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Markoff phenomena Hofstedt, Teresa 1990

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MARKOFF PHENOMENA By Teresa Hofstedt Diploma in General Studies Capilano College B.Sc. (Mathematics) University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE . in THE FACULTY OF GRADUATE STUDIES MATHEMATICS We accept this thesis as conforming to.the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1990 © Teresa Hofstedt, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The purpose of this thesis is to discuss Markoff Numbers, their associated binary quadratic forms, together with the units in the associated real quadratic field. Relations betweeen the Markoff Numbers, the explicit structure of the automorph group of the forms, generators of the Commutator Subgroup V of SL2(Z) = T and the lengths of geodesies on certain Riemann Surfaces are conveyed. A conjecture combining these relations is formulated and expressed at the end of this paper. Table of Contents Abstract ii List of Tables v List of Figures vi Acknowledgements vn 1 Introduction 1 2 Markoff Numbers 4 2.1 Elementary Properties, Descent and the Markoff Tree 4 2.2 Divisibility Properties 6 3 Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 9 3.1 Pell's Equation 13 3.2 The Number of Classes 15 3.3 The Automorphism Group of a Form 17 3.4 Markoff Forms 19 4 Hunting cr 26 4.1 Further Investigation of cr 30 4.2 Solubility of x2 - y2DxD2 = -4 31 5 Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces 36 5.1 Closed Geodesies on H+/G 37 5.2 Closed Geodesies on H+/V 39 5.3 Generators of T' 39 5.4 Simple and Closed Geodesies onJf+/r' 41 5.5 Simple and Closed Geodesies on H+ . . 42 iii 6 A Bold Conjecture Bibliography List of Tables 6.1 Class Numbers and Fundamental Units of Norm +1 for Q(y/DiD2) where D i D 2 < 106 . 45 6.2 Factorization of a Markoff Number r and Discriminant 9r2 — 4 46 6.3 Factorization of a Markoff Number r and Discriminant 9r2 — 4 47 v List of Figures 2.1 The Markoff Tree for all triples (p,q,r) with max(p,g) < 105 8 3.2 The Markoff Form Tree 25 vi Acknowledgements Special thanks to David Boyd, Kee Lam, Peter Sarnak, Asmus Schmidt, and of course my advisor Larry Roberts for both mathematical and emotional support. An especial thanks to my parents, Mary and the motivating force and strength gained from my now able left foot. Chapter 1 Introduction The deep and highly difficult study of binary quadratic forms dates back as early as the 17th century when Fermat stated theorems regarding the representation of classes of prime numbers by certain binary quadratic forms, albeit Euler was the first to publish these facts. In 1801 Gauss greatly broadened the theory conceptually in various directions, and his work still occupies a central position in the literature though many of his methods have been materially simplified, primarily by Dirichlet. A rich gamut of methods: arithmetical, analytical, algebraic, congruencial, as well as geometrical, have been used to approach the study of such topics as the reduction, equivalence, representation of integers, composition, genera, and the class number for binary quadratic forms. The classical and investigative work regarding the upper limits of the minima of a form was performed by the Russian Number Theorist A. A. Markoff (1856 - 1922). Yes, A. A. Markoff is perhaps better known for his later statistical (probabilistic) work in random variables, whence he latinized his name 'Markov' in accordance with changing orthography. To further confuse (mistake) his identity, his son, though logician and algebraist, is also A. A. Markoff, while his younger brother W.A. Markoff is also a Number Theorist. It is clearly imperative, if naive and scanning random literature, to cite both orthography and initials when determining which Markoff is Markoff. About 1879, the study of minimal properties of real quadratic binary forms led (the eldest) Markoff to discover a very special and elite set of forms, called Markoff Forms. Defining m(f) = inf | f(x,y) | where x, y 6 Z2/{0,0} and d(f) to be the discriminant of / , let n(f) = m(f)/y/d(f) where d(f) is in absolute value if / is definite. Markoff showed that for any indefinite binary quadratic form / , 3 < /•*(/) < ^g- All values taken on within this interval is duly called the Markoff Spectrum, being discrete and infinite, accumulating at i . The astounding feature of Markoffs's work, regarding /*(/), revealed that there are uncountably many non-proportional classes (of forms) for which p(f) — \ and furthermore if n(f) > § then / is either equivalent to or derived from a 1 Chapter 1. Introduction 2 Markoff Form. As expected, the situation for definite binary quadratic forms is very much different, less tantalizing and simpler. However the analogous result for definite forms is of course classical, where if {^(/)} is the set of values for all definite / then {p(f)} = (0, -^ =]. A resurgence of interest and progressed study has recently yielded new results birthing a microcosm of implications ranging over many fields, epitomized by the use of geodesies to parallel purely arithmetic results. It is now recognized that the branch of classical number theory pertaining to the study of binary indefinite quadratic forms and their minima enjoys an intimate connection with the theories of Fuchsian groups and hyperbolic Riemann Surfaces. This view was pioneered and nurtured by Cohn through a series of papers he wrote beginning in 1954, in which the central theme is the relationship between Diophantine approximation and the geometry of the subgroup V of the classical modular group SL2(Z) and the associated quotient Riemann Surface H+/V. Asmus Schmidt took a step away from the classical setting of T' by redefining the minimum of a quadratic form with respect to an arbitrary zonal Fuchsian group. He succeeded in giving a proof of Markoff's theorem for all zonal Fuchsian groups topologically conjugate to V'. His proof combines elements of the two classical arguements with Cohns approach via Fricke groups. The more recent ventures in this spirit, which have motivated the geometric viewpoint, have been by Lehner and Sheingorn, Series, and Haas. The purpose of this thesis is to discuss Markoff Numbers, their associated binary quadratic forms, together with the units in the associated real quadratic field. Relations between the Markoff Numbers, the explicit structure of the automorph group of the forms, generators of the Commutator Subgroup V of SL2(Z) = r and the lengths of geodesies on certain Riemann Surfaces are conveyed. A conjecture combining these relations is formulated and expressed at the end of this paper. In more detail, Chapter 2 is concerned with Markoff Numbers and Markoff Discriminants, and gives some (apparently) new results concerning their divisibility. These properties are crucial to the following work, their scope being especial to the results of Chapter 4. Chapter 3 sets up the required theoretical ground work regarding indefinite binary quadratic forms in order to establish results concerning the properties of Markoff Forms and the Markoff Spectrum! For each Markoff Discriminant, an infinite explicit class of Markoff Forms is revealed where it is discovered that each class (except wrt. the Markoff Number one) consists of specifically two reduced Markoff Forms. The automorph group for a Markoff Form is identified. The aim of Chapter 4 is to present some results regarding the nature of cr, the exponent which relates the Chapter 1. Introduction 3 fundamental solution to Pell's equation for +4 corresponding to a Markoff Discriminant to that of the fundamental solution wrt. its square-free part. As in Chapter 2, residue criteria is the predominant tool. Numerous results and restrictions are established regarding the features of cr. Conjecturely a is always one; in order to be greater than one, a must pass a slew of tests relying upon residue symbol criteria, the number of odd squares dividing a Markoff Discriminant, and the number of factors of a Markoff Number. This together with stipulating criteria regarding the solubility of the negative Pell's equation yields a theorem stating when the fundamental unit for a real quadratic (Markoff) field is known. Moreover this fundamental unit appears as the largest eigenvalue of the positive generator of the automorph group of a Markoff Form. Chapter 5 ventures the geometric manifestation of the role of the automorph group for a Markoff Form wrt. simple and closed geodesies of certain hyperbolic Riemann Surfaces. Projections of the fixed axes of these groups are found to arise as a special subspace of the closed geodesies on H+ /T, while accounting for all simple and closed geodesies on H+ /V and H+ /Y(Z). The fact that T(3), V, and T are discrete subgroups of PSIi2(R) and that elements of the automorph group for a primitive quadratic form are hyperbolic is truly pertinant as this information as well as the identification of a generator of the automorph group for a Markoff Form allows an explicit formulation of the lengths of these geodesies. Chapter 6 highlights the major results of this paper in the formulation of a conjecture. The conjecture is especially bold in that a is predicted to be one wrt. all Markoff values. Tables supporting this conjecture are featured. Chapter 2 Markoff Numbers Definition: A Markoff Triple is any triple ( m i , m 2 , 7 7 x 3 ) of positive integers satifying the Diophantine Equation: ml + m2, + m\ = 3mim 2 m 3 (2-1) Definition: 2.1 is called the Markoff Equation. Clearly any permutation of the m; also satisfies the Markoff Equation. Definition: A Markoff Number is any positive integer occuring in a Markoff Triple. If r is a Markoff Number then the number 9r2 — 4 is called a Markoff Discriminant (so named because it arises as the discriminant of an indefinite binary quadratic form associated with r). 