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Rational solutions of \(\Sigma _{i = 1}^3{a_i}x_i^2 = d{x_1}{x_2}{x_3} \)and simple closed geodesics on Fricke surfaces

  • Mark Sheingorn
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 10)

Abstract

Let H be the upper-half plane, Γ(n) = {γ∈ SL(2, ℤ)∣γ ≡ ±I mod n}, Γ′ the commutator subgroup of Γ(1) and Γ3 = 〈γ3∣ γ∈ Γ (1)〉. It has recently been established that the integral solutions of the Markov diophantine equation x 2 +y 2 + z 2 = 3xyz characterize the simple closed geodesics on the modular surface H/Γ(3), H/Γ′, and H3 ([H], [L-S], [S]). In this paper we seek analogous results for more general diophantine equations, specifically equations,
$$ \sum\limits_{{i = 1}}^{3} {{a_{i}}x_{i}^{2} = d{x_{1}}{x_{2}}{x_{3}};\,\left( {{a_{1}},{a_{2}},{a_{3}},d} \right) = 1.} $$
(1.0)

Keywords

Riemann Surface Rational Solution Integral Solution Fuchsian Group Commutator Subgroup 
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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Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Mark Sheingorn
    • 1
  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA

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