Ranks of Jacobians of curves related to binary forms (Q2906833)

Let \(f(x,y)\in\mathbb{Z}[x,y]\) be an irreducible binary form of degree \(n\geq 3\). Let \(C\) denote the nonsingular projective curve defined by the equation \(z^n=f(x,y)\), and let \(C_m:mz^n=f(x,y)\) where \(m\) is an \(n\)-th-power-free integer. Let \(J_m\) denote the Jacobian variety of \(C_m\) and let \(\hat{h}_m\) denote the canonical height on \(J_m\). In this paper the author defines an infinite series, which is similar to the one introduced in [\textit{K. Rubin} and \textit{A. Silverberg}, Exp. Math. 9, No. 4, 583--590 (2000; Zbl 0959.11023)], and investigates the connection between the convergence of the series and the boundedness of the Mordell-Weil rank of \(J_m(\mathbb{Q})\). More precisely, for any positive real numbers \(k,j\), define the infinite series \(T_f(k,j)=\sum'_m1/|m|^k\sum_{P\in J_m(\mathbb{Q})\setminus J_m(\mathbb{Q})_{tor}}1/\hat{h}_m(P)^j\), where \(\sum'\) means that \(m\) runs through \(n\)th-power-free integers. He proves that the following three statements are equivalent: (a) \(\text{rank}J_m(\mathbb{Q})<2j\) for every \(m\in\mathbb{Z}\setminus\{0\}\); (b) \(T_f(k,j)\) converges for every \(k>1\); (c) \(T_f(k,j)\) converges for some \(k>1\). The main ingredients for the proof are Northcott's theorem on the finiteness of algebraic points with bounded height on projective varieties, and the theory of Epstein zeta functions.