The arithmetic of critical values II: critical elliptic curves
Abstract
In this second chapter of the Arithmetic of critical values series (ACV), we study certain double covers Ef→ℙ1 whose branch locus coincides with that of a quartic polynomial f. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves Ef admit a $3$-isogeny. Our methods are Galois-theoretic, and lead us to a thorough analysis of the Galois closure of f:ℙ1→ℙ1. We exploit its rich geometry to prove a Selmer companionship theorem for the family Ef, allowing us to exhibit elements in certain Tate-Shafarevich groups which are visible in an abelian surface. We also give some dynamical and Diophantine applications of our constructions, as well as new examples of Jacobians isogenous to a power of an elliptic curve.
Related
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BibTeX
@misc{naccarato2026arithmetic,
title = {{The arithmetic of critical values II: critical elliptic curves}},
author = {Francesco Naccarato},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.29836},
eprint = {2608.29836},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29836},
}
RIS
TY - UNPB
TI - The arithmetic of critical values II: critical elliptic curves
AU - Francesco Naccarato
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2608.29836
UR - https://arxiv.org/abs/2608.29836
AB - In this second chapter of the Arithmetic of critical values series (ACV), we study certain double covers E_f→ℙ^1 whose branch locus coincides with that of a quartic polynomial f. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves E_f admit a $3$-isogeny. Our methods are Galois-theoretic, and lead us to a thorough analysis of the Galois closure of f:ℙ^1→ℙ^1. We exploit its rich geometry to prove a Selmer companionship theorem for the family E_f, allowing us to exhibit elements in certain Tate-Shafarevich groups which are visible in an abelian surface. We also give some dynamical and Diophantine applications of our constructions, as well as new examples of Jacobians isogenous to a power of an elliptic curve.
ER -