Research on the arithmetic of algebraic curves and jacobian varieties

代数曲线与雅可比簇的算法研究

• 批准号: 09640075
• 负责人: HASHIMOTO Kiichiro
• 金额: $ 1.98万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640075/
• 关键词: elliptic curves; modular curves; Taniyama-Simura Conjecture; jacobian variety; modular forms; abelian varieties; algebraic curvers; Q-curves; modularity; アーベル曲面; Q-曲線; テータ級数; 保型形式

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).

1994年Wiles和Taylor解决了环Q上(半稳定)椭圆曲线的Taniyama-Shimura猜想的证明问题，并将其应用于Fermat大定理的证明，这是本世纪最伟大的成就之一。在以前的研究中，我们推广了Wiles-Taylor证明Q上某些交换变种的模性的结果，包括数域上的Q-曲线和GL(2)型QM-曲线的Jacobian。本研究的目的是提供尽可能多的Q上代数曲线的具体例子，并研究这类曲线的各种算术性质。对于这些例子，我们的雅可比模性准则可以应用于这些例子。我们的一些主要结果是：·我们得到了Q上雅可比变元为GL(2)型的亏格2曲线族，并从数值和理论上验证了它们的模性。反之，对于其傅立叶系数生成二次域K的权2的每个尖点f(Z)，我们试图在QィイD4-ィエD4上找到一条代数曲线，其雅可比簇与附着于f的下村交换曲面AィイD2fィエD2同源。我们已经解决了K=Q(ィイD8-5ィエD8)，Q(ィイD8-1ィエD8)的所有已知情况下的问题。有11个这样的f。·构造了Q上具有7个自由参数的最一般的亏格2曲线族，它们构成了一个椭圆曲线族的双曲线。其中E(J)是椭圆曲线的泰特模型，j(E(J))=j.然后证明了→(J)的简单因子是二次域Q(ィイD8j-12ィイD13ィエD1ィエD8)上的一条Q曲线.

项目成果

Yuji Hasegawa: "Hyperelliptic quotients of modular curves X_O (N)"Tokyo Journal of Mathematics. 22. 105-125 (1999)

长谷川雄二：“模曲线的超椭圆商 ​​X_O (N)”东京数学杂志。

Ki-ichiro Hashimoto: "Inverse Galois Problem for Dihedral Groups" Technical Report Adv.Research Inst.Waseda. 98-4. 1-17 (1998)

Ki-ichiro Hashimoto：“二面体群的逆伽罗瓦问题”技术报告早稻田高级研究所。

Ki-ichiro Hashimoto: "Q-curves of degree 5 and jacobian surfaces of GL_2-type"Manuscripta Mathematica. 98. 165-182 (1999)

Ki-ichiro Hashimoto：“5 次 Q 曲线和 GL_2 型雅可比曲面”Manuscripta Mathematica。

Yuji Hasegawa: "Trigonal modular curves"ACTA ARITHMETICA. 81. 129-140 (1999)

长谷川雄二：“三角模曲线”ACTA ARITHMETICA。

Kiichiro Hashimoto, Yuji HasegawaFumiyuki Momose: "Modularity conjecture for Q-curves and QM-curves"International J. Math. vol.10-7. 1011-1036 (1999)

Kiichiro Hashimoto、Yuji Hasekawa Fumiyuki Momose：“Q 曲线和 QM 曲线的模块化猜想”International J. Math。