Non-trivial Ш in the Jacobian of an infinite family of curves of genus 2
Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 1-13.

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.

Nous donnons une famille infinie de courbes de genre 2 dont la Jacobienne possède des éléments non triviaux du groupe de Tate-Shafarevich pour une descente via l’isogénie de Richelot. Nous le prouvons en effectuant une descente via l’isogénie de Richelot et une 2-descente complète sur la Jacobienne isogène. Nous donnons également un modèle explicite d’une famille associée de surfaces qui violent le principe de Hasse.

Published online:
DOI: 10.5802/jtnb.654
Anna Arnth-Jensen 1; E. Victor Flynn 1

1 Mathematical Institute, University of Oxford 24–29 St Giles’, Oxford OX1 3LB United Kingdom
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Anna Arnth-Jensen; E. Victor Flynn. Non-trivial $\Sha$ in the Jacobian of an infinite family of curves of genus 2. Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 1-13. doi : 10.5802/jtnb.654. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.654/

[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

[2] N. Bruin, E.V. Flynn, Exhibiting Sha[2] on Hyperelliptic Jacobians. J. Number Theory 118 (2006), 266–291. | MR: 2225283 | Zbl: 1118.14035

[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. | MR: 2342713

[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. | MR: 1406090 | Zbl: 0857.14018

[5] E.V. Flynn, Descent via isogeny in dimension 2. Acta Arithm. 66 (1994), 23–43. | MR: 1262651 | Zbl: 0835.14009

[6] E.V. Flynn, On a Theorem of Coleman. Manus. Math. 88 (1995), 447–456. | MR: 1362930 | Zbl: 0865.14012

[7] E.V. Flynn, The arithmetic of hyperelliptic curves. Progress in Mathematics 143 (1996), 167–175. | MR: 1414450 | Zbl: 0874.14012

[8] E.V. Flynn, J. Redmond, Applications of covering techniques to families of curves. J. Number Theory 101 (2003), 376–397. | MR: 1989893 | Zbl: 1119.14026

[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. | MR: 1480542 | Zbl: 0958.11024

[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). | Numdam | MR: 1838086 | Zbl: 1046.11038

[11] E.F. Schaefer, 2-descent on the Jacobians of Hyperelliptic Curves. J. Number Theory 51 (1995), 219–232. | MR: 1326746 | Zbl: 0832.14016

[12] E.F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve. Math. Ann. 310 (1998), 447–471. | MR: 1612262 | Zbl: 0889.11021

[13] M. Stoll, Two simple 2-dimensional abelian varieties defined over with Mordell-Weil group of rank at least 19. C. R. Acad. Sci. Paris 321, Série I (1995), 1341–1345. | MR: 1363577 | Zbl: 0859.11033

[14] M. Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98 (2001), 245–277. | MR: 1829626 | Zbl: 0972.11058

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