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→Riemann hypothesis for curves over finite fields: I discovered this GF! Tag: Reverted |
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A generation later [[Pierre Deligne]] completed the proof.
(See [[étale cohomology]] for the basic formulae of the general theory.)
If {{mvar|E}} is an elliptic curve over a finite field with {{mvar|q}} elements, then the number of points of {{mvar|E}} defined over the field with {{math|''q''<sup>''m''</sup>}} elements is {{math|1 − ''α''<sup>''m''</sup> − ''β''<sup>''m''</sup> + ''q''<sup>''m''</sup>}}, where {{mvar|α}} and {{mvar|β}} are complex conjugates equal to <math>q^\frac{m}{2}e^{\pm i\theta}</math>.
The zeta function is
:<math>\frac{(1-\alpha q^{-s})(1-\beta q^{-s})}{(1-q^{-s})(1-q^{1-s})}</math>.
The zeta product is then
:<math>\frac{\zeta(s-\frac12)^2}{\zeta(s)\zeta(s-1)}</math>
==General formulas for the zeta function==
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