On the $p$-isogenies of elliptic curves with multiplicative reduction over quadratic fields
Number Theory
2023-06-21 v2
Abstract
Let be a prime and a quadratic number field. In this article we extend a previous result of Najman and the author and prove that if is an elliptic curve with potentially multiplicative reduction at all primes , then does not have prime isogenies of degree greater than and different from . As an application to our main result, we present a variant of the asymptotic version of Fermat's Last Theorem over quadratic imaginary fields of class number one.
Keywords
Cite
@article{arxiv.2305.19969,
title = {On the $p$-isogenies of elliptic curves with multiplicative reduction over quadratic fields},
author = {George C. Ţurcaş},
journal= {arXiv preprint arXiv:2305.19969},
year = {2023}
}
Comments
An error in the displayed equation on page 7 affects the validity of the main result (Theorem 2). The point (x,x^{\tau}) is not a rational point on X_0^{w}