Mordell-Weil Lattices of Elliptic Curves and Abelian Varieties
| Project/Area Number |
12640044
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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 | RIKKYO UNIVERSITY |
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Principal Investigator |
SHIODA Tetsuji Rikkyo Univ, Department of Math., Professor, 理学部, 教授 (00011627)
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| Co-Investigator(Kenkyū-buntansha) |
AOKI Noboru Rikkyo Univ, Department of Math., Associate Professor, 理学部, 助教授 (30183130)
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| Project Period (FY) |
2000 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2001: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2000: ¥1,900,000 (Direct Cost: ¥1,900,000)
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| Keywords | Mordell-Weil Lattices / Elliptic Curves / Abelian Varieties / Integral Points / Codes / Elliptic Modular Surfaces / Tate-Shafarevich Group / Hodge Cycles / アーベル多様体 / ABC定理 / Davenportの限界 / コード / 球のつめこみ / 不変式論 |
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| Research Abstract |
(1) Integral Points and Mordell-Weil Lattices.
As an application of Mordell-Weil Lattices, we have developed a method to study integral points in the function field case. In some favorable situation, this method gives a very efficient way for a complete determination of all the integral points of an elliptic curve. [S1]
(2) K3 Surfaces and Sphere Packings
We hare obtained lattice sphere packings in higher dimensional case (especially dimension 16, 17, 18) of fairly large packing density, by means of the Mordell-Weil Lattices of certain elliptic K3 surfaces. [S3]
(3) Invariant theory of plane quartics vs Mordell-Weil Lattices
We hare established a close relationship of the classical invariant theory of plane quartics (moduli of genus three curves) and the invariant theory of the Weyl group of type E_7 (a finite group). [S4]
(4) Some codes arising from the elliptic modular surfaces
For any N. we have constructed a linear code over the residue ring mod N which is associated with the elliptic modular surfaces of level N. If N is a prime number, this linear code over a field of N elements has a remarkable property that every nonzero code-word has a constant Bernoulli norm. The construction is based on the height formula of Mordell-Weil Lattices, [S2]
(5) Tate-Shafarevich group of elliptic curves
Aoki has proven that the 3-part of Tate-Shafarevich group can be arbitrarily large. [A2]
(6) Hodge conjecture of abelian varieties
The Hodge cycles on the Jacobian variety of a Fermat curve are studied from combinatorial viewpoint. By this, the Hodge conjecture is verified for wider class of abelian varieties of Fermat type. [A1], [A3]
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Report
(4 results)
Research Products
(22 results)
