2012 Fiscal Year Final Research Report
Study on the arithmetic theory of automorphic forms of several variables
| Project/Area Number |
22540036
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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 | Kinki University |
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Principal Investigator |
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| Project Period (FY) |
2010 – 2012
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| Keywords | 整数論 / 保型形式 / p 進理論 |
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| Research Abstract |
The p-adic theory of modular forms initiated by J.-P. Serre and H.P.F.Swinnerton-Dyer was established in the case of one variable by several people. I tried to generalize the theory to the case of several variables, for example, Siegel modular forms and Hermitian modular forms, and obtained several results. More specifically, I established a constructing method by taking a p-adic limit of a sequence of ordinary modular forms. Concerning the mod p theory, I studied the structure of the graded ring of Hermitian modular forms and determined the structure in some cases. Moreover, I generalized some congruence properties of Siegel modular forms originated with Ramanujan. This result was applied to show the existence of cusp forms of several variables. I studied the p-adic theory and mod p theory of modular forms of several variables and produced fruits. Moreover, the case of vector-valued was considered. In particular, the p-adic theory of the theta operator, which is a kind of differential operator on modular forms, was developed.
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Research Products
(12 results)
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[Presentation] forms2012
Author(s)
S. Boecherer,長岡昇勇
Organizer
日本数学会秋季総合分科会
Place of Presentation
九州大学
Year and Date
2012-09-18
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