Modular curves and bad reduction

TL;DR

This paper investigates conditions under which elliptic curves associated with points on modular curves exhibit bad reduction at specific primes, with results applied to certain quadratic fields.

Contribution

It provides new results linking modular curves to bad reduction of elliptic curves over number fields, including explicit cases for quadratic fields.

Findings

Elliptic curves with cyclic torsion subgroup of order 20 over certain quadratic fields have bad reduction at primes over 3.

Different proof techniques are used depending on whether the prime is split or inert in the quadratic field.

The paper establishes conditions under which elliptic curves must have bad reduction at specified primes.

Abstract

We prove results that imply, under various hypotheses, that every elliptic curve over a number field $k$ corresponding to a point on a modular curve has bad reduction at a certain prime $p$ of $\mathcal{O}_k$. For example, every elliptic curve with a cyclic torsion subgroup of order 20 defined over $\mathbb{Q}(\sqrt{-11})$ or $\mathbb{Q}(\sqrt{17})$ has bad reduction at all primes lying over $3$. The proofs of these statements are quite different, since $3$ is split in $\mathbb{Q}(\sqrt{-11})$ and inert in $\mathbb{Q}(\sqrt{17})$.