On Twists Of Tuples Of Hyperelliptic Curves

We investigate families of hyperelliptic curves whose Jacobians possess positiveMordell-Weil rank over Q. Given a square-free polynomial f(x) ∈ Q[x] of degreeat least 3 and fixed odd integers m1,m2,m3 ≥ 3, we construct parametric familiesof non-square rational values D such that the Jacobians of the twisted curvesC : Dy2 = f(x) and Ci : y2 = Dxmi +ai for i = 1,2,3, attain positive Mordell-Weilrank over Q(u,v1,v2,v3). Our approach involves solving a Diophantine system reducibleto finding rational points on certain elliptic curves defined by intersectionsof quadratic surfaces. Exploiting explicit rational parametrizations and leveragingSilverman’s specialization theorem, we demonstrate the existence of infinite-orderpoints on these Jacobians. This yields rational functions D that simultaneouslyinduce high-rank twists across the considered families of curves.