Modulo Errors

The procedure I described in my previous post for computing types of zeta functions is informative the first time you try it and tedious thereafter; thus I’ve cobbled together some maple code to do the job for some simple cases.

A call to totalTrace(d,t) will scurry off and determine all degree d polynomials of trace t, provided neither your chosen trace or degree are too high (data tables only exist up to a certain point, and I wasn’t patient enough to implement much of what is known, either!). If you’re confident that the trace is sufficiently small to guarantee a building block from the set of exceptional polynomials S, you can use the fractionally faster totalTraceS(d,t) instead- it’ll tell you if you’re wrong!

Calling smallDefectPol(g) will, for a genus g, display the possible polynomials Q (whose roots are the β_i) corresponding to small defect curves (where small means at most 0.780022g, using the exceptional set S). You can get the types (of the form used e.g. by Serre) instead by calling smallDefectTypes(g).

Some defects that don’t meet the bound for small can nonetheless be computed with totalTrace(d,t), use its output as the argument for zetaTypes(X) to recover their corresponding types. This will become more useful if I add additional cases from the data tables.

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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.

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The abc conjecture is deceptively easy to state considering the deep implications it has for number theory. This E2 entry covers a couple of formulations of the conjecture, examines the connection to the ABC theorem for polynomials, and briefly discusses some of the problems it impacts upon.