Towards the Sato-Tate Groups of Trinomial Hyperelliptic Curves

Abstract

We consider the identity component of the Sato-Tate group of the Jacobian of curves of the form $$C_1\colon y^2=x^{2g+2}+c, C_2\colon y^2=x^{2g+1}+cx, C_3\colon y^2=x^{2g+1} +c,$$ where $g$ is the genus of the curve and $c\in\mathbb Q^*$ is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for $C_1$ 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 $C_1$ 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 $C_1$, $C_2$, and $C_3$. 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 for these families of curves.