Editing Conjecture (section)

===Weil conjectures===
{{main|Weil conjectures}}
In [[mathematics]], the [[Weil conjectures]] were some highly influential proposals by {{harvs|txt|authorlink=André Weil|first=André |last=Weil|year=1949}} on the [[generating function]]s (known as [[local zeta-function]]s) derived from counting the number of points on [[algebraic variety|algebraic varieties]] over [[finite field]]s.

A variety ''V'' over a finite field with ''q'' elements has a finite number of [[rational point]]s, as well as points over every finite field with ''q''<sup>''k''</sup> elements containing that field. The generating function has coefficients derived from the numbers ''N''<sub>''k''</sub> of points over the (essentially unique) field with ''q''<sup>''k''</sup> elements.

Weil conjectured that such ''zeta-functions'' should be [[rational function]]s, should satisfy a form of [[functional equation]], and should have their zeroes in restricted places. The last two parts were quite consciously modeled on the [[Riemann zeta function]] and [[Riemann hypothesis]]. The rationality was proved by {{harvtxt|Dwork|1960}}, the functional equation by {{harvtxt|Grothendieck|1965}}, and the analogue of the Riemann hypothesis was  proved by {{harvtxt|Deligne|1974}}.