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The non-abelian (or non-linear) method of Chabauty

  • Minhyong Kim
Part of the Aspects of Mathematics book series (ASMA)

Abstract

This article is a brief introduction to the ideas surrounding the non-linear Albanese map that provides an approach to Diophantine finiteness theorems in the spirit of the method of Chabauty.

Keywords

Fundamental Group Galois Group Global Point Maximal Abelian Extension Rank Hypothesis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Besser, Amnon Coleman integration using the Tannakian formalism. Math. Ann. 322 (2002), no. 1, 19–48.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Chabauty, Claude, Sur les points rationnels des courbes algébriques de genre supérieur l’unité. C. R. Acad. Sci. Paris 212, (1941). 882–885.MATHMathSciNetGoogle Scholar
  3. [3]
    Coleman, Robert F. Effective Chabauty. Duke Math. J. 52 (1985), no. 3, 765–770.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Deligne, Pierre Le groupe fondamental de la droite projective moins trois points. Galois groups over ℚ (Berkeley, CA, 1987), 79–297, Math. Sci. Res. Inst. Publ., 16, Springer, New York, 1989.Google Scholar
  5. [5]
    Fox, Ralph H. Free differential calculus. I. Derivation in the free group ring. Ann. of Math. (2) 57, (1953). 547–560.CrossRefMathSciNetGoogle Scholar
  6. [6]
    Hain, Richard M. Higher Albanese manifolds. Hodge theory (Sant Cugat, 1985), 84–91, Lecture Notes in Math., 1246, Springer, Berlin, 1987.Google Scholar
  7. [7]
    Jannsen, Uwe On the l-adic cohomology of varieties over number fields and its Galois cohomology. Galois groups over ℚ (Berkeley, CA, 1987), 315–360, Math. Sci. Res. Inst. Publ., 16, Springer, New York, 1989.Google Scholar
  8. [8]
    Kim, Minhyong The motivic fundamental group of P 1 \ {0, 1,∞} and the theorem of Siegel. Inventiones Mathematicae (to be published).Google Scholar
  9. [9]
    Ochi, Yoshihiro; Venjakob, Otmar On the ranks of Iwasawa modules over p-adic Lie extensions. Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 1, 25–43.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Vologodsky, Vadim Hodge structure on the fundamental group and its application to p-adic integration. Mosc. Math. J. 3 (2003), no. 1, 205–247, 260.MATHMathSciNetGoogle Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden 2006

Authors and Affiliations

  • Minhyong Kim
    • 1
  1. 1.Department of MathematicsUniversity of ArizonaTucson

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