In this previous post, we saw the existence of a common eigenvector, namely number of nonzero solutions to modulo . This was not a coincidence. Indeed, it was based on the fact that is a family of self-adjoint and commuting operators on the space of complex-valued functions on .

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

This post is a generalization.

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

Given any holomorphic on the upper half plane and , define

.

It is a fact that for any , there is a double-coset decomposition .

Define for such a decomposition,

.

Observe that so that defines a well-defined action of on the ‘s. There is a vector space called the space of modular forms and a -invariant subspace – – the space of cusp forms (similar to in the previous post) and for varying , the operators (called the **Hecke operators**)

,

.

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 function has an Euler product (similar to the function). This Euler product gives the Ramanujan’s identity –

(Here, is the Ramanujan- 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 ‘s in the earlier post. I haven’t seen any book that mentions about these.

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