Abstract
We shall discuss the idea of finding all rational points on a curve \(\mathcal{C}\) by first finding an associated collection of curves whose rational points cover those of \(\mathcal {C}\). This classical technique has recently been given a new lease of life by being combined with descent techniques on Jacobians of curves, Chabauty techniques, and the increased power of software to perform algebraic number theory. We shall survey recent applications during the last 5 years which have used Chabauty techniques and covering collections of curves of genus 2 obtained from pullbacks along isogenies on their Jacobians.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Batut, C., Belabas, K., Bernardi, D., Cohen, H., Olivier, M.: PARI-GP., Available from ftp://megrez.math.u-bordeaux.fr/pub/pari
Bruin, N.: The Diophantine equations x 2±y 4 = ±z 6 and x 2+y 8 = z 3. Compositio Math. 118, 305–321 (1999)
Bruin, N.: Chabauty methods using covers on curves of genus 2. Report MI 1999-15, Leiden., http://www.math.leidenuniv.nl/reports/1999-15.shtml
Bruin, N.: KASH-based program for performing 2-descent on elliptic curves over number fields., http://www.math.uu.nl/people/bruin/ell.shar
Buchholz, R.H., MacDougall, J.A.: When Newton met Diophantus: A study of rational-derived polynomials and their extension to quadratic fields. To appear in J. Number Theory
Cassels, J.W.S.: Local Fields. LMS–ST, vol. 3. Cambridge University Press, Cambridge (1986)
Cassels, J.W.S., Flynn, E.V.: Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2. LMS–LNS, vol. 230. Cambridge University Press, Cambridge (1996)
Chabauty, C.: Sur les points rationnels des variétés algébriques dont l’irrégularité est supérieure à la dimension. C. R. Acad. Sci. Paris 212, 1022–1024 (1941)
Coleman, R.F.: Effiective Chabauty. Duke Math. J. 52, 765–780 (1985)
Daberkow, M., Fieker, C., Klüners, J., Pohst, M., Roegner, K., Schörnig, M., Wildanger, K.: KANT V4. J. Symbolic Comput. 24(3-4), 267–283 (1997), Available from ftp://ftp.math.tu-berlin.de/pub/algebra/Kant/Kash
Djabri, Z., Schaefer, E.F., Smart, N.P.: Computing the p-Selmer group of an elliptic curve. To appear in Trans. Amer. Math. Soc. (1999) (manuscript )
Flynn, E.V.: A flexible method for applying chabauty’s t eorem. Compositio Math- ematica 105, 79–94 (1997)
Flynn, E.V., Smart, N.P.: Canonical heights on the Jacobians of curves of genus 2 and the infinite descent. Acta Arith. 79, 333–352 (1997)
Flynn, E.V., Poonen, B., Schaefer, E.F.: Cycles of quadratic polynomials and rational points on a genus-two curve. Duke Math. J. 90, 435–463 (1997)
Flynn, E.V., Wetherell, J.L.: Finding Rational Points on Bielliptic Genus 2 Curves. Manuscripta Math. 100, 519–533 (1999)
Flynn, E.V.: On Q-Derived Polynomials. . To appear in Proc. Edinburgh Math. Soc. (2000) (manuscript)
Flynn, E.V., Wetherell, J.L.: Covering Collections and a Challenge Problem of Serre. (2000) (manuscript)
McCallum, W.: On the method of Coleman and Chabauty. Math. Ann. 299(3), 565–596 (1994)
Morton, P.: Arithmetic properties of periodic points of quadratic maps, II. Acta Arith. 87(2), 89–102 (1998)
Schaefer, E.F.: Computing a Selmer group of a Jacobian using functions on the curve. Math. Ann. 310(3), 447–471 (1998)
Serre, J.-P.: Lectures on the Mordell-Weil Theorem Transl. In: Brown, M. (ed.) From notes by Michel Waldschmidt, Vieweg, Wiesbaden (1989)
Sesiano, J.: Books IV to VII of Diophantus’ Arithmetica in the Arabic Translation attributed to Qusta ibn Luqa. Springer, Heidelberg (1982)
Siksek, S.: Infinite descent on elliptic curves. Rocky Mountain J. Math. 25(4), 1501–1538 (1995)
Silverman, J.H.: The Arithmetic of Elliptic Curves. GTM 106. Springer, Heidelberg (1986)
Stoll, M.: On the height constant for curves of genus two. Acta Arith. 90(2), 183–201 (1999)
Stoll, M.: Implementing 2-descent for Jacobians of hyperelliptic curves (1999) (preprint)
Stoll, M.: On the height constant for curves of genus two, II. (2000) (manuscript)
Wetherell, J.L.: Bounding the Number of Rational Points on Certain Curves of High Rank. PhD Dissertation, University of California, Berkeley (1997)
Young, G.C.: On the Solution of a Pair of Simultaneous Diophantine Equations Connected with the Nuptial Numbers of Plato. Proc. London Math. Soc. 23(2), 27–44 (1924)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Flynn, E.V. (2000). Coverings of Curves of Genus 2. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_3
Download citation
DOI: https://doi.org/10.1007/10722028_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67695-9
Online ISBN: 978-3-540-44994-2
eBook Packages: Springer Book Archive