Construction of newform theory for modular forms of halfintegral weight
Project/Area Number 
14540027

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Nara Women's University 
Principal Investigator 
UEDA Masaru Nara Women's University, Faculty of Science, Professor, 理学部, 教授 (80193811)

Project Period (FY) 
2002 – 2004

Project Status 
Completed (Fiscal Year 2004)

Budget Amount *help 
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2004: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)

Keywords  modular form / halfintegral weight / newform / twisting operator / metaplectic group / ニューホーム / ツイスティング作用素 
Research Abstract 
The purpose of our research is to construct a theory of newforms of halfintegral 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 AtkinLehner operators of integral weight. Next, we successfully established a theory of newform of halfintegral weight in the case that levels are powers of 2. In order to get a theory of newforms of halfintegral weight, we need to completely describe spaces of oldforms, and then for that, we must obtain a certain nonvanishing property of Fourier coefficients of cusp forms of halfintegral weight. We got such nonvanishing 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 halfintegral weight for arbitrary levels N. For that, we must extend the above nonvanishing properties for arbitrary rings of residue classes modulo N. We are now investigating such nonvanishing properties.

Report
(4 results)
Research Products
(6 results)