Atkin–Lehner theory

In mathematics, Atkin–Lehner theory is part of the theory of modular forms, in which the concept of newform is defined in such a way that the theory of Hecke operators can be extended to higher level. A newform is a cusp form 'new' at a given level N, where the levels are the nested subgroups

Γ₀(N)

of the modular group, with N ordered by divisibility. That is, if M divides N, Γ₀(N) is a subgroup of Γ₀(M). The oldforms for Γ₀(N) are those modular forms f(τ) of level N of the form g(d τ) for modular forms g of level M with M a proper divisor of N, where d divides N/M. The newforms are defined as a vector subspace of the modular forms of level N, complementary to the space spanned by the oldforms, i.e. the orthogonal space with respect to the Petersson inner product.

The Hecke operators, which act on the space of all cusp forms, preserve the subspace of newforms and are self-adjoint and commuting operators (with respect to the Petersson inner product) when restricted to this subspace. Therefore, the algebra of operators on newforms they generate is a finite-dimensional C*-algebra that is commutative; and by the spectral theory of such operators, there exists a basis for the space of newforms consisting of eigenforms for the full Hecke algebra.

Atkin–Lehner involutions

The congruence subgroup Γ(N) is the normal subgroup of the special linear group of integral matrices congruent to the identity matrix modulo N. Γ(N) is also a normal subgroup of the group Γ₀(N) defined by

$\Gamma_0(N) = \{ \begin{pmatrix}a & b \\c & d \end{pmatrix} \isin \Gamma (N) : c \equiv 0\ (\operatorname{mod} N) \},$

i.e., the integral matrices congruent to upper triangular matrices modulo N.

Consider the integral matrices of the form

$W_e = \begin{pmatrix}ae & b \\ cN & de \end{pmatrix}$

where det W_e = e and e is a Hall divisor of N, which means that not only does e divide N, but also e and N/e are relatively prime (denoted e||N).

Then $W_e^2 = e^2 ,$ and therefore, after normalization to have determinant 1, acts as an involution on cusp forms, and is called an Atkin-Lehner involution. In particular, if N is squarefree, then the newforms are eigenvectors with eigenvalue 1 or -1. If e and f are both Hall divisors of N, then W_e and W_f commute and their product is W_g where

$g = e f/(e,f)^2 .$

Let Γ₀(N)+ be the group generated by Γ₀(N) and the Atkin-Lehner involutions. Then Γ₀(N) is a normal subgroup of Γ₀(N)+ with index 2^s where s is the number of distinct prime divisors of N.

References

• Atkin, A. O. L.; Lehner, J. (1970), "Hecke operators on Γ₀ (m)", Mathematische Annalen 185: 134–160, doi:10.1007/BF01359701, ISSN 0025-5831, MR 0268123
• Koichiro Harada (2010) "Moonshine" of Finite Groups, page 13, European Mathematical Society ISBN 978-3-03719-090-6 MR 2722318