Algebra & Number Theory

Chabauty for symmetric powers of curves

Samir Siksek

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Let C be a smooth projective absolutely irreducible curve of genus g2 over a number field K, and denote its Jacobian by J. Let d1 be an integer and denote the d-th symmetric power of C by C(d). In this paper we adapt the classic Chabauty–Coleman method to study the K-rational points of C(d). Suppose that J(K) has Mordell–Weil rank at most gd. We give an explicit and practical criterion for showing that a given subset C(d)(K) is in fact equal to C(d)(K).

Article information

Algebra Number Theory, Volume 3, Number 2 (2009), 209-236.

Received: 2 April 2008
Revised: 20 January 2009
Accepted: 17 February 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G30: Curves of arbitrary genus or genus = 1 over global fields [See also 14H25]
Secondary: 11G35: Varieties over global fields [See also 14G25] 14K20: Analytic theory; abelian integrals and differentials 14C20: Divisors, linear systems, invertible sheaves

Chabauty Coleman curves Jacobians symmetric powers divisors differentials abelian integrals


Siksek, Samir. Chabauty for symmetric powers of curves. Algebra Number Theory 3 (2009), no. 2, 209--236. doi:10.2140/ant.2009.3.209.

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  • W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system, I, The user language”, J. Symbolic Comput. 24:3-4 (1997), 235–265.
  • N. Bourbaki, Lie groups and Lie algebras, Chapters 1–3, Springer, Berlin, 1989.
  • N. R. Bruin, Chabauty methods and covering techniques applied to generalized Fermat equations, CWI Tract 133, Centrum voor Wiskunde en Informatica, Amsterdam, 2002. 2003i:11042
  • N. Bruin, “Chabauty methods using elliptic curves”, J. Reine Angew. Math. 562 (2003), 27–49.
  • N. Bruin and N. D. Elkies, “Trinomials $ax\sp 7+bx+c$ and $ax\sp 8+bx+c$ with Galois groups of order 168 and $8\cdot168$”, pp. 172–188 in Algorithmic number theory (Sydney, 2002), edited by C. Fieker and D. R. Kohel, Lecture Notes in Comput. Sci. 2369, Springer, Berlin, 2002.
  • N. Bruin and M. Stoll, “Deciding existence of rational points on curves: An experiment”, Experiment. Math. 17:2 (2008), 181–189.
  • J. W. S. Cassels and E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus $2$, London Mathematical Society Lecture Note Series 230, Cambridge University Press, 1996.
  • C. Chabauty, “Sur les points rationnels des variétés algébriques dont l'irrégularité est supérieure à la dimension”, C. R. Acad. Sci. Paris 212 (1941), 1022–1024. 0025.24903
  • R. F. Coleman, “Effective Chabauty”, Duke Mathematical J. 52:3 (1985), 765–770.
  • R. F. Coleman, “Torsion points on curves and $p$-adic abelian integrals”, Ann. of Math. $(2)$ 121:1 (1985), 111–168.
  • P. Colmez, Intégration sur les variétés $p$-adiques, Astérisque 248, Société Mathématique de France, Paris, 1998.
  • O. Debarre and M. J. Klassen, “Points of low degree on smooth plane curves”, J. Reine Angew. Math. 446 (1994), 81–87.
  • G. Faltings, “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Invent. Math. 73:3 (1983), 349–366.
  • G. Faltings, “Diophantine approximation on abelian varieties”, Ann. of Math. $(2)$ 133:3 (1991), 549–576.
  • G. Faltings, “The general case of S. Lang's conjecture”, pp. 175–182 in Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), edited by V. Cristante and W. Messing, Perspect. Math. 15, Academic Press, San Diego, CA, 1994.
  • E. V. Flynn, “Descent via isogeny in dimension $2$”, Acta Arith. 66:1 (1994), 23–43.
  • E. V. Flynn, “On a theorem of Coleman”, Manuscripta Math. 88:4 (1995), 447–456.
  • E. V. Flynn, “A flexible method for applying Chabauty's theorem”, Compositio Math. 105:1 (1997), 79–94.
  • E. V. Flynn and N. P. Smart, “Canonical heights on the Jacobians of curves of genus $2$ and the infinite descent”, Acta Arith. 79:4 (1997), 333–352.
  • E. V. Flynn and J. L. Wetherell, “Finding rational points on bielliptic genus 2 curves”, Manuscripta Math. 100:4 (1999), 519–533.
