1999 Fiscal Year Final Research Report Summary
On arithmetic theory of automorphic forms and special values of automorphic Lfunctions
Project/Area Number 
10640028

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Osaka City University (1999) Hiroshima University (1998) 
Principal Investigator 
FURUSAWA Masaaki Osaka City University, Faculty of Science Professor > 大阪市立大学, 理学部, 教授 (50294525)

CoInvestigator(Kenkyūbuntansha) 
KOMORI Youhei Osaka City University, Faculty of Science Lecturer, 理学部, 講師 (70264794)
IMAYOSHI Yoichi Osaka City University, Faculty of Science Professor, 理学部, 教授 (30091656)
KAMAE Tetsuro Osaka City University, Faculty of Science Professor, 理学部, 教授 (80047258)
MATSUMOTO Keiji Hokkaido University, Graduate School of Science Assistant Professor, 大学院・理学研究科, 助教授 (30229546)
MOCHIZUKI Takuro Osaka City University, Faculty of Science Assistant, 理学部, 助手 (10315971)

Project Period (FY) 
1998 – 1999

Keywords  Siegel modular form / automorphic Lfunction / special value of Lfunction / Deligne's conjecture / relative trace formula / trace formula 
Research Abstract 
We proved the fundamental lemma for the unit element in the Hecke algebra for two relative trace formulas for GSp(4). Our ultimate goal is to prove Bocherer's conjecture on the central critical values of the quadratic twists of the spinor Lfunctions associated to holomorphic Siegel eigen cusp forms of degree two. The announcements of the fundamental lemma have been published in C. R. Acad. Sci. Paris and the details of the proof will appear elsewhere. In the course of the proof of the fundamental lemma, we evaluated certain matrix argument Kloosterman sums explicitly in terms of the classical GL(2) Kloosterman sums. We remark that our Kloosterman sum is a special case of the generalized Kloosterman sum which appears in the Fourier coefficients of the Poincare series for the Siegel modular group. Our result on the Kloosterman sum may be of some independent interest, since it is rare that such generalized Kloosterman is evaluated explicitly. Our second conjectural trace formula is related to the quadratic base charge for GSp (4). Our result suggests that the JacquetYe criterion for the quadratic base change for GL(2) generalizes to GSp(4). This clearly deserves some further investigation. Finally our result implies that it is important to study the whole Lpacket when we study the special values of automorphic Lfunctions. It seems very interesting to clarify the relationship between the period part of the special value expected by our result and Deligne's conjecture.

Research Products
(4 results)