BIBLIOGRAPHIE

a

Les references anterieures 1934 ne figurent pas dans cette bibliographie, car on peut les trouver dans le livre de Deuring [1]. On comp1etera utilement les references ci-dessous avec cel1es du livre de Reiner [1], qui sont complementaires, mis part quelques exceptions.

a

G. AEBERLI [1] Der Zusammenhang zwischen quaternaren quadratischen Formen und Idealen in Quaternionenringen. Comment. Math. Helv. 33 (1959), 212-239. B. BECK [1] Sur les equations polynomia1es dans les quaternions. A paraitre l'Enseignement Mathernatique (1980).

a

G. BENNETON [lJ Sur un probleme d'Euler. C. R. Acad. Sci. Paris 214 ( 1942), 459-461[21

Sur l'arithmetique des quaternions. C. R. Acad. Sci. Paris 214 (1942), 406-408.

[3J

Sur l'arithmetique des quaternions et des biquaternions. Ann. Sci. Ecole Norm. Sup. (3) 60 (1943), 173-214.

[4J

Une arithmetique des biquaternions. C. R. Acad. Sci. Paris 216 (1943), 262-264.

[5J

Arithmetique des quaternions. Bull. Soc. Math. France 71 (1943), 78-111.

M. BERGER, P. GAUDUCHON, M. MAZET [lJ Le spectre d'une variete riemannienne. Springer-Verlag Lecture Notes 194 (1971). A. BLANCHARD [1] Les corps non commutatifs. Presses Universitaires de France, Collection SUP (1972). N. BOURBAKI [ 1] [2J

r3J

Algebre, ch. 8. Hermann, Paris (1972) .

Topologie Generale, ch. 1 Topologie Generale, ch. 5

a a

4

Hermann, Paris (1971) .

8

Hermann, Paris (1971) .

H. BRANDT [lJ Uber die Zerlegungsgesetze der rational en Zahlen in Quaternionenkorpern. Math. Ann. 117 (1941), 758-763. Zur Zahlentheorie der Quaternionen. Jber. Deutsch. Math.-Verein. 53 (1943), 23-57. R. BRAUER [1] Algebra des Hyperkomplexer Zahlensysteme (Algebren). Math. J. Okayama Univ. 21 (1979), 53-89. D.M. BROWN [lJ Arithmetics of rational generalized quaternion algebras. Bull. Amer. Math. Soc. 46 (1940),899-908. J.H.H. CHALK [lJ An estimate for the fundamental solutions of a generalized Pell equation. Math. Annalen 132 (1956) 263-276. [2]

Quelques equations de Pell generalisees. C. R. Acad. Sci. Paris Ser. A-B 244 (1957) 985-988.

158

r

3J

Generating Sets for Fuchsian Groups. Proc. R.S.E. 72, 27 (1974/74), 317-326.

[4J

Gen r a t.o r s of F'uch s ian Groups. Tohoku Math. J. 26, n " 2 (1974), 203-218.

[5]

Diophantine Pellian Equations and Units of Quaternion Algebras. Preprint 1977.

P. CARTIER et D. HEJHAL [l l Sur les zeros de la fonction zeta de Selberg. Preprint Paris, 1979. I.V. CEREDNIK 11J Uniformization of algebraic curves by discrete arithmetic sybgroups of PGL2(kw) with compact quotient spaces. Math. Sb. 100 (1942) (1976), 59-88. H. COHEN r 1 l 7ables numsrjques ces valeurs ces fonct;ons zeta des quadratiques reels aux entiers negatifs. Mathematics of computation, UMT file, et preprint 1974. H. COHEN et J. OESTERLE I1J Dimension des espaces de formes modulaires. Modular Functions of One Variable VI, Springer-Verlag Lecture Notes 627 (1977). M. COXETER [lJ The binary polyhedral groups and other generalizations of the quaternion group. Duke Math. J. 7 (1940), 367-379. [2J

Regular Complex Polytopes. Cambridge University Press (1974).

K. DEDEKIND 11J Uber die Anzahl der Ideal-Klassen in den verschiedinen Ordnungen eines endlicher Korpers. Gesarnmelte Werke 1. J. DIEUDONNE [lJ La geometrie des groupes classiques. Springer-Verlag, Ergebnisse der Math. und ihrer Grenzgebiete, 3e Ed. (1971). [2J

Algebre lineaire et geometrie elementaire, Annexe IV. Hermann, Paris (1964).

