On Sufficient Conditions for Rank Zero in Families of Elliptic Curves
Abstract
In this article, we study the Mordell Weil ranks of elliptic curves in the family ED:y2=x3+Dx, where D∈ℤ∖{0}. Under suitable congruence conditions on D and its prime divisors, we use $2$-descent via a rational $2$-isogeny to study the rank of ED. Under our congruence hypotheses, the ranks of these matrices give sufficient conditions for ED to have rank zero. We give explicit sufficient conditions for the curves ED to have Mordell Weil rank zero over ℚ in the following cases: (i) D is an odd negative integer, (ii) D is an odd positive integer, and (iii) D is a positive even integer. Also we give a quantitative result describing how many square free parameters satisfy the required matrix-rank condition. In particular, our results show that these rank zero criteria hold for a significant proportion of the admissible parameters.
Related
- Torsion points in families of elliptic curves
- On quantitative sufficient second-order optimality conditions for elliptic optimal control problems
- Rank stability of elliptic curves over 𝔽q(t) in residue classes
- Rank stability of elliptic curves over prime cyclic extensions of function fields
- Second derivatives of p-adic L-functions and the Shafarevich--Tate group of rank-two CM elliptic curves
BibTeX
@misc{das2026sufficient,
title = {{On Sufficient Conditions for Rank Zero in Families of Elliptic Curves}},
author = {Shamik Das and Debajyoti De and Arup Mondal},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08029},
eprint = {2611.08029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08029},
}
RIS
TY - UNPB
TI - On Sufficient Conditions for Rank Zero in Families of Elliptic Curves
AU - Shamik Das
AU - Debajyoti De
AU - Arup Mondal
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08029
UR - https://arxiv.org/abs/2611.08029
AB - In this article, we study the Mordell Weil ranks of elliptic curves in the family E_D:y^2=x^3+Dx, where D∈ℤ∖{0}. Under suitable congruence conditions on D and its prime divisors, we use $2$-descent via a rational $2$-isogeny to study the rank of E_D. Under our congruence hypotheses, the ranks of these matrices give sufficient conditions for E_D to have rank zero. We give explicit sufficient conditions for the curves E_D to have Mordell Weil rank zero over ℚ in the following cases: (i) D is an odd negative integer, (ii) D is an odd positive integer, and (iii) D is a positive even integer. Also we give a quantitative result describing how many square free parameters satisfy the required matrix-rank condition. In particular, our results show that these rank zero criteria hold for a significant proportion of the admissible parameters.
ER -