2014/01/03 by Tomasz Jędrzejak, Jędrzejak, Tomasz, Maciej Ulas +1 · 1 citation
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1401.0602
openalex publication_date 2014/01/03 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
Let f\∈ Q[x] be a square-free polynomial of degree \≥ 3 and m\≥ 3\nbe an odd positive integer. Based on our earlier investigations we prove that\nthere exists a function D1\∈ Q(u,v,w) such that the Jacobians of the\ncurves \C1:
;D1y2=f(x),
quad\nC2:
;y2=D1xm+b,
quad C3:
;y2=D1xm+c, have all\npositive ranks over Q(u,v,w). Similarly, we prove that there exists a\nfunction D2\∈ Q(u,v,w) such that the Jacobians of the curves\n\C1:
;D2y2=h(x),
quad C2:
;y2=D2xm+b,
quad\nC3:
;y2=xm+cD2, have all positive ranks over\n Q(u,v,w). Moreover, if f(x)=xm+a for some a\∈ Z\∖ 0 , we\nprove the existence of a function D3\∈ Q(u,v,w) such that the Jacobians\nof the curves \C1:
;y2=D3xm+a,
quad\nC2:
;y2=D3xm+b,
quad C3:
;y2=xm+cD3, have all\npositive ranks over Q(u,v,w). We present also some applications of these\nresults.\n Finally, we present some results concerning the torsion parts of the\nJacobians of the superelliptic curves yp=xm(x+a) and yp=xm(a-x)k\nfor a prime p and 0<m<p-2 and k<p and apply our result in order to prove\nthe existence of a function D\∈ Q(u,v,w,t) such that the Jacobians of the\ncurves \C1:
;Dyp=xm(x+a),
quad Dyp=xm(x+b)\n have both positive rank over Q(u,v,w,t).\n