Hang Lok Edison Au-Yeung

arXiv2026

An upper bound for valuations of elliptic nets at primes of bad reduction

Let $E/K$ be an elliptic curve over a field defined by the short Weierstrass equation and let $\mathbf{P}=(P_1, P_2) \in E(K)^2\setminus \{(O, O), (\pm P, \pm P)\}$ be a pair of non-torsion points, where $O$ denotes the point at infinity. We give a uniform upper bound for the quantity $\min(ν(Φ_{(v_1, v_2)}(\mathbf{P})), 2ν(Ψ_{(v_1, v_2)}(\mathbf{P})))$, where $ν$ is a normalised discrete valuation associated to primes where $E$ has bad reduction and $Ψ^2_{(v_1, v_2)}(\mathbf{P})$, $Φ_{(v_1, v_2)}(\mathbf{P})$ are rational functions for describing $\mathbb{Z}$--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.

math.NT↗

arXiv2026

Higher rank elliptic sieve via CM endomorphisms

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 $\mathbb{Q}(i)$. 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 $P \in E(K)$ 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 $\mathbb{Z}$--linear combination of points on elliptic curves and the usefulness of elliptic nets to counting problems on elliptic curves.

math.NT↗