Local-global principle for 11-isogenies of elliptic curves is true over quadratic fields

TL;DR

This paper proves that the local-global principle for 11-isogenies of elliptic curves holds over quadratic fields, confirming a conjecture and advancing understanding of elliptic curve isogenies over number fields.

Contribution

It establishes the validity of the local-global principle for 11-isogenies over quadratic fields, resolving a conjecture by Banwait and Cremona.

Findings

The local-global principle for 11-isogenies is true over quadratic fields.

Determined the set of quadratic points on the modular curve X_{D_{10}}(11).

Confirmed the conjecture by Banwait and Cremona.

Abstract

In this paper, we prove that the local-global principle of $11$-isogenies for elliptic curves over quadratic fields does not fail. This gives a positive answer to a conjecture by Banwait and Cremona. The proof is based on the determination of the set of quadratic points on the modular curve $X_{D_{10}}(11)$.