1999 Fiscal Year Final Research Report Summary
STUDIES ON MODULAR FORMS, NUMBER THEORY AND ALGEBRAIC STRUCTURES
| Project/Area Number |
10640015
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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 | GIFU UNIVERSITY |
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Principal Investigator |
HATADA Kazuyuki GIFU UNIVERSITY, FACULTY OF EDUCATION, FULL PROFESSOR, 教育学部, 教授 (40144000)
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| Co-Investigator(Kenkyū-buntansha) |
IWATA Keiji GIFU UNIVERSITY, FACULTY OF EDUCATION, FULL PROFESSOR, 教育学部, 教授 (80021327)
TAKEUCHI Shigeru GIFU UNIVERSITY, FACULTY OF EDUCATION, FULL PROFESSOR, 教育学部, 教授 (30021330)
CHUMAN Goro GIFU UNIVERSITY, FACULTY OF EDUCATION, FULL PROFESSOR, 教育学部, 教授 (30115414)
AIKI Toyohiko GIFU UNIVERSITY, FACULTY OF EDUCATION, ASSOCIATE PROFESSOR, 教育学部, 助教授 (90231745)
FUJIMOTO Yoshio GIFU UNIVERSITY, FACULTY OF EDUCATION, ASSOCIATE PROFESSOR, 教育学部, 助教授 (90192731)
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| Project Period (FY) |
1998 – 1999
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| Keywords | p-adic limits / modular forms of nebentypus / l-adic modular forms / Siegel modular forms / common eigen-functions of Hecke operators / The Ramanujan Conjecture / representations of Galois groups / Abelian varieties |
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| Research Abstract |
1. Kazuyuki Hatada's studies: (1) Let g be any integer【greater than or equal】2. Hatada has obtained that there exist infinitely many Siegel cusp eigenforms of degree g which satisfy the Ramanujan Conjecture and that any of them has a Galois representation Gal(QィイD4-ィエD4/Q) →Gspinn(2g+1,QィイD4-ィエD4ィイD2lィエD2) with the expected property. Hatada has obtained also such an infinite family of Siegel cusp eigenforms of degree g which don't satisfy the Ramanujan Conjecture that any of the family has a good Galois representation of the above type. (2) Hatada has applied his method of (1) to differential forms of the first kind on Abelian varieties. (3) Let m be any integer【greater than or equal】0, l be any prime number, and N be any integer with (N,l)=1. Hatada has obtained that any holomorphic modular form of level NlィイD1mィエD1 m of nebentypus is an l-adic modular form of level N. Hatada has obtained that the mod l Galois representation of any primitive form on ΓィイD21ィエD2(NlィイD1mィエD1) is that of a primitive form on ΓィイD21ィエD2(N) for any 【greater than or equal】l2. (4) Hatada has extended Leopoldt's p-adic limit formula for the generalized Bernoulli numbers to the case of algebraic number fields. (5) Hatada has obtained a new method to construct Q without artificiality.
2. Yoshio Fujimoto's studies: Let X be a non-singular projective threefold of Kodaira dimension κ(X)【greater than or equal】0 with a non-isomorphic surjective endomorphism. Fujimoto obtained, the minimal, models of X are non-singular; Some finite etale covering Y of X is isomorphic to the product of an Abelian variety and a non-singular projective variety if κ(X)=0,2.
3. Toyohiko Aiki showed well-posedness for Caginalp and Penrose-Fife types of phase-field equations, and uniqueness of weak solutions of the shape memory alloy problems under a weaker condition than before.
4. Shigeru Takeuchi studied Category of DR/CR spaces/algebras. Goro Chuman and Keiji Iwata studied Education of Mathematics.
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