# Non-trivial $\u0428$ 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

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topArnth-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

top- A. Arnth-Jensen, E.V. Flynn, Supplement to: Non-trivial $\u0428$ 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
- N. Bruin, E.V. Flynn, Exhibiting Sha[2] on Hyperelliptic Jacobians. J. Number Theory 118 (2006), 266–291. Zbl1118.14035MR2225283
- 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
- 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
- E.V. Flynn, Descent via isogeny in dimension 2. Acta Arithm. 66 (1994), 23–43. Zbl0835.14009MR1262651
- E.V. Flynn, On a Theorem of Coleman. Manus. Math. 88 (1995), 447–456. Zbl0865.14012MR1362930
- E.V. Flynn, The arithmetic of hyperelliptic curves. Progress in Mathematics 143 (1996), 167–175. Zbl0874.14012MR1414450
- E.V. Flynn, J. Redmond, Applications of covering techniques to families of curves. J. Number Theory 101 (2003), 376–397. Zbl1119.14026MR1989893
- 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
- 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
- E.F. Schaefer, 2-descent on the Jacobians of Hyperelliptic Curves. J. Number Theory 51 (1995), 219–232. Zbl0832.14016MR1326746
- E.F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve. Math. Ann. 310 (1998), 447–471. Zbl0889.11021MR1612262
- M. Stoll, Two simple 2-dimensional abelian varieties defined over $\mathbb{Q}$ with Mordell-Weil group of rank at least 19. C. R. Acad. Sci. Paris 321, Série I (1995), 1341–1345. Zbl0859.11033MR1363577
- M. Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98 (2001), 245–277. Zbl0972.11058MR1829626

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