Research on the arithmetic of algebraic curves and jacobian varieties

在1994年，威尔斯和泰勒为Taniyama-Shimura的（半）椭圆形曲线设置了证明。这个，以及它对费马特最后一个定理的证明的应用，是本世纪最伟大的成就之一。在我们先前的研究中，我们扩展了Wiles-Taylor的研究结果，证明了Q上的某些非裔变体的模块化，包括Q-曲线在许多字段上，以及Jacobians的QM-GL （2）-类型。目前的研究目标已经提供了许多，但可能的是Q上的代数曲线的混凝土示例，对于我们的变异性标准可以应用于他们的雅各布人，作为一个良好的调查，可以对这样的曲线的各种算术特性进行调查。我们的一些主要结果是：·我们发现了一些基因2曲线家族的Q雅各布人变体是GL （2）-类型，并从数字和理论上检查了它们的模块化。Conversely，对于每一个体重2的交叉f （z）产生了一个四边形字段K，我们试着找到了一个代数曲线在Q- We have settled this pro\in all known cases for K = Q（イイD8-5イD8）、Q （イD8-1イD8）。有11个这样的f。我们用7个自由参数构建了最普通的家庭，由2个基因曲线组成，在Q上形成了一个椭圆曲线家族的双立方。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 （イD8j-12イD13イD1イD8）。

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).