I know how to prove this congruence in two ways (one using basics of modular forms and the other using Hecke operators) and I will be working on proving other such congruences soon.
The congruence states that for $n\geq 1$:
$\tau(n) \equiv \sigma_{11}(n)$ mod $691$.
My main question is why this congruence is so important? I recognise it as a beautiful thing but the only reason I can come up with for it being interesting is that it links a geometrical function with a number theoretical function.
Are there any other reasons why this is interesting/useful?
Best Answer
Congruences between Hecke eigenforms are outward, "physical" manifestations of corresponding relationships between the associated two-dimensional Galois representations.
In this particular case, Ramanujan's congruence is related to the fact that $691$ is an irregular prime (in the sense of Kummer). Roughly the idea is that Eisenstein series relate to reducible two-dimensional Galois representations, and cuspforms to irreducible two-dimensional Galois representations. The existence of a congruence between the two points to the existence of an object that is somewhere between reducible and irreducible: a certain reducible two-dimensional Galois representation which is, however, indecomposable. The existence of this particular reducible, but indecomposable, two-dimensional representation shows that $691$ is an irregular prime.
To see a hint of how this could be, note that $691$ being an irregular prime means, by class field theory for $\mathbb Q(\zeta_{691})$ --- especially, the theory of the Hilbert Class Field --- that there exists an unramified abelian extension of $\mathbb Q(\zeta_{691})$; so irregularity of $691$ is related to the existence of a certain abelian extension of an abelian extension of $\mathbb Q$, and the reducible but indecomposable Galois representation will have such a thing as its splitting field.
For more information on this, and related ideas, you might like to read Mazur's article on the subject; see the entry June 17, 2010: How can we construct abelian extensions on his web-page.