UBC Theses and Dissertations

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UBC Theses and Dissertations

Fuchsian groups associated with certain indefinite quaternary quadratic forms Wright, John Bell 1940

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EUCHSIAN GROUPS ASSOCIATED-WITH CERTAIN INDEFINITE QUATERNARY QUADRATIC FORMS by-John B e l l Y/ri^ht A T h e s i s submitted f o r the Degree of M A S T E R O F A R T S i n the Department of M A T H E M A T I C S . THE UNIVERSITY OE BRITISH COLUMBIA APRIL, 1940 TABLE OF G OH TEi-lTS 1. I n t r o d u c t i o n . 2. The c o r r e s p o n d i n g E u c h s i a n group of t r a n s f o r m a t i o n s . 3. A r e s t r i c t i o n on u. 4. The i n v a r i a n t s of the F u c h s i a n group. 5. The d e t e r m i n a t i o n of the number of c l a s s e s of e l l i p t i c u n i t s . 6. C o n c l u s i o n . 7. . b i b l i o g r a p h y . F U C H S I M GROUPS A S S O C I A T E D WITH C E R T A I N niDSEnriTE QUATERNARY QUADRATIC FORMS 1. Introduc t i o n . The o b j e c t of t h i s paper i s to present a method of de-t e r m i n i n g a l l i n t e g r a l s o l u t i o n s of the e q u a t i o n ' (1) x,2 + xf D(x 3 2 + x* ) = 1, where D i s any i n t e g e r . The procedure f o l l o w s c l o s e l y t h a t of Dr. Ralph H u l l i n h i s p a p e r 1 "On the U n i t s of I n d e f i n i t e Q u aternion A l g e b r a s " . 'For the p a r t i c u l a r c a s e s , D ^ -1, D = 0, the s o l u t i o n s are t r i v i a l . F o r s m a l l p o s i t i v e v a l u e s of D i t i s easy to determine a c t u a l s o l u t i o n s by t r i a l . We s h a l l show t h a t , i n the g e n e r a l case, a l l s o l u t i o n s may be determined from a f i n i t e number .of s p e c i a l s o l u t i o n s or g e n e r a t o r s . We b e g i n by a s s o c i a t i n g our problem w i t h t h a t of f i n d i n g g e n e r a t o r s of a c o r r e s p o n d i n g F u c h s i a n group of l i n e a r f r a c -t i o n a l t r a n s f o r m a t i o n s of the complex p l a n e . In S e c t i o n 2, we show how t h i s i s done and we a l s o develop c e r t a i n f o r m u l a s which w i l l be used l a t e r . F o r t h i s F u c h s i a n group a p r i n c i p a l c i r c l e and a fundamental polygon can be c o n s t r u c t e d , and when t h i s i s done i t i s p o s s i b l e to determine the g e n e r a t o r s of the group. The method of procedure i s shown i n d e t a i l f o r the 1. American J o u r n a l of Mathematics, v o l . L X I , no. 2, A p r i l , 1939, pp. 365-374. f o u r c a s e s , D = 1, D = 3, D = 5, and D = 7, and these cases i l l u s t r a t e the d i f f e r e n t s i t u a t i o n s t h a t may a r i s e . 2 • From the g e n e r a l theory of F u c h s i a n groups i t i s known t h a t tile s t r u c t u r e of such s, ^ roup i s c o m p l e t e l y determined "by the numbers of i t s c l a s s e s of e l l i p t i c and p a r a b o l i c t r a n s f o r m a t i o n s and the genus of the a s s o c i a t e d Riemann s u r f a c e = I t w i l l be shown i n S e c t i o n 2, t h a t the F u c h s i a n groups i n v o l v e d i n the present problem c o n t a i n e l l i p t i c •transformations of order 2 o n l y ; may c o n t a i n h y p e r b o l i c t r a n s f o r m a t i o n s ; and may, or may not, c o n t a i n p a r a b o l i c t r a n s f o r m a t i o n s , a c c o r d i n g to the form of the i n t e g e r D. I n ac c o r d w i t h a r e s t r i c t i o n on D, d e s c r i b e d i n S e c t i o n 3, we s h a l l d e a l c h i e f l y w i t h the case i n which no p a r a b o l i c t r a n s -f o r m a t i o n s o c c u r 0 For these r e s t r i c t e d v a l u e s of D we have only to determine the c l a s s number of e l l i p t i c t r a n s f o r m a t i o n s , m, and the genus number, h. However, i n S e c t i o n 4, we use 3 4 fo r m u l a s of Humbert and K l e i n to show t h a t h can be computed from D and m. Hence our fund.aariental problem becomes the d e t e r m i n a t i o n of m f o r a g i v e n D. In S e c t i o n 5, we e v a l u a t e m by a c t u a l l y l i s t i n g , a c c o r d -i n g to c e r t a i n c o n g r u e n t i a l c o n d i t i o n s , the p o s s i b l e c l a s s e s of e l l i p t i c t r a n s f o r m a t i o n s f o r the r e s t r i c t e d v a l u e s of D. Me f i r s t o b t a i n an upper l i m i t to the number of these c l a s s e s , 2c F r i c k e - K l e i n , Autoraorphe F u n k t i o n e n , v o l . 1. 3. Humbert, Gomptes Rendus, v o l . 166, 1918, 4. F r i c k e - K l e i n , op. c i t . ( v o l . I . and then prove that t h i s number i s a c t u a l l y a t t a i n e d . In the l a s t s e c t i o n the r e s u l t s o b t a i n e d are summarized, and, as a f u r t h e r i l l u s t r a t i o n , c a n o n i c a l g e n e r a t o r s are e x h i b i t e d f o r the cases D = 3-, and D = 7. These are the only two, of the f o u r examples of S e c t i o n 2, that are 6 i v e n by our r e s t r i c t e d v a l u e of D. • 2. The c o r r e s p o n d i n g F u c h s i a n group of t r a n s f o r m a t i o n s . L a t e r i n t h i s paper we s h a l l have o c c a s i o n to c o n s i d e r , a l o n g w i t h s o l u t i o n s of e q u a t i o n ( l ) , those of the e q u a t i o n (2) x f + x 2 - D ( x 2 + x 2 ) = N, where N may have i n t e g r a l v a l u e s other than N = 1. The quat-er n a r y q u a d r a t i c form on the l e f t of (2) may be regarded as the norm form of a c e r t a i n g e n e r a l i z e d q u a t e r n i o n a l g e b r a , and w i t h t h i s c o n n e c t i o n i n mind, we s h a l l r e f e r to a set of numbers X = [ x , , x 2 , x 5 , x^] as an element; t h a t i s , an element of the a s s o c i a t e d q u a t e r n i o n a l g e b r a . In almost a l l cases, the c o o r d i n a t e s , x, , x 2 , x 3 , and x^ , of the elements employed here w i l l be r a t i o n a l i n t e g e r s , and we s h a l l h e n c e f o r t h use the word "element", w i t h o u t a m o d i f i e r , i n t h i s sense. When the c o o r d i n a t e s of an element X s a t i s f y the r e l a t i o n ( 2 ) , we c a l l N the norm of X, and w r i t e IT = f (X) = f£x, , i z , z 3 , x j . The s o l u t i o n s of e q u a t i o n ( l ) are i n t e g r a l elements of norm N = 1, which we s h a l l c a l l u n i t s . In order to d e f i n e the product of two elements, i t i s c o n v e n i e n t to r e p r e s e n t them as m a t r i c e s . Then, to the element X = £x, ,xz , x 5 , x v ] , we l e t c o r r e s p o n d the m a t r i x (3) x3.