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Can anyone explain (heuristically, intuitively is fine) what the importance of the Weil conjectures is? I realize they have motivated much of recent algebraic geometry. I don't really understand why or what one should do with them, now that they have been proven.

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Which Weil conjectures? There are several going by that name. – Gerry Myerson Dec 8 '12 at 6:12
@GerryMyerson are you serious? – Adeel Dec 8 '12 at 6:27
@Adeel, see for yourself --- – Gerry Myerson Dec 8 '12 at 6:31
@GerryMyerson I'm asking about the conjectures for zeta functions of smooth projective varieties over $\mathbb{F}_q$. – Dog 2 Dec 8 '12 at 6:45
Gerry, there is no ambiguity in the term "Weil conjectures" (note the plural), especially in the context of algebraic geometry. – Adeel Dec 8 '12 at 17:21
up vote 4 down vote accepted

One very incredible thing about the Weil Conjectures is what they say about the geometry of a smooth, projective variety $X$ defined over $\mathbb{Q}$ (for example); some of the geometry of $X$ as a complex manifold can be computed by counting points on $X$ modulo $p$ for some (suitable) prime. Specifically, the Weil Conjectures allow you to compute the Betti numbers (dimensions of vector spaces coming from cohomology) of $X/\mathbb{C}$ by computing the Zeta function of $X/\mathbb{F}_p$. This Zeta function has properties quite analogous to the Riemann Zeta function, including a Riemann Hypothesis. It is a rational function of a single variable, and it is constructed with a(n exponential of a) power series, the $n$th coefficient being the number of points on $X$ over the finite field $\mathbb{F}_{q}$, for $q=p^n$. If you're familiar with Elliptic Curves, the Hasse-Weil inequality

$$ \left| \#E(\mathbb{F}_q) -(q+1) \right| \leq 2\sqrt{q} $$

for an elliptic curve $E/\mathbb{F}_q$, is secretly the Riemann Hypothesis in disguise.

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