Approach to Markoff's Minimal Forms Through Modular Functions

Cohn, Harvey

doi:10.2307/1969618

Cited by 85 publications

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“…Remark. It can be seen from the above proof that the result of Lemma 5 can be improved, say, as c > 5ab/2 and b > 5c ′ a/2 if (a, b, c) = (1, 2, 5), (2,5,29); actually, this was already known to Frobenius [9] with a different proof. But for our purposes in this paper the weaker result that c > 2b and b > 2a is enough.…”

Section: (B) Every Odd Markoff Number Is ≡ 1 (Mod 4) (C) Every Even mentioning

confidence: 98%

“…The following result for Markoff numbers which are prime powers or 2 times prime powers was first proved independently and partly by Baragar [1] (for primes and 2 times primes), Button [2] (for primes but can be easily extended to prime powers) and Schmutz [12] (for prime powers but the proof works also for 2 times prime powers) using either algebraic number theory ( [1], [2]) or hyperbolic geometry ( [12]). A simple, short proof using the hyperbolic geometry of the modular torus as used by Cohn in [5] has been obtained a bit later but only recently posted by Lang and Tan [10]. See [15] for a completely elementary proof which uses neither hyperbolic geometry nor algebraic number theory.…”

Section: (B) Every Odd Markoff Number Is ≡ 1 (Mod 4) (C) Every Even mentioning

confidence: 99%

“…(For interpretations of Markoff's work, see [8], [4] and [7]. For its relation to the hyperbolic geometry of the modular torus, see [5] and [13]. )…”

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confidence: 99%

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Congruence and uniqueness of certain Markoff numbers

Zhang

2007

Acta Arith.

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Section: (B) Every Odd Markoff Number Is ≡ 1 (Mod 4) (C) Every Even mentioning

confidence: 98%

Section: (B) Every Odd Markoff Number Is ≡ 1 (Mod 4) (C) Every Even mentioning

confidence: 99%

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Congruence and uniqueness of certain Markoff numbers

Zhang

2007

Acta Arith.

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“…Elle a ensuite été mise en forme au moyen de formes quadratiques par Cassels [4]. Puis une interprétation géométrique plus profonde en a été donnée par Cohn [7,8]. Dans la période récente, de nouveaux développements sont apparus.…”

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L′interprétation matricielle de la théorie de Markoff classique

Perrine¹

2002

International Journal of Mathematics and Mathematical Sciences

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On explicite l'approche de Cohn (1955) de la théorie de Markoff. On montre en particulier comment l'arbre complet des solutions de l'équation diophantienne associée apparasît comme quotient du groupeGL (2,ℤ)des matrices2×2à coefficients entiers et de déterminant±1par un sous-groupe diédralD6à12éléments. Différents développements intermédiaires sont faits autour du groupeAut (F 2)des automorphismes du groupe libre engendré par deux élémentsF 2.We detail the approach followed by Cohn for the Markoff theory. We show particularly how appears the whole tree of solutions for the associated Diophantine equation as a quotient of the groupGL (2,ℤ)of matrices2×2with integer coefficients and determinant±1by its dihedral subgroupD6with12elements. Some developments are made with the groupAut (F 2)of automorphisms of the free groupF 2generated by two elements.

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“…He does this using binary quadratic forms studied by Cohn [1] and Schmidt [6]. For 77/r', his characterization (in the closed case) is the same as that of [4].…”

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Closed Geodesics on a Riemann Surface With Application to the Markov Spectrum

Beardon¹,

Lehner²,

Sheingorn³

1986

Transactions of the American Mathematical Society

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ABSTRACT. This paper determines those Riemann surfaces on which each nonsimple closed geodesic has a parabolic intersection-that is, an intersection in the form of a loop enclosing a puncture or a deleted disk. An application is made characterizing the simple closed geodesic on H/T{3) in terms of the Markov spectrum.The thrust of the situation is this: If we call loops about punctures or deleted disks boundary curves, then if the surface has "little" topology, each nonsimple closed geodesic must contain a boundary curve. But if there is "enough" topology, there are nonsimple closed geodesies not containing boundary curves.1. Let 72 be a Riemann surface of genus g with k punctures and d disks removed; k, d > 0, k + d > 0. In this paper we consider the closed geodesies on 72.The method used is to represent 72 as a quotient, 72 = 77/r, where H = {z = x + iy: y > 0} and T is a fuchsian group acting on 77. Let it : 77 -► 77/r be the projection map. We assume (77, n) is an unramified covering, so T has no elliptic elements. For most of this paper we shall assume k > 0 and d = 0; the remaining cases are considered in §5. Each hyperbolic axis projects to a closed geodesic in 72. Each closed geodesic a in 72 lifts to a conjugacy class [a] in T and under known conditions [a] is hyperbolic.Let 7 G T be hyperbolic with axis A-,. From now on we assume 7 primitive; i.e., 7 generates the stabilizer of A1. It is easily seen that (A^) is simple iff A1 fl ßA1 = 0 for all ß G T -(7).We write A~, A ßA^ to mean that A~¡ fl ßA1 = {z} for some z in 77. Thus (1.1) ""(-A-y) is nonsimple (= self-intersecting) iff A1 A ßA1 for some ß E V -(7).Since aA^ = Aaia-i, conjugate elements are both simple or both nonsimple. We use the phrase a hyperbolic 7 is simple (nonsimple) to mean n(A1) is simple (nonsimple).It is readily checked that if 7 is nonsimple, we can choose ß in (1.1) to be hyperbolic; indeed, we may replace ß by /?7n for large n. However, it is not always possible to choose ß to be parabolic, and in this connection we prove the following result.

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Approach to Markoff's Minimal Forms Through Modular Functions

Cited by 85 publications

References 2 publications

Congruence and uniqueness of certain Markoff numbers

Congruence and uniqueness of certain Markoff numbers

L′interprétation matricielle de la théorie de Markoff classique

Closed Geodesics on a Riemann Surface With Application to the Markov Spectrum

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