Experiments on $3$-isogeny Selmer groups of elliptic curves with a $3$-torsion point
Abstract
Let EA,B: y2 +Axy + By = x3 be an elliptic curve over ℚ. The $3$-torsion point (0,0) induces a $3$-isogeny ϕ: EA,B→ EA,B'. Assuming EA,B has good reduction at $3$, we construct an explicit (m+t)× m matrix MA,B' over 𝔽3, whose kernel encodes the dual isogeny Selmer group Selϕ̂(EA,B') modulo the image of the torsion point (0,0). Here, m = ω(B) − 1, and t encodes the \emph{global Selmer ratio} 3t−2. We compute MA,B' in various regimes for billions of elliptic curves EA,B. Based on our data, and motivated by prevalence of random linear algebraic models throughout number theory, we conjecture that, for fixed m and t, the matrices MA,B' become uniformly distributed, and we formulate a corresponding conjecture for the distribution of Selϕ(EA,B). Our model predicts that, for fixed t, the average size of Selϕ(EA,B) is 1 + 3t.
Related
- Arithmetic statistics of isogeny Selmer groups associated to hyperelliptic curves
- Selmer groups of CM-twists of elliptic curves
- Torsion groups of rational elliptic curves over ℤp-extensions of quadratic fields: the p≤ 5 case
- Models of Elliptic Curves with Maximal 2-adic Image Subject to a Rational 2-Isogeny
- Refined conjectures on Fitting ideals of BDP Selmer groups
BibTeX
@misc{weiss2026experiments,
title = {{Experiments on $3$-isogeny Selmer groups of elliptic curves with a $3$-torsion point}},
author = {Ariel Weiss and Dongchen Zou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31330},
eprint = {2609.31330},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31330},
}
RIS
TY - UNPB
TI - Experiments on $3$-isogeny Selmer groups of elliptic curves with a $3$-torsion point
AU - Ariel Weiss
AU - Dongchen Zou
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31330
UR - https://arxiv.org/abs/2609.31330
AB - Let E_A,B: y^2 +Axy + By = x^3 be an elliptic curve over ℚ. The $3$-torsion point (0,0) induces a $3$-isogeny ϕ: E_A,B→ E_A,B'. Assuming E_A,B has good reduction at $3$, we construct an explicit (m+t)× m matrix M_A,B' over 𝔽_3, whose kernel encodes the dual isogeny Selmer group Sel_ϕ̂(E_A,B') modulo the image of the torsion point (0,0). Here, m = ω(B) − 1, and t encodes the \emph{global Selmer ratio} 3^t−2. We compute M_A,B' in various regimes for billions of elliptic curves E_A,B. Based on our data, and motivated by prevalence of random linear algebraic models throughout number theory, we conjecture that, for fixed m and t, the matrices M_A,B' become uniformly distributed, and we formulate a corresponding conjecture for the distribution of Sel_ϕ(E_A,B). Our model predicts that, for fixed t, the average size of Sel_ϕ(E_A,B) is 1 + 3^t.
ER -