We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form C1 : y2 = x2g+2 + c,C 2 : y2 = x2g+1 + cx,C 3 : y2 = x2g+1 + c, where g is the genus of the curve and c ∈ ℚ∗ is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for C1 curves of genus 4 and 5 and prove what the identity component of the Sato–Tate group is in each case. We then determine the splitting of Jacobians of higher genus C1 curves by finding maps to lower genus curves and then computing pullbacks of differential 1-forms. In using this method, we are able to relate the Jacobians of curves of the form C1, C2 and C3. Finally, we develop a new method for computing the identity component of the Sato–Tate groups of the Jacobians of the three families of curves. We use this method to compute many explicit examples, and find surprising patterns in the shapes of the identity components ST0(C) for these families of curves.

中文翻译：

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朝向三项式超椭圆曲线的 Sato-Tate 群

我们考虑以下形式的雅可比曲线的 Sato-Tate 群的恒等分量 C1 ： 是的2 = X2G+2 + C,C 2 ： 是的2 = X2G+1 + CX,C 3 ： 是的2 = X2G+1 + C, 在哪里G是曲线的属，并且C ∈ ℚ*是恒定的。我们以三种方式解决这个问题。首先，我们使用 Kani-Rosen 定理来确定 Jacobians 的分裂C1属 4 和 5 的曲线，并证明在每种情况下 Sato-Tate 群的恒等分量是什么。然后我们确定更高属的雅可比矩阵的分裂C1通过找到下属曲线的映射，然后计算微分 1 型的回拉来绘制曲线。在使用这种方法时，我们能够关联以下形式的曲线的雅可比行列式C1,C2和C3. 最后，我们开发了一种新方法来计算三个曲线族的雅可比行列的 Sato-Tate 群的恒等分量。我们使用这种方法来计算许多显式示例，并在身份组件的形状中找到令人惊讶的模式小号吨0(C)对于这些曲线族。