Abstract
We give a generalization of the method of “Elliptic Curve Chabauty” to higher genus curves and their Jacobians. This method can sometimes be used in conjunction with covering techniques and a modified version of the Mordell–Weil sieve to provide a complete solution to the problem of determining the set of rational points on an algebraic curve Y. We show how to apply these explicitly by using them to prove that the equation y 2 = (x 3 + x 2 − 1) Φ11(x) has no rational solutions.
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References
Bosma, W., Cannon, J., Playoust C.: The Magma algebra system. I. The user language. J. Symbol. Comput. 24, 235–265 (1997). Computational algebra and number theory (London, 1993)
Bruin, N.: Chabauty methods and covering techniques applied to generalised Fermat equations. PhD thesis, Universiteit Leiden (1999)
Bruin N.: Chabauty methods using elliptic curves. J. Reine Angew. Math. 562, 27–49 (2003)
Bruin, N., Elkies N.D.: Trinomials ax 7 + bx + c and ax 8 + bx + c with Galois groups of order 168 and 8·168, in algorithmic number theory (Sydney, 2002), vol. 2369 of Lecture Notes in Computer Science, pp. 172–188. Springer, Berlin (2002)
Bruin N., Flynn E.V.: Towers of 2-covers of hyperelliptic curves. Trans. Am. Math. Soc. 357, 4329–4347 (2005)
Bruin N., Stoll M.: Deciding existence of rational points on curves: an experiment. Exp. Math. 17, 181–189 (2008)
Bugeaud Y., Mignotte M., Siksek S., Stoll M., Tengely S.: Integral points on hyperelliptic curves. Algebra Number Theory 2, 859–885 (2008)
Chabauty C. (1941) Sur les points rationnels des variétés algébriques dont l’irrégularité est supérieure à la dimension. C. R. Acad. Sci. Paris 212, 1022–1024
Coleman R.F.: Effective Chabauty. Duke Math. J. 52, 765–770 (1985)
Coleman R.F.: Torsion points on curves and p-adic abelian integrals. Ann. Math. (2) 121, 111–168 (1985)
Duquesne S.: Rational points on hyperelliptic curves and an explicit Weierstrass preparation theorem. Manuscr. Math. 108, 191–204 (2002)
Flynn E.V., Wetherell J.L.: Finding rational points on bielliptic genus 2 curves. Manuscr. Math. 100, 519–533 (1999)
Flynn E.V., Wetherell J.L.: Covering collections and a challenge problem of Serre. Acta Arith. 98, 197–205 (2001)
Lorenzini D., Tucker T.J.: Thue equations and the method of Chabauty–Coleman. Invent. Math. 148, 47–77 (2002)
Poonen B.: Heuristics for the Brauer–Manin obstruction for curves. Exp. Math. 15, 415–420 (2006)
Poonen B., Schaefer E.F.: Explicit descent for Jacobians of cyclic covers of the projective line. J. Reine Angew. Math. 488, 141–188 (1997)
Scharaschkin V. Local-global problems and the Brauer–Manin obstruction. PhD thesis, University of Michigan (1999)
Siksek S. Explicit Chabauty over number fields (2011). arXiv:1010.2603v2
Stoll M.: Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98, 245–277 (2001)
Wetherell, J.L.: Bounding the number of rational points on certain curves of high rank. PhD thesis, University of California, Berkeley (1998)
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Mourao, M. Extending Elliptic Curve Chabauty to higher genus curves. manuscripta math. 143, 355–377 (2014). https://doi.org/10.1007/s00229-013-0621-2
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DOI: https://doi.org/10.1007/s00229-013-0621-2