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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  degree at the  thesis  in  University of  partial  fulfilment  of  of  department  by  his  or  her  representatives.  for an advanced  Library shall make it  agree that permission for extensive  this thesis for scholarly purposes may be or  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  It  is  granted  by the  understood  that  head of copying  my 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  4  5  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  Hunting cr  26  4.1  Further Investigation of cr  30  4.2  Solubility of x - y D D 2  2  x  2  = -4  31  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  5.5  Simple and Closed Geodesies on H+  +  onJf+/r'  41 . .  iii  42  6  A Bold Conjecture  Bibliography  List of Tables  6.1  Class Numbers and Fundamental Units of Norm +1 for Q(y/DiD )  6.2  Factorization of a Markoff Number r and Discriminant 9r — 4  46  6.3  Factorization of a Markoff Number r and Discriminant 9r — 4  47  2  2  2  v  where D i D  2  < 10  6  . 45  List of Figures  2.1  The Markoff Tree for all triples (p,q,r) with max(p,g) < 10  3.2  The Markoff Form Tree  5  8 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 17  th  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) n(f)  = inf | f(x,y)  = m(f)/y/d(f)  | where x, y 6 Z /{0,0} and d(f) to be the discriminant of / , let 2  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.  2  Introduction  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 SL (Z) 2  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 SL (Z) 2  = 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  3  Chapter 1. Introduction  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 , 7 7 x 3 ) of positive integers satifying the Diophantine 2  Equation: ml + m , + m\ = 3 m i m m  (2-1)  2  2  Definition:  3  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 9r — 4 is called a Markoff Discriminant (so named because it arises as the 2  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 , m , m 3 ) is a non-exceptional Markoff Triple, then the three triples (m , m , 3 m m 3 — 1  ( ii m  m  2  2  3 i 3mim — m ), and (mi, m , 3mim — 7713) 3  2  2  2  3  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  2  Chapter 2. Markoff Numbers  Uniqueness Conjecture  5  For any Markoff Number r, 3 exactly one and only one Markoff Triple of  height r. Remark: The Uniqueness Conjecture holds Vr < l O  105  . 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...)log z. 2  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  Remark:  6  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 x +y 2  = 0{Q ) then either Q = 2orQ = l(A) where a = 1, 2, 3,...  2  a  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 x + y = 3xyr - r = 0(Q) 2  2  2  (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 9r — 4 has no prime factor Q = 3(4). 2  Proof. Let (x,y, r) be a Markoff Triple and let Q be an odd prime dividing the Markoff Discriminant 9r - 4. As Q ^ 2, exclusively, either 3r - 2 = 0 (Q) or Zr + 2 = 0(Q). So 3r = ±2(Q). 2  x + y + r = Zxyr 2  2  =>  2  (x y) T  2  Therefore  =  ±2xy (Q)  (2.3)  =  -r (Q)  (2.4)  2  Hence —1 is a quadratic residue modulo Q thereby forcing Q = 1(4). Therefore Q / 9 r - 4 if Q =3(4). 2  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).  7  Chapter 2. Markoff Numbers  Therefore x +y +r 2  2  = 2 = Zxyr = 3r (4)  2  Hence 2117- if r is even.  Theorem A If r is even then 2 \\9r —4. 5  Proof.  2  Notate r = 2r . Since 2 \\r, by Theorem 1 r = 1(4). Therefore either ro = 1(8) or ro = 5(8). 0  In either case r = 2 (8);  0  i.e. r j£ 6 (8).  Therefore 3r - 2 = 4(8)  => 2 ||3r - 2 2  (2.5)  In light of Theorem 2, we can now write  ^>  3r-2  =  4(4A + 1)  (2.6)  3r- + 2  =  16/fc + 8 =>2 ||3r + 2 3  Hence 2 ||9r - 4 . 5  2  Corollary 2 If r is even r = 2(16) whence r = 1(8). 0  Proof. Either r is congruent to 2 or 10 modulo 16. If r is congruent to 10, then 3r + 2 = 0(16).  (2.7)  Chapter 2. Markoff Numbers (1,75025,1964)8) (1.28657,75025) ^(28657, 75025, 6449974274) (1,10946,28657) ^^(10946,20657,941038565) (1 ,4181 , 10940) (4161,10946, 137295677) (1,1597,4181) (1597,4181,20031170) (1,610,1597) (610,1597,2922509) (1,233,610)  (233,610,4263e9) ( 1 ,89, 233)-  /  ^"(89,62 210, 1660983 7) •(89, 233,02210)  (1,34,89)  (34,9077,925765)  M34,89,9071 (1,13,34)  "(233,02210,43484701)  ^ - ( 8 9 , 9077 , 2423525) ^—(13,51641,2012674) _(13,1325,51641) ' -(1 3 25,51641,205272962) (13,34,1325) (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) .(2,33461,19=025) -(2,5741,33461) (2,985,5741)  (1,2,5)  • (574 ) , 3346) ,57fc?98b':l :  (985, 5741, 16964653) (2,169,985) (169,985,499393) (2,29,169) _ 129,169,14701)  (29.14701,1 278818)  't169,14701,7453378) (1,1,2)  ^—(5,96557,1441889) ^^(5,6466,96557) — (5,433,6466) * < 6466 , 96557 ,} 87 3( 12*31) (5,29,433) (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) = ax + bxy+cy 2  (3.8)  2  with real coefficients is called indefinite if its discriminant d(f) = b — 4ac > 0. 2  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) = and  a6-/3y  =  f{x,y)  (3.9)  ±1  (3.10)  identically in x and y.  I If g is transformed into / by the unitary substitution  a  0 \ 6 GSL2(Z) of determinant e, then if 6 J  V7  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 GSL (Z) equivalence. / = g will denote GSL2{Z) 2  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'a + b'aj + c '  b =  2a'al3 + b'(a6+f3j) + 2c'y6  c  a'p + b'/36 + c'6  =  2  2  9  (3.11)  2 7  2  (3.12) (3.13)  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, K \ d(f) and the discriminant of the primitive form from which it is 2  derived is d(/)//c . 