In this previous post, we saw the existence of a common eigenvector, namely \phi(n) = a_n = number of nonzero solutions to x^2=d modulo n. This was not a coincidence. Indeed, it was based on the fact that \{ T_p : p \nmid N := 4|d| \} is a family of self-adjoint and commuting operators on the space of complex-valued functions on G = (\mathbb Z/N \mathbb Z)^*.

(Here, by self-adjoint, I’m talking about the inner product

\langle f,g \rangle = \displaystyle\sum_{r\in G} f(r) \overline{g(r)} . \qquad )

This post is a generalization.

Let f be a holomorphic function on the upper half complex plane. We say f is modular if it satisfies a technical condition called “holomorphic at the cusps” and the following.

\displaystyle f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k f(z) \quad \forall \gamma = \begin{pmatrix} a&b\\c&d \end{pmatrix} \in \Gamma := SL(2,\mathbb Z).

Given any f holomorphic on the upper half plane and \gamma \in \Gamma(1), define

\displaystyle (f|_\gamma)(z) := (cz+d)^{-k} (\text{det} \gamma)^{k/2} f\left(\frac{az+b}{cz+d}\right), \quad \forall \gamma = \begin{pmatrix} a&b\\c&d \end{pmatrix} \in \Gamma(1).

It is a fact that for any \alpha \in \text{GL}(2,\mathbb Q)^+, there is a double-coset decomposition \Gamma(1) \alpha \Gamma(1) = \displaystyle\bigcup_{i=1}^l \Gamma(1) \alpha_i.

Define for such a decomposition,

f|_{T_\alpha} := \displaystyle\sum_{i=1}^l f|_{\alpha_i}.

Observe that (f|_{\alpha})|_{\beta} = f|_{\alpha\beta} so that defines a well-defined action of \Gamma(1) on the f‘s. There is a vector space called the space of modular forms and a T_\alpha-invariant subspace – S_k(\Gamma(1)) – the space of cusp forms (similar to V_N in the previous post) and for varying \alpha, the operators T_\alpha (called the Hecke operators)

T_\alpha : S_k(\Gamma(1)) \to S_k(\Gamma(1)),

T_\alpha(f) := f|_{T_\alpha}.

It’s a cool theorem that the Hecke algebra is commutative and the Hecke operators are self-adjoint with respect to an inner product (the Petersson inner product). A standard result in linear algebra tells that these can be diagonalized; there is a common eigenvector, called the Hecke eigenform. When suitably normalized, it’s associated L- function has an Euler product (similar to the \zeta function). This Euler product gives the Ramanujan’s identity –

\displaystyle\sum_{n=1}^\infty \tau(n) n^{-s} = \prod_p \frac{1}{1-\tau(p) p^{-s} + p^{11-2s}}.

(Here, \tau is the Ramanujan-\tau function. )

Pretty cool stuff, eh!


Tailpiece: References (since I am very vague here) –

  • A first course in modular forms – Diamond, Shurman
  • Automorphic forms and representations – Daniel Bump

Also, I was interested in the properties the L-function corresponding to the a_p‘s in the earlier post. I haven’t seen any book that mentions about these.