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


Lution Nite Haas Riene 


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  1. [C]
    Cohn, H., Minimal Geodesics on Fricke’s Torus-Covering, Ann. of Math Studies 97 (1981), 73–86.Google Scholar
  2. [H]
    Haas, A., Diophantine approximation on hyperbolic Rxemann surfaces, Acta Math. (Uppsala) 156 (1986), 33–82.MATHMathSciNetGoogle Scholar
  3. [K]
    Keen, L., On Fricke Moduli, Ann. of Math. Studies 66 (1971), 205–224.MathSciNetGoogle Scholar
  4. [L-S]
    Lehner, J. and Sheingorn, M., Simple closed geodesies on H+ /Γ(3) arise from the Markov spectrum, BAMS (2) 11 (1984), 359–362.Google Scholar
  5. [N]
    Nielsen, J., Unter, zur Topoligie der geschlossenen zweiseitigen Flächen, Acta Math. (Uppsala) 50 (1927), 189–358.MATHGoogle Scholar
  6. [R]
    Rosenberger, G., Uber die Diophantische Gleichung ax2+ by2 + cz2 = dxyz, J. Riene Angew. für Math. 305 (1979), 122–125.Google Scholar
  7. [Sc]
    Schmidt, A.L., Minimum of quadratic forms with respect to Fuchsian groups I, J. Riene Angew. für Math. 286/287 (1976), 341–368.Google Scholar
  8. [S]
    Sheingorn, M., Characterization of simple closed geodesies on Fricke surfaces, Duke Math. J. 52 (1985), 535–545.CrossRefMATHMathSciNetGoogle Scholar

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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