2  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 , w  2  and  ui' be the roots of 2  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 +p/ w +6, )1  1  1  1) = 0  (3.15)  yielding uii = auii +P/ywi + 6 = LJ[ or u' . Distinguishing between proper and improper equivalence 2  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) — ax + bxy + cy m 2  = 2  (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 K \ m. Furthermore if x = 2  KX'  and y = ny' then f(x',y') is a primitive  representation of m//c . 2  When d(f) > 0 / can represent both positive and negative values thus inspiring the term indefinite.  11  Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms  Definition: Remark:  Define m(f) = inf | f(x,y) | where x, y £ Z /{0,0}. 2  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(/).  = -^i=L.  Definition: Define /j,(f)  V (f)  .  d  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  = ~  b +  W  l  ^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) = b — Aac. 2  f is reduced iff the two following equivalent conditions hold: |  W l  |<l,  |w |>l, a  ^i^2<0  (3.17)  0 < Vb - Aac - b < 2 | a | < sjb - Aac + b 2  Note that 3.17  2  (3.18)  & > 0 and 3.18. Furthermore since b < y/d(f), b — d(f) — ac is negative whence a 2  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 / , R f is reduced for some positive n  n. This means that the sequence {R f}  is eventually periodic, and he shows that the periodic part  n  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 = K d. By above 2  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 — Q DQ where DQ = 0 2  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)  = 4P  P  =  -1(4)  (3.20)  Do = 8P  P  =  ±1(4)  (3.21)  D  0  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 Q Do, for any Q £ Z. 2  Chapter 3.  Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms  Proof. If D = Q Do 2  xy — ^ T ^ y Corollary  2  = 0(4) then x — ^y 2  13  is an integral form of discriminant D. Otherwise x +  2  2  is an integral form having discriminant D when D — Q DQ = 1(4). 2  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 x -y D 2  = k  2  (3.22)  for k = ± 1 or ± 4 is called Pells Equation. The recondite problem of characterizing those D for which the Pell equation x - Dy = - 1 , or - 4 2  2  (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= [a oi,...,o ] /  0l  JV  (3.24)  Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms14  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) x — Dy = —lis soluble inintegers. 2  2  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  x -y D 2  = +4  2  (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 (t ,u ), is called the Fundamental Solution. 0  0  Theorem 2 3 an infinite number of solutions to 3.25. All solutions are given by t +u 0  2  \  with n > 0 where (t ,u ) is called the n  th  n  n  (3.26)  0  t  )  solution to 3.25.  Theorem 3 Alternatively, if 3.23 is soluble, wrt. k = —4, with fundamental solution (xo,yo) then t + u y/T> _ ^xp + ypVty^ n  where 2n is replaced by 2 if n = 0.  n  2 n  ($21)  Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms  Theorem 4 / / (t ,u ) n  n  is the n  th  solution to 3.25 then (t ,u ) n  x -y D 2  Remark: For D = 1(A) it is possible that t  n  <=>• (\t , \v, ) is the n  th  n  n  = +l  2  n  solution to (3.28)  = u = 1(2) thereby resulting in ^t , ^u  n  15  n  n  is £ Z. Th  case is the only exception to 3.28 not being diophantine. The following two propositions clarify this case. Proposition 6 If t  n  = u  = 1(2) then D = 1 (4) and to = V.Q = 1(2).  n  Proposition 7 3 an infinite number of integral solutions to 3.28. Remark: Even if D = 1(4) for some n, t  = u  n  n  = 0(2).  Definition: Define = *°  E )  +  {D  ! ° ^  (3.29)  When D = DQ we will denote E(Do) =  g  ( - ) 3  3 0  Remark: E(Do) is the smallest unit > 1 of Norm +1 of the real quadratic field Q(\/D) = Q(\J ?) where  r  = 1 4 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 \ U then E(D) = E(D ). 0  3.2  0  The Number of Classes  Definition:  Define K(D) to be the number of primitive classes of discriminant D.  Expressing D = Q DQ 2  we have the following equalities due to Kronecker:  * , =, ( (D  W  { l - ( ^ } ™  n  <»D  where the product applies to all prime factors of Q which do not divide DoRemark:  If E(D) ^ E(D ) then since (t ,u ) is the least positive solution to 0  0  0  x - QDy 2  2  0  2  = +4  (3.32)  Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms  we have that (to, Quo)  16  is a solution to x -y D 2  2  0  = +4  Because (t ,uo) is the least positive solution to 3.32, (to, Quo) 0  (3.33) is the first solution to 3.33 for which  Q I y = Q^oThat is to say, if we denote to = T<r and Quo — U„ then a identifies the least power of E(DQ)  such  that (E(D )r  = E(D)  0  where (T^, J/^)  (3.34)  is the smallest solution to 3.33 for which Q | V„, that is Vn < a, U ^ 0(Q). n  We can now express ^E(DQ)  _  1  log E(D) ~ a where a - 1 <=> E(D) = E(D ) <=> Q \ U . 0  Proposition  8  0  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> X(D)  =  ^-free then  K(D )^n^-^)  (3.38)  0  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 q and q divide Sij then r  r # 5 . Note si  \U7 lf=l  s  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 \ achieved.  Now since each derived class {sij-, rriij} is obtained from a primitive class {l,mij /sij}  we have | D/s j 2  is always  |= K(D/s?j).  Recall that by Proposition 3, there exists a form of discriminant  and by Corollary 1, K(D/s j)  is non-empty.  