The Steinitz class of the Mordell-Weil group of some CM elliptic curves (Q1907852)

The \(\text{End}(E)\)-module structure of the Mordell-Weil groups \(E(\mathbb{Q}(\sqrt{5}, \sqrt{-10}))\) of two explicitly given elliptic curves \(E\) defined over \(\mathbb{Q}(\sqrt{5})\) is determined. The curves have \(\text{End}(E)= \mathbb{Z}[\sqrt{-10}]\), and therefore a rational point of order 2 (in fact a torsion point for the prime in \(\mathbb{Z}[\sqrt{-10}]\) above 2) exists. The curves considered are related through this: one is the quotient by the point of order 2 of the other. The example is in fact what is called a \(\mathbb{Q}\)-curve; in the present case this refers to the property that the curves are each other's Galois conjugate over \(\mathbb{Q}(\sqrt{5})\). The authors compute the Mordell-Weil groups \(E(\mathbb{Q}(\sqrt{5}))\) using standard techniques (descent by 2-isogeny and a height consideration); of course compared to cases over the rational numbers everything is a little more involved here. Next they describe a technique for computing a complex multiplication on an elliptic curve in terms of Weierstrass coordinates. Finally, since \(E(\mathbb{Q}(\sqrt{5}, \sqrt{-10}))= \text{End}(E)\cdot E(\mathbb{Q}(\sqrt{5}))\), the results above allow the authors to deduce the full Mordell-Weil group over the Hilbert class field and its \(\text{End}(E)\)-structure. The example shows in particular that this structure is not an isogeny invariant. In view of the following lemma which is proven in the paper, this seems not very surprising: if \(M\) is finitely generated and free over \(\mathbb{Z}[\sqrt{-10}]\), and \(N\subset M\) is a submodule of index \(2^m\), then \(N\) is free precisely when \(m\) is even.