1999 Fiscal Year Final Research Report Summary
Research on the arithmetic of algebraic curves and jacobian varieties
| Project/Area Number |
09640075
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Waseda University |
|---|
Principal Investigator |
HASHIMOTO Kiichiro Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90143370)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
HASEGAWA Yuji Waseda Univ. School of Sci. and Eng., (JSPS Fellow), 特別研究員 (30287982)
ADACHI Norio Waseda Univ. School of Sci. and Eng., Professor, 理工学部, 教授 (60063731)
KOMATSU Keiichi Waseda Univ. School of Sci. and Eng., Professor, 理工学部, 教授 (80092550)
KAGAWA Takaaki Waseda Univ. School of Sci. and Eng., Assistant, 理工学部, 助手(平10) (90298175)
OZAKI Manabu Waseda Univ. School of Sci. and Eng., Assistant, 理工学部, 助手(平9) (80287961)
|
|---|
| Project Period (FY) |
1997 – 1999
|
| Keywords | elliptic curves / modular curves / Taniyama-Simura Conjecture / jacobian variety / modular forms / abelian varieties / algebraic curvers / Q-curves |
|---|
| Research Abstract |
In 1994 Wiles and Taylor have settled the proof of Taniyama-Shimura conjecture for (semistable) elliptic curves over Q. This, with its application to the proof of Fermat's Last Theorem, was one of the greatest achievment in this century. In our previous research, we extended the result of Wiles-Taylor proving the modularity of certain abelian varieties over Q, including Q-curves over number fields, and jacobians of QM-curves of GL (2) -type. The aim of the present research has been to provide as many as possible the concrete examples of algebraic curves over Q, for which our modularity criterion for their jacobian can be applied, as well as to investigate various arithmetic properties of such curves. Some of our main results are :
・ We obtained some families of genus 2 curves over Q whose jacobian varieties are of GL (2) -type, and checked their modularity numerically and theoretically.
・ Conversely, for each cusp f (z) of weight 2 whose Fourier coefficients generate a quadratic field K, we tried to find an algebraic curve over QィイD4-ィエD4 shose jacobian variety is isogenous to the Shimura's abelian surface AィイD2fィエD2 attached to f. We have settled this problem in all known cases for K = Q(ィイD8-5ィエD8), Q (ィイD8-1ィエD8). There are 11 such f.
・ We constructed the most general family with 7 free parameters, of genus 2 curves over Q which form a double cuver of a family of elliptic curves. Among them we found a generic family of the covering C (j) → E (j) where E (j) is the Tate's model of elliptic curve with j (E (j) ) = j. Then the simple factor of JacC (j) is shown to be a Q-curve over quadratic field Q (ィイD8j-12ィイD13ィエD1ィエD8).
|
