dimensional l-adic representations of the absolute Galois group, then f has unbounded denominators. Namely there does not exist any integer M such that a(n).M is integral for all n.


Congruence modular forms:

Hecke operators, Hecke eigen forms
Any space of congruence cuspforms has an integral basis
Theory of newform (Atkin-Lehner, Miyake, Li)
l-adic Deligne representations attached to newforms
···

Developments of Noncongruence modular forms:
• Fricke and Klein considered Wurzelmoduls, which are noncongruence modular forms.
• (1971) Atkin and Swinnerton-Dyer (ASD)
    1. ASD congruences: for almost all primes p,  there exists a basis {f =Σa(n)qn}.

a(np) --A(p)a(n)+ B(p)a(n/p)   0 mod p e(k,n).

        The concept of p-adic Hecke operators.
    2. Unbounded denominator property (UBD).
        The UBD conjecture: for any genuine noncongruence modular form with algebraic coefficients, its coefficients have unbounded denominators.
(Mid 80’s-present) A. Scholl
    1.
M-integral basis for noncongruence cuspforms
   
2. l-adic Scholl representations
    3.Collective version of ASD congruences
    4.Modularity of some 2-dim’l Scholl representations
• (Mid 90’s-2000) G. Berger
    1.Proved Atkin’s conjecture on Hecke operators on noncongruence modular forms.
    2.Relations between congruence and noncongruence mod. forms
(2005) Li-Long-Yang; Hoffman-Verrill-et al.
    1.
Simultaneous diagonalization for almost all p-adic Hecke operators
    2.
Modularity of 4-diemsnional representations
(2005) Atkin-Li-Long; (published on March 2008, Math Ann.)
    1.
Semi-diagonalization
   
2. Modularity
(2006) Long: “
newforms”, 3-term ASD type congruences
• (2006) Kurth-Long: partial positive answer for UBD (Journal of Number Theory [LK08a])
(2007) Atkin-Long: ASD congruence for a 6-dimensional case
• (2008) Li-Liu-Long: some general modularity results and partial results on UBD for 1-dimensional cases
• (2008) Kurth-Long: proved more cases where the UBD holds [KL08b]

4. Some Related References

[AL70]  A. O. L. Atkin and J. Lehner, Hecke operators on Γ0(m), Math. Ann. 185 (1970), 134–160.
[ALL08] A. O. L. Atkin, W.C. Li, and L. Long, On Atkin and Swinnerton-Dyer congruence relations (2),
Mathematische Annalen,  Volume 340 No. 2 (2008) pp 335-358.
[ALo07] A. O. L. Atkin and L. Long, On Atkin and Swinnerton-Dyer congruence relations for some noncongruence cuspforms.

[Ber94]  G. Berger, Hecke operators on noncongruence subgroups, C. R. Acad. Sci. Paris S´er. I Math. 319 (1994), no. 9, 915–919.
[Ber00] G. Berger, Relations between cusp forms on congruence and noncongruence groups, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2869–2874.

[F+05] L. Fang, J. W. Hoffman, B. Linowitz, A. Rupinski, and Verrill H., Modular forms on noncongruence subgroups and Atkin-Swinnerton-Dyer relations, preprint (2005).
[KL08a]
On modular forms for some noncongruence subgroups of SL2(Z) (with Chris A. Kurth), Journal of Number Theory
[KL08b] On modular forms for some noncongruence subgroups of SL2(Z) II (with Chris A. Kurth), preprint (2008)
[KL07]
Computations with finite index subgroups of PSL2(
Z) using Farey Symbols, (with Chris A. Kurth), expository paper, 2007
[LLY05a] W.C Li, L. Long, and Z. Yang, Modular forms for noncongruence subgroups, Quarterly Journal of Pure and Applied Mathematics 1 (2005), no. 1, 205–221.
[LLY05b] W.C Li, L. Long, and Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. of Number Theory 113 (2005), no. 1, 117–148.
[Lon06] L. Long, On Atkin and Swinnerton-Dyer congruence relations (3), math.NT/0701310 (2006).
[Lon07] L. Long, The finite index subgroups of the modular group and their modular forms, Survey, (2007) (Please refer to the bibliography of this paper for a list of related literature.)
[Sch85] A. J. Scholl, Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences, Invent. Math. 79 (1985), no. 1, 49–77.
[
Sch88] A. J. Scholl, The l-adic representations attached to a certain noncongruence subgroup, J. Reine Angew. Math. 392 (1988), 1–15.
[Sch04] A. J. Scholl, On some l-adic representations of galois group attached to noncongruence subgroups, math.NT/0402111 (2004).

5. Upcoming workshop

Noncongruence modular forms and modularity 2009 AIM workshop

 

6. Data

1. Kurth's data on the Hauptmoduls of index 24 genus 0 torsion free noncongruence character subgroups

 

If you have questions, comments, or suggestions, please contact Ling Long.