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PreprintCC BY 4.0arXiv

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

  • Miguel Pineda-Martín
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Abstract

Let E/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 ℓ is added, in terms of invariants of the curve. Concretely, the conductor, the prime ℓ and the Kodaira symbol at the primes of bad reduction.

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Cite

BibTeXRIS
BibTeX
@misc{pinedamartin2026ramification,
  title = {{On the Ramification of Quadratic Fields where there is a torsion growth}},
  author = {Miguel Pineda-Martín},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01485},
  eprint = {2609.01485},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01485},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Ramification of Quadratic Fields where there is a torsion growth
AU  - Miguel Pineda-Martín
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01485
UR  - https://arxiv.org/abs/2609.01485
AB  - Let E/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 ℓ is added, in terms of invariants of the curve. Concretely, the conductor, the prime ℓ and the Kodaira symbol at the primes of bad reduction.
ER  -