Rational points of curves over finite fields
January 20th, 2007
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Since the restart of term I’ve been studying rational points of curves over finite fields, trying to see how the geometry and number theory can mesh together to say more than either approach offers in isolation. The Weil Bounds give an upper limit on the number of points in terms of the size of the field and the genus of the curve. There is practical interest in finding curves with a large number of points to get good error-correcting codes; the shortfall between the bound and the actual number of points is described as the defect of the curve.
The ‘types’ of zeta function can be characterised in terms of their roots; number-theoretic arguments demonstrate that only finitely many types give curves of small defect. Geometrically this ‘type’ specifys the eigenvalues of the Frobenius endomorphism on the Jacobian of the curve; and thus under certain conditions some more types can also be eliminated, which may render particular defect levels impossible. Conversely, eliminating small defects by better number-theory bounds means the corresponding types cannot arise from the geometry.
This summary article describes the Weil bounds; some results on algebraic integers and their application to determining the possible types of zeta function (following Serre, with a slight refinement); a worked example for genus 3 plus the types for defect 0,1 and 2; and a brief sketch of some of the geometric arguments/results that arise.