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Chabauty methods using elliptic curves. (English) Zbl 1135.11320
From the text: Since 1983 [G. Faltings, Invent. Math. 73, No. 3, 349–366 (1983; Zbl 0588.14026)], it is known that an algebraic curve of genus $$g\geq2$$ over a number field has only finitely many rational points. The proof of this theorem does not help in actually determining them, however. A much older, partial, proof by C. Chabauty [see C. R. Acad. Sci., Paris 212, 1022–1024 (1941; Zbl 0025.24903 and JFM 67.0105.02)] does suggest a way of bounding the number of rational points.
In this article, we consider algebraic curves over $$\mathbb Q$$ that cover an elliptic curve over some extension of $$\mathbb Q$$. We show how we can use the arithmetic on that elliptic curve to obtain information on the rational points on the cover. We apply this method to curves arising from the Diophantine equations $$x^2\pm y^4= z^5$$ and $$x^8+ y^3= z^2$$ and determine all solutions with coprime, integral $$x$$, $$y$$, $$z$$. To do this, we determine the rational points on several curves of genus 5 and 17.

##### MSC:
 11G05 Elliptic curves over global fields 14G05 Rational points
##### Keywords:
finitely many rational points; elliptic curve
Full Text:
##### References:
  E. Arbarello, M. Cornalba, P. A. Gri ths, and J. Harris, Geometry of algebraic curves, Vol. I, Springer-Verlag, New York 1985.  Beukers Frits, Duke Math. J. 91 (1) pp 61– (1998)  Nils Bruin and E. Victor Flynn, Towers of 2-coversof hyperelliptic curves, Technical Report PIMS-01-12, PIMS, 2001, http://www.pims.math.ca/publications/#preprints. · Zbl 1145.11317  Nils Bruin, Chabauty Methods and CoveringTechniques applied to Generalised Fermat Equations, PhD thesis, Universiteit Leiden, 1999.  Nils Bruin, Chabauty methods using elliptic curves, Technical Report W99-14, University of Leiden, 1999, http://www.math.leidenuniv.nl/reports/1999-14.shtml. · Zbl 1135.11320  Bruin Nils, Compos. Math. 118 pp 305– (1999)  J. W. S. Cassels, Lectures on Elliptic Curves, LMS-ST 24, University Press, Cambridge1991.  J. W. S. Cassels and E. V. Flynn, Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2, LMSLNS 230, Cambridge University Press, Cambridge 1996. · Zbl 0857.14018  Chabauty Claude, R. Acad. Sci. Paris 212 pp 1022– (1941)  Daberkow M., J. Symbolic Comput. 24 pp 267– (1997)  Andrew Granville Henri Darmon, Bull. London Math. Soc. 27 (6) pp 513– (1995)  Johnny Edwards, A complete solution of x2y3z50, http://www.math.uu.nl/people/edwards/ icosahedron.ps, 2001.  Faltings G., Invent. Math. 73 (3) pp 349– (1983)  E., Compos. Math. 105 pp 79– (1997)  Flynn E. V., Duke Math. J. 90 (3) pp 435– (1997)  J. S. Milne, Jacobian varieties, in: Arithmetic geometry (Storrs, Conn., 1984), G. Cornell and J. H. Silverman, eds., Springer, New York (1986), 167-212.  Mumford David, Tata Inst. Fund. Res. Stud. Math. pp 5– (1970)  Silverman Joseph H., GTM pp 106– (1986)  R. Tijdeman, Diophantine equations and Diophantine approximations, in: Number theory and applications (Ban , AB, 1988), Kluwer Acad. Publ., Dordrecht (1989), 215-243.  Joseph L. Wetherell, Bounding the number of rational points on certain curves of high rank, PhD thesis, U.C. Berkeley, 1997. Department of Mathematics, Simon Fraser University.Burnaby BC V5A 1S6, Canada
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