2003 Fiscal Year Final Research Report Summary
SPECIAL VALUES OF AUTOMORPHIC L-FUNCTIONS BY RELATIVE TRACE FORMULAS
| Project/Area Number |
13640037
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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 | OSAKA CITY UNIVERSITY |
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Principal Investigator |
FURUSAWA Masaaki OSAKA CITY UNIVERSITY, GRADUATE SCHOOL OF SCIENCE, PROFESSOR, 大学院・理学研究科, 教授 (50294525)
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| Co-Investigator(Kenkyū-buntansha) |
SUMIOKA Takeshi OSAKA CITY UNIVERSITY, GRADUATE SCHOOL OF SCIENCE, ASSOCIATE PROFESSOR, 大学院・理学研究科, 助教授 (90047366)
KANEDA Masaharu OSAKA CITY UNIVERSITY, GRADUATE SCHOOL OF SCIENCE, PROFESSOR, 大学院・理学研究科, 教授 (60204575)
TSUSHIMA Yukio OSAKA CITY UNIVERSITY, GRADUATE SCHOOL OF SCIENCE, PROFESSOR, 大学院・理学研究科, 教授 (80047240)
HASHIMOTO Yoshiatke OSAKA CITY UNIVERSITY, GRADUATE SCHOOL OF SCIENCE, ASSOCIATE PROFESSOR, 大学院・理学研究科, 助教授 (20271182)
浅芝 秀人 大阪市立大学, 大学院・理学研究科, 助教授 (70175165)
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| Project Period (FY) |
2001 – 2003
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| Keywords | RELATIVE TRACE FORMULA / AUTOMORPHIC L-FUNCTION / SIEGE MODULAR FORM / SPECIAL VALUES OF L-FUNCTIONS / DELIGNE CONJECTURE / TRACE FORMULA |
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| Research Abstract |
We have continued the project concerning the conjecture of Siegfried Boecherer and its generalization on the central critical values of the degree four L-functions for the Siegel eigen cusp form of degree two. Our method is to generalize Jacquet's two relative trace formulas which have given another proof of Waldspurger's theorem on the central critical values of L-functions for GL(2). In order to establish a trace formula, the first and crucial step is to prove the fundamental lemma. In a monograph published as No. 782 of Memoirs of the AMS, we gave a proof of the fundamental lemma for the unit element in the Hecke algebra. Our proof is based on the explicit evaluation of the two by two symmetric matrix argument generalized Kloosterman sum. This evaluation might be of some independent interest because of its relationsbip with the Fourier coefficients of the Poincare series for the Siegel modular forms of degree two. Our next task is to extend the fundamental lemma to the entire Hecke algebra. Inspired by Ye's idea on the quadratic base change for GL(n) case, in a paper In Transactions of the AMS, we proved the inversion formula for the Bessel transform. Because of the inversion formula, now we can express the orbital integrals for an arbitrary element in the Hecke algebra, as a finite sum of degenerate Kloosterman orbital integrals for the unit element. We have computed all degenerate orbital integrals and now we are ready to extend the fundamental lemma to the entire Hecke algebra.
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