An application of Mumford's gap principle.
\textit{K.~Rubin} and \textit{A.~Silverberg} [Exp. Math. 9, No. 4, 583--590 (2000; Zbl 0959.11023)] showed that the boundedness of the ranks of the quadratic twists of a given elliptic curve \(y^2=g(x)\) is equivalent to the convergence of a certain infinite series \(S_g(j,k)\) involving \(h(x)\), where \(h\) is the height of the rational number \(x\neq 0,\pm 1\). Given a monic polynomial \(f(x)\) with integer coefficients and degree at least 5, the authors define an analogous infinite series \(S_f(j,k)\). They then prove that \(S_f(j,k)\) converges for every \(k>1\) and any positive real number \(j\) by relating this convergence to the distribution of rational points on the hyperelliptic curves \(my^2=f(x)\) and applying a result of \textit{D.~Mumford} [Am. J. Math. 87, 1007--1016 (1965; Zbl 0151.27301)] concerning the sparsity of the rational points of a curve of genus at least 2 in the Jacobian of the curve. They remark that their result also follows from the deeper work of \textit{P.~Vojta} [Ann. Math. (2) 133, No. 3, 509--548 (1991; Zbl 0774.14019)] on an effective version of the Mordell-Faltings Theorem.
scientific article; zbMATH DE number 2066766
scientific article