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arXiv · 2609.01485

On the Ramification of Quadratic Fields where there is a torsion growth

Miguel Pineda-Martín
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Abstract

Let $E/\mathbb{Q}$ be a rational elliptic curve whose torsion group grows over a quadratic field $K$. In a previous paper, a relation between the primes of the conductor of $E$ and the primes that divide the discriminant of $K$ was shown. In the present paper, we go further in this study and we compute the ramification of the field $K$, when a torsion point of prime order $\ell$ is added, in terms of invariants of the curve. Concretely, the conductor, the prime $\ell$ and the Kodaira symbol at the primes of bad reduction.

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Miguel Pineda-Martín. 2026-09-01. On the Ramification of Quadratic Fields where there is a torsion growth. https://arxiv.org/abs/2609.01485

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