2.1 Elementary Properties, Descent and the Markoff Tree Proposition 1 In a Markoff Triple the m; are distinct except (1,1,1) and (permutations of) (1,1,2). Proof. See [Cassels 57, page 27]. Definition: Call (1,1,1) and (1,1,2) (and permutations) exceptional. Proposition 2 / / ( m 1 , m 2 , m 3 ) is a non-exceptional Markoff Triple, then the three triples (m2, m 3 , 3 m 2 m 3 — ( m i i m 3 i 3mim 3 — m 2), and (mi, m 2 , 3mim 2 — 7713) are distinct as sets and are Markoff. Proof, loc. cit. Definition: Given a non-exceptional Markoff Triple call its' neighbors the three triples above and call the map which associates a Markoff Triple to one of its' neighbors a Nielsen transformation. Remark: The only neighbor of (1,1,1) is (1,1,2) and the neighbors of (1,1,2) are (1,1,1) and (1, 2, 5). The property of being a neighbor is symmetric and neither transitive nor self-inhering (reflexive). Definition: The height of a Markoff Triple is defined to be the max {mi}. 4 Chapter 2. Markoff Numbers 5 Uniqueness Conjecture For any Markoff Number r, 3 exactly one and only one Markoff Triple of height r. Remark: The Uniqueness Conjecture holds Vr < l O 1 0 5 . In 1975 Borosh was credited for this indicative measure of numerical evidence. It should be noted that in 1976, Rosenberger submitted a fallacious proof of the Uniqueness Conjecture. Since then his 'correction' has not been given much creed. Proposition 3 If M is a non-exceptional Markoff Triple, then two of its neighbors have strictly greater height, and the third has strictly smaller height. The only Markoff Triple without a neighbor of strictly smaller height is (1,1,1). Proof. See [Cassels 57, page 28]. Remark: For x > 0 Zagier proved that if m(x) is the number of Markoff Triples with height < x then m(x) ss (0.180717...)log2z. Proposition 4 (Descent) Every Markoff Triple (other than (1, 1, 1)) can be obtained from (1,1, 1) by taking successive neighbors. Proof. Induction on height. See [Cusick 89, page 18]. Corollary 1 The three integers in a Markoff Triple are pairwise relatively prime. Proof. If d divides any pair of the mi it must divide the third, as well as all the entries of any neighbor. By descent, d must be one. Definition: The set of Markoff Triples (each triple is ordered), together with the relation of being neighbors forms a tree with (1, 1,1) as the root. We call this the Markoff Tree. The Markoff Tree is an ordered binary arrangment of the solution space to the Markoff Equation. It is binary regular upon ascension from (1, 1,2). Definition: A Markoff Triple M with r = min {m,} is an r— node in the Markoff Tree if the height of M is less than the height of all Markoff Triples whose minimum is r. Definition: An r— branch is a path of triples beginning from an r—node and such than any triple along this branch has minimum = r. Chapter 2. Markoff Numbers 6 Remark: For each Markoff Number r there is an r— branch of infinite length and height. There exists exactly one r-branch for r = 1 and r = 2. If the Uniqueness Conjecture is true then for any Markoff Number r (r ^  1,2) there exists exactly two r— nodes and therefore only two r— branches in the Markoff Tree. 2.2 Divisibility Properties Lemma 1 If(x,Q) — 1 = (y, Q) and x2+y2 = 0{Qa) then either Q = 2orQ = l(A) where a = 1, 2, 3,... Theorem 1 A Markoff number r has no prime factor Q = 3(4). Proof. Let (x,y,z) be a'Markoff Triple and Q be an odd prime dividing the Markoff Number r. Then x2 + y2 = 3xyr - r2 = 0(Q) (2.2) But by Lemma 1, Q = 1(4). Hence by argument of descent, every Markoff Number r is such that if Q = 3(4) Qjfr. Theorem 2 A Markoff discriminant 9r2 — 4 has no prime factor Q = 3(4). Proof. Let (x,y, r) be a Markoff Triple and let Q be an odd prime dividing the Markoff Discriminant 9r2 - 4. As Q ^  2, exclusively, either 3r - 2 = 0 (Q) or Zr + 2 = 0(Q). So 3r = ±2(Q). Therefore x2 + y2 + r2 = Zxyr = ±2xy (Q) (2.3) => ( x T y ) 2 = - r 2 ( Q ) (2.4) Hence —1 is a quadratic residue modulo Q thereby forcing Q = 1(4). Therefore Q / 9r 2 - 4 if Q =3(4). Theorem Z If a Markoff Number r is even then 2 || r. Proof. Let (x,y, r) be Markoff Triple where r is even. Then by the Corollary to Descent, 2/x and 2 /y. Therefore by Theorem 1 x = y = 1 (4). Chapter 2. Markoff Numbers 7 Therefore x2 + y2 + r2 = 2 = Zxyr = 3r (4) Hence 2117- if r is even. Theorem A If r is even then 25 \\9r2 —4. Proof. Notate r = 2r0. Since 2 \\r, by Theorem 1 r0 = 1(4). Therefore either ro = 1(8) or ro = 5(8). In either case r = 2 (8); i.e. r j£ 6 (8). Therefore 3r - 2 = 4(8) => 22 ||3r - 2 (2.5) In light of Theorem 2, we can now write 3 r - 2 = 4(4A + 1) (2.6) >^ 3r- + 2 = 16/fc + 8 =>23||3r + 2 (2.7) Hence 25 ||9r 2 - 4 . Corollary 2 If r is even r = 2(16) whence r 0 = 1(8). Proof. Either r is congruent to 2 or 10 modulo 16. If r is congruent to 10, then 3r + 2 = 0(16). Chapter 2. Markoff Numbers (1,75025,1964)8) (1.28657,75025) ^(28657, 75025, 6449974274) (1,10946,28657) ^^(10946,20657,941038565) (1 ,4181 , 10940) (1,1597,4181) (4161,10946, 137295677) (1597,4181,20031170) (1,610,1597) (1,233,610) ( 1 ,89, 233)-/ (1,34,89) M34,89,9071 (610,1597,2922509) (233,610,4263e9) ^"(89,62 210, 1660983 7) "(233,02210,43484701) (34,9077,925765) •(89, 233,02210) (1,13,34) (13,34,1325) ^ - ( 8 9 , 9077 , 2423525) ^—(13,51641,2012674) _(13,1325,51641) ' -(1 3 25,51641,205272962) (34,1325,13513) (5,43261,6460) 8) (5,2897,43261 ) (5, 194, 2897) 2697 , 4 3 261 , 3 7 5981 346 ) ^(194,2897,1686049) -(13, 7561 , 294685) (13,194,7561) (194,7561,4400489) -(2,5741,33461) (1,2,5) (2,985,5741) .(2,33461,19=025) • (574 ) , 3346) ,57fc?98b':l : (2,169,985) (2,29,169) (985, 5741, 16964653) (169,985,499393) _ (29.14701,1 278818) (1,1,2) (5,29,433) 129,169,14701) 't169,14701,7453378) ^—(5,96557,1441889) ^^(5,6466,96557) — (5,433,6466) * < 6466 , 96557 ,} 87 3( 12*31) (433,6466,8399329) (29,3 7 666,3 276509) (29,433,37666) (1,1.1) "(433, 37666, 489281 05) Figure 2.1: The Markoff Tree for all triples [p,q,r) with max(p,g) < 10° Chapter 3 Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms Definition: A binary quadratic form f(x,y) = ax2 + bxy+cy2 (3.8) with real coefficients is called indefinite if its discriminant d(f) = b2 — 4ac > 0. Throughout this chapter attention will be primarily given to integral indefinite forms. Definition: Two quadratic forms f(x,y) and g(x,y) are GSL2(Z) equivalent if 3 integers a,f3,y,5 such that g(ax +0y,~yx+6y) = f{x,y) (3.9) and a6-/3y = ± 1 (3.10) identically in x and y. I a 0 \ If g is transformed into / by the unitary substitution 6 GSL2(Z) of determinant e, then if V 7 6 J v e — +1 the equivalence is called proper, otherwise it is called improper. If known, = will specify proper equivalence. Remark: GSL2(Z) equivalence is symmetric and transitive and further requires two equivalent forms to have the same discriminant. Unless otherwise mentioned equivalence will refer to GSL2(Z) equivalence. / = g will denote GSL2{Z) equivalence between the two forms / and g. Denoting / by (a,b,c) and g by (a',b',c') then with respect to the above transformation we have: a = a'a2 + b'aj + c' 7 2 (3.11) b = 2a'al3 + b'(a6+f3j) + 2c'y6 (3.12) c = a'p2 + b'/36 + c'62 (3.13) 9 Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms .!0 From here it is evident that every common divisor of a', b', c' is a common divisor of a,b,c. As it is only sensible to insist upon a form being integral when determining a common divisor, it is to be understood that the greatest common divisor, gcd, will apply to integral forms only. When preferred, gcd (a, b,c) will be notated as gcd(f) or referred to by K where K = gcd (o, 6,c). Similarly K' will denote gcd (a',b',c') = gcd (g). Definition: If the greatest common divisor of the coefficients of a form is 1, then the form is called primitive. Definition: If a form, / , is not primitive, it is said to be derived from the primitive form (CL/K, 6/«, C/K, ) where / = (a, b,c). Remark: If / is a derived form, K2 \ d(f) and the discriminant of the primitive form from which it is derived is d(/)//c2. Remark: If g = f then g and / are either both primitive or both derived forms if the greatest common divisor is not 1. If g and / are derived forms they are derived from equivalent primitive forms. Assume that / and g are each of non-square discriminant and let « i , w2 and ui'2 be the roots of f{x, 1) and g(x, 1) respectively. If equations 3.9 and 3.10 hold then we have: Sfaoi! +(3, jcjt + 6) = 0 (3.14) That is to say g{au)1+p/1w1+6, 1) = 0 (3.15) yielding uii = auii +P/ywi + 6 = LJ[ or u'2. Distinguishing between proper and improper equivalence this way is due to Dirichlet. Definition: A number m is said to be represented by a form (a,b,c) when there can be found integers x and y such that f(x,y) — ax2 + bxy + cy2 = m (3.16) The representation is primitive or derived according as to whether x, y are relatively prime or otherwise. Clearly if gcd(x>y) = K then K2 \ m. Furthermore if x = KX' and y = ny' then f(x',y') is a primitive representation of m//c2. When d(f) > 0 / can represent both positive and negative values thus inspiring the term indefinite. Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 11 Definition: Define m(f) = inf | f(x,y) | where x, y £ Z2/{0,0}. Remark: Representation of m(f) is clearly primitive, thereby defaulting the infimum to be taken over all pairs of relatively prime integers, not both zero. For integral / , m(f) is always attained and in fact bounded below by gcd(f) as gcd(f) | m(/). Definition: Define /j,(f) = -^ i=L. Vd(f) . Proposition 1 If f = g then a) d(f) = d(g) b) gcd (/) = gcd (g) c) m(f) = m(g) d) M(/) = M(</) e) / and g represent the same numbers f) f(x, 1) and g(x, 1) have equivalent roots Definition: Let W l = ~b + ^a ~ ~ a n d W 2 = ~ & ~ ^2a ~ ~ b e t h e r o o t s o f f(x> ^ w h e r e / = (a,b,c) is an indefinite form of non-square discriminant d(f) = b2 — Aac. f is reduced iff the two following equivalent conditions hold: | W l | < l , | w a | > l , ^ i^2< 0 (3.17) 0 < Vb2 - Aac - b < 2 | a | < sjb2 - Aac + b (3.18) Note that 3.17 & > 0 and 3.18. Furthermore since b < y/d(f), b2 — d(f) — ac is negative whence a and c are of opposite signs. Equation 3.18 requires that —b + y/d(f) and b+ \/d(f) are of like sign and that the former is less that the latter yielding 0 < b < yjd{f). This together with 3.17 yields the inequalities in 3.18. For every discriminant D, the number of distinct (inequivalent) classes is finite. Proof of this fairly deep and fundamental truth is due to the fact that there exists only a finite (limited) number of reduced forms for a given discriminant; 'whence' each class contains at least one reduced form. By means of 9 a very honest and combinatorial method one can construct a complete system of reduced forms for a given discriminant. Upon arranging them into sets of equivalent forms, the number of sets is equal to the number of classes for the discriminant considered. Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 12 While not required for what follows, there is an attractive approach to reduction due to Flath which deserves mention. Given an integral binary quadratic form / , Flath defines a properly equivalent "right neighbor", Rf, which is reduced if / is, and shows that for any / , Rnf is reduced for some positive n. This means that the sequence {Rnf} is eventually periodic, and he shows that the periodic part consists precisely of all reduced forms equivalent to / , and that the set of SLi(Z) orbits of forms with a fixed discriminant D is SL,2(Z) bijective with the set of orbits of the (finite set of) reduced forms of discriminant D under the action of the cyclic group generated by the action of R (R acts bijectively on the reduced forms). This is analogous to the continued fraction expansion of a quadratic irrationality and raises the question of the interpretation of the "period length" of the right neighbor sequence. As GSLi2(Z) is of infinite order, each equivalence class of forms c(D), of discriminant D, consists of an infinite number of equivalent forms, all honoring the properties of Proposition 1. It is crucial to realize that for any integral / £ e{D), gcd (/) = K, m(f) — rn, K \ m(f) and if K is not 1 then D is necessarily not O-free. If D is d-free then necessarily K = 1 for any form of discriminant D, whence any c(D) is a primitive equivalence class. It is therefore reasonable to illuminate the above by denoting c(D) = {re, m} where D = K2d. By above remarks we see that for K. ^ 1, e(D) = { K , m} is derived from a class s(d) = {l,m//c} of primitive equivalent forms of discriminant d. So results the expression c(/cd) = Ke(d). Note that if / £ c{D) is a reduced form then / is derived from the reduced form i / 6 e(d). That is to say positive scalar multiplication of equation 3.18 does not violate the inequalities. Proposition 2 As for an integral form D = 0 or 1(4), we may express D — Q2DQ where DQ = 0 or 1(4) and Do is O-free except for allowing the possible square factor 4 and where \Do is either of the form 4k + 2 or 4k + 3. Thus DQ models one of the 3 following conditions: Do = P P = 1(4) (3.19) D0 = 4P P = -1(4) (3.20) Do = 8P P = ±1 (4 ) (3.21) Definition: If Do satisfies one of the 3 above conditions Do is called a fundamental discriminant. Proposition 3 Given DQ, 3 an integral form of discrimant Q2Do, for any Q £ Z. Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 13 Proof. If D = Q2Do = 0(4) then x2 — ^y2 is an integral form of discriminant D. Otherwise x2 + xy — ^ T ^ y2 is an integral form having discriminant D when D — Q2DQ = 1(4). Corollary 1 For any given (integral) discriminant there exists a primitive class of forms. Proof. The above two integral forms are primitive. Proposition 4 / / d(f) = Do then f is primitive. 3.1 Pell's Equation The Pell Equation plays an unrefutably distinguished role in number theory. Its salience is vital to both the theory of binary quadratic forms and the study of units in a real quadratic field, Q(VD). Definition: For positive nonsquare D, the Diophantine equation x2-y2D = k (3.22) for k = ±1 or ± 4 is called Pells Equation. The recondite problem of characterizing those D for which the Pell equation x2 - Dy2 = -1 , or - 4 (3.23) is soluble has exalted both the interest and emphatic pursuit of mathematicians for the past 200 years. Using residue criteria, Legendre, in 1785, proved that 3.23 is soluble if D is a prime p = 1(4), while if p = 3(4) and p \ Do (i.e. the d-free part of D) then 3.23 is insoluble. Unpredictability is most ubiquitous when D has only prime factors p = 1,2(4). However if D = pq with p = q = 1(4), Dirichlet observed that if (p/q) = (q/p) = — 1 then 3.23 is soluble; while applying methods of class field theory, Scholz revealed that if (p/q) ^ (<z/p) 3.23 has no solutions. However, residue symbol criteria gives no decisive result if (p/q) = (q/p) = 1. Determining those D for which 3.23 is soluble by means of quadratic residue criteria is still presumed an open question and for special types of D explicit residue symbol conditions are still being found. See [Lagarias 80]. It has been well known that decideability can be achieved for any positive non-square D by expanding ^/D as an ordinary continued fraction v /5= [a0loi,...,oJV] (3.24) Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 14 If the period N is odd 3.23 is soluble and if N is even there are no solutions to 3.23. Though the method of expansion guarantees to terminate this probing decideability quest, it deserves mention that for certain D more than ^\/D(\og £ ) ) _ 1 computational operations may be required. We close this discussion of 3.23 with the following proposition due to H.J.S. Smith. Proposition 5 The following are equivalent a) x2 — Dy2 = —lis soluble inintegers. b) The forms [1, 0, —D] and [—1, 0, D] are equivalent For purposes of this chapter we give results primarily concerning k = 4 or 1. Proofs of the following propositions can be found in [Landau 58, Flath 89]. Definition: Call a solution trivial if y = 0. Note this only makes sense if k > 0. Remark: When referring to a solution(s) we will be referring to non-trivial positive solutions x > 0, y>0. Theorem 1 3 a non-trivial solution to x2-y2D = +4 (3.25) Corollary 2 3 a solution to 3.25 for which x > 0 and for which y has smallest positive value. Definition: This minimal solution, which we will denote as (t0,u0), is called the Fundamental Solution. Theorem 2 3 an infinite number of solutions to 3.25. All solutions are given by t0 + u0 2 \ t ) with n > 0 where (tn,un) is called the nth solution to 3.25. (3.26) Theorem 3 Alternatively, if 3.23 is soluble, wrt. k = —4, with fundamental solution (xo,yo) then tn + uny/T> _ ^xp + ypVty^ 2 n ($21) where 2n is replaced by 2 if n = 0. Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 15 Theorem 4 / / (tn,un) is the nth solution to 3.25 then (tn,un) <=>• (\tn, \v,n) is the nth solution to x2-y2D = +l (3.28) i s Remark: For D = 1(A) it is possible that tn = un = 1(2) thereby resulting in ^ tn, ^un £ Z. Thcase is the only exception to 3.28 not being diophantine. The following two propositions clarify this case. Proposition 6 If tn = un = 1(2) then D = 1 (4) and to = V.Q = 1(2). Proposition 7 3 an infinite number of integral solutions to 3.28. Remark: Even if D = 1(4) for some n, tn = un = 0(2). Definition: Define E{D) = *° + ! ° ^ (3.29) E(Do) = g ( 3- 3 0) When D = DQ we will denote Remark: E(Do) is the smallest unit > 1 of Norm +1 of the real quadratic field Q(\/D) = Q(\J ?) where r = 14 if Do E 1 or 0(4) respectively. When the norm is not known then the fundamental unit will be denoted by e. Note that if 3.23 is not soluble then E(Do) is the fundamental unit of Q(y/D). Note that if Q \ U0 then E(D) = E(D0). 