On the average period of an elliptic curve (Q1196408)

The author studies elliptic curves defined over \(\mathbb{Z}\). To each \(m=(m_ 1,m_ 2)\in\mathbb{Z}^ 2\) corresponds the elliptic curve \(E_ m\) with equation \(w^ 2=4z^ 3-m_ 1 z-m_ 2\). -- For positive \(\varepsilon\), \(t\), \(k\) the limit of the sum \(\sum_{m\in{\mathcal N}_{\varepsilon,\pm}(t)} \omega^ k(m)\) is calculated as \(t\to\infty\). The ingredients of this formula are: \({\mathcal N}_{\varepsilon,+}(t)= \{m\in\mathbb{Z}^ 2|\) \(P(m)\in(| m_ 1|^ \varepsilon,t]\}\) with \(P(m)=27m_ 2^ 2-m_ 1^ 3\), \({\mathcal N}_{\varepsilon,-}(t)= \{m\in\mathbb{Z}^ 2|\) \(P(m)\in(-t,-| m_ 1|^ \varepsilon]\}\), \(\omega(m)=\int_{E_ m^ u(\mathbb{R})} dz/ \sqrt{4z^ 3-m_ 1z-m_ 2}\) with \(E_ m^ u(\mathbb{R})\) the unbounded component of \(E_ m(\mathbb{R})\).