Implementing 2-descent for Jacobians of hyperelliptic curves

This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2-Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Q-rational Weierstraa point.

[1]  James S. Milne,et al.  Arithmetic Duality Theorems , 1987 .

[2]  J. Cassels,et al.  Arithmetic on Curves of Genus 1. IV. Proof of the Hauptvermutung. , 1962 .

[3]  J. Neukirch Algebraic Number Theory , 1999 .

[4]  S. Lichtenbaum Duality theorems for curves overP-adic fields , 1969 .

[5]  Siegfried Bosch,et al.  Rational points of the group of components¶of a Néron model , 1998, math/9804069.

[6]  Computing the Mordell-Weil rank of Jacobians of curves of genus two , 1993 .

[7]  J. Serre Une ≪formule de masse≫ pour les extensions totalement ramifiées de degré donné d’un corps local , 2003 .

[8]  A. Mattuck Abelian Varieties over P-Adic Ground Fields , 1955 .

[9]  E. V. Flynn,et al.  Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2: Weddle's surface , 1996 .

[10]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[11]  Bjorn Poonen,et al.  Explicit descent for Jacobians of cyclic coevers of the projective line. , 1997 .

[12]  Bjorn Poonen,et al.  The Cassels-Tate pairing on polarized abelian varieties , 1999 .

[13]  J. Cremona Algorithms for Modular Elliptic Curves , 1992 .

[14]  Edward F. Schaefer 2-Descent on the Jacobians of Hyperelliptic Curves , 1995 .

[15]  Edward F. Schaefer Computing a Selmer group of a Jacobian using functions on the curve , 1998 .

[16]  Bjorn Poonen,et al.  Cycles of quadratic polynomials and rational points on a genus-$2$ curve , 1995 .

[17]  The Mordell-Weil Group of Curves of Genus 2 , 1983 .