Adelic point groups of elliptic curves

Athanasios Angelakis¹ ; Peter Stevenhagen²

¹ Department of Mathematics National Technical University of Athens 9 Iroon Polytexneiou St. 15780 Zografou, Attiki, Greece
² Mathematisch Instituut Leiden Universiteit Postbus 9512 2300 RA Leiden, The Netherlands

Acta Arithmetica, Tome 199 (2021) no. 3, pp. 221-236

Résumé

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.

Détail

DOI : 10.4064/aa171025-27-3

Keywords: elliptic curve defined number field nbsp group mathbf points adele ring mathbf topological group analyzed terms galois representation associated torsion points explicit description mathbf given prove degree almost elliptic curves have adelic point group topologically isomorphic mathbf mathbf times widehat mathbf times prod infty mathbf mathbf there exist infinitely many elliptic curves having different adelic point group