The Ostrowski quotient of an elliptic curve

TL;DR

This paper introduces and studies the Ostrowski quotient for elliptic curves over number field extensions, providing a new structural understanding especially for quadratic extensions.

Contribution

It extends the concept of Ostrowski quotient from number fields to elliptic curves, offering a new approach and a structure theorem for the coarse Ostrowski quotient.

Findings

Established a structure theorem for the coarse Ostrowski quotient of elliptic curves.

Analyzed the structure in the case of quadratic extensions.

Provided a new framework connecting ideal class groups and elliptic curve properties.

Abstract

For $K/F$ a finite Galois extension of number fields, the relative P\'olya group $\Po(K/F)$ is the subgroup of the ideal class group of $K$ generated by all the strongly ambiguous ideal classes in $K/F$. The notion of Ostrowski quotient $\Ost(K/F)$, as the cokernel of the capitulation map into $\Po(K/F)$, has been recently introduced in \cite{SRM}. In this paper, using some results of Gonz\'alez-Avil\'es \cite{Aviles}, we find a new approach to define $\Po(K/F)$ and $\Ost(K/F)$ which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For $E$ an elliptic curve defined over $F$, we define the Ostrowski quotient $\Ost(E,K/F)$ and the coarse Ostrowski quotient $\Ost_c(E,K/F)$ of $E$ relative to $K/F$, for which in the latter group we do not take into account primes of bad reduction. Our main result is a non-trivial structure theorem for the group $\Ost_c(E,K/F)$ and we analyze this theorem, in some detail, for the class of curves $E$ over quadratic extensions $K/F$.