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Padé approximations and diophantine geometry.

D V Chudnovsky1, G V Chudnovsky

  • 1Department of Mathematics, Columbia University, New York, NY 10027.

Proceedings of the National Academy of Sciences of the United States of America
|April 1, 1985
PubMed
Summary

Researchers proved a converse to Eisenstein's theorem using Padé approximations. This advances understanding of algebraic function coefficients and has implications for elliptic curve isogenies.

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Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Complex Analysis

Background:

  • Eisenstein's theorem addresses the boundedness of denominators for algebraic function coefficients.
  • Understanding these coefficients is crucial in number theory and algebraic geometry.
  • Meromorphic functions provide a framework for parameterizing function classes.

Purpose of the Study:

  • To prove a converse to Eisenstein's theorem for specific function classes.
  • To explore the implications of this converse for the Tate conjecture.
  • To provide an effective description of isogenies for elliptic curves.

Main Methods:

  • Application of Padé approximation techniques.
  • Analysis of functions parametrized by meromorphic functions.
  • Utilizing number-theoretic and algebraic geometry principles.

Main Results:

  • A converse to Eisenstein's theorem is established for the studied function classes.
  • The findings offer new insights into the properties of algebraic function coefficients.
  • The results are directly applicable to the Tate conjecture.

Conclusions:

  • The study successfully extends Eisenstein's theorem using Padé approximations.
  • The work provides a foundation for further research into algebraic functions and elliptic curves.
  • This research contributes to the effective description of isogenies in elliptic curve theory.

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