Atkin–Lehner theory explained

In mathematics, Atkin–Lehner theory is part of the theory of modular forms describing when they arise at a given integer level N in such a way that the theory of Hecke operators can be extended to higher levels.

Atkin–Lehner theory is based on the concept of a newform, which is a cusp form 'new' at a given level N, where the levels are the nested congruence subgroups:

\Gamma_0(N)=\left\{\begin{pmatrix}a&b\ c&d\end{pmatrix}\inSL(2,Z):c\equiv0\pmod{N}\right\}

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

Consider a Hall divisor e of N, which means that not only does e divide N, but also e and N/e are relatively prime (often denoted e||N). If N has s distinct prime divisors, there are 2^s Hall divisors of N; for example, if N = 360 = 2³⋅3²⋅5¹, the 8 Hall divisors of N are 1, 2³, 3², 5¹, 2³⋅3², 2³⋅5¹, 3²⋅5¹, and 2³⋅3²⋅5¹.

For each Hall divisor e of N, choose an integral matrix W_e of the form

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

with det W_e = e. These matrices have the following properties:

The elements W_e normalize Γ₀(N): that is, if A is in Γ₀(N), then W_eAW is in Γ₀(N).
The matrix W, which has determinant e², can be written as eA where A is in Γ₀(N). We will be interested in operators on cusp forms coming from the action of W_e on Γ₀(N) by conjugation, under which both the scalar e and the matrix A act trivially. Therefore, the equality W = eA implies that the action of W_e squares to the identity; for this reason, the resulting operator is called an Atkin–Lehner involution.
If e and f are both Hall divisors of N, then W_e and W_f commute modulo Γ₀(N). Moreover, if we define g to be the Hall divisor g = ef/(e,f)², their product is equal to W_g modulo Γ₀(N).
If we had chosen a different matrix W ′_e instead of W_e, it turns out that W_e ≡ W ′_e modulo Γ₀(N), so W_e and W ′_e would determine the same Atkin–Lehner involution.

We can summarize these properties as follows. Consider the subgroup of GL(2,Q) generated by Γ₀(N) together with the matrices W_e; let Γ₀(N)⁺ denote its quotient by positive scalar matrices. Then Γ₀(N) is a normal subgroup of Γ₀(N)⁺ of index 2^s (where s is the number of distinct prime factors of N); the quotient group is isomorphic to (Z/2Z)^s and acts on the cusp forms via the Atkin–Lehner involutions.

References

Mocanu, Andreea. (2019). "Atkin-Lehner Theory of Γ₁(m)-Modular Forms"