On the average number of rational points of bounded height on hyperelliptic curves
After the famous theorem of Faltings, it is known that the number of rational points of a curve of genus \(g\geq 2\) defined over \({\mathbb Q}\) is finite. The question arises about the size of the number of rational points. The so-called Uniformity Conjecture would predict a uniform bound. In the present paper the author proves a statement regarding the average number of rational points of bounded height in hyperelleptic curves of genus \(g\geq 2\). Let \(n=2g+2\) and consider the set of all equations of type \[ y^2=f(x)=f_nx^n+f_{n-1}x^{n-1}+\dots+f_0 \] with \(f_i\in{\mathbb Z}\) and defining an hyperelliptic curve of genus \(g\). Endow this set with the \(L_2\)-norm \[ N(f)=\sqrt{f_n^2+f_{n-1}^2+\dots+f_0^2}, \] and let \(C_N\) be the set of those equations for which \(N(f)\leq N\). Let also \({\mathbb P}^1({\mathbb Q},H) = \{(a:b)\in{\mathbb P}^1({\mathbb Q})\, |\, H(a:b)\leq H\}\), where \(H(a:b)\) is the usual Weil height, and \[ R_N(H) = \{(f,(a:b))\in C_N\times {\mathbb P}^1({\mathbb Q},H)\, |\, f(a,b)=y^2 \text{\;for some} \;y\in {\mathbb Z}\}. \] The main theorem states that \[ \sqrt{N}{\#R_N(H)\over \#C_N} = \gamma(H) + O(N^{-{1\over 2}}H^2 + N^{-1}H^3), \] where \(\gamma(H)\) is a constant that does not depend on \(N\).
scientific article; zbMATH DE number 6010049