+ i x ¥ D ( x 5 - - i x , ) x, - ix.,_ i 2 = -1. The determinant of t h i s m a t r i x (3) i s the norm form ( 2 ) . By means of t h i s r e p r e s e n t a t i o n we o b t a i n the proauct of two elements X and (4) Y = [y, ,jz ,7s ,y/J y,.+ iy± y^ + VJ* D(y 3 - iy*) 7, - iya. by m a t r i x m u l t i p l i c a t i o n . The product of these m a t r i c e s , (3) and ( 4 ) , i s the m a t r i x z, + i z z D'(z3 - i z v ) z, - i z 2 z^ +• i z v where ( 5 ) z, = x,y, - x 2 y 2 I ) ( x 3 y 3 + x„y, ) , zz = x 2y, + x,y 2 - D (x y 5 - x,y^ ) , 3 - x 3 y ; - x y y 2 + x,y 3 + x t y v , } z 2 , z 3 » "</ J To t h i s .-'matrix we l e t cor r e s p o n d the element Z-= £z, and so f o r m u l a s (5) d e f i n e the product Z of two elements X and Y where,- • - -[x, ,x 4 ,3E, , x 4 J . [ y , ,y 2 ,y3 • , y ^ J = [ z , , z z , z 3 , z„]. T h i s product f o r m u l a h o l d s true f o r any va l u e of N f o r we know t h a t , s i n c e the norm i s the de t e r m i n a n t , the product of two norms i s e q u a l to the norm of the product. • .. • The sum of two elements, X and Y, is . r e a d i l y d e f i n e d by t h i s m a t r i x r e p r e s e n t a t i o n a l s o , but we do not have, o c c a s i o n to use i t .In the paper. We c a l l t h a t element, which, when m u l t i p l i e d by the e l e -ment X, g i v e s the i d e n t i t y element ["1,0 , u, o], the i n v e r s e of element X, and w r i t e i t as X"1 = £x, ,x 2 ,x_, , x j ~ ' . T h i s i n v e r s e e x i s t s i f and only i f f (X) = IT ^ 0 , and then from r e l a t i o n s (5) we can v e r i f y t h a t IT' H>~N J where N i s the norm.' Gonsider now the set of a l l elements of norm .N = 1, that i s , the set of a l l u n i t s . I t i s e a s i l y v e r i f i e d , by r e l a t i o n s (b) t h a t the product of any two u n i t s i s i t s e l f a u n i t , and tha.t the i n v e r s e of any u n i t i s a u n i t . From these f a c t s i t f o l l o w s t h at the s e t of a l l u n i t s forms a group w i t h r e s p e c t to the type of m u l t i p l i c a t i o n d e f i n e d above. For each value of D we w i l l o b t a i n a. d i f f e r e n t set of s o l u t i o n s a.nd so a d i f f e r e n t group. We propose to f i n d the s t r u c t u r e of these groups, ana so we l e t any be the group G(D). This group i s e a s i l y r e l a t e d to a group of l i n e a r f r a c t -i o n a l t r a n s f o r m a t i o n s of the complex plane. To form the ass -o c i a t i o n we make the f o l l o w i n g , correspondence, (6) [x, ,x 2 ,x 3 ,x 4] z =•• (x, +- ixa)w + D ( x 3 - i x ^ ) 9 ( x 3 + i x ^ ) w + x j - ixz where z and w are complex v a r i a b l e s . T his set of t r a n s f o r m a t -i o n s of the complex plane forms a F u c h s i a n group, d i f f e r e n t f o r d i f f e r e n t v a l u e s of D, which we s h a l l c a l l F(D) to d i s -t i n g u i s h from our group of s o l u t i o n s G(D). In case D ? u, F(D) i s an i n f i n i t e group. T r a n s f o r m a t i o n s of such, a group f a l l i n t o t h r e e c l a s s e s ^ ; h y p e r b o l i c , e l l i p t i c , and p a r a c o l i c . F o r d , L.R., "Automorphic F u n c t i o n s " , 1929, p.67. 6 From Ford-'sf 5 work we have necessary and s u f f i c i e n t c o n d i t i o n s that a t r a n s f o r m a t i o n he of any one of the three t y p e s . We f i n d that, a t r a n s f o r m a t i o n (6) i s e l l i p t i c i f , and only i f , | x, +- i x 2 +• x, - i x g | < 2. Si n c e x, i s an i n t e g e r t h i s means t h a t x » 0, and so we have the type of e l l i p t i c u n i t Y, where (7) Y = [0,y z , y j ,y^] , y/ - D (y| + y*) •= 1. E q u a t i o n s (5) w i l l show t h a t f o r any e l l i p t i c u n i t Y 2 = [0,y a ,y3 , y j 2 = [-1,0,0,.0J. Ag a i n , a t r a n s f o r m a t i o n i s p a r a b o l i c i f and only i f x, •+ i x 2 + x, - i x 2 = ±2, th a t i s , x, - ±1. But-' from ( l ) t h i s would mean t h a t (8) D = / X a X 3 \ 2 f f x 2 x„ f , Vx* +• x*/ Vx| + x*/ \ or t h a t D must be of the form u 2 + v z , where u and v are r a t i o n a l . A l l t r a n s f o r m a t i o n s which do not s a t i s f y the above c o n d i t i o n s are h y p e r b o l i c t r a n s f o r m a t i o n s . The.Fuchsian group F ( D ) , of the l a s t paragraph has a common 7 f i x e d c i r c l e , or " p r i n c i p a l c i r c l e " , g i v e n by the e q u a t i o n (9) w w = D, where w i s the conjugate of w, and the t r a n s f o r m a t i o n (6) c a r r i e s t h a t p r i n c i p a l c i r c l e i n t o i t s e l f , i t s i n t e r i o r i n t o i t s i n t e r i o r , and i t s e x t e r i o r i n t o i t s e x t e r i o r . We are able to draw the i s o m e t r i c c i r c l e s (10) |(x 3 + \-xu) z - x, - i x 2 | 2 = 1, of the t r a n s f o r m a t i o n , which we s h a l l denote by I ( x ) = 6. F o r d , op. c i t . , theorem 15, p. 23. 7. I b i d . , p. 67. 