[Paper Review] The density of the terms in an elliptic divisibility sequence having a fixed G.C.D. with their index

Seoyoung C. Kim|arXiv (Cornell University)|Aug 28, 2017

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper establishes a structural characterization and asymptotic density formula for the set of indices $\mathscr{A}_{E,k} = \{n \geq 1 : \gcd(n, D_n) = k\}$ in elliptic divisibility sequences (EDS). It proves that the asymptotic density of $\mathscr{A}_{E,k}$ exists and is positive if and only if the set is nonempty, and provides an explicit density formula using the Möbius function: $\mathbf{d}(\mathscr{A}_{E,k}) = \sum_{d=1}^\infty \frac{u(d)}{l(dk)}$, under conditions on the elliptic curve’s Frobenius trace distribution and convergence of a related series.

ABSTRACT

Let $\mathbf{D}=(D_{n})_{n\geq 1}$ be an elliptic divisibility sequence associated to the pair $(E,P)$. For a fixed integer $k$, we define $\mathscr{A}_{E,k}=\{n\geq 1 : \gcd(n,D_{n})=k\}$. We give an explicit structural description of $\mathscr{A}_{E,k}$. Also, we explain when $\mathscr{A}_{E,k}$ has positive asymptotic density using bounds related to the distribution of trace of Frobenius of $E$. Furthermore, we get explicit density of $\mathscr{A}_{E,k}$ using the Möbius function.

Motivation & Objective

• To characterize the set $\mathscr{A}_{E,k} = \{n \geq 1 : \gcd(n, D_n) = k\}$ for a given elliptic divisibility sequence $\mathbf{D}$ associated to an elliptic curve $E$ and point $P$.
• To determine when $\mathscr{A}_{E,k}$ has positive asymptotic density, linking this to the distribution of the trace of Frobenius of $E$.
• To derive an explicit formula for the asymptotic density of $\mathscr{A}_{E,k}$ using the Möbius function and the least common multiple $l(n) = \mathrm{lcm}(n, r_n)$, where $r_n$ is the rank of apparition of $n$ in $\mathbf{D}$.
• To extend results known for Fibonacci sequences to the more complex setting of elliptic divisibility sequences, accounting for varying curve and point structures.
• To establish conditions under which the infinite series $\sum_{d=1}^\infty \frac{\mu(d)}{l(dk)}$ converges and equals the asymptotic density of $\mathscr{A}_{E,k}$.

Proposed method

• Define $\mathscr{A}_{E,k} = \{n \geq 1 : \gcd(n, D_n) = k\}$, where $D_n$ is the denominator of the $x$-coordinate of $[n]P$ on an elliptic curve $E$.
• Introduce $r_n = \min\{r \geq 1 : n \mid D_r\}$, the rank of apparition of $n$, and $l(n) = \mathrm{lcm}(n, r_n)$, which governs divisibility behavior in the EDS.
• Construct the set $\mathscr{L}_k = \{p : p \mid k\} \cup \left\{ \frac{l(kp)}{l(k)} : p \nmid k \right\}$, and define $\mathscr{N}(\mathscr{L}_k)$ as the set of integers not divisible by any $s \in \mathscr{L}_k$.
• Prove that $\mathscr{A}_{E,k} = \{ l(k)m : m \in \mathscr{N}(\mathscr{L}_k) \}$, providing a structural description of the index set.
• Use bounds on the distribution of the trace of Frobenius of $E$ to establish conditions under which $\mathbf{d}(\mathscr{A}_{E,k})$ exists and is positive.
• Derive the density formula $\mathbf{d}(\mathscr{A}_{E,k}) = \sum_{d=1}^\infty \frac{\mu(d)}{l(dk)}$ under the convergence assumption $\sum_{d=1}^\infty \frac{|\mu(d)|}{l(d)} < \infty$, justified via Möbius inversion and absolute convergence.

Experimental results

Research questions

• RQ1 Under what conditions does the set $\mathscr{A}_{E,k} = \{n \geq 1 : \gcd(n, D_n) = k\}$ have positive asymptotic density for an elliptic divisibility sequence?
• RQ2 What is the precise structural form of $\mathscr{A}_{E,k}$, and how can it be described in terms of the least common multiple $l(n)$ and the rank of apparition $r_n$?
• RQ3 Can the asymptotic density of $\mathscr{A}_{E,k}$ be expressed in a closed form using the Möbius function, and under what convergence assumptions does this formula hold?
• RQ4 How does the distribution of the trace of Frobenius of the elliptic curve $E$ influence the existence and positivity of $\mathbf{d}(\mathscr{A}_{E,k})$?
• RQ5 To what extent do the results generalize the known Fibonacci sequence case, and what new challenges arise in the context of elliptic divisibility sequences?

Key findings

• The set $\mathscr{A}_{E,k}$ is nonempty if and only if it has positive asymptotic density, under the assumption that $E$ is not CM or is CM and finitely anomalous (i.e., $E(\mathbb{F}_p) \neq p$ for all but finitely many $p$).
• The asymptotic density of $\mathscr{A}_{E,k}$ is given by $\mathbf{d}(\mathscr{A}_{E,k}) = \sum_{d=1}^\infty \frac{\mu(d)}{l(dk)}$, provided the series $\sum_{d=1}^\infty \frac{|\mu(d)|}{l(d)}$ converges.
• The structural description $\mathscr{A}_{E,k} = \{ l(k)m : m \in \mathscr{N}(\mathscr{L}_k) \}$ fully characterizes the index set in terms of a set of forbidden divisors $\mathscr{L}_k$.
• The convergence of $\sum_{d=1}^\infty \frac{|\mu(d)|}{l(d)}$ is crucial for the validity of the density formula and depends on the group structure of the Néron model fibers of $E$.
• Numerical examples show that the partial sums $\sum_{d=1}^n \frac{|\mu(d)|}{l(d)}$ grow extremely slowly, suggesting the convergence condition is plausible for many curves.
• The density formula generalizes the Fibonacci case: for the Fibonacci sequence, $\mathbf{d}(\mathscr{A}_{F,k}) = \sum_{d=1}^\infty \frac{\mu(d)}{l(dk)}$, and this paper extends that result to EDS under suitable conditions.