Variations on twists of tuples of hyperelliptic curves and related results

Abstract

Abstract. Let f ∈ Q[x] be a square-free polynomial of degree ≥ 3 and m ≥ 3 be an odd positive integer. Based on our earlier investigations we prove that there exists a function D 1 ∈ Q(u, v, w) such that the Jacobians of the curveshave all positive ranks over Q (u, v, w). Similarly, we prove that there exists a function D 2 ∈ Q(u, v, w) such that the Jacobians of the curveshave all positive ranks over Q (u, v, w). Moreover, if f (x) = x m + a for some a ∈ Z \ {0}, we prove the existence of a function D 3 ∈ Q(u, v, w) such that the Jacobians of the curveshave all positive ranks over Q (u, v, w). We present also some applications of these results. Finally, we present some results concerning the torsion parts of the Jacobians of the superelliptic curves y p = x m (x + a) and y p = x m (a − x) k for a prime p and 0 < m < p − 2 and k < p and apply our result in order to prove the existence of a function D ∈ Q(u, v, w, t) such that the Jacobians of the curves C 1 : Dy p = x m (x + a), Dy p = x m (x + b) have both positive rank over Q(u, v, w, t).

A note on the torsion of the Jacobians of superelliptic curves $y^{q}=x^{p}+a$

“…Note also that the characterization of torsion subgroups of Jacobians may have interesting applications to ranks. For example, in the case of twisted Fermat's curves C p m : x p + y p = m, a uniform boundedness of #Jac(C p m )(Q) tors for fixed odd prime p was used to obtain certain information about the behaviour of ranks in the infinite family Jac(C p m )(Q) (see [6]), and some information about ranks of p-twist of the Jacobians of the quotients of Fermat's curves (namely, y p = x m (x + a), see [16]).…”