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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.)
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