A Geometric Approach to the Frobenius Unicity Conjecture for the Markoff Equation

Journal of Lie Theory Volume 18 (2008) 595–600 c 2008 Heldermann Verlag A Geometric Approach to the Frobenius Unicity Conjecture for the Markoff Equation Jos´e M. Tornero∗ Communicated by S. Gindikin Abstract. The long–standing Frobenius conjecture on the unicity of ordered solutions for the Markoff equation is translated in a very simple way into an arithmetic statement on the existence of integral points on certain hyperbolas. Some previous work of Kang and Melville can then be used for relating the problem to a statement concerning rank 2 symmetric hyperbolic Kac–Moody algebras. Mathematics Subject Classification 2000: 11D09 (primary); 11D45, 14G05 (secondary). Key Words and Phrases: Markoff equation, Frobenius unicity conjecture, integral points, hyperbolic Kac–Moody algebras, imaginary roots. 1. Introduction The Markoff equation is the Diophantine equation x2+y2+z2= 3xyz;x, y, z ∈Z+; which was studied first by Markoff in [7, 8]. Markoff proved many interesting properties related to the solutions of this equation. Among other things, he proved there were infinitely many solutions and gave a procedure to construct new solutions from old ones. However, it was Frobenius [4], while studying Markoff equations over Gauss integers, who noticed that, for a given ordered solution x≤y≤z, there was no other ordered solution x0≤y0≤z. This conjecture, widely known as the Frobenius unicity conjecture, has remained open since, despite some important partial results have been settled. To mention the most popular ones: •If either z, 3z−2 or 3z+ 2 has the form 2ap, for a= 0,1,2 and pprime, the conjecture is known to be true [1]. The case z=pwas also solved independently in [2, 9]. •If z=kpr, with pprime and k4< m, the conjecture is also known to hold, as proved in [3]. ∗Supported by FQM–218 (JdA) and MTM2004–07203–C02–01 and FEDER (MEC). ISSN 0949–5932 / $2.50 c Heldermann Verlag 596 Tornero These results, combined with brute force computational attacks have proved the conjecture to be true for z < 10140 so far. Many other mathematicians (I. Borosh, R.T. Mumby, H. Cohn, M.L. Lang, J.H. Silverman, S.P. Tang, D. Zagier,...) have worked in and around this fascinating problem. Also it should be mentioned here that the Markoff equation has appeared in a number of apparently disconnected fields as quadratic forms whose values are one-third their discriminant or more, closed geodesics on certain Riemann surfaces or modular groups. Rather than building over these strong (and, in some cases, quite highbrow) techniques, we decided to approach the problem in a naive, yet not reported, way. Fix a positive integer c, and consider the conic defined by Hc:x2−3cxy +y2+c2= 0, by the way a hyperbola. It is then plain than the Frobenius unicity conjectured can be expressed in the following terms: Conjecture 1.1. There is, at most, one integral point (x, y) in the hyperbola Hcverifying x≤y≤c. This paper is based in the fact that integral points in hyperbolas of the form x2−axy +y2=k, a ≥3, k ≤1 have been studied by S.J. Kang and D.J. Melville as a by–product of their work on certain Kac–Moody algebras [6]. We will apply Kang and Melville results to our hyperbola Hcin order to look for a different version of the Frobenius conjecture. In order to do that, we will first review briefly some results from [6]. 2. Review of the Kang–Melville procedure Consider a rank 2 symmetric hyperbolic Kac–Moody algebra g(a) given by a Cartan matrix 2−a −a2, a ≥3. In [6] Kang and Melville show that the set of roots of g(a) is connected to the diophantine problem of finding points with integral coordinates in certain hyperbolas. More specifically: •There is a one–to–one correspondence between the set of real roots of g(a) and the set of integral points on Hreal :x2−axy +y2= 1. •There is a one–to–one correspondence between the set of imaginary roots of length −2kof g(a) and the set of integral points on Hk:x2−axy+y2=−k, for k≥1. They went further in order to describe the root system of g(a). We will review only the necessary information for what follows, but much more can be found in their paper. Consider the sequence {Bj}j≥0, defined by B0= 0, B1= 1, Bj+2 =aBj+1 −Bj,for j≥0. Tornero 597 Also let Ωkbe the following set Ωk=((m, n)∈Z+×Z+|2√k √a2−4≤m≤rk a−2, n=am −p(a2−4)m2−4k 2). Then the set of all imaginary roots of g(a) with length −2kis given by ∆k= Ωk∪{(mBj+1 −nBj, mBj+2 −nBj+1)|(m, n)∈Ωk}∪ {(nBj+1 −mBj, nBj+2 −mBj+1)|(m, n)∈Ωk} = Ωk∪∆(1) k∪∆(2) k up to permutation of coordinates (Hkis obviously invariant w.r.t. these transformations) and sign changes (the displayed roots being the positive ones, up to permutation of coordinates). For a given pair (a, k) the