staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Michael Stoll
Rational points on curves
Journal de théorie des nombres de Bordeaux, 23 no. 1 (2011), p. 257-277, doi: 10.5802/jtnb.760
Article PDF | Reviews MR 2780629 | Zbl 1270.11030
Class. Math.: 11D41, 11G30, 14G05, 14G25

Résumé - Abstract

This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009.

We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve $C$ over $\mathbb{Q}$. The focus is on practical aspects of this problem in the case that the genus of $C$ is at least $2$, and therefore the set of rational points is finite.

Bibliography

[1] M.J. Bright, N. Bruin, E.V. Flynn, A. Logan, The Brauer-Manin obstruction and $\Sha[2]$, LMS J. Comput. Math. 10 (2007), 354–377.  MR 2342713
[2] N. Bruin, Chabauty methods and covering techniques applied to generalized Fermat equations, CWI Tract 133, 77 pages (2002).  MR 1916903 |  Zbl 1043.11029
[3] N. Bruin, Chabauty methods using elliptic curves, J. Reine Angew. Math. 562 (2003), 27–49.  MR 2011330 |  Zbl 1135.11320
[4] N. Bruin, N.D. Elkies, Trinomials $ax^7+bx+c$ and $ax^8+bx+c$ with Galois groups of order $168$ and $8\cdot 168$, in: Algorithmic number theory, Sydney 2002, Lecture Notes in Comput. Sci. 2369, Springer, Berlin (2002), pp. 172–188.  MR 2041082 |  Zbl 1058.11044
[5] N. Bruin, E.V. Flynn, Towers of 2-covers of hyperelliptic curves, Trans. Amer. Math. Soc. 357 (2005), 4329–4347.  MR 2156713 |  Zbl 1145.11317
[6] N. Bruin, E.V. Flynn, Exhibiting SHA$[2]$ on hyperelliptic Jacobians, J. Number Theory 118 (2006), 266–291.  MR 2225283 |  Zbl 1118.14035
[7] N. Bruin, M. Stoll, Deciding existence of rational points on curves: an experiment, Experiment. Math. 17 (2008), 181–189. Article |  MR 2433884 |  Zbl pre05383586
[8] N. Bruin, M. Stoll, 2-cover descent on hyperelliptic curves, Math. Comp. 78 (2009), 2347–2370.  MR 2521292 |  Zbl pre05813148
[9] N. Bruin, M. Stoll, The Mordell-Weil sieve: Proving non-existence of rational points on curves, LMS J. Comput. Math. 13 (2010), 272–306.  MR 2685127
[10] Y. Bugeaud, M. Mignotte, S. Siksek, M. Stoll, Sz. Tengely, Integral points on hyperelliptic curves, Algebra Number Theory 2 (2008), 859–885.  MR 2457355 |  Zbl 1168.11026
[11] J.W.S. Cassels, Second descents for elliptic curves, J. reine angew. Math. 494 (1998), 101–127.  MR 1604468 |  Zbl 0883.11028
[12] J.W.S. Cassels, E.V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus 2, London Math. Soc., Lecture Note Series 230, Cambridge Univ. Press, Cambridge, 1996.  MR 1406090 |  Zbl 0857.14018
[13] C. Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885.  MR 4484 |  JFM 67.0105.01
[14] C. Chevalley, A. Weil, Un théorème d’arithmétique sur les courbes algébriques, Comptes Rendus Hebdomadaires des Séances de l’Acad. des Sci., Paris 195 (1932), 570–572.  Zbl 0005.21611
[15] R.F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), 765–770. Article |  MR 808103 |  Zbl 0588.14015
[16] J.E. Cremona, T.A. Fisher, C. O’Neil, D. Simon, M. Stoll, Explicit $n$-descent on elliptic curves. I. Algebra, J. reine angew. Math. 615 (2008), 121–155. II. Geometry, J. reine angew. Math. 632 (2009), 63–84. III. Algorithms, in preparation.  MR 2384334 |  Zbl pre05598019
[17] J.E. Cremona, T.A. Fisher, M. Stoll, Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves, Algebra Number Theory 4 (2010), 763–820.  MR 2728489 |  Zbl pre05809198
[18] B. Creutz, Explicit second $p$-descent on elliptic curves, PhD Thesis, Jacobs University Bremen, 2010.
[19] V.A. Dem’janenko, Rational points of a class of algebraic curves (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 1373–1396.  MR 205991 |  Zbl 0181.24001
[20] T. Dokchitser, Computing special values of motivic $L$-functions, Experiment. Math. 13 (2004), 137–149. Article |  MR 2068888 |  Zbl 1139.11317
[21] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366.  MR 718935 |  Zbl 0588.14026
[22] E.V. Flynn, An explicit theory of heights, Trans. Amer. Math. Soc. 347 (1995), 3003–3015.  MR 1297525 |  Zbl 0864.11033
[23] E.V. Flynn, A flexible method for applying Chabauty’s theorem, Compositio Math. 105 (1997), 79–94.  MR 1436746 |  Zbl 0882.14009
[24] E.V. Flynn, The Hasse Principle and the Brauer-Manin obstruction for curves, Manuscripta Math. 115 (2004), 437–466.  MR 2103661 |  Zbl 1069.11023
[25] E.V. Flynn, Homogeneous spaces and degree 4 del Pezzo surfaces, Manuscripta Math. 129 (2009), 369–380.  MR 2515488 |  Zbl 1184.11021
[26] E.V. Flynn, B. Poonen, E.F. Schaefer, Cycles of quadratic polynomials and rational points on a genus-2 curve, Duke Math. J. 90 (1997), 435–463. Article |  MR 1480542 |  Zbl 0958.11024