M. DEURING [lJ Algebren. Springer-Verlag, Ergebnisse der Math. und ihrer Grenzgebiete (1935). !2]

Die Anzahl der Typen von Maximalordnungen in einer Quaternionenalgebra von primer Grundzahl. Nachr. Akad. Wiss. Gottingen Math. Phys. KL (1945), 48-50.

[3J

Die Anzahl der Typen von Maximalordnungen einer definiten Quaternionenalgebra mit primer Grundzhal. Jber. Deutsch. Math. - Verein. 54 (1950), 24-41.

M. EICHLER [lJ Untersuchen in der Zahlentheorie der rational en Quaternionenalgebren. J. Reine Angew. Math. 174 (1936), 129-159. [2]

Uber die Klassenzahl total definiter Quaternionenalgebren. Math. Z. 43 (1937), 102-109.

13J

Bestirnmung der Idealklassenzahl in gewissen normalen einfachen Algebren. J. Reine Angew. Math. 176 (1937), 192-202.

[4J

Uber die Idealklassenzahl hyperkomplexer Systeme. Math. Z. 43 (1938), 481-494.

[5J

Allgemeine Kongruenzklasseneinteilungen der Ideale einfacher Algebren tiber algebraischen Zahlkorpern und ihre L-Reihen. J. Reine Angew. Math. 179 (1938), 227-251.

159

[6J

Arithmeties of orthogonal groups. Proe. of the Internat. Congress of Math. Cambridge (1950) vol. 2, 65-70.

[7J

Quadratisehe Formen und orthogonale Gruppen. Springer-Verlag (1952).

[8J

Zur Zahlentheorie der Quaternionen-Algebren. J. Reine Angew. Math. 195 (1955), 127-151. Berichtigung : J. Reine Angew. Math. 197 (1957), 220.

[9J

Uber die Darstellbarkeit von Modulformen durch Thetareihen. J. Reine Angew. Math. 195 (1955), 156-171. Berichtigung : J. Reine Angew. Math. 196 (1956), 155.

[llJ Quadratische Formen und Modulfunktionen. Acta Arith. IV (1958), 217-239. [12J The Basis Problem for Modular Forms and the Traces of the Heeke Operators. Modular Functions of One Variable, Springer-Verlag Lecture Notes 320 (1973). Corrigenda Springer-Verlag Lecture Notes 476. [13J Theta Functions over and over Variable VI, Springer-Verlag 627 (1977).

Modular Functions of One

[14J On theta functions of real algebraic number fields. Acta Arith. XXXIII (1977), 269-292. H.G. FRANKE Kurven in Hilbertsche Modulflachen und Humbertsche Flachen im Siegel-Raum. Bonner Mathematischen Schriften. Bonn (1978). R. FRICKE, F. KLEIN [lJ Vorlesungen Uber die Theorie der automorphen Funktionen I, II (1897). Teubner reprint (1965). R. FUETER [lJ Zur Theorie der Brandtschen Quaternionenalgebren. Math. Ann. 110 (1935), 650-661. G. FUSIJAKI [lJ On the Zeta-Function of the Simple Algebra over the Field of Rational Numbers. J. Fac. Sci. Univ. Tokyo Sect. IA Math. I.M. GELFAND [lJ Automorphic Functions and the Theory of Representations. Proc. Internat. Congress of Math. Stockholm (1962). I.M. GELFAND, M.I. GRAEV, 1.1. PIATETSKII-SHAPIRO [lJ Representation Theory and Automorphic Functions. W.B. Saunders (1969). R. GODEMENT [lJ Les desalgebres simples I et II. Seminaire Bourbaki 1958/1959, Exposes 171 et 176. [2J

Notes on Jacquet-Langlands'theory. Preprint I.A.S. (1970).

R. GODEMENT, H. JACQUET [lJ Zeta Functions of Simple Algebras. SpringerVerlag Lecture Notes 260 (1972). M.-F. GUEHO [lJ Corps de quaternions et fonction zeta au point -1 . C. R. ACad. Sci. Paris Ser. A-B 274 (1972), 296-298 et these de 3e cycle, Bordeaux (1972). Ki-I HASHIMOTO [lJ Twisted Trace Formula of the Brandt Matrix. Proc. Japan Acad. Ser. A Math. Sci. 53 (1977), 98-102. H. HASSE [lJ Uber die Klassenzahl abelscher Zahlkorper. Akademie Verlag Berlin (1952). H. HASSE und O. SCHILLING [lJ Die Normen aus einer normalen Divisionsalgebra. J. Reine Angew. Math. 174 (1936), 248-252.