  • E. V. Flynn and J. L. Wetherell, “Covering collections and a challenge problem of Serre”, Acta Arith. 98:2 (2001), 197–205.
  • D. J. H. Garling, A course in Galois theory, Cambridge University Press, 1986.
  • D. Grant, “A curve for which Coleman's effective Chabauty bound is sharp”, Proc. Amer. Math. Soc. 122:1 (1994), 317–319.
  • B. H. Gross and D. E. Rohrlich, “Some results on the Mordell–Weil group of the Jacobian of the Fermat curve”, Invent. Math. 44:3 (1978), 201–224.
  • J. Harris and J. Silverman, “Bielliptic curves and symmetric products”, Proc. Amer. Math. Soc. 112:2 (1991), 347–356.
  • S. Kamienny, “Torsion points on elliptic curves over all quadratic fields”, Duke Mathematical J. 53:1 (1986), 157–162.
  • S. Kamienny, “Torsion points on elliptic curves over all quadratic fields, II”, Bull. Soc. Math. France 114:1 (1986), 119–122.
  • S. Kamienny, “Torsion points on elliptic curves and $q$-coefficients of modular forms”, Invent. Math. 109:2 (1992), 221–229.
  • M. J. Klassen, Algebraic points of low degree on curves of low rank, thesis, University of Arizona, 1993.
  • M. Klassen and P. Tzermias, “Algebraic points of low degree on the Fermat quintic”, Acta Arith. 82:4 (1997), 393–401.
  • D. Lorenzini and T. J. Tucker, “Thue equations and the method of Chabauty–Coleman”, Invent. Math. 148:1 (2002), 47–77.
  • “MAGMA Computational Algebra System”, web site, University of Sydney, 2009,
  • W. G. McCallum, “The arithmetic of Fermat curves”, Math. Ann. 294:3 (1992), 503–511.
  • W. G. McCallum, “On the method of Coleman and Chabauty”, Math. Ann. 299:3 (1994), 565–596.
  • W. McCallum and B. Poonen, “The method of Chabauty and Coleman”, preprint, 2006,
  • L. Merel, “Bornes pour la torsion des courbes elliptiques sur les corps de nombres”, Invent. Math. 124:1-3 (1996), 437–449.
  • J. S. Milne, “Jacobian varieties”, pp. 167–212 in Arithmetic geometry (Storrs, CT, 1984), edited by G. Cornell and J. H. Silverman, Springer, New York, 1986. 0604.14018
  • P. Parent, “Torsion des courbes elliptiques sur les corps cubiques”, Ann. Inst. Fourier $($Grenoble$)$ 50:3 (2000), 723–749.
  • P. Parent, “No 17-torsion on elliptic curves over cubic number fields”, J. Théor. Nombres Bordeaux 15:3 (2003), 831–838.
  • B. Poonen and E. F. Schaefer, “Explicit descent for Jacobians of cyclic covers of the projective line”, J. Reine Angew. Math. 488 (1997), 141–188.
  • E. F. Schaefer, “$2$-descent on the Jacobians of hyperelliptic curves”, J. Number Theory 51:2 (1995), 219–232.
  • E. F. Schaefer and J. L. Wetherell, “Computing the Selmer group of an isogeny between abelian varieties using a further isogeny to a Jacobian”, J. Number Theory 115:1 (2005), 158–175.
  • S. Siksek, Descents on curves of genus $1$, thesis, University of Exeter, 1995.
  • S. Siksek, “Infinite descent on elliptic curves”, Rocky Mountain J. Math. 25:4 (1995), 1501–1538.
  • M. Stoll, “On the arithmetic of the curves $y\sp 2=x\sp l+A$ and their Jacobians”, J. Reine Angew. Math. 501 (1998), 171–189.
  • M. Stoll, “Implementing 2-descent for Jacobians of hyperelliptic curves”, Acta Arith. 98:3 (2001), 245–277.
  • M. Stoll, “On the arithmetic of the curves $y\sp 2=x\sp l+A$, II”, J. Number Theory 93:2 (2002), 183–206.
  • M. Stoll, “Independence of rational points on twists of a given curve”, Compos. Math. 142:5 (2006), 1201–1214.
  • M. Stoll, “On the number of rational squares at fixed distance from a fifth power”, Acta Arith. 125:1 (2006), 79–88.
  • P. Tzermias, “Algebraic points of low degree on the Fermat curve of degree seven”, Manuscripta Math. 97:4 (1998), 483–488.
  • P. Tzermias, “Parametrization of low-degree points on a Fermat curve”, Acta Arith. 108:1 (2003), 25–35.
  • P. Tzermias, “Low-degree points on Hurwitz-Klein curves”, Trans. Amer. Math. Soc. 356:3 (2004), 939–951.
  • P. Tzermias, “Improved bounds on the number of low-degree points on certain curves”, Acta Arith. 117:3 (2005), 277–282.
  • J. L. Wetherell, Bounding the number of rational points on certain curves of high rank, thesis, University of California at Berkeley, 1997.