[27] E.V. Flynn, N.P. Smart, Canonical heights on the Jacobians of curves of genus $2$ and the infinite descent, Acta Arith. 79 (1997), 333–352. Article |  MR 1450916 |  Zbl 0895.11026
[28] M. Girard, L. Kulesz, Computation of sets of rational points of genus-3 curves via the Dem’janenko-Manin method, LMS J. Comput. Math. 8 (2005), 267–300.  MR 2193214 |  Zbl 1108.14017
[29] Su-Ion Ih, Height uniformity for algebraic points on curves, Compositio Math. 134 (2002), 35–57.  MR 1931961 |  Zbl 1031.11041
[30] V.A. Kolyvagin, Finiteness of $E(\mathbb{Q})$ and $\Sha(E,\mathbb{Q})$ for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat., Vol. 52 (1988), 522–540.  MR 954295 |  Zbl 0662.14017
[31] A.K. Lenstra, H.W. Lenstra, Jr., L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534.  MR 682664 |  Zbl 0488.12001
[32] A. Logan, R. van Luijk, Nontrivial elements of Sha explained through K3 surfaces, Math. Comp. 78 (2009), 441–483.  MR 2448716 |  Zbl pre05813054
[33] Y. Manin, The $p$-torsion of elliptic curves is uniformly bounded (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 459–465.  MR 272786 |  Zbl 0191.19601
[34] J.R. Merriman, S. Siksek, N.P. Smart, Explicit $4$-descents on an elliptic curve, Acta Arith. 77 (1996), 385–404. Article |  MR 1414518 |  Zbl 0873.11036
[35] L.J. Mordell, On the rational solutions of the indeterminate equations of the 3rd and 4th degrees, Proc. Camb. Phil. Soc. 21 (1922), 179–192.  JFM 48.1156.03
[36] B. Poonen, Heuristics for the Brauer-Manin obstruction for curves, Experiment. Math. 15 (2006), 415–420. Article |  MR 2293593 |  Zbl 1173.11040
[37] B. Poonen, E.F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line, J. reine angew. Math. 488 (1997), 141–188.  MR 1465369 |  Zbl 0888.11023
[38] B. Poonen, E.F. Schaefer, M. Stoll, Twists of $X(7)$ and primitive solutions to $x^2+y^3=z^7$, Duke Math. J. 137 (2007), 103–158. Article |  MR 2309145 |  Zbl 1124.11019
[39] B. Poonen, M. Stoll, A local-global principle for densities, in: Scott D. Ahlgren (ed.) et al.: Topics in number theory. In honor of B. Gordon and S. Chowla. Kluwer Academic Publishers, Dordrecht. Math. Appl., Dordr. 467 (1999), 241–244.  MR 1691323 |  Zbl 1024.11047
[40] 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
[41] V. Scharaschkin, Local-global problems and the Brauer-Manin obstruction, Ph.D. thesis, University of Michigan (1999).  MR 2700328
[42] J.-P. Serre, Algebraic groups and class fields, Springer GTM 117, Springer Verlag, 1988.  MR 918564 |  Zbl 0703.14001
[43] J.-P. Serre, Lectures on the Mordell-Weil theorem. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. Aspects of Mathematics, E15. Friedr. Vieweg & Sohn, Braunschweig, 1989.  MR 1002324 |  Zbl 0676.14005
[44] S. Siksek, M. Stoll, On a problem of Hajdu and Tengely, in: G. Hanrot, F. Morain, and E. Thomé (Eds.): ANTS-IX 2010, LNCS 6197, pp. 316–330. Springer Verlag, Heidelberg, 2010.  Zbl pre05793688
[45] D. Simon, Solving quadratic equations using reduced unimodular quadratic forms, Math. Comp. 74 (2005), 1531–1543.  MR 2137016 |  Zbl 1078.11072
[46] S. Stamminger, Explicit 8-descent on elliptic curves, PhD thesis, International University Bremen (2005).
[47] M. Stoll, On the height constant for curves of genus two, Acta Arith. 90 (1999), 183–201. Article |  MR 1709054 |  Zbl 0932.11043
[48] M. Stoll, Implementing 2-descent on Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), 245–277.  MR 1829626 |  Zbl 0972.11058
[49] M. Stoll, On the height constant for curves of genus two, II, Acta Arith. 104 (2002), 165–182.  MR 1914251 |  Zbl 1139.11318
[50] M. Stoll, Descent on Elliptic Curves. Short Course taught at IHP in Paris, October 2004. arXiv:math/0611694v1 [math.NT].
[51] M. Stoll, Independence of rational points on twists of a given curve, Compositio Math. 142 (2006), 1201–1214.  MR 2264661 |  Zbl 1128.11033
[52] M. Stoll, Finite descent obstructions and rational points on curves, Algebra Number Theory 1 (2007), 349–391.  MR 2368954 |  Zbl 1167.11024
[53] M. Stoll, Rational 6-cycles under iteration of quadratic polynomials, LMS J. Comput. Math. 11 (2008), 367–380.  MR 2465796
[54] M. Stoll, On the average number of rational points on curves of genus 2, Preprint (2009), arXiv:0902.4165v1 [math.NT].
[55] M. Stoll, Documentation for the ratpoints program, Manuscript (2009), arXiv:0803.3165 [math.NT].
[56] A. Weil, L’arithmétique sur les courbes algébriques, Acta Math. 52 (1929), 281–315.  MR 1555278 |  JFM 55.0713.01
[57] J.L. Wetherell, Bounding the number of rational points on certain curves of high rank, Ph.D. thesis, University of California (1997).  MR 2696280