160

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On Units of Indefinite Quaternions Algebras. Amer. J. of Math., 61 (1939), 365-374.

J. IGUSA [ll Class number of a definite quaternion with prime discriminant. Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 312-314. Y. lHARA [1] The congruence monodromy prob Lerns • J. Math. Soc. Japan 20 (1968), 107-12l. On congruence monodromy problems. Math. Univ. Tokyo Lecture Notes 1, 2 (1968). H. JACQUET, R.P. LANGLANDS [1] Automorphic Forms on Verlag Lecture Notes 114 (1970).

GL(2). Springer-

M. KUGA and G. SHIMURA [1] On the zeta function of a fibre variety whose fibres are abelian varieties. Ann. of Math. (2) 82 (1965), 478-539. M. KNESER [1] Approximationssatze fUr algebraische Gruppen. J. Reine Angew. Math. 209 (1962), 96-97. f2]

Starke Approximation in algebraischen Gruppen. J. Reine Angew. Math. 218 (1965) 190-203.

[3J

Stong Approximation. Algebraic Groups and Discontinuous Subgroups. Proc. Sympos. Pure Math. Boulder 1965. Amer. Math. Soc. (1966), IX, 187-196.

A. KURIHARA [1] On some examples of equations defining Shimura curves and the Mumford uniformization. J. Fac. Sci. Univ. Tokyo Sec. IA 25 n03 (1979), 277-300. T.Y. LAM r1l The Algebraic Theory of Quadratic Forms. W.A. Benjamin Inc, (1970) ". S. LANG [lJ Expose au seminaire Delange-Pisot-Poitou (1978). C.G. LATIMER [lJ On the class number of a quaternion algebra with a negative fundamental number. Trans. Amer. Math. Soc. 40 (1936), 318-323. [2J

Quaternion algebras. Duke Math. J. 15 (1948), 357-366.

H. LEPTIN [lJ Die Funktionalgleichung der Zeta-Funktion einer einfacher Algebra. Abh. Math. Sem. Hamburg 19 (1955), 198-220.

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Yu. V. LINNIK [lJ Quaternions and Cayley numbers; some applications of the arithmetic of quaternions. Uspehi Mat. Nauk. 4 n05 (33), (1949), 49-98. [2]

Quaternions and Cayley numbers. Math. Centrum

[3J

Application of the theory of Markov chains to the arithmetic of quaternions. Uspehi Mat. Nauk. 9 n04 (62), (1954), 203-210.

(1959).

[4J

Markov chains in the analytical arithmetic of quaternions and matrices. Vestnik Leningrad Univ. 11 (1956), 63-68.

H. MAAS [lJ Beweis des Normensatzes in einfachen hyperkomplexen Systemen. Abh. Math. Sem. Hamburg 12 (1937), 64-69. [2J

Die Bestimmung der Dirichletreihen mit Grossencharakteren zu den Modulformen n-ten Grades. J. Indian Math. Soc. 19 (1955), 1-23.

W. MAGNUS [lJ Noneuclidean tesselations and their groups. Academic Press (1974). J.-F. MICHON [lJ

Courbes de Shimura hyperelliptiques. Preprint Paris 1980.

J. MILNOR [lJ Eigenvalues of the Laplace Operator of certain manifolds. P.N.A.S. 51 n04 (1964) 542. D. MUMFORD [lJ An analytic construction of degenerating curves over complete local rings. Compositio Math., 24 (1972), 129-174. I. NIVEN [lJ Equations in quaternions. Ann. Math. Monthly 48 (1941), 654-661. [2J

A note on the number theory of quaternions. Duke Math. J. 13 (1946), 397-400.

A. NaBS [lJ Konstruktion von automorphen Funktionen durch Spezialisierung von Siegelschen Modulfunktionen. These Basel (1972). J. OESTERLE [lJ Sur la trace des operateurs de Hecke. These de 3e cycle, Orsay 1977. O.T. O'MEARA [lJ

Introduction to quadratic forms. Springer-Verlag (1963).

T. aNa [lJ On Tamagawa Numbers. Algebraic Groups and Discontinuous Subgroups. Proc. Sympos. Pure Math. Boulder 1965. Amer. Math. Soc. (1966), 122-132. G. PALL [lJ Quaternions and sums of three squares. Amer. J. Math. 64 (1942), 503-513. [2J

On generalized quaternions. Trans. Amer. Math. Soc. 59 (1946), 280-332.

