Selmer groups of CM-twists of elliptic curves
Abstract
Let F be a totally real Galois number field of degree g≤5 and let ℓ be an odd prime. Let E/F be an elliptic curve with an F-rational point of order ℓ. Under explicit arithmetic and local hypotheses, together with an explicit non-vanishing condition modulo ℓ, we prove that there are ≫F,E,ℓX1/(2g)/(log X) totally negative square classes d∈ F×/(F×)2 with $\left|\mathrm{N}_{F/\mathbb{Q}}\bigl(D(F(\sqrt{d})/F)\bigr)\right|<X $ for which Selℓ(Ed,F) is trivial. Such twists have rank zero and trivial ℓ-part of the Shafarevich--Tate group. The condition is verifiable by a finite computation, which we carry out in an example. We lift the method of James and Ono from ℚ to the totally real setting, combining a theorem of Morrow, which relates the Selmer group of such a twist to the class group of the CM extension F(√d), with an indivisibility theorem of Takai for relative class numbers. Takai twists only by quadratic Hecke characters, whereas Morrow's conditions are local; we extend the twisting argument to primitive quadratic residue-class characters to make the two results compatible.
Related
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BibTeX
@misc{beckwith2026selmer,
title = {{Selmer groups of CM-twists of elliptic curves}},
author = {Olivia Beckwith and Tùng Hoàng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23847},
eprint = {2609.23847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23847},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Selmer groups of CM-twists of elliptic curves
AU - Olivia Beckwith
AU - Tùng Hoàng
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.23847
UR - https://arxiv.org/abs/2609.23847
AB - Let F be a totally real Galois number field of degree g≤5 and let ℓ be an odd prime. Let E/F be an elliptic curve with an F-rational point of order ℓ. Under explicit arithmetic and local hypotheses, together with an explicit non-vanishing condition modulo ℓ, we prove that there are ≫_F,E,ℓX^1/(2g)/(log X) totally negative square classes d∈ F^×/(F^×)^2 with $\left|\mathrm{N}_{F/\mathbb{Q}}\bigl(D(F(\sqrt{d})/F)\bigr)\right|<X $ for which Sel_ℓ(E^d,F) is trivial. Such twists have rank zero and trivial ℓ-part of the Shafarevich--Tate group. The condition is verifiable by a finite computation, which we carry out in an example. We lift the method of James and Ono from ℚ to the totally real setting, combining a theorem of Morrow, which relates the Selmer group of such a twist to the class group of the CM extension F(√d), with an indivisibility theorem of Takai for relative class numbers. Takai twists only by quadratic Hecke characters, whereas Morrow's conditions are local; we extend the twisting argument to primitive quadratic residue-class characters to make the two results compatible.
ER -