The class group pairing and $p$-descent on elliptic curves

TL;DR

This paper develops explicit formulas for the class group pairing on elliptic curves over number fields, linking it to descent via cyclic isogenies and the logarithmic class group, enabling computational experimentation.

Contribution

It introduces explicit formulae for the class group pairing and relates it to descent and Selmer groups, providing tools for computational and theoretical analysis.

Findings

Explicit formulas for the class group pairing

Connection between Selmer groups and logarithmic class groups

Framework suitable for computer experimentation

Abstract

We give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field. Then we relate it to the descent relative to a suitable cyclic isogeny. This allows us to connect the resulting Selmer group with the logarithmic class group of the base. These constructions are explicit and suitable for computer experimentation. From a conceptual point of view, the questions that arise here are analogues of "visibility" questions in the sense of Cremona and Mazur.