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Let $\sigma_{11}(n)$ denote the sum of the 11th powers of the positive integral divisors of the positive integer n. Let $\tau(n)$ denote Ramanujan's tau function, which is the coefficient of $q^n$ in the power series $q \Pi_{n=1}^{\infty} (1-q^{n})^{24}$. It was proven by Hecke that if $n$ is a positive integer, the number of vectors of length $\sqrt{2n}$ in the Leech lattice is $\frac{65520}{691}( \sigma_{11}(n) - \tau(n) )$. The integrality of this number implies the congruence of $\sigma_{11}(n)$ and $\tau(n)$ modulo 691, but this congruence is often called Ramanujan's congruence. Ramanujan worked before the discovery of the Leech lattice, so he must have had some other way of proving the congruence. How did he prove it?

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Are you sure about Hecke knowing the Leech lattice ? At least, not under this name (Leech papers are from 60's, and Hecke died in 1947). I see in comments below that Witt discovered it in 1940, but did not publish. –  BS. Jan 30 '11 at 12:01
    
No. I am not sure about Hecke knowing the Leech lattice. What seems more likely now is that I misread the phrase 'it follows from work of Hecke' in Conway's paper (about the automorphism group of the Leech lattice) to say that Hecke had the lattice in mind, instead of reading it to say that Hecke provided the tools for the proof (of the counting formula) without knowing they would be used that way. –  DavidLHarden Feb 7 '11 at 23:10
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2 Answers 2

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Some historical comments should not be out of place. I'm writing them without checking the facts.

Ramanujan published his paper in 1916 under the modest title On some arithmetical functions. He made a number of conjectures about the $\tau$-function such as $\tau(mn)=\tau(m)\tau(n)$ whenever $\gcd(m,n)=1$, and a recursive formula for $\tau(p^{r+2})$ in terms of $\tau(p^{r+1})$ and $\tau(p^r)$, for primes $p$.

These were proved by Mordell in 1918 using methods which prefigure the use of Hecke operators. Now they are a consequence of Hecke's theory when you remark that the $\Delta$-function is a primitive cuspidal eigenform of weight $12$ and level $1$. All this was before the discovery of the Leech lattice by Witt in 1940 (unpublished, but see his Collected Papers).

Ramanujan made a second set of conjectures about congruences satisfied by $\tau(n)$ modulo $2^{11}$, $3^7$, $5^3$, $7$, $23$ and $691$, some of which he proved (for example for $691$). It was thinking about these congruences, and about Swinnerton-Dyer's approach to them (see Modular forms in one variable and Serre's talk in the Delange-Pisot-Poitou seminar) that led Serre in the early 70s to the conjecture that there is an $l$-adic ($l$ prime) representation of $\operatorname{Gal}(\bar Q|Q)$ attached to every primitive cuspidal eigenform --- conjecture proved by Deligne (see his Bourbaki talk).

These congruences also led Serre to formulate the first form of his conjecture about the modularity of odd irreducible representations $\operatorname{Gal}(\bar Q|Q)\rightarrow\operatorname{GL}_2(\bar F_l)$ which has recently been proved by Khare and Wintenberger.

The third conjecture made by Ramanujan was the estimate $|\tau(p)|\leq 2p^{11/2}$; it was finally proved in the 70s by Deligne as a consequence of his proof of the Weil conjectures.

There is one conjecture which even Ramanujan did not make. It concerns the distribution of the numbers $\tau(p)/2p^{11/2}$ in the interval $[-1,1]$. The Sato-Tate conjecture asserts that these numbers are equidistributed with respect to the measure $(2/\pi)\sqrt{1-t^2}\,dt$, and this has finally been proved very recently by T. Barnet-Lamb, D. Geraghty, M. Harris and R. Taylor (see Taylor's homepage at Harvard).

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I haven't actually read Ramanujan's original article, which appears in \bib{MR2280843}{collection}{ author={Ramanujan, Srinivasa}, title={Collected papers of Srinivasa Ramanujan}, note={Edited by G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson; Third printing of the 1927 original; With a new preface and commentary by Bruce C. Berndt}, publisher={AMS Chelsea Publishing, Providence, RI}, date={2000}, pages={xxxviii+426}, isbn={0-8218-2076-1}, review={\MR{2280843 (2008b:11002)}}, }

However, Ramanujan certainly was familiar with modular forms and the "standard" proof of this congruence uses only modular forms. Namely, the function $\Delta(z) = \sum_n \tau(n)q^n$ (where $q = e^{2\pi i z}$) and the Eisenstein series $G_{12}(z) = -\frac{B_{12}}{24} + \sum_n \sigma_{11}(n)q^n$ are both modular forms for $\mathrm{SL}(2,\mathbf{Z})$ of weight twelve. One can show that they satisfy the linear relation $$ \Delta = G_{12} + \frac{691}{156}\left( \frac{E_4^3}{720} +\frac{E_6^2}{1008}\right)$$ by a standard method of modular forms: one computes the dimension of the space of modular forms of given weight (in this case dimension two), then one produces more modular forms of this weight than the dimension, and then it suffices to prove a relationship like the one above for finitely many Fourier coefficients in order to prove it for all of them.

Equating the Fourier coefficients in the above equation and reducing modulo 691 gives the congruence of Ramanujan. (NB: the Fourier coefficients of $E_4$ and $E_6$ are integers.)

A nice reference is Don Zagier's part of the book "The 1-2-3 of Modular Forms".

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Probably the $B_{24}$ should be $B_{12}$. –  Chandan Singh Dalawat Jan 30 '11 at 8:25
    
Absolutely, thanks. –  Dan Petersen Jan 30 '11 at 8:38
    
See also the proof in umpa.ens-lyon.fr/~brunault/recherche/FonctionTau.pdf $$ $$ –  Chandan Singh Dalawat Jan 30 '11 at 10:22
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