2004 Fiscal Year Final Research Report Summary
Study on the arithmetic discontinuous groups
| Project/Area Number |
14540008
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Saitama University |
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Principal Investigator |
TAKEUCHI Kisao Saitama University, Dept.of Math., Professor, 理学部, 教授 (00011560)
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| Co-Investigator(Kenkyū-buntansha) |
SATO Takakazu Saitama University, Dept.of Math., Associate Professor, 理学部, 助教授 (70215797)
SAKAI Fumio Saitama Univ., Dept.of Math., Professor, 理学部, 教授 (40036596)
EBIHARA Madoka Saitama Univ., Dept.of Math., Lecturer, 理学部, 講師 (80213578)
ARAI Michio Saitama Univ., Dept.of Math., Assistant, 理学部, 助手 (40008850)
YANO Tamaki Saitama Univ., Dept.of Math., Professor, 理学部, 教授 (10111410)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Number theory / Discontinuous group / Algebraic number field / Discriminant |
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| Research Abstract |
One of the aims of this project is to determine all arithmetic Fuchsian groups Γ with signature (O;e_1,e_2,e_3,e_4) explicitely.
(1)Let K be a totally real algbraic number field K of dgree n. Let A be a quaternion algebra over K such that A【cross product】_Q R〓M_2(R) 【symmetry】H^<n-1>, where H is the Hamilton quaternion algebra. Let O is an order of A. Let Γ^1(A,O) be a Fuchsian group derived from unit group O^1 of O of norm 1. If a Fuchsian group Γ is commensurable with Γ^1 (A,O), Γ is called K-arithmetic Fuchsian group. If there exists a K-arithmetic Fuchsian group Γ with signature (O;e_1,e_2,e_3,e_4), then the degree [K:Q]【less than or equal】10. Moreover, the explicit upper bound D_0 of the discriminant d(K) of such fields K is given. We have determined the imprimitive field K of degree 10 with minimum discriminant d(K). (c.f.K.Takeuchi [1])
(2)We have studied the case K=Q. Let A be a quaternion algebra over Q and let O(f) be an Eichler order in A with square-free level f. Let Γ^*(A,O(f)) be the normalizer ofΓ^1(A,O(f)) in SL_2(R). We show that if Γ is a Q-arithmetic Fuchsian group. Then Γ is a subgroup of Γ^*(A,O) of finite index, (c.f.K.Takeuchi [2])
Consequently, we can deternime all Q-arithmetic Fuchsian groups Γ with signature (O;e_1,e_2,e_3,e_4) explicitly.
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Research Products
(11 results)