2  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/sl )  (3.39)  3  i= l  Allowing j to vary we therefore have  n n(j)  '( ) = Y.Y. ( / h)  K  D  K  D  ( - °)  s  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 SL (Z) 2  and AUT (f) +  C AUT(f)  C  GSL {Z). 2  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 Forms18  Proposition 10 AUT (f)  ~ Z/2xZ  +  with generators ±jf (to,u ) where (t ,uo) is the Fundamental 0  0  Solution to 3.25 and t - bu 0  _  0  7,(to,«o)=|  ) auo  (•yf(to,uo)) = Jf(tn,u )  where (t ,u )  n  n  n  is the n  n  / t  (3-43)  ^—  solution to 3.25 and  ~bu  n  "  th  u  \  n  i T ^ J  w  (344)  Proof. See [Flath 89, Cassels 57]. To see that  " 2 ^ " G Z it suffices to show that t  n  and  are of like parity. This condition is easily  satisfied (simply accessed) for if D = 1(4) then b = 1 (2) and t  n  = u = 0,1(2) and if D = 0(4) then n  & = t„=0(2). Proposition 11 Tr(jf(t ,u )) n  n  = i  and jf(t ,u )  n  n  n  has eigenvalues  Ai,„  =  * " + ^ ^  A ,n  =  j  2  (3.45) (  3  °)  When n = 0 we will refer to Ai, A , instead of Ai o,A2,o, respectively. 2  i  Corollary 3 Ai = E(D) whence X  Un  =  (E(D)) . n  Proposition 12 Let K = gcd(f) and f = Kg. Then if (£o,"o) is the Fundamental Solution to x -y d(g) 2  and K I UQ then AUT (f) +  =  2  = +A  (3.47)  AUT (g). +  Proof. By the discussion of equations 3.32 - 3.35, if K \ uo then (to,u /K) is the fundamental solution 0  to *  2  - V d(f) = +4  (3.48)  2  where d(f) = K d(g). 2  Since / = (/ca, Kb, KC) it is an easy check to see that 7 y ( £ o , u o / ) K  =  lg(to. o)-u  Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms  3.4  19  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( )  0<k<r  r  (3.49)  and let I be the integer defined by k + 1 = Ir  (3.50)  2  Note that by reducing the Markoff Equation mod r it is clear that r \ k + 1. 2  Definition: The form F (x,y)  defined by  r  F (x,y) = rx + (3r - 2k)xy + (I - M)y 2  (3.51)  2  r  is called a Markoff Form. Remark: The definition of F is asymmetric in x and y. That is to say if F' is the form obtained by T  r  switching roles of x and y in 3.49 then F EE Fj. For suppose r  yk' = x (r)  with  0 < k' < r  (3.52)  where I' is the integer defined by k' + 1 — I'r  (3.53)  2  Then similarly Fj. is defined by Fj(x,y) = rx + (3r - 2k')xy + (/' - 3k')y 2  Now since x  2  = —y (r), we have k 2  2  (3.54)  2  = k' (r) where k = k' = 0 if r = 1 and where otherwise 2  k + k' = r. Therefore we have Fj = F when k = k' = 0 and Fj = F (x + 2y, -y) V other r ± 1. Fj T  T  and rF are clearly equivalent as ad — be — — 1 — 0 = —1. r  Note that by choosing ko = min{A;,A;'}  0 < 2k < r. Upon setting lo = m i n { / , a n d notating 0  F ,o — (r,3r — 2ko,lo — 3&o), then clearly F o is one of the above defined F or Fj. Heeding this T  T:  T  convention, there corresponds a genealogical tree of Markoff Forms F o to the Markoff Tree of triples. r<  As each F o is uniquely constructed from a respective triple (x,y,r) T}  of height r, the map ip :  (x,y,r) —* F o is a bijective map from the Markoff Tree to the Markoff Form Tree. Tt  It deserves  Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms20  mention that the map <p : r —-» F o is not bijective if the Uniqueness Conjecture is false; however, to Tt  Borosh's credit, bijectivity holds W < l O  105  .  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 + 1 = KJ)r  KJ?  where = I  + rj + 2kj 2  and similarly  k'(jf + 1 - l'(j)r where = V + rj + 2k'j. 2  We now obtain the following forms: F (x,y) Td  Fl (x,y) d  Proposition 13 F  Proof. F j,F' r  r  rJ  =  rx + (3r - 2k(j))xy +  - U{j))y  =  rx + (3r - 2(rj + k))xy + (I + rj + 2kj - 3rj - 3k)y  2  2  2  2  2  =  rx + (3r-2k'(j))xy + (l'(j)-3k'(j))y  =  rx + (3r - 2(rj + k'))xy + (I' + rj + 2k'j - 3rj - 3k')y  2  2  2  2  I F = F' i r  2  F^.  T  are transformed into F ,F}. respectively by the proper unitary substitution r  Wig. we will show the former. F (x Tj  + jy,y)  =  r(x + jy) + (3r - 2(rj + k))(x + jy)y + [l + rj + 2kj - 3rj - Zk)y  =  rx + (2rj + 3r — 2rj — 2k)xy  2  2  2  2  + (rj + Zrj - 2rj - 2kj + I + rj + 2kj - 3rj - 3k)y 2  =  2  2  rx + (3?- - 2k)xy + (I - 3k)y 2  2  2  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  F j. r  Theorem 6 Except for the last stipulation on j, the following properties hold for all F j T  a)  d(F .j) = 9r T  Fr.j  2  -4  = ~T ,j r  c)  m(F )  e)  If r is odd e(9r - 4 ) =  f)  If r is even s(§r  g)  and  rJ  = r  {l,r}  2  2  - 4) = {2,r}  < 1 and ui < — 1 if j = 0 2  Proof. See [Cusick 89, pages 20 - 21]. Corollary 4 Expressing 9r — 4 = Q DQ, DO = 1 (4) if r is odd otherwise 2 || Do for even r. 2  3  2  Proof. See Theorems 2 and 4 of Chapter 2 and Proposition 2 of this chapter. Remark: Since K = 2 for F  when r = 2r by (/), we have that  2roJ  0  F  2  T  O  J  is derived from  ^F ,j 2ro  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 F  T  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 F j r  where b = 3r — 2(rj + k).  First note that 3r - 1 < v ^ r - 4 < 3r. 2  Now 0 < Vb — Aac — b < 2 | a | becomes: 2  0 < V$r  2  - 4 - [3T- - 2(T-J + k)\ < 2r  (3.56)  Replacing \/9r — 4 by 3r yields 2  0 < 3r - [3r - 2(rj + k)} < 2r  (3.57)  .  Chapter 3. Theory of Indefinite Binary Quadratic Forms with Special Focus on Markoff Forms22  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  *v  consists of the numbers  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(/)>3b)  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\\ where the inf is taken over all rational q. a)  If v(6) > ^ then 6 is equivalent to a root of F j(x, 1) = 0. r  (3.60)  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 F ,j(x, 1) = 0 then r  "<"> = 7sh%  > \  <- > 3  61  and there are infinitely many solutions of q\\6\\ < v(&). The two roots of F j(x, 1) = 0 are T  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 x - (9r - 4)y = +4 2  2  (3.62)  2  as by Corollary 1, UQ > 1. We therefore have , 7F .