An upper bound for valuations of elliptic nets at primes of bad reduction
Abstract
Let E/K be an elliptic curve over a field defined by the short Weierstrass equation and let P=(P1, P2) ∈ E(K)2∖ {(O, O), (± P, ± P)} be a pair of non-torsion points, where O denotes the point at infinity. We give a uniform upper bound for the quantity min(ν(Φ(v1, v2)(P)), 2ν(Ψ(v1, v2)(P))), where ν is a normalised discrete valuation associated to primes where E has bad reduction and Ψ2(v1, v2)(P), Φ(v1, v2)(P) are rational functions for describing ℤ--linear combination of points [v1]P1+[v2]P2 on E, higher--dimensional analogues of division polynomials ψ2n, ϕn known as net polynomials. They are also elliptic nets, which are n--dimensional arrays with values in K satisfying the same nonlinear recurrence relation that division polynomials do as well. Hence, they are useful in studying counting problems on elliptic curves.
Related
- On Greenberg's conjecture for rational elliptic curves at Eisenstein primes
- On the Iwasawa invariants of admissible quadratic twists of elliptic curves at ordinary Eisenstein primes
- Iwasawa theory of CM elliptic curves at potentially supersingular primes
- An explicit power-saving bound for Sárközy's Theorem on shifted primes, assuming GRH
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BibTeX
@misc{auyeung2026upper,
title = {{An upper bound for valuations of elliptic nets at primes of bad reduction}},
author = {Hang Lok Edison Au-Yeung},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20273},
eprint = {2610.20273},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20273},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An upper bound for valuations of elliptic nets at primes of bad reduction
AU - Hang Lok Edison Au-Yeung
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20273
UR - https://arxiv.org/abs/2610.20273
AB - Let E/K be an elliptic curve over a field defined by the short Weierstrass equation and let P=(P_1, P_2) ∈ E(K)^2∖ {(O, O), (± P, ± P)} be a pair of non-torsion points, where O denotes the point at infinity. We give a uniform upper bound for the quantity min(ν(Φ_(v_1, v_2)(P)), 2ν(Ψ_(v_1, v_2)(P))), where ν is a normalised discrete valuation associated to primes where E has bad reduction and Ψ^2_(v_1, v_2)(P), Φ_(v_1, v_2)(P) are rational functions for describing ℤ--linear combination of points [v_1]P_1+[v_2]P_2 on E, higher--dimensional analogues of division polynomials ψ^2_n, ϕ_n known as net polynomials. They are also elliptic nets, which are n--dimensional arrays with values in K satisfying the same nonlinear recurrence relation that division polynomials do as well. Hence, they are useful in studying counting problems on elliptic curves.
ER -