Editing Local zeta function (section)

==Riemann hypothesis for curves over finite fields==

For projective curves ''C'' over ''F'' that are [[Algebraic curve#Singularities|non-singular]], it can be shown that

:<math>Z(t) = \frac{P(t)}{(1 - t)(1 - qt)}\ ,</math>

with ''P''(''t'') a polynomial, of degree 2''g'', where ''g'' is the [[genus (mathematics)|genus]] of ''C''. Rewriting

:<math>P(t)=\prod^{2g}_{i=1}(1-\omega_i t)\ ,</math>

the '''Riemann hypothesis for curves over finite fields''' states

:<math>|\omega_i|=q^{1/2}\ .</math>

For example, for the elliptic curve case there are two roots, and it is easy to show the absolute values of the roots are ''q''<sup>1/2</sup>. [[Hasse's theorem on elliptic curves|Hasse's theorem]] is that they have the same absolute value; and this has immediate consequences for the number of points.

[[André Weil]] proved this for the general case, around 1940 (''Comptes Rendus'' note, April 1940): he spent much time in the years after that [[Foundations of Algebraic Geometry|writing]] up the [[algebraic geometry]] involved. This led him to the general [[Weil conjectures]]. [[Alexander Grothendieck]] developed [[scheme (mathematics)|scheme]] theory for the purpose of resolving these.
A generation later [[Pierre Deligne]] completed the proof. 
(See [[étale cohomology]] for the basic formulae of the general theory.)