7 , z 2 , z 3 , x j on the accompanying i l l u s t r a t i o n s . These i s o -8 m e t r i c c i r c l e s are a l l o r t h o g o n a l to the p r i n c i p a l c i r c l e . In t h i s way we can c o n s t r u c t a fundamental r e g i o n f o r the group F (D) , where we take the d e f i n i t i o n of fundamental r e 0 i o n 9 from Ford . D e f i n i t i o n 1. Two c o n f i g u r a t i o n s , ( p o i n t s , c u r v e s , r e g i o n s , e t c . ) are s a i d to be c o n 6 r u e n t w i t h r e s p e c t to a, 6 r o u p i f there i s a. t r a n s f o r m a t i o n of the group o t h e r than the i d e n t i c a l ' t r a n s f o r m a t i o n , which c a r r i e s one c onf i g u r a t i o n i n t o the o t h e r . D e f i n i t i o n 2. A r e g i o n , connected or not, no two of whose p o i n t s are congruent w i t h r e s p e c t to a, g i v e n group, and such t h a t the neighbourhood of any p o i n t on the boundary c o n t a i n s p o i n t s congruent to p o i n t s i n the g i v e n r e g i o n , i s c a. l i e a a fundamental r e g i o n f o r the group. W i t h t h i s r e g i o n we may a s s o c i a t e a. Riemann s u r f a c e i n . much the same way as a torus i s d e f i n e d by i d e n t i f y i n g the s i d e s of the fundamental p a r a l l e l o g r a m i n the case of e l l i p t i c f u n -c t i o n s . An important number a s s o c i a t e d w i t h a Riemann s u r f a c e i s i t s genus number^ which we s h a l l re.quire l a t e r . In our case the fundamental r e g i o n i s e n c l o s e d by a r c s of the i s o m e t r i c c i r c l e s , and so we c a l l i t a fundamental polygon. Any one i s o m e t r i c c i r c l e of the polygon i s c a r r i e d i n t o another c i r c l e or i n t o i t s e l f , by a s u i t a b l e t r a n s f o r m a t i o n , and these t r a n s -S. F o r d , op. c i t . , p. 67. 9. I b i d . , p. 37. 1U. I b i d . , p. 227. f o r m a t i o n s y i e l d a set of g e n e r a t o r s ana r e l a t i o n s f o r the group F ( D ) . We i l l u s t r a t e the method of d e t e r m i n i n g g e n e r a t o r s ana r e l a t i o n s from the fundamental polygon f o r the cases D = 1, 3, 5, and 7. The method of procedure i s to a s s i g n such i n -t e g r a l v a l u e s 0, 1, '2,.... , i n th a t order -- to X j + as w i l l y i e l d i n t e g r a l s o l u t i o n s of e q u a t i o n ( l ) f o r the g i v e n D, and then to de termine those s o l u t i o n s . Then we draw the p r i n c i p a l c i r c l e , and the i s o m e t r i c c i r c l e s c o r r e s -ponding to- the d i f f e r e n t s o l u t i o n s or t r a n s f o r m a t i o n s . I t i s to he expected t h a t i n t h i s way the whole of the p r i n c i p a l c i r c l e w i l l he c l o s e d 'off by u s i n g r e l a t i v e l y s m a l l v a l u e s of x 3 + x^ . I n c e r t a i n cases we f i n d a s o l u t i o n whose c o r r e s -ponding i s o m e t r i c c i r c l e does not c o n t r i b u t e to the c l o s i n g o f f of the p r i n c i p a l c i r c l e . This' i s so whenever the c e n t r e of the i s o m e t r i c c i r c l e l i e s w i t h i n a p r e v i o u s l y determined i s o m e t r i c c i r c l e . T h i s happens f o r the u n i t [6, 0, 2, - l j , of D — 7, f o r the c e n t r e of i t s i s o m e t r i c c i r c l e f a l l s w i t h i n the i s o m e t r i c c i r c l e of the u n i t [_b, 2, 2, O j . When t h i s ..... 2 2 happens we proceed w i t h the next v a l u e of x 3 + z. . When the fundaxaental r e g i o n has been c o m p l e t e l y c l o s e d o f f we are able to determine u n i t s ca.rrying one p a r t i n t o the o t h e r . These u n i t s , i n d i c a t e d by arrows i n the accompanying f i g u r e s , form the s et of g e n e r a t o r s f o r the group F (!)) . The arrow i n d i c a t e that the i s o m e t r i c c i r c l e where i t s t a r t s i s c a r r i e d by the u n i t i n t o that c i r c l e to which i t points.The v e r t i c e s of the polygon are d i v i d e d up i n t o complete s e t s of congruent v e r t i c 9 vrtiich we s h a l l c a l l c y c l e s . In some cases the c y c l e has a s i n g l e v e r t e x , but i n o t h e r s there are s e v e r a l v e r t i c e s . The sum of the a,ngles i n the o r d i n a r y 'sense' of any one c y c l e may add to 27T, 7f, or 0 radia,ns. On m u l t i p l y i n g t o g e t h e r u n i t s necessary to complete the above c y c l e s we w i l l ^ e t u n i t s which are h y p e r b o l i c , e l l i p t i c , or p a r a b o l i c i n the r e s p e c t i v e c ases. The three types are i l l u s t r a t e d by the f o l l o w i n g u n i t s : A , of case D - 7, where « - TT, G/GG^'B"', of case D .= 7, where v, + £x + Y 3 + ^ = 2 7?7 By , of case I) = 5, where j6T-Q. I t w i l l be found t h a t two d i s t i n c t cases a r i s e a c c o r d i n g to the nature of the i n t e g e r D. The examples D — 3 or 7, and D = .5, i l l u s t r a t e the two s i t u a t i o n s . The groups F ( 3 ) , F ( G ) , a.nd F (7) can a l l be generated by a. f i n i t e number of u n i t s . The group F(5) w i l l c o n t a i n p a r a b o l i c , as w e l l as e l l i p t i c and h y p e r b o l i c u n i t s , but the groups F(3) ana F ( 7 ) . w i l l have no p a r a b o l i c u n i t s . The v e r t i c e s of the polygon f o r J'(3) ^ i v e r i s e to e l l i p t i c c y c l e s o n l y , w h i l e those of the polygon f o r F(7) have e l l i p t i c - a n d h y p e r b o l i c cycles.