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The logo for this year's ICM show the inequality $ |\tau(n)| \leq n^{11/2} d(n)$ where $\sum \tau(n)q^n = q \prod_{n \neq 1} (1 - q^n)^{24}$ is the tau function. Wikipedia says this bound was conjectured by Ramanujan (appropriate for a conference in Hyderabad) and proven by Deligne in '74 in the process of proving as a corollary of the Weil Conjectures (which I also don't get). The background of the ICM logo looks like Ford circles (or sun rays). What is the hyperbolic geometry behind the Tau Conjecture and its proof?

EDIT: It would also be nice to see the proof of this bound, but the Weil conjectures and l-adic cohomology are a topic in themselves.

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There is no hyperbolic geometry behind the conjecture... So please explain what do you like to have: the precise link between Weil conjectures and the growth of the coefficients of cusp forms? – Wadim Zudilin Apr 17 '10 at 14:42
I am not too familiar with modular forms. However, I know the generating function of the tau function is the Dedekind eta function, which is a modular form of weight 1/2. This is the type of geometric relation I was expecting. Cusp forms are defined by the vanishing of the 0-th Fourier coefficient and I guess the tau function is an example of this. The Weil conjectures do not involve hyperbolic geometry and I don't know why they come into play here. – john mangual Apr 17 '10 at 15:38
I would only correct that the generating function of Ramanujan's tau is the 24th power of Dedekind's function, so it's a cusp form of weight 12. A simple argument due to Hecke (see, e.g., Serre's Course in Arithmetic) shows that the $n$-th Fourier coefficient of a cusp form $f(z)$ of weight $2k$ is $O(n^k)$. If $f$ admits an additional arithmetic structure, namely if its $L$-series admits an Euler-product expansion, then one gets $O(n^{k-1/2}d(n))$. – Wadim Zudilin Apr 17 '10 at 23:29
up vote 18 down vote accepted

The logo has a piece of the complex upper half plane divided into fundamental domains for the action of $SL_2(\mathbb{Z})$ by Möbius transformations (which are hyperbolic isometries - see the appropriate entry in McMullen's gallery or Wikipedia). There are no Ford circles in sight, but you may have been confused by the semicircular regions on the bottom. Those regions are unions of fundamental domains, and are cut out by geodesics in the $SL_2(\mathbb{Z})$ orbit of $1/2 \leftrightarrow \infty$ (which forms part of the boundary of a few fundamental domains). Ford circles come from the $SL_2(\mathbb{Z})$ orbit of the horocycle $\operatorname{Im}(\tau) = 1$, and horocycles have constant nonzero geodesic curvature (imagine driving with your steering wheel turned a bit to the right, but never returning to where you started).

The generating function for $\tau$ is the 24th power of Dedekind's $\eta$ function (often written as the discriminant form $\Delta$), and it is a function on the upper half plane that is invariant under the weight 12 action of $SL_2(\mathbb{Z})$. Up to normalization, it is the unique lowest weight nonzero level 1 cusp form, and this makes it automatically a Hecke eigenform. Both Mordell's proof of the earlier Ramanujan conjectures and Deligne's proof of the growth conjecture use this fact in an essential way. I should note that Deligne proved the Ramanujan conjecture as a corollary of the Weil conjectures, not "in the process".

The connection between hyperbolic geometry and the Ramanujan conjecture is not particularly strong, as far as I know (but I would be happy to be shown the errors of my ways).

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One way to answer this question is as follows: Ramanujan's conjecture is a special case of a much more general conjecture that any cuspdial automorphic representation of $GL_n$ over a number field is tempered. This is a technical but fundamental notion, which in the special case of the automorphic representation of $GL_2$ attached to the $\Delta$ function, reduces to Ramanujan's original conjecture. In fact, many people working in the theory of automorphic representations refer to this very general conjecture simply as the Ramanujan conjecture.

When applied to other cuspforms on $GL_2$ (namely Maass forms) it includes Selberg's conjecture that on congruence quotients of the upper half-plane, the spectrum of the hyperbolic Laplacian is bounded below by $1/4$.

The appearance of hyperbolic geometry can be understood in the following way: the quotient of $SL_n(\mathbb R)$ by $SO(n)$ is a non-compact symmetric space, which in the particular case of $SL_2$ is the hyperbolic plane. So, while the particular appearance of hyperbolic geometry may be a bit of a red herring, the appearence of highly symmetric geometry is a reflection of the group representation theory that is underlying the picture.

As of the current moment, no purely representation-theoretic approach to the (general form of) Ramanujan's conjecture is known. (Or rather, a proof strategy involving what is called symmetric power functoriality is known, but the requisite results on symmmetric power functoriality seem very much out of reach at the moment). The only cases that are proved at the moment are cases when one can relate the group theoretic picture of automorphic forms to algebraic geometry (first over $\mathbb C$, then over a number field, and then ultimately over finite fields, so that the Weil conjecture apply). This is how Deligne's proof proceeds. This connection between the geometry of symmetric spaces and arithmetic and geometry over finite fields is one of the profound points of investigation of modern number theory, but despite many positive results related to it, it remains essentially mysterious, even to experts.

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