(*o,uo)  3  r - ( 3 r - 2 ( r j + fe)) _  =  riJ  ( / +  3  (  r  j  r +  ^  2+  (  3  r  _^ 2  _  ( r j + k))  3 f c )  \ ^- ) 3  63  "2 rj + k  -(l + rj + 2kj - 3rj -3k)  r  3r — (rj + k)  2  (3.64)  For j = 0 7F,(*OIUO) = j  \  Corollary 5 For r = 2r , AUT (\F j)  =  +  0  r  ^  r  3r — k  M  I  (3-65)  AUT (F j). +  T  Proof. (3r, 2) is a solution to x -y {9r 2  where d(\F ) rd  - §r\ - l . T f u = 1 then 0  2  2  - 1 ) =+4  (3.66)  = 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 Forms24  Theorem  The automorph group of a Markoff Form satisfies the following properties.  9  a)  AUT(F j)  b)  ±j .(3r, 1) generate  d)  T  =AUT {F ).  Fr  A2=  +  rJ  AUT(F ). rJ  3 -y4^r4 r  Proof. Only (a) needs clarification. Suppose 3.23 is soluble, wrt. k = — 4 and where D = 9r — 4. 2  Expanding equation 3.27 we have that Discriminant.  <  ^ ^ x<  0  — \ thereby forcing D — 8 which is not a Markoff f  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  y D = +4  x -  2  2  by  (to,uo)  and define the  n  solution  th  by  (t ,u ) n  n  t + u Vb~ _ (t + U VD\ n  0  n  ~  ~  0  \  2  .  )  (4-67)  By expanding 4.67 and gathering appropriate terms we have the following explicit formulas for t and n  If n = 21 then  Y U +l) ,k=0 ^ n  t  /  n 0  -  { 2 k + 1 ) )  u  2  K + L 0  D  (4.68)  K  '  If n = 21 + 1 then  E ^  + l)  (4.69)  t^^ul^D"  In both cases  J2(2 k)t - ^u D n  n  2  2K  (4.70)  K  0  I  n V /  Remark: UQ \ u Vn. If n is odd then to \t . n  n  Proposition 1 If n — jk then  VD  T  Remark:  t + UQVD 0  2  tj + UA VD"  If k is odd tj \ t and to | tk • However unless j is odd to ftj, n  (4.71)  tk }ft and tofan• Regardless n  of parity, Uj | u for any j | n. 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 t where n = kj and k is odd. n  26  Chapter 4. Hunting a  27  Lemma 1 If (u,v) and (u',v') are positive integer solutions of x —y D = + 4 where D is O-free, with 2  u | u', then 3 fixed positive integers Dx,D  2  such thai D\D = D,  2  2  u+ 2  =  bD^l  u' + 2  =  6Div[  u-2  u'/  2  6D v , 2  2  6D v'  -2  2  2  where 5, ui, v , v[, v' are positive integers satisfying 2  2  5= <  1,  2 j/u  2  4I u  4, Proof.  (^D D X  2  t  2  =3(4)) <£> Z>iZ» = 2(4) 2  2||u  6viv = v 6v[v' =v'  = 1(4))  2  (  or =>• £>!D =1(4) 2  )  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\D  = 1,2,3(4) yields S and the above powers of 2 for all three cases. For  2  D = 3(4) it was found that 4 | u always, as in view of 3.28 2 | u is forced if x — 3y = 1 (4) is to hold. 2  2  Remark: Lemma 1 applies in general to any positive •-free D. From here on the convention will be to denote D by D\D where the factors Di and D are determined by to ± 2 respectively. Here to and 2  2  K o are u and v, respectively, as in Lemma 1 above. Accordingly, (to,uo) will denote the fundamental solution to x -y D D 2  2  l  2  (4.72)  = +4.  We can now express to + 2 and to ~ 2 as 6D\v\ and <5D-uf, respectively, where Uo = 6viv . Note that 2  6 | iio | u where (t ,u ) is the n  th  n  n  n  2  solution to 4.72 and (4.73)  For the purposes of this chapter, unless otherwise mentioned, D D X  2  will denote the O-free part of a  Markoff Discriminant. If 9r —4 is •-free then r is necessarily odd, since by Theorem 4 of Chapter 1, 2 ||9r —4 if r is even. As 2  5  2  2/9r —4 if r = 1(4), allowing for odd squares, the convention will be to notate 9r — 4 = n™=iPiE>iD , 2  2  2  if r is odd and to notate 9r — 4 = 4 T/T"_ p Di£'2 if r is even and where 2 || D\D . 2  2  2  1  2  Chapter 4. Hunting o  28  For r = 1 (4) 6 \ UQ \ W* Pi since (3r-, <5n"=iP«) ^  solution to 4.72, thereby forcing 5 = 1.  sa  =l  For r = 2(4) we have 5 = 4 since D D 1  = 2(4).  2  When referring to either parity we will conveniently notate n  9r -A  = 6 Y[p D D .  2  (4.74)  2  2  i  1  2  t=i  This will also allow inclusive reference to (3r, 5rj™ pi) as a solution to 4.72. =1  Note also that in either case, by Corollary 4 of Chapter 3, we have Do = In both cases 9r - 4 = 2 (3) => D D X  2  Theorem 1 / / (to,uo) = the fundamental solution to x —y D\D 2  X  2  2  = +4 then 3 | to.  2  2  Proof. As D D  bD\D .  = 2 (3) (as a • = 1 (3) if 3 / • ) .  2  = 2(3), for any solution (t,u), t - u D D 2  = t - 2u = 1(3).  2  1  2  2  2  Clearly not both 3 | t and 3 | u. If 3 jft and 3 jfu then t - 2u = 1 - 2 x 1 = -1 £ 1 (3). Therefore 2  2  either 3 | t and 3 / u or 3 jft and 3 | u: If 3 \t and 3 / u then t - 2u 2  If 3 jft and 3 | u then t - 2u 2  Now since (3r, <5n™ Pi) i  solution to E  s a  =1  2  =0-2x1=1(3)  2  — y D\D  = 1 - 2 x 0 = 1 (3)  2  — +4, if (3r, <5ll™ Pi) = (io^o) we have 3 ||to  2  2  =1  as by Theorem 1 of Chapter 2, 3 jfr. If (3r, 5f|™ pi) = =1  3/pi  (tpjUcr)  ^  then since UQ \ u  (to.'Uo),  a  VCT, 3 jfuo since by Theorem 2 of Chapter 2  V i and 3/5. Hence 3 | to -  Theorem 2 / / (3r, 5fT™ pi) = (t ,u ) ^ (t ,uo)) ^ =1  CT  a  e  0  fundamental solution to x — y D\D 2  2  then 2 jfo and 3/cr. Proof. Case 1:  Suppose cr = 2k. Then as by 4.71  *r + SYE^piy/Dtt  =  ^t + uo^D^D~ y 2  0  k  ^t +  =  u -jD^D "^  k  k  we obtain  f or  6r  =  =  t^+upxi?, t\ + ulDiD  2  a n d  and  2  ^n^ . 2 t ^ P  =  <5r]" pj =1  =  tu. k  k  2  = +4,  Chapter 4. Hunting cr  29  As by Theorem 1 3 | t or 3 | u . But 3j/6 and 3/p, Vi. Hence 2 / a is forced. k  Remark:  k  For' r = 2ro, one can alternatively prove that  2 fcr by observing that  6r + 4 = t\ +  •u^r>iD +4 = 2*2., thereby implying that 3r + 2 = t . = • . But by equation 2.7 of Chapter 2 2 ||3r + 2 2  3  2  thus annihilating the possibility that 2 | cr. Note, t  + 2 = t\ is always true.  