Non-trivial Ш in the Jacobian of an infinite family of curves of genus 2

Anna Arnth-Jensen[1]; E. Victor Flynn[1]

  • [1] Mathematical Institute, University of Oxford 24–29 St Giles’, Oxford OX1 3LB United Kingdom

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 1, page 1-13
  • ISSN: 1246-7405

Abstract

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We give an infinite family of curves of genus 2 whose Jacobians have non-trivial members of the Tate-Shafarevich group for descent via Richelot isogeny. We prove this by performing a descent via Richelot isogeny and a complete 2-descent on the isogenous Jacobian. We also give an explicit model of an associated family of surfaces which violate the Hasse principle.

How to cite

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Arnth-Jensen, Anna, and Flynn, E. Victor. "Non-trivial $\Sha$ in the Jacobian of an infinite family of curves of genus 2." Journal de Théorie des Nombres de Bordeaux 21.1 (2009): 1-13. <http://eudml.org/doc/10871>.

@article{Arnth2009,
abstract = {We give an infinite family of curves of genus 2 whose Jacobians have non-trivial members of the Tate-Shafarevich group for descent via Richelot isogeny. We prove this by performing a descent via Richelot isogeny and a complete 2-descent on the isogenous Jacobian. We also give an explicit model of an associated family of surfaces which violate the Hasse principle.},
affiliation = {Mathematical Institute, University of Oxford 24–29 St Giles’, Oxford OX1 3LB United Kingdom; Mathematical Institute, University of Oxford 24–29 St Giles’, Oxford OX1 3LB United Kingdom},
author = {Arnth-Jensen, Anna, Flynn, E. Victor},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Jacobian; Tate-Shafarevich group},
language = {eng},
number = {1},
pages = {1-13},
publisher = {Université Bordeaux 1},
title = {Non-trivial $\Sha$ in the Jacobian of an infinite family of curves of genus 2},
url = {http://eudml.org/doc/10871},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Arnth-Jensen, Anna
AU - Flynn, E. Victor
TI - Non-trivial $\Sha$ in the Jacobian of an infinite family of curves of genus 2
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 1
SP - 1
EP - 13
AB - We give an infinite family of curves of genus 2 whose Jacobians have non-trivial members of the Tate-Shafarevich group for descent via Richelot isogeny. We prove this by performing a descent via Richelot isogeny and a complete 2-descent on the isogenous Jacobian. We also give an explicit model of an associated family of surfaces which violate the Hasse principle.
LA - eng
KW - Jacobian; Tate-Shafarevich group
UR - http://eudml.org/doc/10871
ER -

References

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  1. A. Arnth-Jensen, E.V. Flynn, Supplement to: Non-trivial Ш in the Jacobian of an infinite family of curves of genus 2. Available at: http://people.maths.ox.ac.uk/flynn/genus2/af/artlong.pdf Zbl1179.14030
  2. N. Bruin, E.V. Flynn, Exhibiting Sha[2] on Hyperelliptic Jacobians. J. Number Theory 118 (2006), 266–291. Zbl1118.14035MR2225283
  3. N. Bruin, M. Bright, E.V. Flynn, A. Logan, The Brauer-Manin Obstruction and Sha[2]. LMS J. Comput. Math. 10 (2007), 354–377. Zbl1222.11084MR2342713
  4. J. W. S. Cassels, E.V. Flynn, Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2. LMS-LNS, Vol. 230, Cambridge University Press, Cambridge, 1996. Zbl0857.14018MR1406090
  5. E.V. Flynn, Descent via isogeny in dimension 2. Acta Arithm. 66 (1994), 23–43. Zbl0835.14009MR1262651
  6. E.V. Flynn, On a Theorem of Coleman. Manus. Math. 88 (1995), 447–456. Zbl0865.14012MR1362930
  7. E.V. Flynn, The arithmetic of hyperelliptic curves. Progress in Mathematics 143 (1996), 167–175. Zbl0874.14012MR1414450
  8. E.V. Flynn, J. Redmond, Applications of covering techniques to families of curves. J. Number Theory 101 (2003), 376–397. Zbl1119.14026MR1989893
  9. E.V. Flynn, B. Poonen, E. Schaefer, Cycles of Quadratic Polynomials and Rational Points on a Genus 2 Curve. Duke Math. J. 90 (1997), 435–463. Zbl0958.11024MR1480542
  10. B. Poonen, An explicit algebraic family of genus-one curves violating the Hasse principle. J. Théor. Nombres Bordeaux, 13 (2001), 263–274. 21st Journées Arithmétiques (Rome, 2001). Zbl1046.11038MR1838086
  11. E.F. Schaefer, 2-descent on the Jacobians of Hyperelliptic Curves. J. Number Theory 51 (1995), 219–232. Zbl0832.14016MR1326746
  12. E.F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve. Math. Ann. 310 (1998), 447–471. Zbl0889.11021MR1612262
  13. M. Stoll, Two simple 2-dimensional abelian varieties defined over ℚ with Mordell-Weil group of rank at least 19. C. R. Acad. Sci. Paris 321, Série I (1995), 1341–1345. Zbl0859.11033MR1363577
  14. M. Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98 (2001), 245–277. Zbl0972.11058MR1829626

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