M. PETERS [lJ Ternare und quaternare quadratische Formen und Quaternionenalgebren. Acta Arith. 15 (1968/69), 329-365. A. PIZER [lJ Type numbers of Eichler orders. J. Reine Angew. Math. 264 (1973), 76-102. Dissertation, Yale Univ. 1971. [2J

On the arithmetic of quaternion algebras I. Acta Arith. 31 (1976) n01, 61-89.

[3J

On the arithmetic of quaternion algebras II. J. Math. Soc. Japan 28 (1976), n04, 676-688.

162

:4]

The representability of modular forms by theta series. J. Math. Soc. Japan 28 (1976), 689-698.

[5;

A note on a conjecture of Heeke. Pacific J. of Math. 72 541-548.

[6J

An algorithm for computing modular forms.

(1978),

Preprint (1979), University of Rochester. To appear in J. of Algebra.

H. POINCARE [1] Oeuvres. Gauthiers-Villars, Paris (1916) tome II. Theorie des groupes fuchsiens, 108-168. Memoire sur les groupes kleineens, 258-299. B. POLLAK [1 J The equation J. 27 (1960), 261-271.

tat

=b

in a quaternion algebra. Duke Math.

P. PONOMAREV [lJ Class numbers of positive definite quaternary forms. Bull. Amer. Math. Soc. 76 (1970), 261-271. [2J

Class number of definite quaternary forms with nonsquare discriminant. Bull. Amer. Math. Soc. 79 (1973), 594-598.

[3J

Class Numbers of Definite Quaternary Forms with Square Discriminant. J. Number Theory 6 (1974), 291-317.

:4 1

Arithmetic of quaternary quadratic forms. Acta Arith. 29 (1976), 1-28.

[5]

A correspondence between quaternary quadratic forms. Nagoya Math. J. 62 (1976), 125-140.

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Maximal Ord8rs. Academic Press (1975).

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163

[2J

Cours d'arithmetique. Presses Universitaires de France, collection PUF (1970).

[3]

Arbres, amalgames, SL 2 . Asterisque 46 (1977).

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On traces of Hecke operators. J. FaC. Sci. Univ. Tokyo Sect. IA 10 (1963), 1-19.

[3l

On zeta functions of quaternion algebras. Ann. of Math. (1965), 166-193.

(2) 81

G. SHIMURA [ll On the theory of automorphic functions. Ann. of Math. 70 (1959) 101-144.

(2)

[2]

On the zeta-functions of the algebraic curves uniformized by certain automorphic functions. J. Math. Soc. Japan 13 (1961), 275-331.

[3]

Class-fields and automorphic functions. Ann. of Math. 444-463.

[4J

On the field of definition for a field of automorphic functions II. Ann. of Math. 81 (1965), 124-165.

[5J

Construction of class fields and zeta functions of algebraic curves. Ann. of Math. (2) 85 (1967), 58-159.

[6]

Introduction to the theory of automorphic functions. Princeton University Press (1971).

C.L. SIEGEL

(2) 80 (1964),

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[1]

The volume of the fundamental domain for some infinite groups (1936). Vol. 1 , 459-468.

[2]

Discontinuous Groups (1943). Vol. 2 (1943), 390-405.

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SL 2(R). J. Fac. Sci.

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Commensurability classes of arithmetic triangle groups. J. Fac. Sci. Univ. Tokyo Sect. 1A 24 (1977), 201-212. T. TAMAGAWA [lJ On the zeta function of a division algebra. Ann. of Math. 77 (2) (1963), 387-405. [2]

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1M

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[3J

Varietes riemanniennes isospectrales et non isometriques. Preprint Paris (1978). A paraitre aux Annals of Math.

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[3]

Simplification pour les ordres de corps de quaternions totalement definis. C. R. Acad. Sci. Paris Ser. A-B 279 (1974), 537-540. J. Reine Angew. Math. 286-287 (1976) 257-277.

A. WElL [1] [2J

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INDEX pages .

59

(th. fondamental)

.

61

Approximation forte (th. d')

.

81

Adeles . . . . . . . . • • . . . . . . . . . . . . . . . . . . . . . . '"

Ca r a c t.e r-e .•••...•••••••••••••••.•.•...•.•.•••

51,61,64,67

Classe d' ideaux

.

25,87,144

de conjugaison modulo un groupe

.

27,95-98

.

31

Classification (th. de) local

.

74

Commutateurs

global

.

13

Conducteur relatif d'un ordre

.

44

Conjugaison • • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Corestriction . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . .

10,101

Corps neutralisant

.

4,9,76-78

Cycle

.