Case 2:  Suppose cr = 3fe . Note that by Case 1, k EE 1(2). Then as  2k  Sr  + ^nLiPiV-DiAj  _  =  /to+  ^  7  5  7  ^  ^  *  (t  _  +  k  u VthlT k  (t + 3tjt^i9il>2 + {U u +  ulD D )\/D D ))/&  2  k  k  2  k  1  2  1  2  we obtain 12r  =  t^ + Zt u D D  (4.75)  4«5n? Pi  =  3*2^+^0^2  (4.76)  =1  k  k  1  2  Now since k is odd, t | t . Therefore as 3 11 by Theorem 1, if 4.75 is to hold, 3 | (tf + 2  0  k  0  U u DiD ). 2  k  k  2  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 ft <=> 6 = 1 and u = v[v' = Y[i=iPi0  Forr = 2r  0  2\\t  0  2  a  <=> <5 = 4,  U o  . = 4v[v' = A\\^ pi, 2  and 2\\D as 2 ||3r- + 2. 3  =1  1  Heeding 5 = 1 or 8 = 4 wrt. the above two cases we now have the following expressions: to+ 2  =  SDxvx  2  and 3r + 2  =  SDxVi'  =  6D v '  2  where vi | vi and to-2  =  6D v 2  2  2  and 3r - 2  where v \ v 2  2  Corollary 2 D = 2(3), D = 1(3) and vf = v' = v\ EE V' = 1(24). 2  x  2  2  2  2  2  Chapter 4. Hunting a  4.1  30  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(9r -4)= 2  3 r  ^  +  r 2  " =Jg(i? £> )  (4.77)  4  1  2  where = ^  E(DiD ) 2  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\D ^ 5 then cr = 1. 2  Proof  Since cr is odd, to = 3 if to ^ 3r. By the relation in Corollary 1 this would force D\D — Di = 5. 2  Theorem 4 for r = 2r , if r is prime and D D 0  0  X  2  ^ 2 tA.en cr = 1.  Proof. 6 11 . Mimic the above arguement with t = 6. This results in D\D = D\ = 2. 0  0  2  Theorem 5 If v\, v' , or n = 1 £A.e7i <r = 1. 2  Proof. Since cr is odd t>i | t;'. If v[ = 1 then trivially Vi = 1 thereby forcing cr = 1. t  If n = 1 then one of the v[ — 1. Corollary 3 If n = 2 and a ^ 1 i/ien Definition:  = v =l *>—' uo = 5 and Z?i = D + 1. 2  2  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)  w = <5 and 0  r(r) = r(<r) <=> t = 3 or 6 => D\D — 5 or 2 depending upon whether r = 1 or 2 (4) respectively. 0  2  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 + U ul(D D ) ] lo  (4.79)  2  0  6  W=iPi  >u  5  = ~[b4u  0  x  2  + lOtlv^DiDs + ug(I>iD ) ] 2  2  (4.80)  Chapter 4. Hunting a  The  31  following proposition is a Markoff Setting of equations 3.37 and 3.38. Here 9r — 4 = Q DQ 2  Do = 6 D i D  2  and Q  = ^]ir=iP»  2  with  where k = y/6.  Proposition 2 Let gcd(kT\™ Pi,6D D2) =1  = K. and Q = Q'Y\i=iPi where Yli=iPi ^ 1  1  •  L S  -I^rll(«-('V))  O-free and  <«•«>  i=l  = ^^^^^[(ft-pf )) 2  X(9T- -4) 2  (4-82)  i=l  Proof. See Propositions 8 and 9 of Chapter 3.  4.2  Solubility of x - y D D 2  = -4  2  x  2  Lemma 2 Suppose x — y D\D 2  = —4 is soluble with fundamental solution (TQ,UQ). Let (to,uo) be the  2  2  fundamental solution to x — y D\D 2  2  2  = +4.  Then either T = kv 0  or TQ = kv  and Uo = kv  x  2  2  and  Uo = kvi , where k = y/6. Proof. Since to + it y D D " /  1  0  2  (T +  _  2  -  I  0  _  U S/D^LT \ 0  2  2  Tg +  -  USDiD 3  2  , +  2T r/o D D " /  0  V  1  2  3  we have the following equalities: (o  uo  q + viKD,  =  =  ( 4 8 3 )  T Uo  (4.84)  0  Case 1:. 2 / t . 0  Then by Lemma 1 and 4.84 6 = 1 so Uo = i>iu = T ?7o- Moreover (Diuf,D x;|) = 1, for if m — 2  (to + 2,to — 2) then to = ±2(m)  0  2  => m = 2 or 4. But this is impossible as to = 1(2). Therefore  (ui,v ) = 1. 2  Now suppose Vi we have:  = ab and t i  2  = cd. Wig. suppose TQ = ac and f/ = bd. Then by 4.83 and Lemma 1 0  Chapter 4. Hunting cr  32  io + 2  =  = a?c + b d D D 2  2  +4  =  2D b  T + 4  =  Ur D D  2  1  Since  2  2  0  a c +4 2  2  bdDD  d= a= D  2  2  2  2  x  = b = Uo and v  2  (4.85)  2D b 2  2  ia  =  a  (4.86)  =  1  (4.87)  2  =1  2  2  Therefore v  2  = 2  2  2  1  2  as (ab,cd) = 1 since (Dxa b , D c d )  2  1  2  2  dD  2  2  2  l  2  x  ia  =  2  2b d D D  Therefore  2  Dab  — c = To  Note, by considering t — 2 the same result ensues. 0  Case 2: 2 || toThen by Lemma 1 and 4.84, 8 — 4 so u = 4v v = T ?7o, whence 4 | Tot^o- Observing T - U D D 2  0  x  2  2  0  1  -4 = 0(4), it is clear that 2 | T and 2 | U as 4/Z)iZ» as r>iD is D-free where D D 0  0  2  2  X  2  =  EE 1,2(4).  2  Moreover (4Z>iu , 4D v$) — 4, for if m = ( i + 2,t - 2) then t EE ± 2 ( m ) =^> m = 2 or 4. Here m = 4 2  2  0  as <5 = 4. Hence (D v ,D v$) 1  0  0  = 1.  2  2  Now as in the above case suppose v = ab and v = cd. Wig. suppose To = 2ac and Uo = 2bd; (here x  2  k = 2). Following the steps in the above case we arrive at 4.85 - 4.87 , thereby getting 2v = 2b = Uo x  and 2-u = 2c = T . 2  0  Similarly, considering to — 2 the same result ensues. Remark: It is a theorem that the Pell equation x — y D\D 2  2  2  = —4 has no solution in integers if D D X  2  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 x — y D D 2  x  Proof.  = -4 is soluble, 3r — 2 = • <^=> D = 1.  2  2  2  Since 2/to, by Case 1 of Lemma 2, u = Tof7o = viv 0  interchangeably, either To = v and Uo = v . x  Case 1:  2  Suppose To = v and Uo — v . x  2  2  with (vi,v ) = 1 = (To,f/o) as 2  33  Chapter 4. Hunting a  Then v Since D vi  - 4= t - 2= Dv ,  2  0  2  = -4  2  2  l  2  (4.88)  4.88 becomes :  2  x  -v D D  2 x  2  ui - D ^ D i ^ i - 4 ) 2  2  =  -4  =>  V(l-D )  =  -4(Z?! + 1)  =>  V(l--Di)  =  -4  2  =>•  £>i  = 5 and  V i  =1  or  D\ = 2 and  Ui  =2  But 2 / D i u i as r = 1(4). Therefore if To = v\ and Uo = v , then  = 1 and D — 5.  2  x  Furthermore t  0  £>iui  + 2 =  =  2  5 =>  => t - 2 = D u 0  2  2 2  t  0  =  3  =1  D = u =1 2  Hence 3r - 2 = D u 2  Case 2:  2 2  2  = u = •. 2  2  Suppose To = v and C7o = v . 