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Elliptic curve rank growth in multi-quadratic extensions
Abstract
In this article we give an algorithm to produce infinitely many explicit quadratic twists of a given elliptic curve over ℚ with a rational point and fixed root number. Moreover, under the Parity Conjecture (or other technical hypotheses), for each n≥ 1, we produce explicit multi-quadratic extensions Ln of ℚ, of degree 2n, such that the growth in rank from ℚ to Ln is at least max{2n+2n−1−1, 2n−1} when the root number of E is −1, and at least max{2n, 2n−1} when it is +1. As a result, assuming Parity, we exhibit elliptic curves over ℚ and extensions of degree 2n where the rank after base-extension is greater than or equal to $34$ or 30+max{2n+2n−1, 2n}.
Related
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BibTeX
@misc{costa2026elliptic,
title = {{Elliptic curve rank growth in multi-quadratic extensions}},
author = {Dylan T. Costa and Álvaro Lozano-Robledo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31925},
eprint = {2610.31925},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31925},
}
RIS
TY - UNPB
TI - Elliptic curve rank growth in multi-quadratic extensions
AU - Dylan T. Costa
AU - Álvaro Lozano-Robledo
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31925
UR - https://arxiv.org/abs/2610.31925
AB - In this article we give an algorithm to produce infinitely many explicit quadratic twists of a given elliptic curve over ℚ with a rational point and fixed root number. Moreover, under the Parity Conjecture (or other technical hypotheses), for each n≥ 1, we produce explicit multi-quadratic extensions L_n of ℚ, of degree 2^n, such that the growth in rank from ℚ to L_n is at least max{2n+2^n−1−1, 2^n−1} when the root number of E is −1, and at least max{2n, 2^n−1} when it is +1. As a result, assuming Parity, we exhibit elliptic curves over ℚ and extensions of degree 2^n where the rank after base-extension is greater than or equal to $34$ or 30+max{2n+2^n−1, 2^n}.
ER -