3.2 The Number of Classes Definition: Define K(D) to be the number of primitive classes of discriminant D. Expressing D = Q2DQ we have the following equalities due to Kronecker: * ( D , = , ( W n { l - ( ^ } ™ <»D where the product applies to all prime factors of Q which do not divide Do-Remark: If E(D) ^  E(D0) then since (t0,u0) is the least positive solution to x2 - Q2D0y2 = +4 (3.32) Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 16 we have that (to, Quo) is a solution to x 2 - y 2 D 0 = +4 (3.33) Because (t0,uo) is the least positive solution to 3.32, (to, Quo) is the first solution to 3.33 for which Q I y = Q^o-That is to say, if we denote to = T<r and Quo — U„ then a identifies the least power of E(DQ) such that (E(D0)r = E(D) (3.34) where (T^, J/^ ) is the smallest solution to 3.33 for which Q | V„, that is Vn < a, Un ^ 0(Q). We can now express ^E(DQ) _ 1 log E(D) ~ a where a - 1 <=> E(D) = E(D0) <=> Q \ U0. Proposition 8 If Q = Q' FJiLi H where the qi are the different odd prime factors not dividing DQ, then a | 1cm | Q ' , 9 I - ( ^ ) .-.9A - ( ^ ) } (3.36) A fortiori a will divide Remark: If every prime factor of Q divides Do, then all that can be concluded is that a | 2Q. As a consequence of the preceding discussions we have the following proposition Proposition 9 // gcd(Q, Do) = Q' ^  Q and Q/Q' = n£=i<7> ^-free then X(D) = K ( D 0 ) ^ n ^ - ^ ) (3.38) See [Mathews 70, Dickson 52] for an account of cr and K(D). Definition Define K'(D) to be the number of imprimitive classes of discriminant D = W^^liDo-In order to formulate K'(D) we require the following discussion. Define Sij to be the product of j of the qi, where it is understood that if qr and qs divide Sij then r # 5 . Note si \U7=llf-Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 17 Let | {si.j,mi.j} | be the number of imprimitive classes of discriminant D and K = s^ -, where it is understood that rtiij = m(f) will vary due to the inequivalence of forms, although S{j \ is always achieved. Now since each derived class {sij-, rriij} is obtained from a primitive class {l,mij /sij} we have | |= K(D/s?j). Recall that by Proposition 3, there exists a form of discriminant D/s2j and by Corollary 1, K(D/s2j) is non-empty. Denote ^j^j by n(j). Now as 3 n(j) possible values for s^j, then assuming the Si.j are unique, the number of all imprimitive classes {s^j,rrii,j} where 1 < i < n(j) is the sum "(>) J2K(D/sl3) (3.39) i = l Allowing j to vary we therefore have n n(j) K'(D) = Y.Y.K(D/sh) (3-4°) j=i i=i 3.3 The Automorphism Group of a Form Definition: A matrix 7 6 GSL2(Z) is said to be an automorphism of an integral binary form / = (a, b, c) <=> 7 / = / or equivalently, iff X ' a b/2 \ I (3-41) We say that 7 is proper iff det(j) — +1 and that 7 is improper iff det(j) = —1. Definition: Let AUT(f) be the group of automorphisms of / and AUT+(f) be the group of proper automorphisms of / . Remark: AUT+{f) C SL2(Z) and AUT+(f) C AUT(f) C GSL2{Z). Theorem 5 / / equation 3.23 is inso luble for D = d(f) then AUT(f) = AUT+(f). That is to say an improper equivalence of f with — f. Proof. This condition is in concordance with Proposition 5. See [Cassels 57]. Definition: Let / = (o,b,c) be an integral form of non-square discrimant D ^ 0. Let (t,u) be a solution to 3.25. Define 7/(t,u) G AUT+(f) by the formula f t-bu _ \ Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 18 Proposition 10 AUT+(f) ~ Z/2xZ with generators ±jf (to,u0) where (t0,uo) is the Fundamental Solution to 3.25 and t0 - bu0 _ 7 , ( t o , « o ) = | ) (3-43) auo u ^— (•yf(to,uo))n = Jf(tn,un) where (tn,un) is the nth solution to 3.25 and / tn ~bun \ " w i T ^ J (3 44) Proof. See [Flath 89, Cassels 57]. To see that " 2 ^ " G Z it suffices to show that tn and are of like parity. This condition is easily satisfied (simply accessed) for if D = 1(4) then b = 1 (2) and tn = un = 0,1(2) and if D = 0(4) then & = t „ = 0 ( 2 ) . Proposition 11 Tr(jf(tn,un)) = i n and jf(tn,un) has eigenvalues Ai,„ = * " + ^ ^ (3.45) A2,n = j ( 3 °) When n = 0 we will refer to Ai, A 2 , instead of Ai io,A2,o, respectively. Corollary 3 Ai = E(D) whence XUn = (E(D))n. Proposition 12 Let K = gcd(f) and f = Kg. Then if (£o,"o) is the Fundamental Solution to x2 -y2d(g) = +A (3.47) and K I UQ then AUT+(f) = AUT+(g). Proof. By the discussion of equations 3.32 - 3.35, if K \ uo then (to,u0/K) is the fundamental solution to *2 - V2d(f) = +4 (3.48) where d(f) = K2d(g). Since / = (/ca, Kb, KC) it is an easy check to see that 7y(£o,uo/ K ) = lg(to.uo)--Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 19 3.4 Markoff Forms Let (x,y,r) be a Markoff Triple of height r. We construct a Markoff Form associated with the Markoff Number r as follows: Let k be the least integer for which xk = y(r) 0<k<r (3.49) and let I be the integer defined by k2 + 1 = Ir (3.50) Note that by reducing the Markoff Equation mod r it is clear that r \ k2 + 1. Definition: The form Fr(x,y) defined by Fr(x,y) = rx2 + (3r - 2k)xy + (I - M)y2 (3.51) is called a Markoff Form. Remark: The definition of FT is asymmetric in x and y. That is to say if F'r is the form obtained by switching roles of x and y in 3.49 then Fr EE Fj. For suppose yk' = x (r) with 0 < k' < r (3.52) where I' is the integer defined by k'2 + 1 — I'r (3.53) Then similarly Fj. is defined by Fj(x,y) = rx2 + (3r - 2k')xy + (/' - 3k')y2 (3.54) Now since x2 = —y2 (r), we have k2 = k'2 (r) where k = k' = 0 if r = 1 and where otherwise k + k' = r. Therefore we have Fj = FT when k = k' = 0 and Fj = FT(x + 2y, -y) V other r ± 1. Fj and rFr are clearly equivalent as ad — be — — 1 — 0 = —1. Note that by choosing ko = min{A;,A;'} 0 < 2k0 < r. Upon setting lo = m i n { / , a n d notating FT,o — (r,3r — 2ko,lo — 3&o), then clearly FT:o is one of the above defined FT or Fj. Heeding this convention, there corresponds a genealogical tree of Markoff Forms Fr<o to the Markoff Tree of triples. As each FT}o is uniquely constructed from a respective triple (x,y,r) of height r, the map ip : (x,y,r) —* FTto is a bijective map from the Markoff Tree to the Markoff Form Tree. It deserves Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 20 mention that the map <p : r —-» FTto is not bijective if the Uniqueness Conjecture is false; however, to Borosh's credit, bijectivity holds W < l O 1 0 5 . Allowing k,k' > r all solutions to 3.49 and 3.52 are respectively k(j) = rj + k and k'(j) = rj + k' (3.55) Modifying 3.50 and 3.53 we have KJ? + 1 = KJ)r where = I + rj2 + 2kj and similarly k'(jf + 1 - l'(j)r where = V + rj2 + 2k'j. We now obtain the following forms: FTd(x,y) = rx2 + (3r - 2k(j))xy + - U{j))y2 = rx2 + (3r - 2(rj + k))xy + (I + rj2 + 2kj - 3rj - 3k)y2 Fld(x,y) = rx2 + (3r-2k'(j))xy + (l'(j)-3k'(j))y2 = rx2 + (3r - 2(rj + k'))xy + (I' + rj2 + 2k'j - 3rj - 3k')y2 Proposition 13 FrJ I Fr = F'T i F^. Proof. Frj,F'r are transformed into Fr,F}. respectively by the proper unitary substitution Wig. we will show the former. FTj(x + jy,y) = r(x + jy)2 + (3r - 2(rj + k))(x + jy)y + [l + rj2 + 2kj - 3rj - Zk)y2 = rx2 + (2rj + 3r — 2rj — 2k)xy + (rj2 + Zrj - 2rj2 - 2kj + I + rj2 + 2kj - 3rj - 3k)y2 = rx2 + (3?- - 2k)xy + (I - 3k)y2 Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 21 When convenient we will (wig.) notate a Markoff Form by Frj. Theorem 6 Except for the last stipulation on j, the following properties hold for all FTj and . a) d(FT.j) = 9r2 - 4 Fr.j = ~Tr,j c) m(FrJ) = r e) If r is odd e(9r2 -4) = {l,r} f) If r is even s(§r2 - 4) = {2,r} g) < 1 and ui2 < — 1 if j = 0 Proof. See [Cusick 89, pages 20 - 21]. Corollary 4 Expressing 9r2 — 4 = Q2DQ, DO = 1 (4) if r is odd otherwise 23 || Do for even r. Proof. See Theorems 2 and 4 of Chapter 2 and Proposition 2 of this chapter. Remark: Since K = 2 for F2roJ when r = 2r 0 by (/), we have that F 2 T O J is derived from ^F2ro,j where ^ i ^ r c j ) = 9r^ - 1 and e(9r^ - 1) = {l,r-0}. Note that by Theorem (g) and equation 3.17 we have that FT and Fj. are reduced forms (if k = k', then r = 1). The following theorem crystalizes property (g). Proposition 14 For j ^ 0, Fr,j and F^j are not reduced. Proof. It suffices to show that either of the first two inequalities of equation 3.18 do not hold unless 3=0. Wig. we will consider Frj where b = 3r — 2(rj + k). First note that 3r - 1 < v^r 2 - 4 < 3r. Now 0 < Vb2 — Aac — b < 2 | a | becomes: 0 < V$r2 - 4 - [3T- - 2(T-J + k)\ < 2r (3.56) Replacing \/9r 2 — 4 by 3r yields 0 < 3r - [3r - 2(rj + k)} < 2r (3.57) Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 22 thereby forcing 0 < 2rj + 2k < 2r (3.58) As k > 0, if j > 0 the second inequality does not hold. Similarly, if j < 0 the first inequality will not hold. The same results are deduced by considering 0 < 2rj + 2k — 1 < 2r. Definition: The Markoff Spectrum is all numbers u(f) as / range over all real binary quadratic indefinite forms. Proposition 15 The Markoff Spectrum (above consists of the numbers *v - TO ( 3 ' 5 9 ) Remark: MSpec will denote the Markoff Spectrum above ^. Proposition 16 ^ < MSpec < . Moreover MSpec is countable and discrete accumulating at j . Theorem 7 Except for (d) the following properties hold for indefinite binary quadratic forms. a) M(/)>3-b) If p.