Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Jędrzejak, Tomasz

doi:10.4064/aa161-3-1

Cited by 6 publications

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“…T n , y 2 = pT n , y 3 = qT n , where u, v are rational parameters and p, q, T have to be determined. After this substitution the system (7) simplifies and we get (8) f…”

Section: Quadratic Twist and Two 2m-twistsmentioning

confidence: 99%

Variations on twists of tuples of hyperelliptic curves and related results

Jędrzejak

Ulas

2014

Journal of Number Theory

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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).

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“…T n , y 2 = pT n , y 3 = qT n , where u, v are rational parameters and p, q, T have to be determined. After this substitution the system (7) simplifies and we get (8) f…”

Section: Quadratic Twist and Two 2m-twistsmentioning

confidence: 99%