论文信息 - Descent via (3,3)-isogeny on Jacobians of genus 2 curves

Descent via (3,3)-isogeny on Jacobians of genus 2 curves

We give a parametrization of curves\nonbreakingspace C of genus 2 with a maximal isotropic (Z/3)2 in J[3], where J is the Jacobian variety of\nonbreakingspace C, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where it is shown that non-reducible Jacobians have non-trivial 3-part of the Tate–Shafarevich group.

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

[2] B. Poonen,et al. GENERALIZED EXPLICIT DESCENT AND ITS APPLICATION TO CURVES OF GENUS 3 , 2012, Forum of Mathematics, Sigma.

[3] E. V. Flynn,et al. Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2: Index rerum et personarum , 1996 .

[4] Jetta Carol Culpepper,et al. Electronic Resources , 2000 .

[5] E. V. Flynn. Descent via (5,5)-isogeny on Jacobians of genus 2 curves , 2015 .

[6] S. Weintraub,et al. The Siegel modular variety of degree two and level four , 1998 .

[7] I. Dolgachev,et al. On isogenous principally polarized abelian surfaces , 2007, 0710.1298.

[8] E. V. Flynn,et al. Exhibiting SHA[2] on hyperelliptic Jacobians , 2006 .

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

[10] M. Arslan. Integral cohomology of the Siegel modular variety of degree two and level three , 2006 .

[11] Everett W. Howe,et al. On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field , 2000, math/0002205.

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