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 $G = (\mathbb Z/N \mathbb Z)^*$.

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

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!

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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.