What are some applications of the Weil conjectures for algebraic curves?

Question

I have been interested in the Weil conjectures for some time, and the easiest place to start has been in studying them for elliptic curves. I've been able to see some of their applications and consequences, but are there any interesting applications of the Weil conjectures for algebraic curves that are not elliptic?

The Riemann hypothesis for elliptic curves is equivalent to Hasse's theorem $|q + 1 - N_q| \le 2\sqrt{q}$ for bounding the error in counting points on elliptic curves over finite fields. I would guess that this can be extended to counting points on any algebraic curve over $\mathbb{F}_q$ once the RH has been proven for curves. But is there anything further one can do with this generalization? Is it worth trying to dive deeper into the curve case, or just appreciate the elliptic curve case before trying to head to higher dimensional varieties?

The applications I like most outside of pure mathematics are in cryptography, but I also would love anything self-contained, i.e. a neat theorem one can prove after having the WC for curves.

Thank you.

Answer 1

One application is to the Ramanujan--Petersson conjecture for weight two modular forms: the Eichler--Shimura congruence relation relates the $p$th Hecke correspondences on the modular curve to the Frobenius correspondence in char. $p$, and RH (applied to the reduction mod $p$ of the modular curve) then gives what you want.

As well as being of interest in its own right, this result has its own nice consequences: e.g. one can use it to prove the Manin-Drinfeld theorem, which states that degree zero divisors supported on the cusps of the modular curve are necessarily torsion in the Jacobian of the modular curve.

Answer 2

This might be slightly out there, but you can calculate the Betti numbers of a toric variety using the Weil conjectures. See page 94 of Fulton's "Introduction to Toric Varieties." edit: Also, you can use it to define the Hasse-Weil L-function. This will help you do things like state Taniyama-Weil, modularity, and work on the Langlands program. For these last ones, I took some very rough (i.e. probably full of errors) notes my first year of grad school see page 13,14 here: http://divisibility.files.wordpress.com/2012/09/fbntremix.pdf

Also, the Weil conjectures imply the generalized Ramanujam conjecture.

Answer 3

I describe a coding theoretical application as promised.

Assume that $q=2^m$, and let $$ tr:\Bbb{F}_q\to\Bbb{F}_2, x\mapsto \sum_{i=0}^{m-1}x^{2^i} $$ be the trace function. It is relatively easy to show that (ask if you haven't seen it) for a fixed element $a\in\Bbb{F}_q$ the equation $$ y^2+y=a $$ has two solution $y\in\Bbb{F}_q$, if $tr(a)=0$ and no solutions in $\Bbb{F}_q$, if $tr(a)=1$. This result is sometimes called Additive Hilbert 90. Let us fix a polynomial $f(x)\in\Bbb{F}_q[x]$ of an odd degree. To such a polynomial we can associate a binary vector of length $q-1$ with component $tr(f(x))$ for all $x\in\Bbb{F}_q^*$, i.e. the vector $$ \vec{c}_f=(tr(f(1)),tr(f(g)),tr(f(g^2)),\ldots,tr(f(g^{q-2})))\in\Bbb{F}_2^{q-1}, $$ where $g$ is a generator of the group $\Bbb{F}_q^*$.

Let the Hamming weight of $\vec{c}_f$ (i.e. its number of components equal to $1$) be $w_f$. By the earlier considerations the number of solutions of $y^2+y=f(x) $ for a fixed $x$ is $2$ if $tr(f(x))=0$ and $0$ otherwise. So letting $x$ vary over all the non-zero values the number $N_f$ of solutions $(x,y)\in\Bbb{F}_q^2, x\neq0$ of the equation $$ y^2+y=f(x)\qquad(*) $$ is tied to $w_f$ by the equation $$ N_f=2(q-1-w_f). $$ This gives us a formula for the Hamming weight $$ w_f=q-1-\frac12 N_f. $$ Hence an upper bound for $N_f$ gives a lower bound for the Hamming weight.

Assuming that $\deg f=2t+1$ it is not hard to show that the genus of the curve $(*)$ is $g=t$. Therefore (counting for the two points on the curve with $x=0$ and for the point at infinity) Hasse-Weil bound gives us the estimate $$ w_f\ge\frac12(q+1-t\sqrt q).\qquad(**) $$

We get the dual of the (narrow sense) $t$-error-correcting BCH-code by letting $f$ range over the set of polynomials with only odd degree terms of degree $\le 2t+1$. Therefore $(**)$ is a lower bound to the minimum Hamming distance of that code. The bound is reasonably accurate in most cases. Some improvements to it are known for carefully constructed subcodes in special cases.

Answer 4

You can use the Hasse-Weil bounds to prove that the Hasse-Weil L-function $L(C,s)$ of a curve $C$ converges for $\Re(s)>3/2$. See page 312 in Husemoller's "Elliptic Curves".