(f) > 3 then p.(f) 6 MSpec <=> / is derived froma Markof f Form. c) There are non-enumerably many forms, none of which is equivalent to a multiple of any other, such that /x( / ) = d) If f is definite then p,(f) £ (0, -7=)-v3 Proof. See [Cassels 57, page 39] and [Cusick 89]. The following theorem illuminates MSpec within the context of irrational numbers, thereby drawing an analogy between the properties of forms in Theorem 7 and their (irrational) roots. Theorem 8 Let 6 be irrational and define v{6) = liminf q\\q6\\ (3.60) where the inf is taken over all rational q. a) If v(6) > ^ then 6 is equivalent to a root of Frj(x, 1) = 0. Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 23 b) Conversely if 6 is equivalent to a root of Fr,j(x, 1) = 0 then "<"> = 7sh% > \ <3-61> and there are infinitely many solutions of q\\6\\ < v(&). The two roots of FTj(x, 1) = 0 are equivalent. c) There are non-enumerably many inequivalent 6 such that u(6) = Proof. See [Cassels 57, page 41]. The conspicuity of the Markoff Discriminant readily facilitates identification of the group of proper automorphisms for a Markoff Form. Easily, (3r, 1) = (to,uo) is the Fundamental Solution to x2 - (9r2 - 4)y2 = +4 (3.62) as by Corollary 1, UQ > 1. We therefore have , 3 r - ( 3 r - 2 ( r j + fe)) _ ( / + r j 2 + ^ _ ^  _ 3 f c ) \ 7FriJ.(*o,uo) = 3 r + ( 3 r 2 (rj + k)) ^3-63) "2 ( rj + k -(l + rj2 + 2kj - 3rj -3k) r 3r — (rj + k) For j = 0 (3.64) 7F , (*OIUO ) = j ^ M (3-65) \ r 3r — k I Corollary 5 For r = 2r0, AUT+(\Frj) = AUT+(FTj). Proof. (3r, 2) is a solution to x2 -y2{9r2 -1) =+4 (3.66) where d(\Frd) - §r\ - l .Tf u0 = 1 then = 9r% + 3. However as 3 /3rg + 1, uo = 2 is forced. Now apply Proposition 12. We conclude our discussion of Markoff Forms with the following theorem. Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms 24 T h e o r e m 9 The automorph group of a Markoff Form satisfies the following properties. a) AUT(FTj) =AUT+{FrJ). b) ±jFr.(3r, 1) generate AUT(FrJ). d ) A 2 = 3 r -y4^r4 Proof. Only (a) needs clarification. Suppose 3.23 is soluble, wrt. k = — 4 and where D = 9r2 — 4. Expanding equation 3.27 we have that < ^ x < ^ 0 — \ f thereby forcing D — 8 which is not a Markoff Discriminant. Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms Figure 3.2: The Markoff Form Tree Chapter 4 Hunting a As in Chapter 3, we will denote the fundamental solution to Pell's Equation x2 - y2D = +4 by (to,uo) and define the nth solution (tn,un) by tn + unVb~ _ (t0 + U0VD\ ~ ~ \ 2 ) By expanding 4.67 and gathering appropriate terms we have the following explicit formulas for tn and . (4-67) If n = 21 then If n = 21 + 1 then In both cases Y/Un+l) t 0 n - { 2 k + 1 ) ) u 2 0 K + LD K ,k=0 ^  ' E ^ + l) t^^ul^D" J2(2nk)t0n-2^u2KDK I n V / Remark: UQ \ un Vn. If n is odd then to \tn. Proposition 1 If n — jk then VD T t 0 + UQVD 2 tj + UA VD" (4.68) (4.69) (4.70) (4.71) Remark: If k is odd tj \ tn and to | tk • However unless j is odd to ftj, tk }ftn and to fan • Regardless of parity, Uj | un for any j | n. The following lemma illustrates the contextual eminence of the above remark for d-free D. u can be thought of as tj, u' as tn where n = kj and k is odd. 26 Chapter 4. Hunting a 27 Lemma 1 If (u,v) and (u',v') are positive integer solutions of x2 —y2D = + 4 where D is O-free, with u | u', then 3 fixed positive integers Dx,D2 such thai D\D2 = D, u + 2 = bD^l u-2 6D2v2, u' + 2 = 6Div[2 u' / - 2 6D2v'2 where 5, ui, v2, v[, v'2 are positive integers satisfying 6viv2 = vt 6v[v'2 = v' 1, 2 j/u (^DXD2 = 1(4)) =3(4)) <£> Z>iZ»2 = 2(4) or =>• £>!D 2 =1(4) 2 4 I u 5 = < ( ) 4, 2||u Proof. See [Le Maohua 89, Lemma 3]. Lemma 3 in Le Maohua's paper is stated for equation 3.28 where D is square-free. The above lemma is specific to equation 3.25 for square-free D and is a modified version of lemma 3. Applying Theorem 4 and Proposition 6 of Chapter 3 and reducing equation 3.25 modulo 4 wrt. D — D\D2 = 1,2,3(4) yields S and the above powers of 2 for all three cases. For D = 3(4) it was found that 4 | u always, as in view of 3.28 2 | u is forced if x2 — 3y2 = 1 (4) is to hold. Remark: Lemma 1 applies in general to any positive •-free D. From here on the convention will be to denote D by D\D2 where the factors Di and D2 are determined by to ± 2 respectively. Here to and K o are u and v, respectively, as in Lemma 1 above. Accordingly, (to,uo) will denote the fundamental solution to We can now express to + 2 and to ~ 2 as 6D\v\ and <5D2-uf, respectively, where Uo = 6viv2. Note that 6 | iio | un where (tn,un) is the nth solution to 4.72 and For the purposes of this chapter, unless otherwise mentioned, DXD2 will denote the O-free part of a Markoff Discriminant. If 9r2 —4 is •-free then r is necessarily odd, since by Theorem 4 of Chapter 1, 25||9r2 —4 if r is even. As 2/9r2—4 if r = 1(4), allowing for odd squares, the convention will be to notate 9r 2 — 4 = n™=iPiE>iD2, if r is odd and to notate 9r2 — 4 = 42T/T"_1p2Di£'2 if r is even and where 2 || D\D2. x2 -y2DlD2 = +4. (4.72) (4.73) Chapter 4. Hunting o 28 For r = 1 (4) 6 \ UQ \ W*=lPi since (3r-, <5n"=iP«) ^s a solution to 4.72, thereby forcing 5 = 1. For r = 2(4) we have 5 = 4 since D1D2 = 2(4). When referring to either parity we will conveniently notate n 9r2-A = 62Y[p2iD1D2. (4.74) t=i This will also allow inclusive reference to (3r, 5rj™=1pi) as a solution to 4.72. Note also that in either case, by Corollary 4 of Chapter 3, we have Do = bD\D2. In both cases 9r2 - 4 = 2 (3) => DXD2 = 2 (3) (as a • = 1 (3) if 3 / • ) . Theorem 1 / / (to,uo) = the fundamental solution to x2 —y2D\D2 = +4 then 3 | to. Proof. As DXD2 = 2(3), for any solution (t,u), t2 - u2D1D2 = t2 - 2u2 = 1(3). Clearly not both 3 | t and 3 | u. If 3 jft and 3 jfu then t2 - 2u2 = 1 - 2 x 1 = -1 £ 1 (3). Therefore either 3 | t and 3 / u or 3 jft and 3 | u: If 3 \t and 3 / u then t2 - 2u2 = 0 - 2 x 1 = 1 ( 3 ) If 3 jft and 3 | u then t2 - 2u2 = 1 - 2 x 0 = 1 (3) Now since (3r, <5n™=1Pi) i s a solution to E 2 — y2D\D2 — +4, if (3r, <5ll™=1Pi) = (io^o) we have 3 ||to as by Theorem 1 of Chapter 2, 3 jfr. If (3r, 5f|™=1pi) = (tpjUcr) ^ (to . 'Uo), then since UQ \ ua VCT, 3 jfuo since by Theorem 2 of Chapter 2 3/pi Vi and 3/5. Hence 3 | to -Theorem 2 / / (3r, 5fT™=1pi) = (tCT,ua) ^ (t0,uo)) ^ e fundamental solution to x2 — y2D\D2 = +4, then 2 jfo and 3/cr. Proof. Case 1: Suppose cr = 2k. Then as by 4.71 *r + SYE^piy/Dtt = ^t0 + uo^D^D~2yk = ^tk + uk-jD^D2"^ we obtain f = t^ +upxi?, a n d ^n^P. = 2 t ^ or 6r = t\ + ulDiD2 and <5r]"=1pj = tkuk. Chapter 4. Hunting cr 29 As by Theorem 1 3 | tk or 3 | uk. But 3j/6 and 3/p, Vi. Hence 2 / a is forced. Remark: For' r = 2ro, one can alternatively prove that 2 fcr by observing that 6r + 4 = t\ + •u^r>iD2+4 = 2*2., thereby implying that 3r + 2 = t2. = • . But by equation 2.7 of Chapter 2 23||3r + 2 thus annihilating the possibility that 2 | cr. Note, t2k + 2 = t\ is always true. Case 2: Suppose cr = 3fe . Note that by Case 1, k EE 1(2). Then as Sr + ^ nLiPiV-DiAj _ / t o + ^ 7 5 7 ^ ^ * _ (tk + ukVthlT2 = (tk + 3tjt^i9il>2 + {U2kuk + ulD1D2)\/D1D2))/& we obtain 12r = t^  + ZtkukD1D2 (4.75) 4«5n?=1Pi = 3 * 2 ^ + ^ 0 ^ 2 (4.76) Now since k is odd, t 0 | tk. Therefore as 3 110 by Theorem 1, if 4.75 is to hold, 32 | (tf + Uku2kDiD2). But 3 || 12r. Hence 3/cr. Corollary 1 If to ^ 3r, then since cr is odd to \ t„ = 3r. Moreover 311 to - Combining previous results on 6 and invoking Lemma 1 we therefore have the following relations: For r EE 1 (4) 2 ft0 <=> 6 = 1 and ua = v[v'2 = Y[i=iPi-Forr = 2r0 2\\t0 <=> <5 = 4, U o . = 4v[v'2 = A\\^=1pi, and 2\\D1 as 23||3r- + 2. Heeding 5 = 1 or 8 = 4 wrt. the above two cases we now have the following expressions: to+ 2 = SDxvx2 and 3r + 2 = SDxVi'2 where vi | vi and to-2 = 6D2v22 and 3r - 2 = 6D2v2'2 where v2 \ v2 Corollary 2 Dx = 2(3), D2 = 1(3) and vf = v'2 = v\ EE V'2 = 1(24). Chapter 4. Hunting a 30 4.1 Further Investigation of a This section investigates criteria for both a = 1, and a ^  1. The salient and excitingly relevant fact is that if a = 1, then g ( 9 r 2 - 4 ) = 3 r + ^ r 2 " 4 = J g ( i ? 1 £ > 2 ) (4.77) where E(DiD2) = ^ n ^ v ^ (4.78) is the first unit > 1 of the real quadratic field Q ( v V - 4 ) = Q(\/.Di£)2) of norm +1. Theorem 3 For odd r, if r is prime and D\D2 ^ 5 then cr = 1. Proof Since cr is odd, to = 3 if to ^ 3r. By the relation in Corollary 1 this would force D\D2 — Di = 5. Theorem 4 for r = 2r 0, if r0 is prime and DXD2 ^ 2 tA.en cr = 1. Proof. 6 110. Mimic the above arguement with t0 = 6. This results in D\D2 = D\ = 2. Theorem 5 If v\, v'2, or n = 1 £A.e7i <r = 1. Proof. Since cr is odd t>i | t;t'. If v[ = 1 then trivially Vi = 1 thereby forcing cr = 1. If n = 1 then one of the v[ — 1. Corollary 3 If n = 2 and a ^  1 i/ien = v2=l *>—' uo = 5 and Z?i = D 2 + 1. Definition: Let r(x) = the number of odd prime factors of x. Corollary 4 For cr ^  1, a > 5 r ( < r ) , r(r) > r(cr), r(n) > 2r(o-), r(n) = 2r(cr) w0 = <5 and r(r) = r(<r) <=> t 0 = 3 or 6 => D\D2 — 5 or 2 depending upon whether r = 1 or 2 (4) respectively. Obsession with a is incomplete if we do not reveal £5 and 145 since a > > 5 if a ^  1. If this is the case, then expanding 4.70 and 4.69 with n = 5 3r- > U = 4[*o + 16*o"o-Di^ a + U0ul(DxD2)2] (4.79) lo 6W=iPi >u5 = ~[b4u0 + lOtlv^DiDs + ug(I>iD2)2] (4.80) Chapter 4. Hunting a 31 The following proposition is a Markoff Setting of equations 3.37 and 3.38. Here 9r2 — 4 = Q2 DQ with Do = 6DiD 2 and Q = ]^ir=iP» where k = y/6. Proposition 2 Let gcd(kT\™=1Pi,6D1D2) = K. and Q = Q'Y\i=iPi where Yli=iPi ^ 1 L S O-free and • - I ^ r l l ( « - ( ' V ) ) <«•«> i = l X(9T - 2 -4 ) = ^ ^^^^[(ft-pf 2)) (4-82) i = l Proof. See Propositions 8 and 9 of Chapter 3. 