117

(anneau de)

.

19

(formule de nombre de classes)

Dedekind .

95

Dictionnaire q Loba Lv-ade Li.que •••••••••••••••••

87

Differente

23

.

Dirichlet (formule analytique de nombre de classes)

142

(th. des unites)

139

.

Discriminant r edui.t. ..•••..•.....•••..•......•.••.•...

23

r edu i t, local

.

35,40

local

.

52

r edu i. t, global

.

59,84

global

.

65

Domaine fondamental d'un s.-g. discret de

SL(2,R)

.

du groupe de Picard . . . . . . . . . . . . . . . . . • . . .

116,123 135

166

Eichler (condi tion d' -)

.

82

(th. des no rme s )

.

80,90

(th. des progressions arithmetiques)

.

90

(matrices d' )-Brandt

.

100,144

.

19

Entier (quaternion) (ideal) .. ,

' .,

Equations polynomiales en quaternions

.

20

.

29

.

52

Fonction canonique locale.............. globale. ... .. . ..... .

. . .. .

.. . . .. . .

67

Frobenius (th. d e ) . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Fujisaki (th. d e ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

Genre de surfaces de Riemann.................

119

Graphe ari t.hrne t i.que . . . . . . . . . . . . . . . . . . . . . . . . . .

146

Groupes finis de rotations......................

14

finis de quaternions reels..............

17

commensurables, arithmetiques, de quaternions de picard...............................

105

108,135

modulaire de Hilbert .

107,109,120

Hamilton (quaternions de)

.

4

de congruence

Hasse (invariant de)

.

32

Hasse Minkowski (principe de)

.

75

Homographie

.

111

(multiplicateur, norme)

.

115

primitive

.

128

Hyperbolique (metrique)

.

112,133

Ideal

.

20-22,86

.

59

.

62

(cercle d')

.

111

(groupe des) de (groupe des) de

. .

112 133

Ideles ( th. fondamental) Isometrie

):13

167

Leech (reseau de)............................

149

Lobachevsky (fonction de)....................

134

Masse

141

'"

Mesures normalisees locales.....................

49

normalisees globales....................

65

compatibles. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Module d' un isomorphisme.....................

49

Niveau d'un ordre d'Eichler local

.

global

'"

39 84

Normalisateur .•..............................

29

d'un ordre d'Eichler

.

39,99

d'un groupe de congruence

.

121

reduite d'un element

.

1

reduite d'un ideal

.

24

d 'un ideal •.............................

47

dans les ad e Le s

60

Norme

.

Ordre . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19,20

d 'Eichler local .......................•.

39,55

(arbre des) maximaux ................•...

40

euclidien

.

90.156

.

57

,

Place

,

Plongement maximal dans les ordres ..........• (formules nurne r i que s )

.

23,36,42-44 92

Pointe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . .

117

Po Lyriome minimal .•...........................

2

Produit restreint

.

59

tensoriel

.

9

.

79,85,98-99

Quaternions sur

(exemples)

(ordres, discriminants)

(groupes de quaternions) ...............•

108,120

(e.E. n'est pas verifiee)

143,161

.

168

Ramification locale

.

31,35

globale

.

58

Regula':eur

.

138,139

Representation (F-) de quaternions

.

4

Re s e a u •••••••••••••••••••••••••••••••••••••••

(passage local-global)

.

19 83

Residuel (corps)

.

34

(degre, extension)

.

35

Riemann (surface de) hyperelliptique............

123

(surface)-ienne, isospectrale non isometrique

127

Schwartz-Bruhat (espace de) local

.

51

global

.

66

.

6

de Hilbert

.

32,36,76

d'Artin,d'Eichler

.

43,94,102

.

52

.

71

Skolem-Noether (th. de) Symbole

Tamagawa (mesure de) locale globale

'

'

de

SL(2,1R) •.••••.••••••••••

de

SL(2,q:) •.•••••••.••••••.

(nombres de)

.

" .. '

.

Trace

114

134 71 50

r edu i t e ••....•.•.....•..••.••...•.....••

1

(formules de)

.

92-99,142

Uni t.e s d' un ordre

.

28,136

Uni formi san te

.

33

Valuation

.

33

va r i e t.e a.be Li.erme •.•••.••.•..•..••..••...••.•

28

Volumes (calculs de)

.

49,54,108-110,120

Wedderburn (th. de)

.

7

169

Zeta (fonction) locale

.

globale

.

48,52 64

(formule mul tiplicati ve )

.

65,66

(Equation fonctionnelle)

.

67