Ranks in the family of hyperelliptic Jacobians of y2 = x5 + ax II

Abstract

This paper is a continuation of our previous one under the same title. In both articles, we study the hyperelliptic curves [Formula: see text] defined over [Formula: see text], and their Jacobians [Formula: see text] (without loss of generality [Formula: see text] is a nonzero [Formula: see text]th power free integer). Previously, we considered the case when the polynomial [Formula: see text] is irreducible in [Formula: see text] and obtained (under certain conditions on the quartic field [Formula: see text]) upper bounds for the [Formula: see text]; in particular, we found infinite subfamilies of [Formula: see text] with rank zero. Now we consider all cases when [Formula: see text] is reducible in [Formula: see text] and prove analogous results. First we obtain (under mild conditions on some quadratic fields) upper bounds for the ranks in a rather general situation, then we restrict to a several infinite subfamilies of [Formula: see text] (when [Formula: see text] has two primes divisors) and get the best possible bounds or even the exact value of rank (if it is zero). We deduce as conclusions the complete lists of rational points on [Formula: see text] in such cases.