-From the f o r e g o i n g d i s c u s s i o n we have the f o l l o w i n g Theorem I , I f D i s any p o s i t i v e i n t e g e r , the F u c h s i a n & r o u p F(D) can be generated by a f i n i t e number of u n i t s , s a t i s f y i n g c e r t a i n r e l a t i o n s . These u n i t s may be h y p e r b o l i c , e l l i p t i c , or p a r a b o l i c , the l a t t e r o c c u r r i n g i f and only i f D . i s e x p r e s s i b l e i n the form u 2 + v 2 , u and v r a t i o n a l . When the g e n e r a t o r s of F (D) are o b t a i n e d , the c o r r e s -l u pondence „(6) g i v e s the l i k e f o r G (D) . I t must he noted however t h a t the two u n i t s £l, 0 , u , o] and [-1, u , u , u] b o t h D i v e the i d e n t i t y t r a n s f o r m a t i o n i n F (D) , but are d i s t i n c t i n G ( l ) ) . Hence when the g e n e r a t o r s of F ( D ) have been determined we must a d j o i n the u n i t £-1, 0 , 0 , uj to o b t a i n the group G (D ) . T h i s g i v e s Theorem I I . The group, G ( D ) , of i n t e g r a l s o l u t i o n s of e q u a t i o n ( l ) , i s o b t a i n e d from the a s s o c i a t e d F u c h s i a n group F (D) , a c c o r d i n g to the correspondence ( 6 ) , by a d j o i n i n g to the g e n e r a t o r s of F ( D ) the u n i t [-1,0,0,0], G e n e r a t o r s . A:[0,1,0,0] B: [ l . l . l . o ] R e l a t i o n s . <X ; A 2= -1 q : P a r a b o l i c FUEDAltSlTTAL POLYGON F(D) = P(3) I : [ 3 , 2 ^ I*. [5 , ^ 2 4 •A I : [ 2 s 0 . ! 0 , - l ] • I : [Ju^S,0 I : [3i 2t0,.-lrQ." G e n e r a t o r s . A; [ 0 , 1 , 0 , 0 ] - B: [2,0,1,0] C: "[0,2,1,0] D; [3,2,2,0] E: [2,3,2 ,0] : R e l a t i o n s , « : A 2 • s : ( c - ) 2 r : ( G E - f 1 : (ED"' ) 2 s •: (DB~' ) 2 (BA-')2 FUND AH5NT AL POLYGON F(D) = F(o) G e n e r a t o r s , R e l a t i o n s . A: •[0,1,0,0]- DC : A 2 = -1 B;: [-1,5,1,2] f. : P a r a b o l i c [-1,5,2,1] TS ' . \J = J-[-1,5,2,-1] s : (DA") 2 . - -1 [-1,5,1,-2] t : (B^B^G^'Bi'D")* C : [0,9,4,0] D: [9,0,4,0] FUNDAMENTAL POLYGON F(D) r= F(7) G e n e r a t o r s , R e l a t i o n s , A: [0,1,0,0]- « : A 2 = ' -1 B : [3,0,3,0] : (BA") 2 - -1 C,: [9,1.2,4,4] -r : G, GC'a'B" = -1 <V C-9,12,4,-4] : (DG^F-'c;')2 -D: [5,2,.2,o] £ : (ED") 2 - -1 E: [2,2,1,0] A : (FE"') i - -1 F: [2*5,2,0] : (GT - - I " G: CO,8,3,0] 16 3. A r e s t r i c t i o n on D. I t i s the purpose of the remainder of t h i s work to d e t e r -mine the number of g e n e r a t o r s necessary to g i v e the ^roup, F ( D ) , f o r a g i v e n value of D. To t h i s end we f i n d i t conven-i e n t here to i n t r o d u c e a r e s t r i c t i o n on D. We s h a l l hence-f o r t h c o n s i d e r only those cases f o r which (11) D = p,-.pa. p A , pA- prime, p^ = 3 (mod 4 ) , * - 1> V* ^ f o r i ^ j . T h i s omits a l l cases i n which F (D) has p a r a b o l i c t r a n s f o r m -a t i o n s and i n p a r t i c u l a r the case D = 1. We s h a l l show, however, t h a t i t i s the p r i n c i p a l case to be c o n s i d e r e d . We s h a l l f i r s t l o o k at some t r i v i a l c a s e s . For a l l v a l u e s of D we have the s o l u t i o n s f o r e q u a t i o n ( l ) ^ i v e n by the u n i t s [±1,0,0,0] and [0,±l,0,oJ. For D < -1, or f o r D = u, these are the only s o l u t i o n s . For''D = -1 we have ei&ht s o l u t i o n s o n l y ; [±l,0,0,0j, [ o,il,0 , o J , £o,0,£I,o], and £ o , 0 , U , ^ l ] . Hence we need not c o n s i d e r f u r t h e r the case i n which D has n e g a t i v e v a l u e s or the value z e r o . Next, suppose D i s e x p r e s s i b l e as the product 3D', wnere S can be w r i t t e n i n the form <xa + , « and p i n t e g e r s , and where I)' i s d e f i n e d as we have d e f i n e d D i n (11) . Tnen from e q u a t i o n ( l ) we have x,2 + x£ - SD'(xJ + x 2 ) = 1, x * * -4 - " D ' ( « 2 * V ) U / + -4) = i , x * + x 2 - D'{(«x3 -/»x^) 2 + (^Xj + « x ¥ f ] ~ 1, y,1 f y/ - D'(y/ * y 2 ) = 1, where Thus every s o l u t i o n of x,2 + x / - SI) ' ( x * + ) - 1, corresponds to a s o l u t i o n of y,2 +• y^2 - D' (y^" - y f ) =- 1. Now, i f D i s a.ny p o s i t i v e i n t e g e r , v;e may w r i t e D = SI)', where S can he w r i t t e n as *z +• p2- , « and p i n t e g e r s , and where D' i s 1 or i s of the form (11). These c o n s i d e r a t i o n s l e a d at once to Theorem I I I . I f D i s any p o s i t i v e i n t e g e r we may w r i t e D — SD' , where S = <xa + pz, « and p "being i n t e g e r s not both z e r o , and where D' may be 1 or of the form ( l l ) . Then the group G (l)) i s a subgroup of the group G (D ') . •It must be noted here t h a t we do not attempt, i n t h i s paper, to show how to determine the g e n e r a t o r s f o r any v a l u e s of D o t h e r than those of the form ( l l ) . However when the present case has been completed i t should be a r e l a t i v e l y s imple matter to extend i t to cover a l l other cases. In S e c t i o n 2 we have i n c l u d e d the complete polygon and g e n e r a t o r s f o r the" specia.l ca.se D = 1, s i n c e Theorem 3 shows i t to be an important case not covered by our r e s t r i c t i o n on D. I t s group of s o l u t i o n s F (D) - F ( l ) i s generated by the two u n i t s A - [0,1 , 0 , 0 ] , ana. B = [1,1,1,0], where A 2 = -1, and B i s a. p a r a b o l i c u n i t . • 4. The i n v a r i a n t s of the F u c h s i a n group. We have a l r e a d y mentioned that the s t r u c t u r e of the group F (D) i s determined c o m p l e t e l y by the number of i t s c l a s s e s of e l l i p t i c u n i t s and the genus number. Let m be the number of c l a s s e s of e l l i p t i c c y c l e s , n the number of c l a s s e s of h y p e r b o l i c c y c l e s , and h the genus of the a s s o c i a t e d s u r -f a c e . The v a l u e s m and h are i n v a r i a n t s of the group F (D), depending only on the group and not on any p a r t i c u l a r way 'of r e p r e s e n t i n g i t . The value n on the other hand i s not i n v a r i a n t , ana i s only used to e s t a b l i s h a r e l a t i o n c o n n e c t i n m and h t h a t we r e q u i r e . D i s p l a c e m e n t s of the i n t e r i o r of the p r i n c i p a l c i r c l e brought about by the l i n e a r f r a c t i o n a l t r a n s f o r m a t i o n s are d i s p l a c e m e n t s of h y p e r b o l i c geometry. From t h i s c o n s i d e r -a t i o n the non-Duelidean a r e a , GL, of our fundamental polygon i s g i v e n by"*"-1" a = (2t - 2)ir - z , where 2t i s the number of sides- to the polygon, counting, the X a x i s as two s i d e s , and 2 i s the sum of the a n g l e s . But by a n . a n a l y t i c proof " of Humbert the n o n - E u c l i d e a n a r e a i s a l s o g i v e n by a = ^D//(I - y^) = 7TG>(D), where <p i s the u s u a l E u l e r f u n c t i o n . Combining these two r e s u l t s we have (2t - 2) 7T - 1 = 71 (0(D) . 