Higher rank elliptic sieve via CM endomorphisms
Abstract
A theorem of Landau and Ramanujan states that almost all integers are not sums of two squares (when ordered by their absolute value). Equivalently, almost all integers are not norms from the Gaussian rationals ℚ(i). In this paper, we prove a version of this norm result for elliptic curves E. Let E be an elliptic curve over a number field K with complex multiplication, such that K contains the endomorphism ring of E. For a fixed non-torsion point P ∈ E(K) and an abelian extension L/K, we show that subject to a technical condition, the y-coordinate of almost all multiples of P does not belong to the norm group of L. The proof is an adaptation of an argument of Bhakta et al. , whose main tools are elliptic divisibility sequences and a sieve technique known as the elliptic sieve. To account for the points in E(K) arising from the additional endomorphisms, we apply the elliptic sieve to elliptic nets, a higher-rank generalisation of elliptic divisibility sequences. This approach demonstrates the possibility of applying the elliptic sieve to ℤ--linear combination of points on elliptic curves and the usefulness of elliptic nets to counting problems on elliptic curves.
Related
- Supersingular Elliptic Curves Without Inseparable Small Degree Endomorphisms
- A decomposition theorem for elliptic structures of minimal rank on compact Lie groups and applications
- Rank stability of elliptic curves over 𝔽q(t) in residue classes
- Elliptic curve rank growth in multi-quadratic extensions
- Rank-Two Drinfeld Modules over Elliptic Coordinate Rings
BibTeX
@misc{auyeung2026higher,
title = {{Higher rank elliptic sieve via CM endomorphisms}},
author = {Hang Lok Edison Au-Yeung},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18281},
eprint = {2610.18281},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18281},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Higher rank elliptic sieve via CM endomorphisms
AU - Hang Lok Edison Au-Yeung
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18281
UR - https://arxiv.org/abs/2610.18281
AB - A theorem of Landau and Ramanujan states that almost all integers are not sums of two squares (when ordered by their absolute value). Equivalently, almost all integers are not norms from the Gaussian rationals ℚ(i). In this paper, we prove a version of this norm result for elliptic curves E. Let E be an elliptic curve over a number field K with complex multiplication, such that K contains the endomorphism ring of E. For a fixed non-torsion point P ∈ E(K) and an abelian extension L/K, we show that subject to a technical condition, the y-coordinate of almost all multiples of P does not belong to the norm group of L. The proof is an adaptation of an argument of Bhakta et al. , whose main tools are elliptic divisibility sequences and a sieve technique known as the elliptic sieve. To account for the points in E(K) arising from the additional endomorphisms, we apply the elliptic sieve to elliptic nets, a higher-rank generalisation of elliptic divisibility sequences. This approach demonstrates the possibility of applying the elliptic sieve to ℤ--linear combination of points on elliptic curves and the usefulness of elliptic nets to counting problems on elliptic curves.
ER -