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

Abstract

Consider the hyperelliptic curves Ca:y2=x5+ax defined over Q, and their Jacobians Ja. Without loss of generality a is a non-zero 8th power free integer. Our aim is to obtain upper bounds for ra:=rankJa(Q). In particular, we would like to find infinite subfamily of Ja with rank 0. We show that under certain assumptions on the quartic field Q(−a4), ra≤ω(2a)+1 (if a>0), and ra≤ω(2a)+2 (if a<0) where ω(m) is the number of distinct prime divisors of m. We also generalize this result to the ranks of the Jacobians of y2=x(xn+a). Moreover, we prove that if p≡3(mod8) is a prime then rp,r−p≤1, r−2p≤2, and if p≡11,19(mod32) then rp=0, and consequently Cp(Q)={∞,(0,0)} for such primes. We also make numerical computations of ranks and rational points, and put a few conjectures.