2018 Fiscal Year Final Research Report
Shimura lifting of Hilbert modular forms and its applicaion
| Project/Area Number |
16K05056
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Mie University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | Hilbert modular form / Shimura lifting / quadratic form / L function |
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| Outline of Final Research Achievements |
(1) It is shown that Hilbert modular forms of half integral weight have Shimura lifting, provided that the level is divisible by 16 and the weight is at least 5/2. The lifting map is described explicitly in terms of Fourier coefficients. Though the condition is not satisfied, the third power of theta series has lift. As its application, the condition that algebraic integers in the quadratic field K containing square root of 2 are the sums of three squares, is obtained. Also the class numbers of the imaginary quadratic extensions of K is ordained.
(2) Let f,g be elliptic modular forms of level N, and let a(n),b(n) be their n-th Fourier coefficients respectively. Let L(s;f,g) be the Dirichlet series whose n-th coefficient is a product of a(n) and the complex conjugate of b(n). Then it is shown that L(s;f,g) extends meromorphically to the whole s plane, and that it has a functional equation. Also the applications to quadratic forms are shown.
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| Free Research Field |
代数学
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| Academic Significance and Societal Importance of the Research Achievements |
(1) Hilbert保型形式のShimura liftingが尖点形式に限定せず，Fourier係数を使って記述されたのは初めてである．応用としてルート２を含む２次体の総虚２次拡大体の類数が初等的な計算のみで求められるようになった．ルート２を含む２次体以外でも限定的ではあるがこれが可能である．（2）全複素平面に有理的に解析接続し，関数等式を持つL関数の種類を増やした．2次形式への応用は既に得られているが，これを始め，様々な応用があると期待される．
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