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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?

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The statement of the congruence or a link to it wouldn't harm anyone... –  Giovanni De Gaetano Jun 6 '12 at 8:54
Oh, I am sorry about this, fixed. –  fretty Jun 6 '12 at 8:57
Philosophically, you could argue that it's interesting because it shows that a cusp form (a deep sophisticated object) can be congruent mod p to an Eisenstein series (a much more explicit and tractable thing). –  David Loeffler Jun 6 '12 at 9:24
Ah right, so this congruence kind of gives more precision to the fact that Eisenstein series generate the space? It tells us a more specific connection between some cusp forms and Eisenstein series? –  fretty Jun 6 '12 at 9:37
When viewed "locally" some highly nontrivial modular forms may become a little bit more trivial? –  fretty Jun 6 '12 at 9:39

3 Answers 3

up vote 7 down vote accepted

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.

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The congruence $$ \tau(n) \equiv \sigma_{11}(n) \pmod{691} $$ is one of several congruences that have been proven.

Now it is well known that $$ \Delta(z) = 2 \pi^2 \sum_{n = 1}^\infty \tau(n)q^n. $$

Lehmer conjectured that $\tau(n) \neq 0$ for all positive integers $n$. As far as I know this has been checked for $n < 10^{11}$. The congruences play a role in showing part of his conjecture.

As to why is it interesting to consider why $\tau(n) \neq 0$ for all positive integers $n$, see this.

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Ah, that makes a bit more sense now. I guess the congruence tells you that $\tau(n)$ is almost always non-zero (I've never viewed it in this way before). It only remains to check $n$ such that $\sigma_{11}(n) \equiv 0$ mod $691$ to prove Lehmer's conjecture. And as your link says, this has powerful meanings in terms of analytic continuations. –  fretty Jun 6 '12 at 12:45
I would recommend you to a paper of Gandhi's about the nonvanishing of $\tau(n)$ if it weren't so full of errors. –  Eugene Jun 6 '12 at 12:48
Well to be honest I have only ever seen the importance of the $\Delta$ function in helping to study other spaces of modular forms. The significance of it by itself is not obvious either! I certainly couldn't just dream up such a function and see it as important. –  fretty Jun 6 '12 at 12:53
@fretty Indeed it is very useful to study spaces of modular forms. For instance, the proof that all even weight modular forms of level $1$ is generated by Eisenstein series depends on the $\Delta$ modular form. Also you can show that the even weight modular forms on $\Gamma_0(N)$ can be generated by Dedekind $\eta$-functions using $\Delta$. –  Eugene Jun 6 '12 at 13:04
Yes I know this, but is that basically the reason for inventing this function? Is there any reason why the $\Delta$ function is interesting by itself? –  fretty Jun 6 '12 at 13:05

This is an extract to Manin's paper "Periods of parabolic forms and p-adic Hecke Series":

"This congruence is so far, our only clue to understanding the 11-dimensional étale cohomology of the so-called Sato variety"

Even though I don't know what he means, he gives some references to Serre's articles.

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You can also take a look at this link math.stackexchange.com/questions/11720/… –  student Jun 18 '13 at 17:53

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