Sato-Tate distributions of twists of y^2=x^5-x and y^2=x^6+1
Sato-Tate distributions of twists of y^2=x^5-x and y^2=x^6+1
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y^2=x^5-x 和 y^2=x^6 1 的扭曲 Sato-Tate 分布
DOI：
10.2140/ant.2014.8.543
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发表时间：
2012
影响因子：
1.3
通讯作者：
Andrew V. Sutherland
中科院分区：
文献类型：
--
作者：
Francesc Fit'e;Andrew V. Sutherland
我们确定了定义在数域k上的阿贝尔曲面A的归一化欧拉因子的极限分布，当A与定义在k上的椭圆曲线的平方同构时，用复数乘法。作为应用，我们证明了曲线y^2=x^5-x和y^2=x^6+1的Q-扭Jacobian的Sato-Tate猜想，它给出了定义在Q上的交换曲面的Sato-Tate群的34种可能性中的18种.事实上，这两条曲线的扭曲，会遇到阿贝尔曲面的Sato-Tate群的所有18种可能性，该阿贝尔曲面与具有复数乘法的椭圆曲线的平方是同构的。这些结果的关键是扭曲的Sato-Tate群的曲线，我们引入，以研究扭曲的影响，其雅可比矩阵的Sato-Tate群。
We determine the limiting distribution of the normalized Euler factors of an abelian surface A defined over a number field k when A is isogenous to the square of an elliptic curve defined over k with complex multiplication. As an application, we prove the Sato-Tate Conjecture for Jacobians of Q-twists of the curves y^2=x^5-x and y^2=x^6+1, which give rise to 18 of the 34 possibilities for the Sato-Tate group of an abelian surface defined over Q. With twists of these two curves one encounters, in fact, all of the 18 possibilities for the Sato-Tate group of an abelian surface that is isogenous to the square of an elliptic curve with complex multiplication. Key to these results is the twisting Sato-Tate group of a curve, which we introduce in order to study the effect of twisting on the Sato-Tate group of its Jacobian.