Descending Rational Points on Elliptic Curves to Smaller Fields

Abstract

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve $E$ defined over a number field $K$ whose Mordell-Weil rank over a Galois extension $F$ is 1, 2 or 3. We show that $E$ acquires a point (points) of infinite order over a field whose Galois group is one of ${{C}_{n}}\times {{C}_{m}}(n=1,\,2,\,3,\,4,\,6,\,m\,=\,1,\,2),\,{{D}_{n}}\times {{C}_{m}}(n=2,\,3,\,4,\,6,\,m=\,1,\,2),\,{{A}_{4}}\times {{C}_{m}}(m=1,\,2),\,{{S}_{4}}\,\times \,{{C}_{m}}(m=1,\,2)$ . Next, we consider the case where $E$ has complex multiplication by the ring of integers $\mathcal{O}$ of an imaginary quadratic field $\Re $ contained in $K$. Suppose that the $\mathcal{O}$-rank over a Galois extension $F$ is 1 or 2. If $\Re \ne \mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$ and ${{h}_{\Re }}$ (class number of $\Re $) is odd, we show that $E$ acquires positive $\mathcal{O}$-rank over a cyclic extension of $K$ or over a field whose Galois group is one of $\text{S}{{\text{L}}_{2}}(\mathbb{Z}/3\mathbb{Z})$ , an extension of $\text{S}{{\text{L}}_{2}}(\mathbb{Z}/3\mathbb{Z})$ by $\mathbb{Z}/2\mathbb{Z}$, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of $L$-functions.