Adelic point groups of elliptic curves

Volume 199 / 2021

Athanasios Angelakis, Peter Stevenhagen Acta Arithmetica 199 (2021), 221-236 MSC: Primary 11G05, 11G07; Secondary 11F80. DOI: 10.4064/aa171025-27-3 Published online: 20 July 2021


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.


  • Athanasios AngelakisDepartment of Mathematics
    National Technical University of Athens
    9 Iroon Polytexneiou St.
    15780 Zografou, Attiki, Greece
  • Peter StevenhagenMathematisch Instituut
    Leiden Universiteit
    Postbus 9512
    2300 RA Leiden, The Netherlands

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