\(\mathbb{Q} (T)\)-rank elliptic curves and certain limit coming from the local points
Let \({\mathcal E}\) denote an elliptic curve defined by a Weierstrass equation with \(\mathbb{Z}[T]\)-coefficients over \(\mathbb{Q}(T)\), the field of rational functions in one variable over the rationals. For \(t\in \mathbb{Z}\), let \(E_t\) denote the curve defined over \(\mathbb{Z}\) by specializing \(T\) to \(t\). For \(p\) a prime, let \(E_{t,p}\) denote the curve over \(\mathbb{F}_p\) defined by reducing the coefficients of \(E_t\) modulo \(p\). Put \[ A_p({\mathcal E})={1\over p}\sum_{t=0}^{p-1}p+1-\# E_t(\mathbb{F}_p). \] In a previous article [Kobe J. Math. 11, No. 2, 211-219 (1994; Zbl 0848.11026)] \textit{K. Nagao} considered the function \[ S(N,{\mathcal E})=-{1\over N}\sum A_p({\mathcal E})\cdot \log p, \] where the sum is over all primes less than or equal to \(N\), and conjectured that \(S(N,{\mathcal E})\) converges to the \(\mathbb{Q}(T)\)-rank of \({\mathcal E}\) as \(N\rightarrow\infty\). In the present article, the author shows that this conjecture holds for several families of elliptic curves over \(\mathbb{Q}(T)\), including curves of the form \(Y^2=f(X)+T^i\), where \(i=1\) or 2 and \(f(X)\in \mathbb{Z}[X]\) is a monic cubic polynomial without double roots.
scientific article; zbMATH DE number 980714