2  x  Then v Since Z? u + 4 = t + 2 = 2  2  0  Z?iU  2  - v DD  2  2  2  x  x  2  = -4  (4.89)  4.89 becomes: v - D {D v 2  2  =>  2 U2  2  2  + A)  ( l - D ) - 4D 2  2  =  -4  (4.90)  =  -4  (4.91)  =>  v (l-D )  =  -4(1-D )  =>  u|(l +  =  -4  2  2  2  2? ) 2  2  (4.92)  34  Chapter 4. Hunting cr  But D > 0. Therefore in view of 4.90 - 4.92, only D = 1 is possible. Hence 3r - 2 = v' = •. 2  2  2  2  Note that in Case 1, (to,uo) = (3,1) was completely determined as D\ = 5, v\ = I = v  2  = D  2  were all  forced. Case 2 is far less restrictive. However from the fact that DiD Case 2) DiD  2  = 24(3A: + 1) + 5, if Di ^ 5, then (wrt.  2  = Di = 12k + 29 and as always with the restriction that for any p \ Di, p = 1 (4).  Theorem 7 For even r, if x — y D\D 2  — —4 is soluble, 3r — 2 = •  2  2  <==> D = 1. 2  Proof. By Corollary 1 2||i , 6 = 4, 2||Z>i and by Lemma 2 * = 2. 0  Case 1:  Suppose To = 2ui and Uo = 2u . 2  Then 4v - 4vlD D  = -4  2  1  2  (4.93)  Since 4D v\ - 4 = 4D v\, 4.93 becomes: x  2  4v - £>i(4£>i-u - 4) 2  2  =•  4v\[\-D\)  =>  uj(l-^i)  =>  =  -4  =  - 4 ( l + £>!)  =  -  1  Z?i = 2 and vi = 1  Therefore T = 2 and 4.93 becomes 4 - 8v^D = -4 thereby forcing D = v = 1 => I7 = 2. 0  Hence 3r - 2 = 4u Case 2:  2  2  2  0  =  2 2  Suppose To = 2u and Uo = 2v\. 2  Then 4v| - 4 w D D = -4  (4.94)  2  1  2  Since 4Z? T;| + 4 = t + 2 = 4D v _, 4.94 becomes: 2  0  2  1  4v| - D (4v 2  =•  2  + 4)  ^ (i-rj )-£> 2  2  2  2  =  -4  = - l  Clearly the only possibility is that D = 1. 2  Hence 3r - 2 = 4u  2 2  = •.  Note that in Case 1, (T ,Uo) = (2,2) and therefore (t ,u ) = (6,4) were completely determined. 0  0  0  Chapter 4. Hunting cr  35  Theorem 8 / / 9r — 4 has no odd square factors and x — y D\D 2  2  = —4 is soluble then r = 1 if  2  2  r = 1 (4) and r = 2 if r = 2(4). If 97" — 4 has no odd square factors then UQ = 6. Moreover a is forced to be one thereby  Proof.  2  forcing £ = 3r. By 4.84 of Lemma 2, Uo = TQUQ = k . 2  0  Case 1: r is odd. For odd r, 9r - 4 is O-free. Here D i D 2  TU 0  0  = 9r - 4. By Corollary 1 5 = 1 . 2  2  = 1 by Lemma 2. Therefore T = U = 1 0  Therefore k = 1 and  1 - (9r - 4) = -4  r = 1 and therefore  2  0  9 r - 4 = 5 = Di. 2  Case 2: r is even. Here 9r - 4 = 4 Z » i D . By Corollary 1 <5 = 4. Therefore k = 2 and T = U = 2 by Lemma 2, 2  2  2  0  yielding 4 - 4£>iD = -4  0  = D = 2. Hence 9r - 4 = 32 => r = 2. 2  2  x  It deserves mention that the only other case found in the data of Tables 6.2 and 6.3 for which x — y D\D 2  -4 is soluble is when D D 1  6Yli=iPi)  2  = D = 26. This occurs when r = 34. Here 9r - 4 = 10400 = 2 5 (2 x 13). 2  2  4  2  x  As ro = 17 is prime, and D\D (3r,  =  2  /  2  2 (and n = 1), a = 1 by Theorem 4 (and 5).  = (*o,wo) = (102,20) and by applying 4.83 and 4.84 we have (T ,U ) 0  0  Therefore  = (10,2).  We conclude this chapter with the following summarizing theorem. Theorem 9 // D  2  ^ 1 and if a = 1 then E(9r  2  — 4) = E(DiD )  — A i , iAe largest eigenvalue of  2  where (t ,uo) = (3r, 5rj"=1Pi), is the Fundamental Unit of Q(y/9r — 4) =  7F j{to, o)  2  u  0  T  Proof. If D ^ 1 then by Theorem 6 and 7, equation 3.23 is insoluble. Therefore any unit of 2  has norm +1. Now if cr = 1 then by equation 3.34, £'(9T- - 4) = E(D D ). case 6D\D  Q(y/DiD ) 2  2  = Do, and <5 | 9r — 4. Now by equations 3.29, 3.30 and the (immediate) following remark 2  2  2  Recall that in the Markoff  2  X  Q(y/DxD ).  2  and Theorem 9, (c) of Chapter 3 e = E(9r - 4) = E( D ) 2  Dl  is the fundamental unit of  Q(y/D D ). X  2  2  =  t+M^y^lE  2  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 t i l ! °f f( i !)• e c  v e  v  x  Definition: PSL (C)* is defined to be the group of all conformal homeomorphisms of the complex 2  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 PSL (C). 2  is a point z and an infinite sequence {V } n  The point a is called a limit point of G provided there  of different elements of G such that V z => a. If a is not a n  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 {V z} need not be distinct. n  Definition:  G is said to be discontinuous at a point ct 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)) =  b)  0(G) C 0(F).  0(G).  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 of  PSL (R). 2  Definition: If 77 is never a non-trivial power of another element of G then 77 is said to be primitive in G. Proposition 4 Let v be a hyperbolic element of the discrete subgroup G C PSL (R)2  an element a £ PSL (R)  Then there exists  such that conjugation with a transforms 77 into the following form  2  (5.95)  where t > 1. Definition: The number t is called the norm of 77, denoted ^ ( 7 7 ) . 2  Definition:  Let u>i and u) be the fixed (real) points of 77. We define its axis A„ to be the (half) 2  circle with center, (uii +ui2)/2, on R and connecting u>i and ui2-  Chapter 5. Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces  Remark: A  can be viewed as a geodesic of H , where H +  v  +  38  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) and UJ . Then x  r\ will take the geodesic A  back to itself. Furthermore A  v  2  will project to a closed geodesic ir(A ) of  n  v  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> £  £{AUT+{f)).  2  b) 7y(to,Uo) is a primitive hyberbolic transformation of T. c)  3 a £ PSL (R) 2  such that cr^to.uo)*  =  ( E{d(f)) V (  0 t + 0  N( (t ,uo)) Jf  0  = E(d(f))> =  Corollary 2 3 a £ PSL (R) 2  x  n  n  n  0  0  (5.97)  0  (5.98)  V n  u  t - u ydTf)  (  V  Uoy/dff)  +  I  Here N( (t ,u ))  (5.96)  1  such that  a j [t ,u )a f  t 2  \  E{d(f))~ 2  ^ d)  0  = *(«*(/))*» =  (h^/M^  0  {E(d(f))-  n  t +U y/d{f) n  n  2 2 —" Q  o =  0  t  -  - — ^ n  (5.99)  U yjd(f)  ^ n + ^ v ^  n  ( r e f e r  t  o  4  .  