Ranks of quadratic twists of Jacobians of generalized Mordell curves

Abstract

Consider a two-parameter family of hyperelliptic curves $C_{q,b}:y^2=x^q-b^q$ defined over $Q$, and their Jacobians $J_{q,b}$ where q is an odd prime and without loss of generality b is a non-zero squarefree integer. The curve $C_{q,b}$ is a quadratic twist by b of $C_{q,1}$ (a generalized Mordell curve of degree q). First, we obtain a few upper bounds for the ranks e.g., if the class number of $Q\left({\zeta }_q\right)$ is odd, $q\equiv 1\left(\mathrm{mod}4\right)$ and any prime divisor of $2b$ not equal to q is a primitive root modulo q then $\mathrm{rank}{J}_{q,b}\left(Q\right)\le (q-1)/2$. Then we focus on $q=5$ and get the best possible bound (by 1) or even the exact value of rank (0). In particular, we found infinitely many b with any number of prime factors such that $\mathrm{rank}{J}_{5,b}\left(Q\right)=0$. We deduce as conclusions the complete list (or the bounds for the number) of rational points on $C_{5,b}$ in such cases. Finally, we found for any given q infinitely many non-isomorphic curves $C_{q,b}$ such that $\mathrm{rank}{J}_{q,b}\left(Q\right)\ge 1$.