Finite Index Subgroups of the Modular Group and Their Modular Forms

1. **Finite Index Subgroups of the Modular
Group**

The special linear group *SL _{2}*(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

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.

Expository
paper:
*Computations with finite index subgroups of
PSL _{2}(*

3. **Noncongruence Modular Forms**

** Relations between congruence and noncongruence modular forms at a first
glance: **Given a elliptic curve

Suppose Theorem (W.C. Li, Tong Liu, and L. Long 2008): fΣ=a(n)q^{n}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)

•

•

** 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,

*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.

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

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, *Heck*e *operator*s *on* Γ_{0}(*m*), Math. Ann. 185 (1970), 134–160.

[ALL08]
A. O. L. Atkin, W.C. Li, and L. Long,
*On* *Atkin* *and* *Swinnerton-Dyer* *congruence* *relation*s *(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* *relation*s *for* *some* *noncongru**ence cuspforms. *

[Ber94]
G. Berger, *Hecke* *operator*s *on* *noncongruence* *subgroups*, C. R. Acad. Sci. Paris S´er. I Math. 319 (1994),
no. 9, 915–919.

[Ber00]
G. Berger,* Relation*s *between* *cusp* *form*s *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* *form*s *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
SL _{2}(Z) II *(with
Chris A. Kurth), preprint (2008)

[KL07]

[LLY05a] W.C Li, L. Long, and Z. Yang,

[LLY05b] W.C Li, L. Long, and Z. Yang,

[Lon06] L. Long,

[Lon07] L. Long,

[Sch85] A. J. Scholl,

[Sch88] A. J. Scholl,

[Sch04] A. J. Scholl,

**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.