2002 Fiscal Year Final Research Report Summary
GENERATORS AND DEFINING EQUATIONS OF MODULAR FUNCTION FIELDS
| Project/Area Number |
12640036
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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 Prefecture University |
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Principal Investigator |
ISHII Noburo Osaka Prefecture University, Department of Mathematics and Information Sciences, Professor, 総合科学部, 教授 (30079024)
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| Co-Investigator(Kenkyū-buntansha) |
YAMAMOTO Yoshihiko Graduate school of Science, Osaka University Department of Mathematics, Professor, 大学院・理学研究科, 教授 (90028184)
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| Project Period (FY) |
2000 – 2002
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| Keywords | Modular function field / Generators / Defining equation / Elliptic crve / Elliptic curve cryptosystems / Frobenius endomorphism / Trace |
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| Research Abstract |
Our results are as follows.
(1) We constracted new generators of the modular function field K(N)of the modular group Γ_1(N) by Weierstrass P-functuins and obatained a plain defining equation of the modular function field.
(2) We gave some examples of the canonical power series solutions of the elliptic curve by constructing the generator of the genus 1 subfields of K(N) from the generators in (1) in the cases K(N) has genus 2.
(3) We obatined an algorithm to determine the j-invariant of the elliptic curve corresponding to the solution of the defining equation. But this algorithm is *neffective for large N.
In the course of studing the properties of the defining equation over the finite fields, the following two results were obtained.
(4) We offered a method of constracting a family of elliptic curves over finite fields of cyclic rational point group of large order using the elliptic curve,rational over an algebraic number field, with complex multiplication.
(5) We determined the trace of Frobenius endomoprphism of the elliptic curves with complex multipication R, where R is an order of discriminant divided by 3, 4,5 and its class number is 2 or 3.
The results (4), (5) are applicable to the elliptic curve cryptosysytems.
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