The Arithmetic of Quaternion Algebra

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Contents 1 Quaternion Algebra over a field 1 1.1 Quaternion Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The theorem of automorphisms and the neutralizing fields . . . . 5 1.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Orders and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Quaternion algebra over a local field 23 2.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Study of M(2, K) . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Orders embedded maximally . . . . . . . . . . . . . . . . . . . . 32 2.4 Zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Quaternion algebra over a global field 43 3.1 Adeles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Zeta function, Tamagawa number . . . . . . . . . . . . . . . . . . 48 3.3 Cassification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4 Norm theorem and strong approximation theorem . . . . . . . . 61 3.5 Orders and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4 Applications to Arithmetic Groups 79 4.1 Quaternion groups . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2 Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Examples and Applications . . . . . . . . . . . . . . . . . . . . . 92 5 Quaternion arithmetic in the case where the Eichler condition is not satisfied any more 105 5.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2 Class number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 i
Page 1: The Arithmetic of Quaternion Algebr
Page 5 and 6: Chapter 1 Quaternion Algebra over a
Page 7 and 8: 1.1. QUATERNION ALGEBRA 3 We shall
Page 9 and 10: 1.2. THE THEOREM OF AUTOMORPHISMS A
Page 11 and 12: 1.2. THE THEOREM OF AUTOMORPHISMS A
Page 13 and 14: 1.3. GEOMETRY 9 Corollary 1.3.2. Th
Page 15 and 16: 1.3. GEOMETRY 11 Theorem 1.3.6. The
Page 17 and 18: 1.4. ORDERS AND IDEALS 13 the isotr
Page 19 and 20: 1.4. ORDERS AND IDEALS 15 Definitio
Page 21 and 22: 1.4. ORDERS AND IDEALS 17 Under the
Page 23 and 24: 1.4. ORDERS AND IDEALS 19 (3)O and
Page 25 and 26: 1.4. ORDERS AND IDEALS 21 Prove D(z
Page 27 and 28: Chapter 2 Quaternion algebra over a
Page 29 and 30: 2.1. CLASSIFICATION 25 Theorem 2.1.
Page 31 and 32: 2.1. CLASSIFICATION 27 F ⊗ Lnr is
Page 33 and 34: 2.2. STUDY OF M(2, K) 29 where n, m
Page 35 and 36: 2.2. STUDY OF M(2, K) 31 and Li+1/(
Page 37 and 38: 2.3. ORDERS EMBEDDED MAXIMALLY 33 T
Page 39 and 40: 2.4. ZETA FUNCTION 35 This number i
Page 41 and 42: 2.4. ZETA FUNCTION 37 Definition 2.
Page 43 and 44: 2.4. ZETA FUNCTION 39 , where U is
Page 45 and 46: 2.4. ZETA FUNCTION 41 From this , w
Page 47 and 48: Chapter 3 Quaternion algebra over a
Page 49 and 50: 3.1. ADELES 45 where p.p. means ”
Page 51 and 52: 3.1. ADELES 47 2)(theorem of approx
Page 53 and 54:
3.2. ZETA FUNCTION, TAMAGAWA NUMBER
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
3.3. CASSIFICATION 57 Verify the fo
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3.3. CASSIFICATION 59 Corollary 3.3
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3.4. NORM THEOREM AND STRONG APPROX
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3.5. ORDERS AND IDEALS 63 We can pr
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3.5. ORDERS AND IDEALS 65 From II.1
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3.5. ORDERS AND IDEALS 67 Proof. Ac
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3.5. ORDERS AND IDEALS 69 If x �=
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3.5. ORDERS AND IDEALS 71 is equiva
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3.5. ORDERS AND IDEALS 73 Card(G\O
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3.5. ORDERS AND IDEALS 75 Exercise
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3.5. ORDERS AND IDEALS 77 c) The mu
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Chapter 4 Applications to Arithmeti
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4.1. QUATERNION GROUPS 81 set of T
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4.1. QUATERNION GROUPS 83 The expli
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4.2. RIEMANN SURFACES 85 6. Hilbert
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4.2. RIEMANN SURFACES 87 Corollary
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4.2. RIEMANN SURFACES 89 double poi
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4.2. RIEMANN SURFACES 91 (2) The nu
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4.3. EXAMPLES AND APPLICATIONS 93 t
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4.3. EXAMPLES AND APPLICATIONS 95 h
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4.3. EXAMPLES AND APPLICATIONS 97 3
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4.3. EXAMPLES AND APPLICATIONS 99 f
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4.3. EXAMPLES AND APPLICATIONS 101
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4.3. EXAMPLES AND APPLICATIONS 103
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Chapter 5 Quaternion arithmetic in
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5.1. UNITS 107 Exercise 1. Structur
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5.2. CLASS NUMBER 109 Corollary 5.2
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5.3. EXAMPLES 111 Corollary 5.2.5.
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5.3. EXAMPLES 113 isomorphic to Q
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5.3. EXAMPLES 115 that the differen
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5.3. EXAMPLES 117 The class number
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The Arithmetic of Quaternion Algebra

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