Averages of the number of points on elliptic curves (Q399369)

Let \(E\) be an elliptic curves defined over \(\mathbb{Q}\) and \(p\) be a prime number of a good reduction for \(E\). Next, let \(E(\mathbb{F}_{p})\) denotes the set of \(\mathbb{F}_{p}\)-rational points lying on \(E\). Moreover, for a given non-negative integer \(N\) define the function \(M_{E}(N)=\sharp\{p:\;\sharp E(\mathbb{F}_{p})=N\}\). Recently, \textit{C. David} and \textit{E. Smith} studied the average of the function \(M_{E}(N)\) over certain regions of elliptic curves. In particular, they showed that for each integer \(N\), the average of \(M_{E}(N)\) over sufficiently large box is \(\sim K^{*}(N)\log N\), where \(K^{*}(N)\) is an explicitly given function of \(N\) [Compos. Math. 149, No. 2, 175--203 (2013); corrigendum ibid. 150, No. 8, 1347--1348 (2014; Zbl 1300.11057)]. Motivated by the results obtained by David and Smith the authors of the paper under review prove several results concerning the statistical properties of the function \(K^{*}(N)\). In particular, they compute the mean value of \(K^{*}(N)\) over all \(N\), odd \(N\) and prime \(N\). Moreover, they show that \(K^{*}(N)\) has a distribution function.