2004 Fiscal Year Final Research Report Summary
Arithmetic of Cubic Fields and Elliptic Curves associated to them
| Project/Area Number |
14540037
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Tokyo Metropolitan University |
|---|
Principal Investigator |
MIYAKE Katsuya Tokyo Metropolitan University, Department of Mathematics, Professor, 理学研究科, 教授 (20023632)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
NAKAMULA Ken Tokyo Metropolitan University, Department of Mathematics, Professor, 理学研究科, 教授 (80110849)
KURIHARA Masato Tokyo Metropolitan University, Department of Mathematics, Assoc. Professor, 理学研究科, 助教授 (40211221)
TOKUNAGA Hiroo Tokyo Metropolitan University, Department of Mathematics, Assoc. Professor, 理学研究科, 助教授 (30211395)
MATSUNO Kazuo Tokyo Metropolitan University, Department of Mathematics, Assist. Professor, 理学研究科, 助手 (40332936)
NAKANO Shin Gakushuin University, Department of Mathematics, Assoc. Professor, 理学部, 助教授 (40180327)
|
|---|
| Project Period (FY) |
2002 – 2004
|
| Keywords | Cubic Field / Mordell Curve / Elliptic Curve / Cubic Generic Polynomial / Mordell-Weil Group / Mordell-Weil Rank / Elliptic Surface / Miranda-Persson's Problem |
|---|
| Research Abstract |
For, an irreducible cubic polynomial P(X):=X^3+aX^2+bX+c over the rational number field Q, define a cubic curve E : w^3=P(u), and let E[Q] be the set of all rational points of E over Q (including the point at infinity; there are three points at infinity, and only one of them is rational over Q). Let ξ be a root of P(X), and define W(ξ):={α=qξ+r|q, r^∈Q, N_<Q(ξ)/Q>(α)=1}. Then we have a natural bijective map from W(ξ) to E[Q]. The subset W(ξ) of the cubic field Q(ξ) is stable under affine transformations of form ξ→sξ+t, s, t^∈Q, s≠0. By using such a transformation the curve is isomorphically mapped to one of two Mordell curves, y^2=x^3-2^43^3A^2, y^2=x^3-B^2(B+3), A, B^∈Q. The former is a short form of the pure cubic twist of the Fermat curve X^3+Y^3+AZ^3=0 as is well know. As for the latter, we showed that the Mordell-Weil rank is positive unless either B=-4 or -8/3. We could also obtain such a subfamily as the ranks of the members are at least 2 with a few exceptions. The subfamily was constructed by using the above presentation of E[Q] by W(ξ).
In case where P(X) is a generic cyclic polynomial of degree 3, namely, P(X)=X^3-(s-3)X^2-sX-1, the short form is determined. In this cyclic case, W(ξ) allowed us to construct an elliptic curve by using Hilbert's theorem 90. An isomorphism between the two curves over Q was also obtained.
In this project we deal with families of elliptic curves with one parameter in arithmetic viewpoint. It is also natural to see them as elliptic surfaces and to handle them in the way of algebraic geometry. It may be noteworthy that one of the investigator H. Tokunaga could solve Miranda-Persson's problem on the Mordell-Weil group of an extremal elliptic K3 surface.
|
