Shimara Cerresponchence of Hilbort modular forms
Project/Area Number  09640028 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Algebra

Research Institution  Mie University 
Principal Investigator 
TSUYUMINE Shigeaki Mie University, Faculty of Education, Professor, 教育学部, 教授 (70197763)

CoInvestigator(Kenkyūbuntansha) 
KOSEKI Harutaka Mie University, Faculty of Education, Associate Professor, 教育学部, 助教授 (60234770)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,800,000 (Direct Cost : ¥2,800,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1997 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  totally real algebraic number field / Hilbert Modular form / Lfunction / quadratic form / Eisenstein series / 総実代数体 / Hilbert保型形式 / L関数 / 二次形式 / Eisenstein級数 / 志村対応 / 保型形式 / Hilberc保型形式 
Research Abstract 
Let K be a totally real algebraic number field. We consider the HilbertEisenstein series on K.At first let .K be real and quadratic. Then we discovered some relation between the elliptic modular forms obtained by restricting the HilbertEisenstein series to the diagonal, and modular forms of half integral weight which are products of theta series and Eisensteinseries. By this it is shown that all modular forms of weight at least 5/2can be lifted to modular forms of integral weight in Shimura's sense (notethat this has been known only for cusp forms). As the application, we canobtain formulas for special values of the Dirichlet Lfunctions by computing Fourier coefficients of the modular forms, as well as relations between some arithmetic functions. Secondly let K be a general totally real algebraic number field. Let F bea totally real algebraic number filed which is a quadratic extension of K.We consider the Hilbert modular forms on K obtained by restricting the HilbertEisenstein series on F to K, and investigate how they work on number theory of a totally algebraic number fields K.When the structure of graded ring of Hilbert modular forms of K is known, this method works well as in the case of elliptic modular forms. We discovered several formulas for special values for Dede kind zeta functions or the number of representations of positive quadratic forms over K.We investigate also the Shimura correspondence of Hilbert modular forms over K.As a result it is prove that the Hilbertmodular form in the form of (theta series) x Eisenstein series, can be lifted to a Hilbert modular form of integral weight. In the elliptic case it follows from this that all modular form of half integral weight can be lifted, however in this case we need further investigation.

Report
(4results)
Research Output
(9results)