Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
Which Weil conjectures? There are several going by that name. –  Gerry Myerson Dec 8 '12 at 6:12
2  
@GerryMyerson are you serious? –  Adeel Dec 8 '12 at 6:27
    
@Adeel, see for yourself --- en.wikipedia.org/wiki/Weil_conjecture_(disambiguation) –  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
4  
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

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.