On the CM Character of the Curvesy2=xq−1 Original Research Article

For most odd primesqwe present an explicit formula for the number of points on the curvey2=xq−1 modulop. The description is similar to Gaussʹs 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≡2 mod q2.