Well-approximated points on linear extensions of elliptic curves
HTML articles powered by AMS MathViewer
- by Deanna M. Caveny and Robert Tubbs
- Proc. Amer. Math. Soc. 138 (2010), 2745-2754
- DOI: https://doi.org/10.1090/S0002-9939-10-10334-7
- Published electronically: March 10, 2010
- PDF | Request permission
Abstract:
We employ a result on linear forms in logarithms of algebraic points on commutative algebraic groups, a study initiated by Philippon and Waldschmidt, a so-called “local nullstellen inequality” of Brownawell, and some elementary analytic estimates to study the approximation properties of coordinates of non-generic points on a linear (algebraic) group extension of an elliptic curve.References
- W. Dale Brownawell, Local Diophantine Nullstellen inequalities, J. Amer. Math. Soc. 1 (1988), no. 2, 311–322. MR 928261, DOI 10.1090/S0894-0347-1988-0928261-3
- Deanna Caveny and Robert Tubbs, The arithmetic of well-approximated numbers, Number theory with an emphasis on the Markoff spectrum (Provo, UT, 1991) Lecture Notes in Pure and Appl. Math., vol. 147, Dekker, New York, 1993, pp. 53–59. MR 1219324
- M. Hindry, Groupes algébriques commutatifs, exemples explicites, Sem. d’Arithmétique Saint–Etienne, Université de Saint-Etienne, France, 1988–89, pp. 9–42.
- Noriko Hirata-Kohno, Formes linéaires de logarithmes de points algébriques sur les groupes algébriques, Invent. Math. 104 (1991), no. 2, 401–433 (French). MR 1098616, DOI 10.1007/BF01245082
- Patrice Philippon and Michel Waldschmidt, Formes linéaires de logarithmes sur les groupes algébriques commutatifs, Illinois J. Math. 32 (1988), no. 2, 281–314 (French). MR 945864
- Éric Reyssat, Approximation algébrique de nombres liés aux fonctions elliptiques et exponentielle, Bull. Soc. Math. France 108 (1980), no. 1, 47–79 (French, with English summary). MR 603340
- Michel Waldschmidt, Nombres transcendants et groupes algébriques, Astérisque, vol. 69, Société Mathématique de France, Paris, 1979 (French). With appendices by Daniel Bertrand and Jean-Pierre Serre; With an English summary. MR 570648
- Michel Waldschmidt, Diophantine approximation on linear algebraic groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 326, Springer-Verlag, Berlin, 2000. Transcendence properties of the exponential function in several variables. MR 1756786, DOI 10.1007/978-3-662-11569-5
- Michel Waldschmidt, Nombres transcendants et groupes algébriques, Astérisque, vol. 69, Société Mathématique de France, Paris, 1979 (French). With appendices by Daniel Bertrand and Jean-Pierre Serre; With an English summary. MR 570648
Bibliographic Information
- Deanna M. Caveny
- Affiliation: Department of Mathematics, College of Charleston, 66 George Street, Charleston, South Carolina 29424
- Email: cavenyd@cofc.edu
- Robert Tubbs
- Affiliation: Department of Mathematics, Campus Box 395, University of Colorado, Boulder, Colorado 80309
- Email: tubbs@euclid.colorado.edu
- Received by editor(s): December 13, 2009
- Published electronically: March 10, 2010
- Additional Notes: The authors would like to thank the referee(s) for valuable and insightful feedback, which contributed to substantial improvements in the manuscript and its results.
- Communicated by: Wen-Ching Winnie Li
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 2745-2754
- MSC (2010): Primary 11J89
- DOI: https://doi.org/10.1090/S0002-9939-10-10334-7
- MathSciNet review: 2644889