The Southwestern Center for Arithmetical Algebraic Geometry

Arizona Winter School 1999
Poonen's Project Description

Project: To compute the rank of the Mordell-Weil group of the Jacobian of the genus 2 curve

y2 = x6 - 2x5 - x4 + 4x3 + 3x2 + 2x + 1

over Q. If there is extra time, one can try also to find all the rational points on the curve, compute the rational torsion subgroup of the Jacobian, or the endomorphism ring of the Jacobian.

References:

  1. Flynn, Poonen, Schaefer, "Cycles of quadratic polynomials and rational points on a genus-2 curve", Duke Math J. 90 (1997), 435-463.
  2. Schaefer, "2-descent on the Jacobians of hyperelliptic curves", J. Number Theory 51 (1995), no. 2, 219--232.

The part most directly relevant to the project is probably pp. 443-452 of (1).