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)


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,
$$ \sum\limits_{{i = 1}}^{3} {{a_{i}}x_{i}^{2} = d{x_{1}}{x_{2}}{x_{3}};\,\left( {{a_{1}},{a_{2}},{a_{3}},d} \right) = 1.} $$


Riemann Surface Rational Solution Integral Solution Fuchsian Group Commutator Subgroup 
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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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