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 Ca:y2=x5+ax defined over ℚ, and their Jacobians Ja (without loss of generality a is a nonzero 8th power free integer). Previously, we considered the case when the polynomial x4+a is irreducible in ℚ[x] and obtained (under certain conditions on the quartic field ℚ(4√−a)) upper bounds for the rankJa(ℚ); in particular, we found infinite subfamilies of Ja with rank zero. Now we consider all cases when x4+a is reducible in ℚ[x] 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 Ja (when 2a 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 Ca in such cases.
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