7 1 )  .  Chapter 5. Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces  5.2  Closed Geodesies on  39  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 PSL (Z) 2  Proof  so that f = g iff Jf{to,v>o) is conjugate to 7 (t ,u ). 9  0  0  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) 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  jF (io^o). ryj  Theorem 1 For each Markoff Number r, there exist closed geodesies on H /T of length +  2 log ^3r + %/9r - 4^ of multiplicity K(§r 2  2  - 4), K(9r^ - 1) if r is odd or r = 2r , respectively. 0  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~ B~ 1  1  2  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  I ] \ 1 2 /  and  1  1 1 /  (5.100)  1  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: x + y + z = xyz 2  2  (5.101)  2  The following proposition is due to Fricke. Proposition 8 If A and B generate Y' then (Tr(A))  2  + (TR(B))  (  a  2  l,l  + (Tr(AB)) a  l,2 \  2  = Tr(A)Tr(B)Tr(AB)  (5.102)  we define the form f^ by  0,2,2 J  0-2,1  /A(33,y) = a ,ix + (a , - a )xy - a y 2  2  2  2  2  ltl  lj2  (5.103)  Proposition 9 Let f = (a,b,c) be an integral form of non-square discriminant. If UQ = 1 then f(*,y) =  f (t ,u ){ ,y)x  lf  0  0  Proof. Easily if U Q = 1 then 7,(*o,l)=(^ Theorem 3 p.(f) £ MSpec  t^bj-  / is either equivalent to or derived from a form  generator of Y . 1  Proof. See [Cohn 55, Haas 85]. Theorem 4 For each Markoff Number r, 7 F ( i o > u o ) is a generator of Y'. rj  ^ where A is a  Chapter 5. Explicit Lengths of Simple and Closed Geodesies on certain Riemann Surfaces  41  Proof. For purposes of simplification we will denote jF j (to, UQ) by A. r  By Proposition 15 and Theorem 7 (b) of Chapter 3, p,(F j) £ MSpec. By discussion of equation 3.62 r  of Chapter 3, UQ = 1 wrt. Qr — 4 = d(F j).  Thus by Proposition 9 we have  2  T  j(x,y) =  F (x,y)  (5.105)  rJx  Hence by Theorem 4, A is a generator of T'. Let (p,q,r) be a Markoff Triple and denote JF . (3p, 1), jF (3c, 1), 7F,,J (Zr, 1) by A j, A -, and A j P j  p  tj q  GJ  T  respectively. Remark: (3p, Zq, Zr) = (Tr(A j),Tr(A ,j),Tr(A j)) p  Conjecture  5.4  A .j and ( A ' ) p  q  _ 1  g j  is a solution to equation 5.101.  Tt  generate V with A (Agj)" PJ  1  = A j•. rt  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 £, axis of any element of AUT (g). +  will be denoted by A . Note that A g  We define the height of  g  is the fixed  by  We define the quantity H(A^^) by H(A ) U  = sup{ht(B(A )) | B £ T'} u  Remark: H(A^.c) is constant on the T' orbit of the geodesic ^4^ and is related to the minimum of a form by Lemma 1 H(A ) = 2^|g) ^ ^ * derived from a Markoff Form. 5  g  Proof. See [Haas 85, Cohn 71]. See Chapter 3, for a definition of fi. Corollary 4 If g = K.F  rJ  then H(A ) = g  H(A ). Frij  Definition: A is called a Markoff Geodesic if g is derived from a Markoff Form. g  Theorem 5 A is a Markoff Geodesic g  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(AF )  is a simple and closed geodesic on the Riemann  RJ  Surface H /V +  of length  Proof. By Theorem 4 jp . (to, Uo) is a generator of T'. By Lemma 1 and Theorem 6 and 7 definition r  Ap  rj  H /V. +  is a Markoff Geodesic. Hence by Theorem and Ap  rj  projects to a simple and closed geodesic on  By Proposition 6 (b), it follows that 7F (io; o) is also a primitive hyperbolic transformation u  rj  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  6  J=±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  Proposition 10 Let p > 3 be a prime. ±jf(to,uo)  43  £ T(p) iff p \u . 0  Proof. See [Sarnak 82].  Corollary 5 Let f be a primitive indefinite binary quadratic form with distinct roots LJI and w - If 2  n is the smallest power of E(d(f)) such that p \ u  then ±jf(t ,u )  n  n  is the T(p) primitive hyperbolic  n  matrix fixing u)i and u) . 2  Theorem 9 jF j (t , u ) 2  r  is the T(3) primitive hyperbolic matrix fixing 6 j  2  r  and 8' j, the roots of T  F {x,l). rd  Proof. First note that jp (to,uo)  £ T(3) as u = 1. As  Tj  h  0  +  (3r + V 9 r - 4 ^  ^2—A  2  U2  2  •  =  2  [  J  ( 5  -  1 0 8 )  we have t — 9r — 2 and u = 3r. Hence n = 2 and 2  2  2  9r -2-3r(3r-2(r k))  _  2  j+  7^(9r -2,3r)=l  ^  2  3  r  {  l+  rj  2  +  2  k  j  _ 3,3 - 3k)  9 , -- 2 + + 3,(3r 9r 3r( - 2 ( r j + fc)) I 2 2  2  ^  is a primitive hyperbolic element of T(3). It needs to be remarked that although 7F (9r  2  2ToJ  Fj  is not primitive for r = 2r$, 7±F  T  - 2 , 3 r ) as by Corollary 5, Chapter 3, AUT  To simplify notation, denote 7 F  (9r — 2, 3r) by 2  RI  (§F -) =  +  (Sr ~ 2,6r 2  2r3  AUT (F ). +  rj  rJ  y(F j). T  Theorem 10 A ^p .) projects to a closed geodesic Tr(A (F .)) of length 1  r  y  „,  / 9r - 2 + 3TV9T- - 4 2  2 log ^ on the Riemann Surface  r  2  -  H /T(3). +  Proof. By Corollary 2, N(j(F ))  = ^  rJ  2  ~ + 3ry/9r 2  2  -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(F j) is a primitive hyperbolic element of T(3) by Theorem 9, we now T  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) • Then TT(A„) is simple iff 2  = v(u ) > \. 2  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) <  T. 2  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  + 3rV5^4^  ( 5  n  o  )  Proof. By Theorem 8, of Chapter 3, v(6 ) = u(9' ) > | . Now by Theorems 10 and 11 we have the T  desired result.  T  Chapter 6  A Bold Conjecture  Conjecture —  Let r be a Markoff Number not equal to 1, 2, or 34.  j ^*  — - of a generator ~fp . (3r, 1) of the Commutator Subgroup V' C T is the Fundar  mental Unit of the real Quadratic Field Q(\/9r H /V +  The largest eigenvalue Ai =  and H /T(3) +  — 4) = Q(\/DiD ).  