explicit computation of Ωkis fairly easy, and so is the obtained recurrent formula for the set of integral points on Hk. The elements Bjcan also be expressed in terms of a close formula (`a la Fibonacci) which we will not be concerned about. This algorithm is, according to the authors, much faster than the usual number–theoretic option which involves finding a point and then using a stereographic projection. The process will be called KM process from now on. We must remark that the Kang–Melville procedure sketched above is an application of the much more general results on positive imaginary roots one can find in [5], where the set Ωkis actually introduced. It is also shown that any positive imaginary root lies in the orbit of some element in Ωkby the Weyl group From that it is plain that the set Ωkhas an interesting connection with the root multiplicities. In fact, if |Ωk|= 1 then clearly all roots of length −2kmust have the same multiplicity. This might not be the case if |Ωk|>1, as [6] show with specific counterexamples. 3. The KM procedure for Hc Our aim is to apply the KM process to Hc, hence we are considering a rank 2 hyperbolic symmetric Kac–Moody algebra g(3c), with c∈Z+and we are interested primarily on the imaginary roots of length −2c2, which correspond to integral points on Hc. In our case the situation of the previous section can be now written as: B0= 0, B1= 1, Bj+2 = 3cBj+1 −Bj,for j≥0; and Ωc2=(m, n)∈Z+×Z+|2c √9c2−4≤m≤c √3c−2, n=3cm −p(9c2−4)m2−4c2 2), 598 Tornero where some rather popular expressions (as 3c−2 or 9c2−4) from the literature concerning Markoff equation appear. Now we will devote ourselves some time for showing some special features of Ωc2and {Bj}in this case. Lemma 3.1. 0≤m≤n≤c. Proof. We will only do m≤n, as the other two inequalities follow easily. n≥m⇐⇒ 3cm −p(9c2−4)m2−4c2≥2m ⇐⇒ (3c−2)m≥p(9c2−4)m2−4c2 ⇐⇒ (9c2−12c+ 4)m2≥(9c2−4)m2−4c2 ⇐⇒ m2≤c2 3c−2 which is guaranteed by the definition of Ωk. In order to make the statements simpler, for a fixed pair (m, n)∈Ωc2, let us write: αj=mBj+1 −nBj βj=nBj+1 −mBj so that ∆(1) c2={(αj, αj+1)|j≥0},∆(2) c2={(βj, βj+1)|j≥0} Lemma 3.2. The sequence {Bj}j≥0is a strictly increasing sequence. Proof. Straightforward (since c > 0). Lemma 3.3. The sequence {βj}j≥0is a strictly increasing sequence of positive terms. Proof. From Lemmas 1 and 2 it is straightforward βj=nBj+1 −mBj>0. On the other hand Bj+2 −Bj+1 = (3c−1)Bj+1 −Bj≥Bj+1 −Bj; hence βj=nBj+1 −mBj= (n−m)Bj+1 +m(Bj+1 −Bj) <(n−m)Bj+2 +m(Bj+2 −Bj+1 =nBj+2 −mBj+1 =βj+1 So the result is proved. Tornero 599 Lemma 3.4. The sequence {αj}j≥0is a strictly increasing sequence of positive terms. Proof. Obviously α0=mB1−nB0=m > 0. In the general case αj=m(3cBj−Bj−1)−Bj 3cm −p(9c2−4)m2−4c2 2 =Bj 3cm +p(9c2−4)m2−4c2 2−mBj−1 > mBj−nBj−1=αj−1 taking into account Lemma 1 and c > 0. Corollary 3.5. ∆(1) c2and ∆(2) c2have no points (x, y)with 0≤x≤y≤c. That is to say, the interesting points from the point of view of the unicity conjecture lie in Ωc2. Proof. After Lemmas 3 and 4 it is enough proving α1> c and β1> c. But: α1= 3cm −n =3cm +p(9c2−4)m2−4c2 2 ≥3c 2·2c √9c2−4 =3c √9c2−4·c>c β1= 3cn −m≥c(3n−1) > c Hence the result is proved. The main result of the paper goes then as follows: Proposition 3.6. The Frobenius conjecture for the Markoff equation is equivalent to |Ωc2| ≤ 1for all c∈N. 4. Acknowledgments The author wants to thank D. Fern´andez–Ternero and L. Narv´aez for their cooperation in the early stages of this work. Special thanks are due to Prof. Duncan Melville for his help, understanding and patience with my vast lack of knowledge. References [1] Baragar, A., On the unicity conjecture for Markoff numbers, Canad. Math. Bull. 39 (1996), 3–9. [2] Button, J. O., The uniqueness of the prime Markoff numbers, J. London Math. Soc. 58 (1998), 9–17. 600 Tornero [3] —, Markoff numbers, principal ideals and continued fraction expansions, J. Number Theory 87 (2001), 77–95. [4] Frobenius, G., ¨ Uber die Markoffschen Zahlen, Preuss. Akad. Wiss. Sitzungsberichte (1913), 458–487. [5] Kac, V. G., “Infinite dimensional Lie algebras (3rd edition),” Cambridge Univ. Press, Cambridge, 1990. [6] Kang, S. J., and D. J. Melville, Rank 2 symmetric hyperbolic Kac–Moody algebras, Nagoya Math. J. 140 (1995), 41–75. [7] Markoff, A. A., Sur les formes binaires quadratiques ind´efinies 1, Math. Ann. 15 (1879), 381–409. [8] —, Sur les formes binaires quadratiques ind´efinies 2, Math. Ann. 17 (1880), 379–399. [9] Schmutz, P., Systoles of arithmetic surfaces and the Markoff spectrum, Math. Ann. 305 (1996), 191–203. Jos´e M. Tornero Departamento de ´ Algebra Facultad de Matem´aticas Universidad de Sevilla P.O. 1160. 41080 Sevilla Spain [email protected] Received January 10, 2008 and in final form April 24, 2008