2005 Fiscal Year Final Research Report Summary
Algebraic points on shimura curves
| Project/Area Number |
16540041
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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 | Chuo University |
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Principal Investigator |
MOMOSE Fumiyuki Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (80182187)
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| Project Period (FY) |
2004 – 2005
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| Keywords | Shimura curves / algebraic points / isogeny character |
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| Research Abstract |
We studied the problem of classification of isogenies of "prime" degree of QM-abelian surfaces over given algebraic number field of finite degree. In this form, we can apply the methods of the classification of isogenies of prime degree of elliptic curves. But, we studied the classification of algebraic points on the Shimura curves with prime level, which is the coarse moduli space of the QM-abelian surfaces. Then, there appeared difficult problems. Because, in this case, we can not have a good model for an point over given algebraic number field k.
Using the theory of the Brauer group, we take a quadratic extension K of k over which the corresponding element becomes 0, and we used the composite of isogeny character with the transfer map associated with the quadratic extension. Then, the 12 th power of the composite map is unramified outside the prime level p, and 4 th power of it is independent of the choice of the model. Using this map, we classified our objects into two type I, II, except for some explicit exceptions. A key point is that there are so many choices of the quadratic extensions.
For type II, we got similar results as for the elliptic curves, using isogeny character over K. For type I, using the study of the reduction of Shimura curves, we showed that p is congruent to 3 mod 4. Then, we got similar results as before. For the above study, we needed a condition for a prime number. But, the proof before allow at most two exceptional primes, and we could use of a results of the analytic number theory.
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