Infinite products of automorphic forms
Project/Area Number  09640024 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Algebra

Research Institution  Shizuoka University 
Principal Investigator 
ASAI Tetsuya Shizuoka Univ.Science Professor, 理学部, 教授 (50022637)

CoInvestigator(Kenkyūbuntansha) 
NIWA Shinji Nagoya City Univ.Art & Tech.Professor, 芸術工学部, 教授 (00123323)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥1,200,000 (Direct Cost : ¥1,200,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1997 : ¥700,000 (Direct Cost : ¥700,000)

Keywords  modular function / automorphic form / infinite product / circle method / Kloosterman sum / lifting of automorphic form / Siegel modular / Eisenstein series / モジュラー関数 / 保型形式 / 無限積 / サークルメソッド / クルスターマン和 / 保型形式の持上げ / ジーゲルモジュラー / アイゼンシュタイン級数 
Research Abstract 
The research project has been pursued by two authors mainly on number theory of automorphic forms, especially on the infinite products of modular functions and on lifting theory of various automorphic forms. 1. Concerning the modular function with real coefficients of the qexpansion, it is known that the sequences of signs of the coefficients are very often periodic. The cases of Thompson series are observed by McKay Strauss, and some special cases including 1/j are treated by the first author and KanekoNinomiya. On many cases of the infinite products type or so, the sign patterns of McKayStrauss' type can be explained by the circle method of HardyRamanajan. In fact it can be shown the signs of coefficients coincide with the signs of Kloosterman sums and so they are periodic. The first author treated the more sign patterns of many other infinite products of general type, and in particular it was found that it sometimes happen the corresponding coefficients are all zero periodically, which relates to the vanishing problem of a certain general Kloosterman sum. 2. It is known there exists a lifting from GL(2)automorphic forms to GL(3)forms. In fact, Gelbart & Jaquet already gave the first construction by using the functional equations method of Lfunctions. The second author has succeeded in reconstruction of the lifting by theta correspondence method, where he uses new integral expression of Eisensten series. This remarkable success of quite new method will shed new light on many other liftings. On another direction, the second author also executed a big calculation of Fourier coefficients of some Sigel modular forms of degree 3, which supports strongly a conjecture of Miyawaki lifting.

Report
(4results)
Research Output
(12results)