11. C o o l i d g e , J . L o w e l l , The Elements of Non-Euclidean Geo-metry , 1909, theorem 5, p. 173. 12. Humbert op. c i t . . p . ? 7 0 . 18 But' 2= ITJTT + 2nrr, s i n c e the sum of the angles of each c y c l e of e l l i p t i c v e r t i c e s i s 7T, and the sum of the a n 0 l e s of a h y p e r b o l i c c y c l e i s 27»\ Then ( 2t - 2) 7r - m w - 2n W = ir <D (D ) , (12) 2 ( t - n) = <p (D) +• 2 +- m. Now the genus number h, of the a s s o c i a t e d s u r f a c e i s g i v e n by the f o r m u l a 2 h ~ l — t - n - m . •'Hence t - n = 2 h - l - » - m , and on s u b s t i t u t i n g t h i s v alue of t - n i n (12) we have 2(2h - 1 + m) = 41 (D) -f 2 +• m, (13) 4h cp (D) + 4 - m. This f o r m u l a (13) g i v e s us the r e q u i r e d r e l a t i o n , s i n c e i t enables us to e v a l u a t e -h when m" has been determined. To f i n d the number of g e n e r a t o r s f o r the group F(D) we go to the g e n e r a l theory"*"^ of Fuchs i a n groups. The l i n e a r f r a c t i o n a l group, F ( D ) , has a c a n o n i c a l s e t of m + 2h g e n e r a t o r s (14) u,;u 2, , u m , v, ,v a , v„, V,\Yz\ .... .\n , where U- or V. r e p r e s e n t the u n i t s U.- = f u- , u• , u..„ a. 1 and _ [ v-, . Y,i > Y^ 3 > Yp] - These g e n e r a t o r s #ust s a t i s f y the r e l a t i o n s U,z = U 2 Z= = i C - - i , u, .TL, u j // v. .v; M- X: I = -1. 13. F r i c k e - K l e i n , op. c i t . , c h a p t e r 3, f o r m u l a (2)., p. 262. 14. I b i d . , pp. 186 - 187. 5. The d e t e r m i n a t i o n of the number of c l a s s e s of e l l i p t i c u n i t s . The v e r t i c e s of an e l l i p t i c c y c l e are f i x e a p o i n t s of e l l i p t i c t r a n s f o r m a t i o n s - of a l l of which b e l o n 0 to trie same c l a s s . We r e c a l l t h a t two e l l i p t i c s u b s t i t u t i o n s of a Fuchs i a n group are s a i d to be i n the same c l a s s i f one i s the t r a n s f o r m .of the other by a s u b s t i t u t i o n of the group. In o t h e r words, two u n i t s , A and B, of the group F ( D ) , are i n the same c l a s s i f there e x i s t s a u n i t X, of the group such t h a t X~'AX = B . Our t a s k i s to determine how many of these separate c l a s s e s e x i s t f o r a g i v e n D. The u n i t [.0,1,0,0'] i s present f o r every I) ana i t de-termines one c l a s s which i s r e p r e s e n t e d by a s i n g l e v e r t e x . I t w i l l appear t h a t any e l l i p t i c u n i t , B = £0,b a,b 3,b^J, can be transformed i n t o t h i s one by some element X = [ x , ,x 2 , x3 , x j , of norm IT. I t i s necessary t h a t we determine the d i f f e r e n t v a l u e s that M can assume f o r our group F (l)) . Then by s t u d y i n g c e r t a i n c o n g r u e n t i a l c o n d i t i o n s imposed on b z we can determine m. Suppose t h e n - t h a t A and B are two u n i t s such that B i s the t r a n s f o r m of A by some element X of norm IT > 0, Then (15) X AX = B . T h i s means t h a t a 2 ». a 3 ' a 15. F o r d , op. c i t . , pp. 60-61. Then on m u l t i p l i c a t i o n as d e f i n e d by the r e l a t i o n s (b) we have (16) Fb, - Na ( , b, — a, 9 F b z = x z 2 + D ( x | + x / ) j a., + 2 D ( x / x y - x 2 x J ) a 3 . -2D(x,x 3 + . X A X y ) a < / , . • " Fb^ =^  2 ( x A x y 4- x_ zx_ ?)a a -/- (x,-2 - x 2 - Dx/ +- l ) x < f ) a 3 - 2 (x^ Xj Dx_j x ) a ^ , Nb^ = '2,(xa_xv - x 7 X j )a. a +- 2 ( x / x 2 ^ - D x 3 x ^ ) a 3 - f ^ x , 2 -x/ + DXj 2 - Dx 2 j a ^ . f e wish' to determine the. •minimum value of I to g i v e an element X which t r a n s f o r m s any u n i t A i n t o a u n i t B. S i n c e the u n i t £O,1,G,GJ i s p r e s e n t f o r every value of D we s i m p l i f y our problem by t a k i n g " . . A = [a, , a-L , a 3 , a^J — [0 ,1, 0 , o j . We must now determine- the minimum V a l u e of H to g i v e i n t e g r a l s o l u t i o n s x, , xx, X3 ,:':and of ( 2 ) , s a t i s f y i n B the new r e l -a t i o n s o b t a i n e d from ('16'), (17) .  b A = 0 , Wox = x,2 +- x^ -h"D{-x.f +• -X-*) , Hb 3 2 ( x / x v *• x ^ j ) , Nby = 2 ( X j X y - X/x_j ) . ' From r e l a t i o n s (17) i t f o l l o w s t h a t , f o r B to be i n r. the same c l a s s as A, t h a t i s IT = 1, we must have ••0/= X , V X / + D ( X 3 4 * XJ) = l + 2 D ( X / > X * ) • Then i t f o l l o w s t h a t b a — l(mod 2D), p o s i t i v e . Now c o n s i d e r any e l l i p t i c u n i t B = [0 , b t , h3 , b w ] , where The n (19) b * =s l(mod D ) . S i nce D i s d e f i n e d as i n ( l l ) the above congruence i s equiv a l e n t to the set of congruences (20) b / = l(mod gx) , i - 1, 2, , r . Hence (21) b^ H 1 or -1 (mod p , ) , b x = 1 or -1 (rood p a) , b a s 1 or -1 (mod p ^ ) . ¥e then have 2 R d i s t i n c t p o s s i b i l i t i e s f o r b^, But we nave a l s o the f u r t h e r two p o s s i b i l i t i e s b_, s 0 or 1 (mod 2 ) , and so i n a l l we have 2 R + 1 p o s s i b l e ways oi c hoosing b A . •Prom f o r m u l a s (17) a necessary c o n d i t i o n that XT'AX = B i s t h a t (22) Uhx =. x, a * x / + DU;2"-*. x * ) . Since-IT i s chosen p o s i t i v e then b^ i s a l s o p o s i t i v e . Prom e q u a t i o n (2) we