Skip to main content
×
Home
    • Aa
    • Aa

ON SELMER RANK PARITY OF TWISTS

  • MAJID HADIAN (a1) and MATTHEW WEIDNER (a2)
Abstract

In this paper we study the variation of the $p$ -Selmer rank parities of $p$ -twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$ -rational $p$ -torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

Copyright
Corresponding author
hadian@caltech.edu
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

Z. Klagsbrun , B. Mazur and K. Rubin , ‘Disparity in Selmer ranks of quadratic twists of elliptic curves’, Ann. of Math. (2) 178(1) (2013), 287320.

K. Kramer , ‘Arithmetic of elliptic curves upon quadratic extensions’, Trans. Amer. Math. Soc. 264 (1981), 121135.

B. Mazur , ‘Rational points of abelian varieties with values in towers of number fields’, Invent. Math. 18(3–4) (1972), 183266.

B. Mazur and K. Rubin , ‘Finding large Selmer rank via an arithmetic theory of local constants’, Ann. of Math. (2) 166(2) (2007), 579612.

B. Mazur and K. Rubin , ‘Ranks of twists of elliptic curves and Hilbert’s tenth problem’, Invent. Math. 181(3) (2010), 541575.

B. Mazur , K. Rubin and A. Silverberg , ‘Twisting commutative algebraic groups’, J. Algebra 314(1) (2007), 419438.

B. Poonen and E. Rains , ‘Random maximal isotropic subspaces and Selmer groups’, J. Amer. Math. Soc. 25(1) (2012), 245269.

B. Poonen and M. Stoll , ‘The Cassels–Tate pairing on polarized Abelian varieties’, Ann. of Math. (2) 150(3) (1999), 11091149.

P. Swinnerton-Dyer , ‘The effect of twisting on the 2-Selmer group’, Math. Proc. Cambridge Philos. Soc. 145(3) (2008), 513526.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 19 *
Loading metrics...

Abstract views

Total abstract views: 115 *
Loading metrics...

* Views captured on Cambridge Core between 28th September 2016 - 22nd June 2017. This data will be updated every 24 hours.