Local zeta-function

In number theory, a local zeta-function is a generating function

Riemann hypothesis for curves over finite fields

For projective curves C over F that are non-singular, it can be shown that

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:Z(t) = P(t)/{(1 − t)(1 − qt)},

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with P(t) a polynomial, of degree 2g where g is the genus of C. The Riemann hypothesis for curves over finite fields states that the roots of P have absolute value

Related Topics:
Genus - Absolute value

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:q−1/2,

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where q = |F|.

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For example, for the elliptic curve case there are two roots, and it is easy to show their product is q−1. Hasse's theorem is that they have the same absolute value; and this has immediate consequences for the number of points.

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Weil proved this for the general case, around 1940 (Comptes Rendus note, April 1940): he spent much time in the years after that, writing up the algebraic geometry involved). This led him to the general Weil conjectures, finally proved a generation later. See étale cohomology for the basic formulae of the general theory.

Related Topics:
Weil - Algebraic geometry - Weil conjectures - étale cohomology

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