have t h a t x / *• x / = IT .J)(x/> X*) , and so we may w r i t e f o r (22) Nb a - IT +• 2D ( x / + xj 1) , (23) N(ba. - 1) = 2D(x/+ x£). How suppose t h a t ( 2 4 ) D - D,D A, where D, « p,.p^, Ps , • I> A = ?A » ( / £ S and a l s o sup, ose that (25) b ^ = 1 (mod D, ) , b^ =. -1 (mod ) . Then To x + 1 i s d i v i s i b l e by D A and b^ - 1 i s prime to . In such a case e q u a t i o n (23) shows that IT must be d i v i s i b l e by D . In the case b. = 0 (mod 2 ) , b, - I i s odd and so IT xuust be d i v i s i b l e by 2DZ. In the case b^ = 1 (mod 2 ) , b^ - 1 i s even and we can only say th a t IT i s d i v i s i b l e by D^ . Bef o r e we can co n t i n u e w i t h a theorem r e g a r d i n g the ex-i s t e n c e of elements X of norm IT s a t i s f y i n g , the pr e v i o u s cond-i t i o n s , we must prove a lemma: which- i s e s s e n t i a l to the proof of the theorem. The lemma and proof f o l l o w . Lemma. I f m and n are- r e l a t i v e l y prime p o s i t i v e m t e 0 e r s , and i f x/ and are i n t e g e r s such t h a t x, *V x* =* mn, then there e x i s t s a s e t of i n t e g e r s y, '±yA ,z, , and zx such that (26) y,**. y / ^ m, z / % z* = n, (27) X, = y , z a +y,_z ( J x a y 1 z i - y ( Z | . This lemma, i s a consequence of the e x t e n s i v e theory of the r e p r e s e n t a t i o n of p o s i t i v e i n t e g e r s as sums of two i n t e -g r a l squares. I t may a l s o be p r o v e d 1 6 as f o l l o w s by the means of the i d e a l theory of the q u a d r a t i c number f i e l d R ( i ) , ' w u e r e i 2 " - -1, R i s the r a t i o n a l f i e l d . The 0 i v e n i n t e g e r s x, and x z determine a p r i n c i p a l i d e a l ( ( x f . + i x ^ ) ) of the f i e l d R.(i), of norm mn. The g r e a t e s t common d i v i s o r s of the i d e a l ( ( x , +- i x 4 ) ) w i t h the i d e a l s ( ( E ) ) and ( (n) ) , r e s p e c t i v e l y , are i d e a l s of norms m and n. Since a l l i d e a l s of R.(i) are 16. T h i s proof of the-lemma has been s u g 0 e s t e d to me by Dr. R a l p h H u l l . 2 3 p r i n c i p a l , i a e a i s , these conuaon d i v i s o r s are p r i n c i p a l i a e a i s , say ( (y, -f i y ^ )) and ( ( z z - i z / ) ) , r e s p e c t i v e l y , where y/ , yx , 7,t , and z- s a t i s f y (.26). Moreover, s i n c e m and n are r e l a t i v e -l y prime, ( ( x , r i x x ) ) = ( (y, +- i y a ) ) ( (z ^  - i z , ) ) = ( (y z^ -f -y^z, -f. i y A z A - i y , z ,)) . From t h i s e q u a l i t y of i a e a i s i t f o l l o w s t h a t x , +- i x x - e (y, z z + y A z , 4 i y , z 2 - iy, z ,) , where e i s one of the f o u r u n i t s 1, -1, i , , - i of R ( i ) . I f & =1 we have (27) as d e s i r e d . I f , f o r example, € = i , we o b t a i n (27), w i t h o u t a l t e r i n g (26), by trie replacement of y ( ,yx,z,, and z^ by y, ,yAlzx ,and - z , , r e s p e c t i v e l y . S i m i l a r l y , i f e = -1 or - i , we o b t a i n (27),- w i t h o u t a l t e r i n g (26), by s u i t a b l e i n t e r c h a n g e s 'of y( ,y x ,.z ,, and zx, and t h e i r s i 0 n s . T h i s com-p l e t e s the proof of the 'lemma. We are now ready to prove , " Theorem M. Let D' = D/D,, where D, = p, .p 4 . . . . p 5 , ana Da - p^ .'p,^ . . ...pA. I f b A > 0, l)z ~ 1 (mod D, ) , ana b^ B -1 (mod Dx) , there e x i s t s an element X of norm IT > 6, which transforms the u n i t A = £ o,l,0,Oj i n t o the u n i t B = £u,b x,b^,bj, such t h a t IT = f (X) = when b-^  i s odd, and IT = f (x) = 2L-, when b^ i s even. Gase I . b 2 odd, i . e . , b^ s. 1 (mod 2 ) . Let b a ~ 1 + 2kD, , where lc ^  0 because b a > 0, and k i s prime to Dz. S i n c e B i s a u n i t we may w r i t e 4 - D(b/- + Tr^) = 1, = *>a ~ 1 = 1 * 4 k D * + 4k* 3}1 " 1 •=. 4kD , (1 + kD , ) , . B , B 2 ( b / -^ b j ) = . 4kD, ( l +• M), ), D i ( V + = 41c(l + kTD ,.) . Since I>z i s odd, by* -t b^ i s d i v i s i b l e by 4. Let by - 2bj , k/z - 2b'. Then + »/) ^ k ( l + kD, ) . Since k i s prime to Dz; then 1 +. kD, must be a m u l t i p l e of D z so ^ 1 + ^ 0 f j i s an i n t e g e r . Then Here k > 0, / V-llA/3'J>0, and the two are r e l a t i v e l y prime. The lemma t h a t we have j u s t proved 0 i v e s the e x i s t e n c e oi' i n t e g e r s Yt tVs- >zi > a n (i z*. such t h a t y, ..+ y. - ( - s r j . . V = ^ y ^ . = ya v ~ y, z7-Take x, s L\y ( , x a = D^y, , x 3 . . z/, x v'= z < # Then ' \ x - . x / - D.(x/.r x*) - 2 ) j ( y / - - y/) - D U / r - z?) ' = D a = N,. x * 4- x / + D(x/--^ xj") =. D a ( l + kD, ) +- Dk = D ab, = .Nb a, 2 ( x 7 x y * x,x^) = 2D a(y y ?^ + y a z , ) - 2D ab; - Nb^, 2 ( x A - x y x ^ ) = 2 D j ( y a z , - y v z , ) = . 2 ^ = 1 ^ . . Hence the r e l a t i o n s (17) are s a t i s f i e d and so i n t h i s case we have proved the e x i s t e n c e of a u n i t X, of norm N = f ( x ) - D a , which t r a n s f o r m s A i n t o B. Case I I . b ^ even, i . e . , b^ ~ 0 (mod 2 ) . Let b 2 = 1+D, +•' 2kD, , where k > 0. Since B i s a u n i t we have -D(b/^- b*) •- b / - 1 = D, (l), +- 4k*D, -/- 2 + 4k 4kD,j D '(!)'/ b^ ') = + 2k)* + 2(1 2k) -= (1 t..2k)(2 f D, + 2kD,h How o b v i o u s l y 1 + 2k i s prime to , or we should have' b - 1 d i v i s i b l e by D z i n c o n t r a d i c t i o n to our h y p o t h e s i s . Hence-2 +.D/ + SkD, must be a m u l t i p l e of'D . So we w r i t e b / 4- b a = (1 + 2k) ( 2 D/ rf. 2k"D, ). Since .k - 0, the 'two f a c t o r s are p o s i t i v e . A l s o -D,(l -h 2k) +-(2 .+ D, +• 2KD, ) = 2 , so the g r e a t e s t common . d i v i s o r of ( l -+ 2k) and (2 + D, -h 2kD, ) is 1 or 2. But ( l + 2k) i s odd. "ence the two f a c t o r s are r e l a t i v e l y prime. The' c o n d i t i o n s here are i n a.ccofd with the hypotheses of. the lemma so we have the e x i s t e n c e of i n t e g e r s Y/ ,Y^,zt, and z a such that. (23) \ y / + y / = ( ^ D ^ k D J . ^ •-•z,3'+ = .l..