Adelic point groups of elliptic curves
Volume 199 / 2021
Acta Arithmetica 199 (2021), 221-236
MSC: Primary 11G05, 11G07; Secondary 11F80.
DOI: 10.4064/aa171025-27-3
Published online: 20 July 2021
Abstract
We show that for an elliptic curve $E$ defined over a number field $K$, the group $E(\mathbf{A} _K)$ of points of $E$ over the adele ring $\mathbf{A} _K$ of $K$ is a topological group that can be analyzed in terms of the Galois representation associated to the torsion points of $E$. An explicit description of $E(\mathbf{A} _K)$ is given, and we prove that for $K$ of degree $n$, ‘almost all’ elliptic curves over $K$ have an adelic point group topologically isomorphic to $$ (\mathbf{R} /\mathbf{Z} )^n\times \widehat {\mathbf{Z}} ^n\times \prod _{m=1}^\infty \mathbf{Z} /m\mathbf{Z} . $$ We also show that there exist infinitely many elliptic curves over $K$ having a different adelic point group.
