New algebraic points on covers of elliptic curves
Abstract
For a smooth projective curve C/ℚ of genus ≥ 2 and L/ℚ an extension, we write C(L)new={P∈ C(L):ℚ(P)=L}. Recent work of Khawaja and Siksek conjectures that this set is empty for 100% of degree n number fields L, when ordered by absolute discriminant. Moreover, they bring evidence towards this conjecture when C is a degree n cover of ℙ1. We complement their work by proving analogous results for degree n covers ψ:C→ E of elliptic curves E. Our main result shows that, under suitable hypotheses, the number of distinct absolute discriminants at most X of primitive degree n fields L with C(L)new≠∅ is O(X1/2) or O(X/(log X)α), for some α>0. In degrees $2,3,4$ and $5$ we show that these fields have density $0$ among all fields of the same degree (in degree $4$, also among the primitive ones). The novelty is for degrees $4$ and $5$, where we use work of Bhargava--Shankar--Wang and McGown--Thorne--Tucker to count fields with specified local constraints. Moreover, we give concrete examples of $8$ bielliptic modular curves X0(N), for which X0(N)(L)new=∅ for 100 % of quadratic fields L. Lastly, we point out modular covers of degrees $3$ and $5$ in the LMFDB for which similar conclusions hold.
Related
- Nakajima-extremal Artin--Schreier covers of ordinary elliptic curves in characteristic $2$
- Torsion points in families of elliptic curves
- Likely and unlikely ramification of algebraic points on curves
- Ideal Classes of Order 2n in a Family of Totally Real Cubic Fields and Rational Points on Associated Elliptic Curves
- Hopf-Galois Structures on Extensions Arising from Division Points on Elliptic Curves
BibTeX
@misc{mocanu2026algebraic,
title = {{New algebraic points on covers of elliptic curves}},
author = {Diana Mocanu and George C. Turcas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08753},
eprint = {2610.08753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08753},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - New algebraic points on covers of elliptic curves
AU - Diana Mocanu
AU - George C. Turcas
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08753
UR - https://arxiv.org/abs/2610.08753
AB - For a smooth projective curve C/ℚ of genus ≥ 2 and L/ℚ an extension, we write C(L)_new={P∈ C(L):ℚ(P)=L}. Recent work of Khawaja and Siksek conjectures that this set is empty for 100% of degree n number fields L, when ordered by absolute discriminant. Moreover, they bring evidence towards this conjecture when C is a degree n cover of ℙ^1. We complement their work by proving analogous results for degree n covers ψ:C→ E of elliptic curves E. Our main result shows that, under suitable hypotheses, the number of distinct absolute discriminants at most X of primitive degree n fields L with C(L)_new≠∅ is O(X^1/2) or O(X/(log X)^α), for some α>0. In degrees $2,3,4$ and $5$ we show that these fields have density $0$ among all fields of the same degree (in degree $4$, also among the primitive ones). The novelty is for degrees $4$ and $5$, where we use work of Bhargava--Shankar--Wang and McGown--Thorne--Tucker to count fields with specified local constraints. Moreover, we give concrete examples of $8$ bielliptic modular curves X_0(N), for which X_0(N)(L)_new=∅ for 100 % of quadratic fields L. Lastly, we point out modular covers of degrees $3$ and $5$ in the LMFDB for which similar conclusions hold.
ER -