+-2k,.' ^ 3 = y, za- + y^z,-, b y=, y-az^ - y ( z.,.-Take x y D^y, , x^ = Byy, , x 3 ^ z, , x v - z A, and we can e a s i l y show that r e l a t i o n s (17.) are s a t i s f i e d by the u n i t X = '£*,_, , *4 ,#VJ of norm N.= f ( x ) =• 2D^. T h i s completes the proof of Theorem 4„ There remains now only the d e t e r m i n a t i o n of the value m. Let B - [p , b i ,b 3 ,ba] and B' ^ [u ,hj ,b/,b/j be two e l l i p t i c u n i t s , of the group F ( D ) . By the p r e v i o u s theorem there e x i s t s a u n i t X and a u n i t X 1 such that N = f ( X ) » f (X') = B u f o r \ odd, II =: f (X) = f ( X T ) = 2LV, f o r b^even, t r a n s f o r m i n g B and B' i n t o A. T h i s means t h a t X"BX - A , X^'B'X* = A. Then A ^ X"'BX - Z R / B 'X' , and so B' = X'X-'B XX' _ / =• (XX ," /)~ /B(XX ," /) . I f we can show t h a t (XX'"') i s a u n i t then we have shown t h a t B and B' are e q u i v a l e n t . Now - X x^x/ +• x^ x^ - x,x 3' - x A x ^ , - x 3 x j + x^x; - x y x ; + x 2 x ; J , T h i s product i s of norm N =. f (XX' _ /) = 1, so a l l t h a t remains to show t h a t i t i s a u n i t i s to show t h a t each of the c o o r d m ates i s d i v i s i b l e by N, where N = , f o r odu, and N — 2I)X f o r liz even. In- the proof of Theorem 4 our i n t e g e r s x ; and x^ were chosen to be m u l t i p l e s - of D . Hence f o r the case b 2 odd, the c o o r d i n a t e s a,re d i v i s i b l e by D A and so XX'~/ i s a u n i t . In the case of b x even we must show a l s o d i v i s i b i l i t y of each c o o r d i n a t e by 2. I t f o l l o w s from r e l a t i o n s (28) that i f b^ i s odd, b^ even, then y, 5 '?, , y^ s z^mod 2 ) . Then f o r the c o o r d i n a t e s to be d i v i s i b l e by 2 we must have y/ = z/ Xz.' ~ z l (moA 2 ) 9 i . e . , by odd, b^ ' even. This c o n d i t i o n i s expressed by (29) b 3 = b/ , b„ - lo< (Mod 2 ) . This then i s the c r i t e r i a t h a t XX'"' be a u n i t i n the case b^ even. T h i s a l l o w s two d i s t i n c t p o s s i b i l i t i e s f o r t h i s l a t t e r ca.se f o r we have one u n i t i f b 3 and b,' are b o t h odd, b^ and b,' b o t h even, and a.nother u n i t when b, and b' a.re both even w h i l e b^ and b j are odd. We have t h e r e f o r e , shown t h a t f o r b^ oda or even there e x i s t s a. u n i t t r a n s f o r m i n g B i n t o B f . Hence the two u n i t s B ana B' are i n the same c l a s s of e l l i p t i c u n i t s . Now i f b ^ i s oad we have the 2 ^ p o s s i b i l i t i e s to cnoose from and, s i n c e there w i l l be a c l a s s of e l l i p t i c u n i t s assoc i a t e d w i t h each c h o i c e , we have 2^ p o s s i b l e c l a s s e s of e l l i p t u n i t s . On the other hand i f b i s even we have 2 c l a s s e s of e l l i p t i c u n i t s , d i s t i n g u i s h e d by c e r t a i n congruences (29), a s s o c i a t e d w i t h every one of the 2^ c h o i c e s of b^ ..-- Hence i n t h i s case we have 2.2 p o s s i b l e c l a s s e s of e l l i p t i c u n i t s . A-In a l l then we have 3.2 p o s s i b l e c l a s s e s of e l l i p t i c u n i t s and hence m ^ 3 . 2 F U. We have here a c q u i r e d an upper l i m i t to the value of m. We s h a l l show that t h i s l i m i t i s a c t u a l l y a t t a i n e d by p r o v i n g the e x i s t e n c e of s o l u t i o n s of e q u a t i o n (2) where N = 1 , 2 8 Dj_, or 2L\ ; Dj. d e f i n e d as i n (24). S o l u t i o n s of e q u a t i o n (2) are e a s i l y o b t a i n e d f o r the cases N = 1, or 2, f o r i n these cases we have the u n i t s [0,1,0,0] and [ l , l , G , 0 j r e s p e c t i v e l y . The cases f o r N = , or N = 2D A are more d i f f i c u l t . We s h a l l f i r s t prove the e x i s t e n c e of s o l u t i o n s of the e q u a t i o n (30) x,1 + X j 1 - I)(x/ + x^) = p^, p^ d e f i n e d as i n (11), 1 £ i £ r. W r i t e D = P^B'- Then (31) x ; A + x / - p^D'(x/+- x/) = p;, x/ +- 4 = p, { l 4 D ' (x3*- + x ; ) ] , x,1 r = 0 (mod p„j) . Since i s a prime - 3 (mod 4 ) , we may w r i t e (32) x , = p^x; ., x ^ = p A x | and at the same time x 3 = X3 »' x ^ = x</ • The'n on s u b s t i t u t i n g these v a l u e s i n (31) we get. P * ( x ; * + x ; 4 ) - P , D ' ( x j ^ t -x^) = p^-} (33) P,-(x;*+ x i 2 - ) - D ' ( x ; a - f x j * ) = 1 . I f we can show the e x i s t e n c e of s o l u t i o n s of (3o) the t r a n s f o r mation (32) w i l l g i v e the s o l u t i o n s of (31) and of (30). 17 We now make use of the work of Humbert on the b i n a r y H e r m i t i a n form axx + bxy + bxy + cyy, i n which a a,nd c are r e a l , and where x denotes the conjugate of x. Humbert proves that a l l such forms, h a v i n g the same d i s c r i m i n a n t £> - bb - ac > 0, are e q u i v a l e n t . In other words, any two such forms, h a v i n g the same d i s c r i m i n a n t , can be c a r r i e d the one i n t o the o t h e r , by a l i n e a r t r a n s f o r m a t i o n on the v a r i a b l e s . Our quaternary q u a d r a t i c forms ( l ) ana (33) are. of t h i s form w i t h a - 1, b = 0, c = -D ,-«$' = D, and a ^ n< , b = 0, c = -JJ 1 , B = p, D ' = D, i n the r e s p e c t i v e c a s e s . The d i s c r i m i n a n t s are e q u a l and p o s i t i v e . Hence the two forms are e q u i v a l e n t . Then there e x i s t s a l i n e a r t r a n s f o r m a t i o n of the form (34) x/ = a„x, +• a / ax^ + a / 3x 3 + a / Vx^ , 17. Humbert, op. c i t . , v o l . 166, 1918, pp. 86o-370. 