1995 Fiscal Year Final Research Report Summary
Arithmetics on Jacobian Varieties
| Project/Area Number |
06452004
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
Algebra
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| Research Institution | Osaka University |
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Principal Investigator |
YAMAMOTO Yoshihiko Osaka University , Dept.of Math.Prof., 理学部, 教授 (90028184)
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| Co-Investigator(Kenkyū-buntansha) |
OGAWA Hiroyuki Osaka University , Dept.of Math.Assit., 理学部, 助手 (70243160)
MURAKAMI Jun Osaka University , Dept.of Math.Assoc.Prof., 理学部, 助教授 (90157751)
KIYOKAZU Nagatomo Osaka University , Dept.of Math.Assoc.Prof, 理学部, 助教授 (90172543)
HIRAMINE Yutaka Osaka University , Dept.of Math.Assoc.Prof., 理学部, 助教授 (30116173)
MATSUMURA Akitaka Osaka University , Dept.of Math.Prof., 理学部, 教授 (60115938)
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| Project Period (FY) |
1994 – 1995
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| Keywords | Algebraic curve / Jacobian variety / Abelian variety / Elliptic curve / Division points / Galois group |
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| Research Abstract |
1. Classification of Jacobian varieties of hyperelliptic curves defined over finite fields : We tried to (1) classify hyperelliptic cureves of genus g defined overfinite field of order q then (2) classify Jacobian varieties of curves classifyed by (1) in specific cases. In case q = 3,5 all hyperelliptic curves of genus 2 are classified and get isogeny classes by computing congruence zeta fanction of each curve. Then we determied the endmorphism rings of each Jacobians.
2. Construction of abelian varieties of with a rational point of large order : (1) Orders of specific points on Jacobian varieties of hperelliptic curves can be given by the periods of continued fractional expansion of corrsponding functions on the curves. (2) We found a series of hyperelliptic curves of geneus 2 over the rationalfield, whose Jacobian varieties have a point of order 23.
3. Computation of Galois groups of the n-division fields of abelian varieties : For the Jacobian varieties of hyperelliptic curves of gunus 2,2 and 3-division equations are determied by computational-procession.
4. Construction of algebraic curves with many ration points : We got some series of algebraic curves over funcion fields with many rational points, by using the n-division equation of Jacobian varieties.
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Research Products
(17 results)
