PreprintarXiv licencearXiv
Torsion growth of rational elliptic curves over Zp-extensions of quadratic fields
Abstract
Let E/Q be an elliptic curve and let K̃p be the compositum of all Zp-extensions of a quadratic field K. We prove that E(K̃p)tors=E(K)tors for p≥5. For p=3 and imaginary quadratic K≠Q(√(−3)), torsion on each extension is determined by its intersections with the cyclotomic extension and the $2$-division field. Over Q(√(−3)), we construct infinitely many non-CM curves with full $3$-torsion in the first anticyclotomic layer and compute the $3$-primary torsion on every slope for eight CM curves. For p=2, we bound the odd-primary torsion and exclude all primes greater than $7$. We also give uniform bounds for non-CM primary torsion and correct two assertions in Li's preprint about noncyclotomic Z3-extensions.
Related
- Torsion groups of rational elliptic curves over ℤp-extensions of quadratic fields: the p≤ 5 case
- Elliptic curve rank growth in multi-quadratic extensions
- On the Ramification of Quadratic Fields where there is a torsion growth
- Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions
- Torsion points in families of elliptic curves
BibTeX
@misc{xu2026torsion,
title = {{Torsion growth of rational elliptic curves over $\Z_p$-extensions of quadratic fields}},
author = {Shisong Xu and Haoyang Yuan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.17303},
eprint = {2607.17303},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.17303},
}
RIS
TY - UNPB
TI - Torsion growth of rational elliptic curves over Z_p-extensions of quadratic fields
AU - Shisong Xu
AU - Haoyang Yuan
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2607.17303
UR - https://arxiv.org/abs/2607.17303
AB - Let E/Q be an elliptic curve and let K̃_p be the compositum of all Z_p-extensions of a quadratic field K. We prove that E(K̃_p)_tors=E(K)_tors for p≥5. For p=3 and imaginary quadratic K≠Q(√(−3)), torsion on each extension is determined by its intersections with the cyclotomic extension and the $2$-division field. Over Q(√(−3)), we construct infinitely many non-CM curves with full $3$-torsion in the first anticyclotomic layer and compute the $3$-primary torsion on every slope for eight CM curves. For p=2, we bound the odd-primary torsion and exclude all primes greater than $7$. We also give uniform bounds for non-CM primary torsion and correct two assertions in Li's preprint about noncyclotomic Z_3-extensions.
ER -