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Ramanujan's $\tau$ conjecture states that $$\tau(n)=O_\epsilon(n^{\frac{11}2+\epsilon}),$$ which is a consequence of Deligne's proof of Weil conjectures. Answers in https://math.stackexchange.com/questions/1205419/status-of-taun-before-deligne/1205516 tell that best exponent before Deligne reached $\frac{29}5$.

What was it that cohomological approach gave that broke the traditional approach barrier to reach $5.5$ in exponent?

Posted originally in here https://math.stackexchange.com/questions/1206558/etale-cohomology-approach-on-taun where comments suggested to post in MO.

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One of the goals of the development etale cohomology was to generate a cohomology theory that could successfully count points on varieties over finite fields, with one of the main goals of proving the conjecture that Andre Weil laid out in 1949 (see his paper here). Indeed, the formulation of the conjectures already suggests Weil was thinking along these lines. The most striking of these conjectures is the Riemann hypothesis for the zeta function of a variety over a finite field. This conjecture was proven by Deligne in 1974 (available here).

The simplest case had been known for a quite while, namely that if $E/\mathbb{F}_{p}$ is an elliptic curve, then $$ p + 1 - 2 \sqrt{p} < |E(\mathbb{F}_{p})| < p + 1 + 2\sqrt{p}.$$ This result is as sharp as it gets. For every integer $k$ satisfying $p+1 - 2 \sqrt{p} < k < p+1 + 2 \sqrt{p}$, there is an elliptic curve $E/\mathbb{F}_{p}$, with exactly $k$ points. In general, the bounds on the number of points on a variety (and character sums in general) that one obtains using the Riemann hypothesis are very powerful. (For example, in 1981 Hooley proved an asymptotic formula for the number of representations of an integer as a sum of two squares and three non-negative cubes. Hooley's work required estimates sharp bounds on certain character sums, and Milne proved these estimates by relating them to etale cohomology and using Deligne's proof of the Weil conjectures. Hooley and Milne's papers appear back-to-back in the same issue of Crelle.)

What does this have to do with the Fourier coefficients of modular forms? Deligne constructed algebraic varieties whose etale cohomology groups afford the Galois representations (conjectured to exist by Serre and Swinnerton-Dyer) attached to modular forms. Applying the Riemann hypothesis to these gives the "Deligne bound" that $|\tau(n)| \leq d(n) n^{11/2}$, where $d(n)$ is the number of divisors of $n$. (This application is explicitly mentioned in Deligne's 1974 paper - see Theoreme 8.2.)

Most interestingly, Rankin's 1939 bound $\tau(n) = O(n^{29/5})$ played a role in Deligne proving the Riemann hypothesis for varieties over finite fields. In particular, Langlands observed that knowledge of the poles of symmetric power $L$-functions attached to $\Delta$ would be sufficient to conclude that $|\tau(p)| \leq 2p^{11/2}$, and Deligne observed that there was a nice translation of this idea into Grothendieck's cohomology theory. By making this work, Deligne overcame the fact that (at the time), knowledge was only had of the poles of $L({\rm Sym}^{2} \Delta, s)$.

Finally, I will note that Kapil Paranjape and Dinakar Ramakrishnan have constructed an explicit Calabi-Yau 11-fold that gives rise to the Galois representation attached to $\Delta$. (See their paper here.)

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  • $\begingroup$ "Langlands observed that knowledge of the poles of symmetric power L-functions attached to Δ would be sufficient to conclude that |τ(p)|≤2p11/2, and Deligne observed that there was a nice translation of this idea into Grothendieck's cohomology theory" Is there a way to make this more explicit? $\endgroup$ – T.... Mar 26 '15 at 7:18
  • $\begingroup$ Yes. One way of describing this is saying that Langlands's observation is that the Ramanujan conjecture would follow from knowledge of the poles of $L_{n}(s) = L(\Delta \otimes \Delta \otimes \cdots \otimes \Delta, s)$ (the $n$-fold Rankin-Selberg convolution of $\Delta$ - which factors as a product of symmetric power $L$-functions). The reason is that if $\tau(p) = p^{11/2} (\alpha_{p} + \alpha_{p}^{-1})$ with $\alpha_{p} \in \mathbb{C}$, then $(1 - \alpha_{p}^{n} p^{-s})^{-1}$ is one of the local factors of $L_{n}(s)$. $\endgroup$ – Jeremy Rouse Mar 26 '15 at 12:51
  • $\begingroup$ Convergence for $s > 1$ gives that $|\alpha_{p}^{n}| < p$ and this inequality for all $n$ gives $|\alpha_{p}| = 1$. On the etale cohomology side, Deligne proves a bound on the Frobenius eigenvalues of a variety $X$ of the strength that is roughly the same as $|\tau(p)| \ll p^{6}$. Applying this bound to the variety $X^{n}$ for all $n$ yields the Riemann hypothesis. (See page 301 of Deligne's paper.) $\endgroup$ – Jeremy Rouse Mar 26 '15 at 12:53
  • $\begingroup$ So this is not case of mysterious cancellations improving on random averaging argumentation? $\endgroup$ – T.... Mar 26 '15 at 12:54
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    $\begingroup$ I think Qiaochu's heuristic can be refined a little bit. In particular, fix a number $x$. When $\sum_{k \in \mathbb{Z}} (-1)^{k} q^{k(3k-1)/2}$ is raised to the $12$th power, there are a total of $O(x^{12})$ terms $q^{r}$ for all $r$ with $x \leq r \leq 2x$. Heuristically, one would expect about the same number of terms $q^{r}$ for each $r \in [x,2x]$ and so the coefficient of $q^{r}$ would be a sum $O(x^{11})$ signs. The variance is then $O(x^{11/2})$. So it is a case of "cancellations" achieving averaging augmentation. $\endgroup$ – Jeremy Rouse Mar 26 '15 at 16:16

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