The complex AGM, periods of elliptic curves over \(\mathbb C\) and complex elliptic logarithms (Q2445012)

The paper under review is devoted to the study of the complex arithmetic-geometric mean (AGM) and its connection with elliptic curves over \({\mathbb C}\), their period lattices and complex parametrisation. The authors present efficient algorithms for computing bases for the period lattices and elliptic logarithms of points for arbitrary elliptic curves defined over \({\mathbb C}\). As an example of the obtained results, the algorithm for computing a basis of a period lattice is given below. Let \((a,b)\in{\mathbb C}^2\) be a pair of complex numbers satisfying \(a,b\neq 0\) and \(a\neq \pm b\). The pair \((a,b)\) is \textit{good} if \(|a-b|\leq |a+b|\). An AGM sequence is a sequence \(((a_n,b_n))_{n=0}^\infty\), whose pairs \((a_n,b_n)\in {\mathbb C}^2\) satisfy \(2a_{n+1}=a_n+b_n\) and \(b_{n+1}^2=a_nb_n\) for all \(n\geq 0\). An AGM sequence is \textit{optimal} if all \((a_n,b_n)\), \(n>0\), are good. For an optimal AGM sequence \(((a_n,b_n))_{n=0}^\infty\) starting with \((a_0,b_0)\) set \(M(a_0,b_0):=\lim_{n\rightarrow \infty}a_n=\lim_{n\rightarrow \infty}b_n\,.\) { Algorithm} (Computation of a period lattice basis) { Input:} An elliptic curve \(E\) defined over \({\mathbb C}\), and roots \(e_1, e_2, e_3\in {\mathbb C}\). { Output:} Three primitive periods of \(E\), which are minimal coset representatives, any two of which form a \({\mathbb Z}\)-basis for the period lattice of \(E\). {\parindent=6mm \begin{itemize}\item[1.] Label one of the roots as \(e_1\), and the other two arbitrarily as \(e_2\), \(e_3\). \item[2.] Set \(a_0=\sqrt{e_1-e_3}\) with arbitrary sign, and then \(b_0=\pm\sqrt{e_1-e_2}\) with the sign chosen such that \(|a_0-b_0|\leq|a_0+b_0|\). \item[3.] Output \(w=\pi/M(a_0,b_0)\), using the optimal value of the AGM (see the paper for the details). \item[4.] Repeat with each root \(e_j\) in turn playing the role of \(e_1\). \end{itemize}} The paper is nicely written and contains a number of illustrative examples.