Non-existence of elliptic curves with everywhere good reduction over some real quadratic fields

TL;DR

This paper proves that certain real quadratic fields do not admit elliptic curves with good reduction everywhere, resolving a specific case in the classification of elliptic curves over number fields.

Contribution

It establishes the non-existence of such elliptic curves over some real quadratic fields, filling a gap in the understanding of elliptic curves with everywhere good reduction.

Findings

No elliptic curves with good reduction everywhere over specified real quadratic fields

Completes classification for these fields

Provides methods applicable to similar non-existence proofs

Abstract

We prove the non-existence of elliptic curves having good reduction everywhere over some real quadratic fields.