Integral bases for TQFT modules and unimodular representations of mapping class groups

We construct integral bases for the SO(3)-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus 3 and p=5, we still give an explicit basis

On the CM Character of the Curvesy2=xq−1

AbstractFor most odd primesqwe present an explicit formula for the number of points on the curvey2=xq−1 modulop. The description is similar to Gauss's formula for the caseq=3. The result follows from a determination of the CM character of the curve. The onlyqthat are excluded are those rare ones for which 2q≡2modq2