4.2 Solubility of x2 - y2DxD2 = -4 Lemma 2 Suppose x2 — y2D\D2 = —4 is soluble with fundamental solution (TQ,UQ). Let (to,uo) be the fundamental solution to x2 — y2D\D2 = +4. Then either T0 = kvx and Uo = kv2 or TQ = kv2 and Uo = kvi , where k = y/6. Proof. Since to + it0 y/D 1D 2" _ (T0 + U0S/D^LT2\ _ Tg + USDiD2 , 2T 0 r / o V / D 1 D 2 " 2 - I 2 - 3 + 3 we have the following equalities: (o = q + viKD, ( 4 8 3 ) uo = T0Uo (4.84) Case 1:. 2 / t 0 . Then by Lemma 1 and 4.84 6 = 1 so Uo = i>iu2 = T0?7o- Moreover (Diuf,D 2x;|) = 1, for if m — (to + 2,to — 2) then to = ±2(m) => m = 2 or 4. But this is impossible as to = 1(2). Therefore (ui,v 2) = 1. Now suppose Vi = ab and t i 2 = cd. Wig. suppose TQ = ac and f/0 = bd. Then by 4.83 and Lemma 1 we have: Chapter 4. Hunting cr 32 io + 2 = = Dxa2b2 Since a?c2 + b2d2D1D2 + 4 = 2Dia2b2 T 0 2 + 4 = Ur2D1D2 a2c2+4 = b2d2D1D2 Therefore 2b2d2DlD2 = 2Dia2b2 d2D2 = a2 (4.86) (4.85) d = a = D2 = 1 (4.87) as (ab,cd) = 1 since (Dxa2b2, D2c2d2) = 1 Therefore vx = b = Uo and v2 — c = To Note, by considering t0 — 2 the same result ensues. Case 2: 2 || to-Then by Lemma 1 and 4.84, 8 — 4 so u0 = 4vxv2 = T0?7o, whence 4 | Tot^ o- Observing T2 - U2D1D2 = -4 = 0(4), it is clear that 2 | T 0 and 2 | U0 as 4/Z)iZ» 2 as r>iD2 is D-free where DXD2 EE 1,2(4). Moreover (4Z>iu2, 4D2v$) — 4, for if m = (i 0 + 2,t0 - 2) then t0 EE ± 2 ( m ) =^> m = 2 or 4. Here m = 4 as <5 = 4. Hence (D1v2,D2v$) = 1. Now as in the above case suppose vx = ab and v2 = cd. Wig. suppose To = 2ac and Uo = 2bd; (here k = 2). Following the steps in the above case we arrive at 4.85 - 4.87 , thereby getting 2vx = 2b = Uo and 2-u2 = 2c = T 0 . Similarly, considering to — 2 the same result ensues. Remark: It is a theorem that the Pell equation x2 — y2D\D2 = —4 has no solution in integers if DXD2 has a prime factor Q = 3(4); refer to section 3.1 for a discussion of equation 3.23. Therefore the case where 4 | to in Lemma 1 was not considered here. Theorem 6 If r is odd and x2 — y2DxD2 = -4 is soluble, 3r — 2 = • <^=> D2 = 1. Proof. Since 2/to, by Case 1 of Lemma 2, u 0 = Tof7o = viv2 with (vi,v2) = 1 = (To,f/o) as interchangeably, either To = vx and Uo = v2. Case 1: Suppose To = vx and Uo — v2. Chapter 4. Hunting a 33 Then vx2 -v22DlD2 = -4 (4.88) Since Dxvi2 - 4 = t0 - 2 = D2v22, 4.88 becomes : u i 2 - D ^ D i ^ i 2 - 4 ) = -4 => V ( l - D 2 ) = -4(Z?! + 1) => V ( l - - D i ) = -4 =>• £>i = 5 and V i = 1 or D\ = 2 and U i = 2 But 2/Diui as r = 1(4). Therefore if To = v\ and Uo = v2, then = 1 and Dx — 5. Furthermore t 0 + 2 = £ > i u i 2 = 5 => t 0 = 3 => t 0 - 2 = D 2 u 2 2 = 1 D 2 = u 2 = 1 Hence 3r - 2 = D 2 u 2 2 = u 2 2 = • . Case 2: Suppose To = v2 and C7o = vx. Then v22 - vx2DxD2 = -4 (4.89) Since Z? 2u2 + 4 = t 0 + 2 = Z?iU 2 4.89 becomes: v2 - D2{D2v2 + A) = -4 (4.90) => U 2 2 ( l - D 2 ) - 4D 2 = -4 (4.91) => v 2 ( l - D 2 2 ) = - 4 ( 1 - D 2 ) (4.92) => u|(l + 2? 2) = -4 Chapter 4. Hunting cr 34 But D2 > 0. Therefore in view of 4.90 - 4.92, only D2 = 1 is possible. Hence 3r - 2 = v'22 = •. Note that in Case 1, (to,uo) = (3,1) was completely determined as D\ = 5, v\ = I = v2 = D2 were all forced. Case 2 is far less restrictive. However from the fact that DiD2 = 24(3A: + 1) + 5, if Di ^ 5, then (wrt. Case 2) DiD2 = Di = 12k + 29 and as always with the restriction that for any p \ Di, p = 1 (4). Theorem 7 For even r, if x2 — y2D\D2 — —4 is soluble, 3r — 2 = • <==> D2 = 1. Proof. By Corollary 1 2||i 0, 6 = 4, 2||Z>i and by Lemma 2 * = 2. Case 1: Suppose To = 2ui and Uo = 2u2. Then 4v2 - 4vlD1D2 = -4 (4.93) Since 4Dxv\ - 4 = 4D2v\, 4.93 becomes: 4v2 - £>i(4£>i-u2 - 4) = -4 =• 4v\[\-D\) = -4(l + £>!) => u j ( l - ^ i ) = - 1 => Z?i = 2 and vi = 1 Therefore T 0 = 2 and 4.93 becomes 4 - 8v^D2 = -4 thereby forcing D2 = v2 = 1 => I70 = 2. Hence 3r - 2 = 4u 2 2 = Case 2: Suppose To = 2u2 and Uo = 2v\. Then 4v| - 4 w 2 D 1 D 2 = -4 (4.94) Since 4Z?2T;| + 4 = t 0 + 2 = 4D1v2_, 4.94 becomes: 4v| - D2(4v2 + 4) = -4 =• ^ 2 ( i - r j 2 2 ) - £ > 2 = - l Clearly the only possibility is that D2 = 1. Hence 3r - 2 = 4u 2 2 = • . Note that in Case 1, (T0,Uo) = (2,2) and therefore (t0,u0) = (6,4) were completely determined. Chapter 4. Hunting cr 35 Theorem 8 / / 9r2 — 4 has no odd square factors and x2 — y2D\D2 = —4 is soluble then r = 1 if r = 1 (4) and r = 2 if r = 2(4). Proof. If 97"2 — 4 has no odd square factors then UQ = 6. Moreover a is forced to be one thereby forcing £ 0 = 3r. By 4.84 of Lemma 2, Uo = TQUQ = k2. Case 1: r is odd. For odd r, 9r2 - 4 is O-free. Here D i D 2 = 9r2 - 4. By Corollary 1 5 = 1 . Therefore k = 1 and T0U0 = 1 by Lemma 2. Therefore T 0 = U0 = 1 1 - (9r2 - 4) = -4 r = 1 and therefore 9 r 2 - 4 = 5 = Di . Case 2: r is even. Here 9r2 - 4 = 4 2 Z»iD 2 . By Corollary 1 <5 = 4. Therefore k = 2 and T 0 = U0 = 2 by Lemma 2, yielding 4 - 4£>iD 2 = -4 = Dx = 2. Hence 9r 2 - 4 = 32 => r = 2. It deserves mention that the only other case found in the data of Tables 6.2 and 6.3 for which x2 — y2D\D2 = -4 is soluble is when D1D2 = Dx = 26. This occurs when r = 34. Here 9r2 - 4 = 10400 = 2452(2 x 13). As ro = 17 is prime, and D\D2 / 2 (and n = 1), a = 1 by Theorem 4 (and 5). Therefore (3r, 6Yli=iPi) = (*o,wo) = (102,20) and by applying 4.83 and 4.84 we have (T0,U0) = (10,2). We conclude this chapter with the following summarizing theorem. Theorem 9 // D2 ^ 1 and if a = 1 then E(9r2 — 4) = E(DiD2) — A i , iAe largest eigenvalue of 7FTj{to,uo) where (t0,uo) = (3r, 5rj"=1Pi), is the Fundamental Unit of Q(y/9r2 — 4) = Q(y/DxD2). Proof. If D2 ^ 1 then by Theorem 6 and 7, equation 3.23 is insoluble. Therefore any unit of Q(y/DiD2) has norm +1. Now if cr = 1 then by equation 3.34, £'(9T-2 - 4) = E(DXD2). Recall that in the Markoff case 6D\D2 = Do, and <52 | 9r2 — 4. Now by equations 3.29, 3.30 and the (immediate) following remark and Theorem 9, (c) of Chapter 3 e = E(9r2 - 4) = E(DlD2) = t+M^y^lE2 is the fundamental unit of Q(y/DXD2). Chapter 5 Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces For the purposes of this chapter H+ will denote the upper half plane with its standard hyperbolic geometry. This will allow PSLi2(R) to act as isometries on H+. The Modular Group, SL,2(Z), will be denoted by T. Unless otherwise mentioned, a form / , or g, will be understood to be a primitive indefinite binary quadratic form of non-square discriminant. Requiring that d(f) ^ • will allow reference to the real and distinct roots wi,u>2 = ^ 2a^~ ' r e s P e c t i v e l v ! °f f(xi !)• Definition: PSL2(C)* is defined to be the group of all conformal homeomorphisms of the complex sphere. Each element of PSL2(C) is a fractional linear transformation , az + b x' - Tx = cz + d where a, b, c, d are complex numbers with ad — be = 1. Definition: Let G be a subgroup of PSL2(C). The point a is called a limit point of G provided there is a point z and an infinite sequence {Vn} of different elements of G such that Vnz => a. If a is not a limit point, it is an ordinary point of G. Remark: A point a which is a fixed point of infinitely many different V G G is a limit point of G; i.e. the images {Vnz} need not be distinct. Definition: G is said to be discontinuous at a point c t if a is an ordinary point of G. It is said to be discontinuous in a set S if it is discontinuous at each point of S. A group is called a discontinuous group if it is discontinuous somewhere. That is, G has at least one ordinary point. Remark: Discontinuity of G at a implies that the orbit Gz does not accumulate at a no matter what point of the complex sphere z may be. Denote the set of limit points of G by C(G) and by 0(G) the set of ordinary points. By definition £ and O are complements. 36 Chapter 5. Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces 37 Proposition 1 Let F C G, then a) For each V £ G, V{C(G)) = £(G) and V(0(G)) = 0(G). b) 0(G) C 0(F). Corollary 1 A subgroup of a discontinuous group is discontinuous. Definition: A subgroup G C PSL2(C) is said to be discrete if and only if each element of G is an isolated point of G. Proposition 2 A discontinuous group is discrete. Proposition 3 The Modular Group, T, is a discontinuous group. Proofs of the above theorems can be found in Lehner. Definition: Let G be a discrete subgroup of PSL2(R). An element r\ £ G is called hyperbolic if when viewed as a linear fractional transformation its fixed points are real and distinct. 