Local zeta function: Difference between revisions

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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''^''m''}} elements is {{math|1 − ''α''^''m'' − ''β''^''m'' + ''q''^''m''}}, 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>.

==General formulas for the zeta function==