2  2  All simple and closed geodesies on  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 is the square of the fundamenx  tal unit of the real quadratic fields Q(y/E), Q{V32) = Q(y/2), and Q( 10400) = <?(-\/26) respectively. /  v  The following table includes all Markoff D\D < 1,000,000. H(DiD ) 2  and fundamental unit of Q(^/L\D ). E(DiD ) 2  and e denote the class number  The norm, N(e) = ^1, indicates whether equation 3.23 is soluble  2  or not, respectively.  2  will denote the first unit of Norm +1. As the column <5 [7_"=1  is  2  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. DiD  2  r 1 2 5 13 34 89 169 194 233 610 6466  < 10 . Data for the 5 6  97-2 - 4  5 32 221 1517 10400 71285 257045 338720 488597 3348896 376282400  th  column was found in [Mollin and Williams 89].  » ll?=iP? 2  0  42  0 0  45 2  2  0 0  42  0  42  45 2  2  DD X  N(e)H{D D )  2  1  5 2 221 1517 26 71285 257045 21170 488597 209306 940706  -1 -1 2 2 -2 16 16 16 24 64 112  2  E{D D ) X  2  (3 + V5)/2 3 + 2V2  (15 + y/22\)/2 (39 + 71517)72 51 + 10^26 (267 + ^71285)72 (507 + V257045)/2 291 + 2^/21170 (699 + V488597) /2 915 + 2v/209306 9699 + 10^940706  Table 6.1: Class Numbers and Fundamental Units of Norm +1 for Q{^/L\D~ ) where D D 2  45  X  2  < 10  6  46  Chapter 6. A Bold Conjecture  D = 1 for r = 1, 2, and 34. Except for these three cases, for all other Markoff Numbers in Tables 6.2 2  and 6.3, the data reveals that E(DiD ) 2  = e as D ^ 1 and n < 1 (see Theorem 5 and 9 of Chapter 4). 2  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 2 5 13 29 34 89 169 194 233 433 610 985 1325 1597 2897 4181 5741 6466 7561 9077 10946 14701 28657 33461 37666 43261 51641 62210 75025 96557 135137  1 2 5 13 29 2, 17 89 13, 13 2, 97 233 433 2, 5, 61 5, 197 5, 5, 53 1597 2897 37, 113 5741 2, 53, 61 7561 29, 313 2, 13, 421 61, 241 28657 33461 2, 37, 509 43261 113, 457 2, 5, 6221 5, 5, 3001 96557 337, 401  1 2, 2 13 37 5, 17 2j 2^ 5) 5 5, 53 5, 101 2) 2j 5j 29 17, 41 1297 2, 2, 457 2953 29, 137 4789 8689 12541 17, 1013 2, 2, 13, 373 37, 613 73, 373 2, 2, 8209 44101 13, 17, 389 37, 2713 2, 2, 13, 41, 53 233, 557 13, 17, 701 2, 2, 13, 37, 97 173, 1301 289669 37, 10957  5 2, 2, 2 17 41 89 2, 2, 2, 13 269 509 2, 2, 2, 73 701 1301 2j 2^ 2^ 229 2957 41, 97 4793 8693 5, 13, 193 5, 5, 13 53 2, 2, 2, 5, 5, 97 5, 13, 349 113, 241 2, 2, 2, 5, 821 5, 8821 149, 577 5, 17, 1181 2, 2, 2, 5, 5, 5, 113 5, 101, 257 5, 5, 6197 2, 2, 2, 41, 569 225077 37, 7829 405413  Table 6.2: Factorization of a Markoff Number r and Discriminant 9r — 4 2  47  Chapter 6. A Bold Conjecture  r — Markoff Number  Factorization of r  3r - 2  3r + 2  195025 196418 294685 426389 499393 646018 925765 1278818 1441889 1686049 2423525 2922509 3276509 4400489 7453378 8399329 11485154 16609837 16964653 20031170 21531778 43484701 48928105 137295677 205272962 253191266 375981346 576298801 647072098 941038565 1475706146 1873012681 6449974274 6684339842 65995009186  5, 5, 29, 269 2, 17, 53, 109 5, 58937 426389 73, 6841 2, 323009 5, 185153 2, 193, 3313 17, 89, 953 1686049 5, 5, 13, 7457 2922509 3276509 29, 41, 3701 2, 17, 219217 397, 21157 2, 5742577 29, 601, 953 3001, 5653 2, 5, 29, 69073 2, 10765889 13, 3344977 5, 9785621 7901, 17377 2, 29, 29, 122041 2, 126595633 2, 13, 29, 37, 13477 13, 157, 373, 757 2,137, 2361577 5, 11597, 16229 2, 737853073 1433, 1307057 2, 29, 317, 350809 2, 3342169921 2, 32997504593  585073 2, 2, 41, 3593 101, 8753 5, 17, 101, 149 569, 2633 2, 2, 661, 733 2777293 2, 2, 41, 149, 157 5, 109, 7937 5, 109, 9281 7270573 5, 5, 13, 53, 509 5, 5, 393181 5, 317, 8329 2, 2, 149, 37517 . 5, 41, 101, 1217 2, 2, 5, 13, 89, 1489 2269, 21961 113, 233, 1933 2, 2, 1733, 8669 2, 2, 977, 16529 130454101 13, 11291101 29, 313, 45377 2, 2, 1997, 77093 2, 2, 113, 433, 3881 2, 2, 4073, 69233 1728896401 2, 2, 73, 6648001 17, 29, 241, 23761 2, 2, 13, 353, 433, 557 37, 8353, 18181 2, 2, 5, 6221, 155521 2, 2, 5013254881 2, 2, 1093, 5437, 8329  585077 2, 2, 2, 73, 1009 884057 137, 9337 41, 36541 2, 2, 2, 242257 809, 3433 2, 2, 2, 13, 37, 997 29, 149161 1109, 4561 17, 427681 17, 515737 269, 36541 17, 776557 2, 2, 2, 37, 75541 3037, 8297 2, 2, 2, 17, 253349 49829513 50893961 2, 2, 2, 89, 84401 2, 2, 2, 13, 41, 15149 5, 2909, 8969 146784317 17, 24228649 2, 2, 2, 937, 82153 2, 2, 2, 5, 5, 29, 173, 757 2, 2, 2, 5, 28198601 1728896405 2, 2, 2, 557, 435641 2823115697 2, 2, 2, 5, 173, 639757 5, 1123807609 2, 2, 2, 2418740353 2, 2, 2, 17, 29, 5084437 2, 2, 2, 5, 4949625689  Table 6.3: Factorization of a Markoff Number r and Discriminant 9r — 4 2  Bibliography  [Borosh 75]  Borosh, I. [1975], More Numerical Evidence on the Uniqueness of Markov Numbers. BIT 15 (1975), 351 - 357.  [Cassels 57]  Cassels, J.W.S. [1957], An Introduction to Diophantine Approximation, Chapters I, II. Cambridge Univ. Press, 1957.  [Cassels 78]  Cassels, J.W.S. [1978], Rational Quadratic Forms. Academic Press Inc. (London) Ltd. 1978.  [Cohn 55]  Cohn, Harvey [1955], Approach to Markoff's Minimal Forms through Modular Functions, Ann. of Math. (2) 61 (1955), 1 - 12.  [Cohn 71]  Cohn, Harvey [1971], Representation of Markoff's Binary Quadratic Forms by Geodesies on a Perforated Torus, Acta Arith. 18 (1971), 125 - 136.  [Cohn 80]  Cohn, Harvey [1980], Advanced Number Theory, Dover Pub. Inc. N.Y. (1980).  [Cusick 89]  Cusick, Thomas W., Flahive, Mary E. [1989], The Markoff and Lagrange Spectra, Amer. Math. Soc, Math. Surveys and Monographs 30 (1989).  [Dickson 52]  Dickson, Leonard Eugene [1952], History of the Theory of Numbers, Volume III, Chelsea, N.Y. (1952) .  [Flath 89]  Flath, Daniel E. 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