2006 Fiscal Year Final Research Report Summary
Number Theoretic Study of Elliptic Curves
Project/Area Number 
16540006

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Tohoku University 
Principal Investigator 
NAKAMURA Tetsuo Tohoku University, Graduate School of Science, Professor > 東北大学, 大学院理学研究科, 教授 (90016147)

CoInvestigator(Kenkyūbuntansha) 
SATOH Atsusi Tohoku University, Graduate School of Science, Research Assistant, 大学院理学研究科, 助手 (30241516)

Project Period (FY) 
2004 – 2006

Keywords  elliptic curve / complex multiplication / Abelian variety / class number / quadratic field 
Research Abstract 
We intended to solve several number theoretic problems concerning elliptic curves defined over number fields. 1. Torsion on elliptic curves. We consider an elliptic curves defined over a number field and its isogeny class. We studied the behavior of the torsion group of elliptic curves on the isogeny class. We got several information of the structure of the torsion groups. 2. Quadratic fields with class number divisible by 5. We treated the problem expressing concretely quadratic fields with class number divisible by 5. We proposed a problem to expressing such fields by using parameters satisfying certain conditions and discussed some examples. 3. Abelian varieties associated with an imaginary quadratic field. A higher dimensional abelian varietiy A is called singular if A is isogenous to a direct product of an elliptic curve with complex multiplication. We studied them in the following aspects. (1)We investigated how singular abelian surfaces defined over the rational number field are constructed from a Qcurve. We showed that they are obtained by a Galois extension satisfying some conditions and by restriction of scalars of a Qcurve with respect to the extension. (2)We consider singular abelian varieties over the rationals such that they have complex multiplication over the imaginary quadratic field K and they have exact dimension the class number of K. We completed the classification of them and gave a characterization of their Hecke characters over K.

Research Products
(7 results)