5.1 Closed Geodesies on H+JG In the following definitions, 77 will be understood to be a hyperbolic element of a discrete subgroup G Definition: If 77 is never a non-trivial power of another element of G then 77 is said to be primitive in Proposition 4 Let v be a hyperbolic element of the discrete subgroup G C PSL2(R)- Then there exists an element a £ PSL2(R) such that conjugation with a transforms 77 into the following form where t > 1. Definition: The number t2 is called the norm of 77, denoted ^ ( 7 7 ) . Definition: Let u>i and u)2 be the fixed (real) points of 77. We define its axis A„ to be the (half) circle with center, (uii +ui2)/2, on R and connecting u>i and ui2-of PSL2(R). G. (5.95) Chapter 5. Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces 38 Remark: Av can be viewed as a geodesic of H+, where H+ is equipped with its standard hyperbolic geometry. Definition: Let ir > H+/G be defined to be the projection map from H+ to the quotient surface H+/G. Proposition 5 Let n £ G be primitive and hyperbolic with distinct real fixed points u)x and UJ2. Then r\ will take the geodesic Av back to itself. Furthermore An will project to a closed geodesic ir(Av) of length logN(rj) on the quotient surface H+/G. AU conjugates of rj in G gives rise to the same closed geodesies on H+jG. Moreover, all closed geodesies are accounted for in this way. Proof. See [Sarnak 82, Huber 59] Proposition 6 Let f be a primitive indefinite binary quadratic form with distinct roots u>i and u>2. Then a) U l , u>2 £ £{AUT+{f)). b) 7y(to,Uo) is a primitive hyberbolic transformation of T. c) 3 a £ PSL2(R) such that ( E{d(f)) 0 \ c r ^ t o . u o ) * = (5.96) V 0 E{d(f))~1 ( t0 + Uoy/dff) 2 u ^ 0 t 0 - u0ydTf) (5.97) d) N(Jf(t0,uo)) = E(d(f))> = t 2 + Corollary 2 3a £ PSL2(R) such that a xjf[tn,un)a - (5.98) I 0 {E(d(f))-n ( tn+Uny/d{f) 2 " V o - — ^ — 0 Q tn - Unyjd(f) (5.99) Here N(V(tn,un)) = *(«*(/))*» = (h^/M^ = ^ n + ^ v ^ ( r e f e r t o 4 . 7 1 ) . Chapter 5. Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces 39 5.2 Closed Geodesies on H+/T The following Proposition reveals that the primitive hyperbolic elements of T are exactly the fundamental automorphs of indefinite primitive binary quadratic forms. Proposition 7 Define the map <p by <p(f) = 7/(to, UQ); then a) <j> is a one-to-one map of primitive quadratic forms onto primitive hyperbolic elements of Y. b) cj) commutes with the action of PSL2(Z) so that f = g iff Jf{to,v>o) is conjugate to 7 9(t 0,u 0). Proof See [Sarnak 82]. Remark: Note that here conjugacy classes correspond to classes of forms. Definition: Denote by T> the set of positive ring discriminants. That is V = {D > 0 : D = 0, 1 (4), D ^ Corollary 3 The norms of the conjugacy classes of primitive hyperbolic transformations ofY are E(D)2 where D £"D, with multiplicity K(D). Or, put another way, the lengths of the closed geodesies on H+ /T are the numbers 2\og E(D) with multiplicity K(D). Proof. See [Sarnak 82]. See Chapter 3 for definition of K(D). Remark: H+/T is of genus zero and topologically equivalent to the one punctured sphere. We can now apply the above results to jFryj(io^o). Theorem 1 For each Markoff Number r, there exist closed geodesies on H+ /T of length 2 log ^3r + %/9r2 - 4^ of multiplicity K(§r2 - 4), K(9r^ - 1) if r is odd or r = 2r0 , respectively. Remark: Note that knowledge of cr is required when determining the above multiplicity. See Chapter 4, Proposition 2. 5.3 Generators of V Definition: We define the Commutator Subgroup T' of the Modular Group T to be the set of all finite products of commutators ABA~1B~1 with A,B £ f. Chapter 5. Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces 40 Theorem 2 T' is a free group of rank 2 generated freely by 2 1 ] and I 1 1 ] (5.100) 1 1 / \ 1 2 / The genus of T' is 1. Proof. [Lehner 64, page 363] . Remark: H+ /Y' is a Riemann Surface topologically equivalent to a punctured torus. Definition: A matrix A £ V is called a generator of V if there is another matrix B £ V which together with A generates V. Definition: Fricke's Equation is defined to be the following Diophantine Equation: x2 + y2 + z2 = xyz (5.101) The following proposition is due to Fricke. Proposition 8 If A and B generate Y' then (Tr(A))2 + (TR(B))2 + (Tr(AB))2 = Tr(A)Tr(B)Tr(AB) (5.102) ( a l , l al,2 \ we define the form f^ by 0-2,1 0,2,2 J /A(33,y) = a2,ix2 + (a 2, 2 - altl)xy - alj2y2- (5.103) Proposition 9 Let f = (a,b,c) be an integral form of non-square discriminant. If UQ = 1 then f(*,y) = flf(t0,u0){x,y)-Proof. Easily if U Q = 1 then 7 , ( * o , l ) = ( ^ t^bj- ^ Theorem 3 p.(f) £ MSpec / is either equivalent to or derived from a form where A is a generator of Y1. Proof. See [Cohn 55, Haas 85]. Theorem 4 For each Markoff Number r, 7F r j ( i o > u o ) is a generator of Y'. Chapter 5. Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces 41 Proof. For purposes of simplification we will denote jFrj (to, UQ) by A. By Proposition 15 and Theorem 7 (b) of Chapter 3, p,(Frj) £ MSpec. By discussion of equation 3.62 of Chapter 3, UQ = 1 wrt. Qr2 — 4 = d(FTj). Thus by Proposition 9 we have Hence by Theorem 4, A is a generator of T'. Let (p,q,r) be a Markoff Triple and denote JFP j. (3p, 1), jFq tj (3c, 1), 7F,,J (Zr, 1) by Apj, AGJ-, and ATj respectively. Remark: (3p, Zq, Zr) = (Tr(Apj),Tr(Aq,j),Tr(ATtj)) is a solution to equation 5.101. Conjecture A p.j and ( A g j ' ) _ 1 generate V with AP J(Agj)" 1 = Artj•. 5.4 Simple and Closed Geodesies on H+ /V Define the geodesic to be the (half) circle with center, (£ +()/2, on R and endpoints £ and (. When referring to a form g with roots £ and £, will be denoted by Ag. Note that Ag is the fixed axis of any element of AUT+(g). We define the height of by Remark: H(A^.c) is constant on the T' orbit of the geodesic ^4^ and is related to the minimum of a form by j(x,y) = FrJx(x,y) (5.105) We define the quantity H(A^^) by H(AU) = sup{ht(B(A u)) | B £ T'} Lemma 1 H(Ag) = 2 |^g) ^ ^ *5 derived from a Markoff Form. Proof. See [Haas 85, Cohn 71]. See Chapter 3, for a definition of fi. Corollary 4 If g = K.FrJ then H(Ag) = H(AFrij). Definition: Ag is called a Markoff Geodesic if g is derived from a Markoff Form. Theorem 5 Ag is a Markoff Geodesic Chapter 5. Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces 42 Proof. See [Haas 85]. Definition: Let TT • H+/V be defined to be the projection map from H+ to the Riemann Surface H+/V. Definition: A simple geodesic is one that does not intersect itself. Theorem 6 •K(A^(%) is a simple closed geodesic on the Riemann Surface H+ /V <=>• is the fixed axis of a generator of V. Proof. See [Haas 85, Nielsen 18]. The above theorem in combination with Theorem 4 yields Theorem 7 A geodesic on H+/T' is simple and closed iff it is the projection of a Markoff Geodesic in H+. Theorem 8 For each Markoff Number r, ir(AFRJ) is a simple and closed geodesic on the Riemann Surface H+/V of length Proof. By Theorem 4 jpr . (to, Uo) is a generator of T'. By Lemma 1 and Theorem 6 and 7 definition Aprj is a Markoff Geodesic. Hence by Theorem and Aprj projects to a simple and closed geodesic on H+/V. By Proposition 6 (b), it follows that 7F r j(io; uo) is also a primitive hyperbolic transformation of T' C T. As r' is discrete, by Corollary 1 and Propositions 2 and 3, Proposition 5 together with Proposition 6 (d) yield the above lengths. Remark: For an account of the behavior of simple and closed geodesies on H+/V see [Haas 85, Haas 86]. 5.5 Simple and Closed Geodesies on H+/T(3) Definition: We define the principal congruence subgroup of level n in the Modular Group, T = T(l) by r(n)=|f a 6J=±f 1 ° j {mod n);a,b,c,d £Z,ad-bc= 1 I (5.107) Chapter 5. Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces 43 Proposition 10 Let p > 3 be a prime. ±jf(to,uo) £ T(p) iff p \u0. Proof. See [Sarnak 82]. Corollary 5 Let f be a primitive indefinite binary quadratic form with distinct roots LJI and w2- If n is the smallest power of E(d(f)) such that p \ un then ±jf(tn,un) is the T(p) primitive hyperbolic matrix fixing u)i and u)2. Theorem 9 jFr j (t2, u2) is the T(3) primitive hyperbolic matrix fixing 6rj and 8'Tj, the roots of Frd{x,l). Proof. First note that jpTj(to,uo) £ T(3) as u0 = 1. As h + U2^2—A (3r +V9r 2 - 4 ^ 2 • = [ 2 J ( 5 - 1 0 8 ) we have t2 — 9r2 — 2 and u2 = 3r. Hence n = 2 and 9r2-2-3r(3r-2(rj+k)) _ 3 r { l + rj2 + 2 k j _ 3,3 - 3k) 7 ^(9r 2 - 2 ,3 r )= l ^ 9, 2 - 2 + 3,(3r - 2( r j + fc)) I ^ 2 r 2   r  is a primitive hyperbolic element of T(3). It needs to be remarked that although FTj is not primitive for r = 2r$, 7±F2r 3 (Sr2 ~ 2,6r 7F2ToJ(9r2 -2,3r) as by Corollary 5, Chapter 3, AUT+ ( § F r j - ) = AUT+(FrJ). To simplify notation, denote 7F R I (9r2 — 2, 3r) by y(FTj). Theorem 10 A1^pr .) projects to a closed geodesic Tr(Ay(Fr.)) of length „, / 9r 2 - 2 + 3TV9T-2 - 4 2 log ^ -on the Riemann Surface H+/T(3). Proof. By Corollary 2, N(j(FrJ)) = ^ 2 ~ 2 + 3ry/9r2 - 4 ^ A s r ^ i g d i s c r e t e ] b y Corollary 1, and Propositions 2 and 3, and f(FT j) is a primitive hyperbolic element of T(3) by Theorem 9, we now have the desired result by Proposition 5. Remark: H+/T(3) is of genus zero and topologically equivalent to a four punctured sphere. Chapter 5. Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces 44 Theorem 11 Let rj £ T(3) be hyperbolic with real fixed points u>i and u)2 • Then TT(A„) is simple iff = v(u2) > \. Proof. See [M.S and J.L 84]. See Chapter 3, Theorem 8, for a definition of v. Corollary 6 Let Njj+ir{3)(T) be the number of simple closed geodesies on H+ /T(3) of hyperbolic length < T. Then T < N H + / r ( 3 ) (T) < T2. Proof. See [M.S and J.L 84]. Theorem 12 For each Markoff Number r, ?r(Ay(F •)) is a simple and closed geodesic on the Riemann Surface H+ /T(3) of length 21og ^ 2 - 2 + 3 r V 5 ^ 4 ^ ( 5 n o ) Proof. By Theorem 8, of Chapter 3, v(6T) = u(9'T) > | . Now by Theorems 10 and 11 we have the desired result. Chapter 6 A Bold Conjecture Conjecture Let r be a Markoff Number not equal to 1, 2, or 34. The largest eigenvalue Ai = — j *^ —- of a generator ~fpr . (3r, 1) of the Commutator Subgroup V' C T is the Funda-mental Unit of the real Quadratic Field Q(\/9r2 — 4) = Q(\/DiD2). All simple and closed geodesies on H+/V and H+/T(3) have lengths 21ogAi and 41ogAi respectively. 21ogAi is the length of a closed geodesic on H+/T. Remark For r — 1, 2, and 34, the above conjecture holds except that A x is the square of the fundamen-tal unit of the real quadratic fields Q(y/E), Q{V32) = Q(y/2), and Q(v/10400) = <?(-\/26) respectively. The following table includes all Markoff D\D2 < 1,000,000. H(DiD2) and e denote the class number and fundamental unit of Q(^/L\D2). The norm, N(e) = ^1, indicates whether equation 3.23 is soluble or not, respectively. E(DiD2) will denote the first unit of Norm +1. As the column <52 [7_"=1 is concerned with non-trivial squares, if 6 = 1 and n = 0 a column entry will be 0. Note this occurs for all 6 = 1 wrt. DiD2 < 106. Data for the 5th column was found in [Mollin and Williams 89]. r 97-2 - 4 » 2ll?=iP? DXD2 N(e)H{D1D2) E{DXD2) 1 5 0 5 -1 (3 + V5)/2 2 32 42 2 -1 3 + 2V2 5 221 0 221 2 (15 + y/22\)/2 13 1517 0 1517 2 (39 + 71517)72 34 10400 4 25 2 26 -2 51 + 10^26 89 71285 0 71285 16 (267 + ^71285)72 169 257045 0 257045 16 (507 + V257045)/2 194 338720 42 21170 16 291 + 2^ /21170 233 488597 0 488597 24 (699 + V488597) /2 610 3348896 42 209306 64 915 + 2v/209306 6466 376282400 4 25 2 940706 112 9699 + 10^940706 Table 6.1: Class Numbers and Fundamental Units of Norm +1 for Q{^/L\D~2) where DXD2 < 106 45 Chapter 6. A Bold Conjecture 46 D2 = 1 for r = 1, 2, and 34. Except for these three cases, for all other Markoff Numbers in Tables 6.2 and 6.3, the data reveals that E(DiD2) = e as D2 ^ 1 and n < 1 (see Theorem 5 and 9 of Chapter 4). It is intriguing that in all cases, if n — 1 then p = 5. All data satsfies the Conjecture. It is to be noted that Table 6.3 does not include all r, 195025 < r < 65995009186. r = Markoff Number Factorization of r 3r - 2 3r + 2 1 1 1 5 2 2 2, 2 2, 2, 2 5 5 13 17 13 13 37 41 29 29 5, 17 89 34 2, 17 2j 2^  5) 5 2, 2, 2, 13 89 89 5, 53 269 169 13, 13 5, 101 509 194 2, 97 2) 2j 5j 29 2, 2, 2, 73 233 233 17, 41 701 433 433 1297 1301 610 2, 5, 61 2, 2, 457 2j 2^  2^  229 985 5, 197 2953 2957 1325 5, 5, 53 29, 137 41, 97 1597 1597 4789 4793 2897 2897 8689 8693 4181 37, 113 12541 5, 13, 193 5741 5741 17, 1013 5, 5, 13 53 6466 2, 53, 61 2, 2, 13, 373 2, 2, 2, 5, 5, 97 7561 7561 37, 613 5, 13, 349 9077 29, 313 73, 373 113, 241 10946 2, 13, 421 2, 2, 8209 2, 2, 2, 5, 821 14701 61, 241 44101 5, 8821 28657 28657 13, 17, 389 149, 577 33461 33461 37, 2713 5, 17, 1181 37666 2, 37, 509 2, 2, 13, 41, 53 2, 2, 2, 5, 5, 5, 113 43261 43261 233, 557 5, 101, 257 51641 113, 457 13, 17, 701 5, 5, 6197 62210 2, 5, 6221 2, 2, 13, 37, 97 2, 2, 2, 41, 569 75025 5, 5, 3001 173, 1301 225077 96557 96557 289669 37, 7829 135137 337, 401 37, 10957 405413 Table 6.2: Factorization of a Markoff Number r and Discriminant 9r2 — 4 Chapter 6. A Bold Conjecture 47 r — Markoff Number Factorization of r 3r - 2 3r + 2 195025 5, 5, 29, 269 585073 585077 196418 2, 17, 53, 109 2, 2, 41, 3593 2, 2, 2, 73, 1009 294685 5, 58937 101, 8753 884057 426389 426389 5, 17, 101, 149 137, 9337 499393 73, 6841 569, 2633 41, 36541 646018 2, 323009 2, 2, 661, 733 2, 2, 2, 242257 925765 5, 185153 2777293 809, 3433 1278818 2, 193, 3313 2, 2, 41, 149, 157 2, 2, 2, 13, 37, 997 1441889 17, 89, 953 5, 109, 7937 29, 149161 1686049 1686049 5, 109, 9281 1109, 4561 2423525 5, 5, 13, 7457 7270573 17, 427681 2922509 2922509 5, 5, 13, 53, 509 17, 515737 3276509 3276509 5, 5, 393181 269, 36541 4400489 29, 41, 3701 5, 317, 8329 17, 776557 7453378 2, 17, 219217 2, 2, 149, 37517 2, 2, 2, 37, 75541 8399329 397, 21157 . 5, 41, 101, 1217 3037, 8297 11485154 2, 5742577 2, 2, 5, 13, 89, 1489 2, 2, 2, 17, 253349 16609837 29, 601, 953 2269, 21961 49829513 16964653 3001, 5653 113, 233, 1933 50893961 20031170 2, 5, 29, 69073 2, 2, 1733, 8669 2, 2, 2, 89, 84401 21531778 2, 10765889 2, 2, 977, 16529 2, 2, 2, 13, 41, 15149 43484701 13, 3344977 130454101 5, 2909, 8969 48928105 5, 9785621 13, 11291101 146784317 137295677 7901, 17377 29, 313, 45377 17, 24228649 205272962 2, 29, 29, 122041 2, 2, 1997, 77093 2, 2, 2, 937, 82153 253191266 2, 126595633 2, 2, 113, 433, 3881 2, 2, 2, 5, 5, 29, 173, 757 375981346 2, 13, 29, 37, 13477 2, 2, 4073, 69233 2, 2, 2, 5, 28198601 576298801 13, 157, 373, 757 1728896401 1728896405 647072098 2,137, 2361577 2, 2, 73, 6648001 2, 2, 2, 557, 435641 941038565 5, 11597, 16229 17, 29, 241, 23761 2823115697 1475706146 2, 737853073 2, 2, 13, 353, 433, 557 2, 2, 2, 5, 173, 639757 1873012681 1433, 1307057 37, 8353, 18181 5, 1123807609 6449974274 2, 29, 317, 350809 2, 2, 5, 6221, 155521 2, 2, 2, 2418740353 6684339842 2, 3342169921 2, 2, 5013254881 2, 2, 2, 17, 29, 5084437 65995009186 2, 32997504593 2, 2, 1093, 5437, 8329 2, 2, 2, 5, 4949625689 Table 6.3: Factorization of a Markoff Number r and Discriminant 9r2 — 4 Bibliography [Borosh 75] [Cassels 57] [Cassels 78] [Cohn 55] [Cohn 71] [Cohn 80] [Cusick 89] [Dickson 52] [Flath 89] [Haas 85] [Haas 86] [Huber 59] [Lagarias 80] [Landau 58] [Lehner 64] [Le Maohua 89] [Markoff 1879] Borosh, I. [1975], More Numerical Evidence on the Uniqueness of Markov Numbers. BIT 15 (1975), 351 - 357. Cassels, J.W.S. [1957], An Introduction to Diophantine Approximation, Chap-ters I, II. Cambridge Univ. Press, 1957. Cassels, J.W.S. [1978], Rational Quadratic Forms. Academic Press Inc. (Lon-don) Ltd. 1978. Cohn, Harvey [1955], Approach to Markoff's Minimal Forms through Modular Functions, Ann. of Math. (2) 61 (1955), 1 - 12. Cohn, Harvey [1971], Representation of Markoff's Binary Quadratic Forms by Geodesies on a Perforated Torus, Acta Arith. 18 (1971), 125 - 136. Cohn, Harvey [1980], Advanced Number Theory, Dover Pub. Inc. N.Y. (1980). Cusick, Thomas W., Flahive, Mary E. [1989], The Markoff and Lagrange Spec-tra, Amer. Math. Soc, Math. Surveys and Monographs 30 (1989). Dickson, Leonard Eugene [1952], History of the Theory of Numbers, Volume III, Chelsea, N.Y. (1952) . Flath, Daniel E. [1989], Introduction to Number Theory, John Wiley and Sons, Inc. (1989). Haas, Andrew [1985], The Geometry of Markoff Forms, (1985). Haas, Andrew [1986], Diophantine Approximation on Hyperbolic Riemann Surfaces, Acta Math. 156 (1986), 33 - 82. Huber, H. [1959], Zur analytischen Theorie hyperbolischen Raumformen und Bewequngruppen, Math. Ann. 138 (1959), 1 - 26. Lagarias, J.C. [1980], On the Computational Complexity of Determining the Solvability or Unsolvabilty of the Equation x2 — Dy2 = —I, Trans. Amer. Math. Soc. 260,2, (1980), 485 - 508. Landau, Edmund [1958], Elementary Number Theory, Chelsea, N.Y. (1958). Lehner, Joseph [1964], Discontinuous Groups and Automorphic Functions, Math. Surveys, 8, Amer. Math. Soc, Providence, R.I. (1964). Le Maohua [September 1989], A Note on the Diophantine Equation x2p — Dy2 = 1, Proc. Amer. Math. Soc. Volume 107 1 (September 1989), 27 - 34. Markoff, A.A. [1879], Sur les formes quadratiques binaires indefinies, Math. Ann. 17 (1879), 381 - 406. 48 Bibliography 49 [Mathews 70] Mathews, G.B. [1970] Theory of Numbers, (second edition), Chelsea, N.Y. (1970). [Mollin and Williams 89] Mollin, Richard A., Williams, H.C. [1989] Computation of the Class Number of a Real Quadratic Field, (1989). [Mollin and Williams 89] Mollin, Richard A., Williams, H.C. [1989] Class Numbers of Real Quadratic Fields to One Million. (1989). [Nielsen 18] Nielsen, J. [1918] Die isomorphismen der allgemeinen unendlichen gruppe mit zwei erzeugenden, Math. Ann. 78 (1918), 385 - 397. [Rosenberger 76] Rosenberger, Gerhard [1976], The Uniqueness of the Markoff Numbers, Math. Comp 30, (1976), 361 - 365. [Sarnak 82] Sarnak, Peter [1982] Class Numbers of Indefinite Binary Quadratic Forms, J. Number Theory 15, (1982), 229 - 247. [Schmidt 76] Schmidt, Asmus L. [1976] Minimum of Quadratic Forms with respect to Fuch-sian Groups I.J. Reine Angew. Math. 286, (1976), 341 - 368. [Series 85] Series, Caroline [1985] The Geometry of Markoff Numbers, Math. Intelligencer, Volume VII, 3. (1985). [M.S and J.L 84] Sheingorn, M., Lehner, J. [1984], Simple Closed Geodesies on #+/T(3) Arise from the Markoff Spectrum, Bull. Amer. Math. Soc, Volume II, 2, (1984). [Sheingorn 85] Sheingorn, M. [1985], Characterization of Simple Closed Geodesies on Fricke Surfaces, Duke Math. Journal 52, (1985), 535 - 545. [Zagier 82] Zagier, Don [1982], On the Number of Markoff Numbers Below a Given Bound, Math, of Comp. Volume 39, 160, (1982), 709 - 723. 

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