A generalization of Mazur's theorem (Ogg's conjecture) for number fields

TL;DR

This paper extends Mazur's theorem, originally for rational fields, to general number fields by modifying Stevens' theorem, enabling new insights into cuspidal subgroups without relying on Shimura subgroups.

Contribution

It generalizes Ogg's conjecture for number fields by adapting Stevens' theorem for congruence subgroups, removing the dependence on Shimura subgroups.

Findings

Generalization of Mazur's theorem to number fields.

Modification of Stevens' theorem for $ ext{GL}_2$ congruence subgroups.

Identification of cuspidal subgroup components independently of Shimura subgroups.

Abstract

In this article, we prove a generalization of a theorem (Ogg's conjecture) due to Bary Mazur for arbitrary $N\in \N$ and for {\it number fields}. The main new observation is a modification of a theorem due to Glenn Stevens for the congruence subgroups of the form $\Ga_0(N)$ for any $N \in \N$. This in turn help us to determine the relevant part of the cuspidal subgroups without dependence on Shimura subgroups.