2004 Fiscal Year Final Research Report Summary
Construction of newform theory for modular forms of half-integral weight
| Project/Area Number |
14540027
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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 | Nara Women's University |
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Principal Investigator |
UEDA Masaru Nara Women's University, Faculty of Science, Professor, 理学部, 教授 (80193811)
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| Project Period (FY) |
2002 – 2004
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| Keywords | modular form / half-integral weight / newform / twisting operator / metaplectic group |
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| Research Abstract |
The purpose of our research is to construct a theory of newforms of half-integral weight. We investigated this subject from April 2002 to March 2005. And we have the following results.
First, we proved trace identities of the twisted Hecke operators in the case of arbitrary even levels and any even conductors. As we already conjectures, the traces of the twisted Hecke operators are represented by linear combinations of traces of Hecke operators and Atkin-Lehner operators of integral weight.
Next, we successfully established a theory of newform of half-integral weight in the case that levels are powers of 2.
In order to get a theory of newforms of half-integral weight, we need to completely describe spaces of oldforms, and then for that, we must obtain a certain non-vanishing property of Fourier coefficients of cusp forms of half-integral weight. We got such non-vanishing properties by using representation theory of Metaplectic groups over quotient rings modulo powers of 2.
Our final purpose is to get a theory of newform of half-integral weight for arbitrary levels N. For that, we must extend the above non-vanishing properties for arbitrary rings of residue classes modulo N. We are now investigating such non-vanishing properties.
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