2004 Fiscal Year Final Research Report Summary
Kac-Moody Lie algebra and Hilbert modular forms
| Project/Area Number |
14540022
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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 | Mie University |
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Principal Investigator |
TSUYUMINE Shigeaki Mie University, Faculty of Education, Professor, 教育学部, 教授 (70197763)
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| Co-Investigator(Kenkyū-buntansha) |
KOSEKI Harutaka Mie University, Faculty of Education, Professor, 教育学部, 教授 (60234770)
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| Project Period (FY) |
2002 – 2004
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| Keywords | theta series / totally real algebraic number field / quadratic forms / automorphic function / Weyl group |
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| Research Abstract |
Its is known that by integrating modular forms for SL_2(Z) with the kernel function which is theta series of indefinite quadratic form with polynomial, one obtain automorphic forms with singularities on Grassmann manifolds. There is a series of papers by R.E.Borcherds on this topic. Let A be the nonsingular integral symmetric matrix corresponding to the real part of variable of theta series, and let A_+ be the positive definite real symmetric matrix corresponding to the imaginary part. The matrix A_+ salifies the condition A_+A^<-1>A_+=A, and the whole of A_+ form the Grassmann manifold. Under some conditions of A, the singularities of the automorphic form determines Weyl chamber on the Grassmannians. It follows the relation between the automorphic forms and the Weyl groups of some Kac-Moody Lie algebras, and their denominator functions.
Let K be a totally real algebraic number field, and let O_K be the ring of integers. In this research, we try to extend the all of the above argument to the case of Hilbert modular group SL_2(O_K). We show the inversion formula and transformation formulas for theta series with polynomial, of symmetric matrix A with coefficients in K. Further in the case that A has an anisotropic vector, we extend the theta series to the series involving theta series of quadratic forms of lower degree. This result is corresponding to Theorem 5.2 of Borcherds' paper "Automorphic forms with singularities oh Grassmanns", which is the key to obtaining automorphic form on Grassmanns. This result may be useful to study automorphic forms associated with Weyl chamber which is rational over K.
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