Counting points on geometrically hyperelliptic curves of genus three

Abstract: 

Let C/QC/Q be a curve of genus 3, given as a double cover of a plane conic.  Such a curve is hyperelliptic over the algebraic closure of QQ, but may not have a hyperelliptic model of the usual form over QQ. For a prime pp of good reduction, let CpCp be the reduction of CC modulo pp. We discuss an algorithm that counts points on the curves CpCp simultaneously at all good pp up to a prescribed bound NN. We briefly report on our implementation and compare its performance to previous algorithms for the ordinary hyperelliptic case.

Joint work with David Harvey and Andrew Sutherland.