Computing L-series of geometrically hyperelliptic curves of genus three

Abstract

Let C/Q be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of Q, but may not have a hyperelliptic model of the usual form over Q. We describe an algorithm that computes the local zeta functions of C at all odd primes of good reduction up to a prescribed bound N. The algorithm relies on an adaptation of the "accumulating remainder tree" to matrices with entries in a quadratic field. We report on an implementation, and compare its performance to previous algorithms for the ordinary hyperelliptic case.

15 pages, to be presented at ANTS XII

References in corpus (3)

• A database of genus 2 curves over the rational numbers
• Sato-Tate groups of y^2=x^8+c and y^2=x^7-cx
• Fast Jacobian arithmetic for hyperelliptic curves of genus 3

Cited by in corpus (4)

• A database of nonhyperelliptic genus 3 curves over Q
• Fast Jacobian arithmetic for hyperelliptic curves of genus 3
• Computing L-Polynomials of Picard curves from Cartier-Manin matrices
• Counting points on smooth plane quartics