29 . x_' ^ a x, + a x, -f- a x, -+- a x , x' =• a x, a x_, + a x a x , whose determinant i s not z e r o , which, w i l l c a r r y (33) i n t o ( l ) . But we can determine a s o l u t i o n jjc, ,x A ,x^ , x v j of ( l ) . Sub-s t i t u t i n g these v a l u e s f o r x y , x^, x 5 , and x^ , i n (34J w i l l g i v e v a l u e s x,', x^ , x] ,- and x j which w i l l form a s o l u t i o n jjsJ ,x a',x^ ,xy'J of (33). Hence we have proved the e x i s t e n c e of s o l u t i o n s of (33) and so of (30) as d e s i r e d . How we know that there e x i s t s o l u t i o n s , £xf , ,x 3 ,x^J and [y, ,y A ,y 3 , y ^ j , of norms p^ - and p- r e s p e c t i v e l y , of the I two e q u a t i o n s x , a x / - D(x/"-y x/") = p^, y/" + y* - D(y/ + y/) « P^ . Then the product [ x ; , x t , T L 3 , X J J - ["y, ,y* ,y^ . y j a c c o r d i n g to the formula. (5) i s an element [z, , z 1 ( z 5 , z^J of norm p^ - p-which s a t i s f i e s the r e l a t i o n 1 - - 1 - D ( z j L +- z/) = p^p •. z, z a By p r o c e e d i n g i n t h i s way we can show s o l u t i o n s of the e q u a t i o n s x/" + x / - D ( x 3 ^ x^) - D, or 2D 2, where D a i s a product of the form (24.). Then t h i s completes the proof of the f o l l o w i n g Theorem 7. If D - p, .p 4 p A, r ^ 1, p L ^ p^ f o r i ^ j , and p^ s 3 (mod 4 ) , then the F u c h s i a n group F (D) , of t r a n s f o r -mations z =r (x; -f- ix'Qw + D ( X 3 - i x ¥ ) ( x 3 +- i x 4 ) w + x, - i x z > of the complex plane, has e x a c t l y 3.2'v d i s t i n e t c l a s s e s of e l l i p t i c t r a n s f o r m a t i o n s . 30 6. C o n c l u s i o n . We have now completed the d i s c u s s i o n f o r the case t h a t we have chosen. Y/e have determined the number of c a n o n i c a l g e n e r a t o r s f o r the F u c h s i a n group, F ( D ) , of t r a n s f o r m a t i o n s of the complex p l a n e , where D i s s u b j e c t to the r e s t r i c t i o n s of f o r m u l a (11) . From the'se g e n e r a t o r s we can determine gener-a t o r s of our group, G-(D), of s o l u t i o n s of the e q u a t i o n ( l ) by the use of the correspondence ( 6 ) , and by a d j o i n i n g the one u n i t [-1,0 , 0 , 0] as mentioned p r e v i o u s l y . We may sum up our work i n the form of Theorem V I . I f D = p,. p a p A , r * l , p f ¥ f or i * j , and p c prime = 3 (mod 4 ) , then the group G ( D ) of s o l u t i o n s i n i n t e g e r s of the e q u a t i o n x / + 4 ~ D(x 3 i + x,1) - 1, i s generated by a s e t of u n i t s c o n s i s t i n g of the s i n g l e u n i t [-1,0,0,0], and a set of m + 2h -canonical g e n e r a t o r s , TJ, , U 4 , U^, V,, V„ , V(' , V; , .... Y; , s u b j e c t to the c o n d i t i o n s C = u/« , u„* = - l , D , . u . . . . . . . n „ [ 7 T 7-Vv v/' j - - i , v/here rr - 3 • ^ 4h = q> (B) + 4 - m. To i l l u s t r a t e t h i s f i n a l r e s u l t we s h a l l r e f e r to our i l l u s t r a t i o n s of the cases D = 3, and D = 7, which, f u l f i l the hypotheses of the theorem. On the i l l u s t r a t i o n s we l i s t e d g e n e r a t o r s of the F u c h s i a n groups, F ( D ) S but these are not 31 the r e q u i r e d c a n o n i c a l g e n e r a t o r s of G ( D ) . We s h a l l a c t u a l l y show these s p e c i a l g e n e r a t o r s and show how they are determmed from the g i v e n ones. Case -- D — 3. D = p - 3, so r = 1. in — 3.2^ = 3.2 = 6. 4h - (? ( D ) -+ 4 - m = 2 -+-4 - 6 =- U, s o h = 0. Then we must have' m - f -2h -6 c a n o n i c a l g e n e r a t o r s . By a c t u a l t r i a l we are abl e to o b t a i n these from the u n i t s g i v e n . We f i n d them to be ' U, = A = To, 1,0,0], u^ = £0,-2,-1,0], C l " = [0,4,2,1], ED" = £0,5,2,2], DB' = & , 4 , l , - 2 ] , BA-/= £0,-2,0,-1] Since h =• 0 there w i l l be no ' s. These g e n e r a t o r s s a t i s f y the r e l a t i o n s of our theorem f o r 2 2 . 2 2 z 2 n, = ua = = u v *= u> = u 6 = - i , and TJ, .U2 ,U 5 .TJV .IV-Ui =/l.C'"/.CE~/.SD~/.DB"/.BA-/ = -1.. To the c a n o n i c a l g e n e r a t o r s we ado. the u n i t £-l,U,u,oJ . Hence 'we have the group G(D) g i v e n by G(3) = J £-1,0, 0,0] , [ 0 , 1 , 0 , ' 0 ] , [0,-2,-2,oJ, £0,4, 2,1] , [0,5,2,2], [ 0 , 4 , l , - 2 j , [v,-2,0,-1] J . Case -- D — 7. D - P i - 7 , so r = 1. m = 3 . 2 A = 3 . 2 = 6 . 4h = 8} (D) + 4 - m = 6 -h 4 - 6 = 4, so h = 1. Here we have m 2h - 6 + 2 - 8 c a n o n i c a l g e n e r a t o r s . S i x of these v / i l l be e l l i p t i c and the oth e r two h y p e r b o l i c . Again by t r i a l we f i n d them to be U, = G"' = [0,-8,-3,0] U 2 = FE"' = [o,6,2,l] Uj = ED"' = [u,6,l,2] U„ = D C 2F~ ' c ; '' = [0,-27,-2, -lo] Uj. = • BA"' = [0,-8,0,-3] U 6 = A = [o,l,0 ,oJ V, = C a~ y. = [-9,-12,^4,4] V/ = G^ 'F ^ [-2,-16,-6,l] A g a i n these c a n o n i c a l g e n e r a t o r s s a t i s f y the r e l a t i o n s of the theorem f o r u , . nx. u 3 . LV . u4_. u 6 . v/'. v / . v, . v/ = G"/.FE_/.ED~/.DCzF"/G;'.BA-/.A.G;i .G^F .G^'.F^G = G"'F CiF"/G;/BC_2G"/FC;/F~/G.. But from the r e l a t i o n s on the i l l u s t r a t i o n we have that C/;B =: GG1"/, and so u s i n g t h i s we have - (T/FCiF~/GC^/CaG:/FC;/F~'/G- = -1. To these e i g h t c a n o n i c a l g e n e r a t o r s we ado. the s i n g l e u n i t [-1,0,0,0] and we then have the group G(D) determined as f o l l o w s [0,-8,0,-3], [0,1,0,0], [-9,-12,-4, 4 ] , [ 0 ^ 7 , - ^ ] ? a. 2 a- 1 i 2 U, = U 4 = U 3 -- 1^ = = U 6 a. 0 ] , [0,-8,-3,0], [ 0 , 6 , 2 , l ] , [ 0 , 6 , l , 2 j , 7. B i b l i o g r a p h y . C o o i i d g e , J . L o w e l l , The Elements of Non-Euclidean Geometry, The Clarendon P r e s s , Oxford, 1909. Eord, L e s t e r R., Automorphic Func t i o n s , McGraw - H i l l Book Company, I n c . , New Y o r k , 1929. F r i c k e , R. and K l e i n , F., Automorphe F u n k t i o n e n . H u l l , Ralph, On the U n i t s of I n d e f i n i t e Q uaternion A l g e b r a s , American J o u r n a l of Mathe-m a t i c s , v o l L X I , no. 2, A p r i l , 1939, pp. 365 - 374. Humbert, G., Comptes Rendus P a r i s , v o l . 166, 1918. 

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