Large orders of automorphisms of smooth curves in ${\mathbb{P}}^1\times {\mathbb{P}}^1$

For $a,b\ge 3$, we calculate the orders of automorphisms of smooth curves with bidegree $(a,b)$ in the product $\text{pp}$ of the projective line ${\mathbb{P}}^1$. We identify smooth curves in $\text{pp}$ which have automorphisms with the largest orders. In addition, we study the relationship between symmetry and geometric structure of curves. We provide a sufficient condition for the quotient space by an automorphism to be ${\mathbb{P}}^1$.