1 Introduction↩︎

The study of perfect powers occurring among the terms, or among the products of terms, of various recurrence sequences has a long and rich history in Diophantine number theory. A classical result of Erdős and Selfridge [1] states that Diophantine equation \[n(n+1)\cdots(n+s-1)=m^{\ell},\] admits no solutions in integers \(n\ge 0\) and \(s,\ell\ge 2\). Of course, to study such questions, always it is enough to assume that \(\ell\) is a prime. Thus, throughout this whole paper, the exponent \(\ell\) is reserved as a prime. The related questions for products of the terms of an arithmetic progression have also been extensively studied; see for example [2]–[5], and the references therein.

The case in which the factors arise from terms of linear recurrence sequences has also been extensively studied. In the special case of a single term, that is determining whether a term of a linear recurrence sequence can be a perfect power, the problem has received considerable attention. Notable contributions include those of Pethő [6], Shorey and Stewart [7], Corvaja and Zannier [8], and Fuchs and Tichy [9]. To the best of our knowledge, the most general result in this direction, building on several previous works, is due to Bugeaud and Kaneko [10].

On the other hand, Luca and Shorey [11] have obtained an effective upper bound for the solutions of equations in which a product of the terms of a Lucas sequence is a perfect power. This has been extended by Bravo, Das, Guzmán, and Laishram [12] to certain special cases, specifically for the Pell and Pell-Lucas sequences, where they explicitly determined all the solutions. For further historical background and additional developments, we refer the reader to these works and the references therein.

We focus on Diophantine equations of the types \[f(n_1)\cdots f(n_s)=m^{\ell},\] where \(f(n)\) is an arithmetic function taking integer or rational values, and each \(n_i\) is an integer in an interval \((M,M+N]\). In the natural setting \(f(n)=n\), such equations have received considerable attention. In particular, the case \(s=3\) and \(\ell=3\) has been studied by de la Bretèche [13], Fouvry [14], and Heath-Brown and Moroz [15]. Further work includes Tolev [16] for \(s=2\) and \(\ell>2\), and more recently de la Bretèche, Kurlberg, and Shparlinski [17] in the general case. In this paper, we turn to sparse sequences \(\left(f(n)\right)_{n\ge 1}\) arising from certain nonlinear recursions, notably elliptic divisibility sequences, as well as the sequences associated with rational points on elliptic curves.

1.1 Set up↩︎

For an elliptic curve \(E\) given by a short Weierstrass equation \[\label{eq:Weier} y^2 = x^3 + a x + b\tag{1}\] with integral coefficients \(a\) and \(b\), we denote by \(E({\mathbb{Q}})\) the group of rational points of \(E\), and \(O\) denotes the point at infinity, see [18] for background. We can write any point \(P \in E({\mathbb{Q}})\), in the lowest form \[\label{eqn:P} P =(x(P),y(P))=\left(\frac{a_P}{d_P^2},\frac{b_P}{d_P^3}\right),\tag{2}\] where \(d_P \in{ \mathbb{N}}\), and \(\gcd(a_Pb_P , d_P)=1\).

Everest, Reynolds, and Stevens [19] have shown that, for any fixed \(\ell\), there are only finitely many perfect \(\ell\)th power among the denominators \(d_P\). They have also showed the same uniformly over \(\ell\), assuming that abc-conjecture holds. Unconditionally, first Reynolds [20] and then Alfaraj [21] have studied this uniformly over the exponent \(\ell\), for certain elliptic curves. More recently, Nowroozi and Siksek [22] further extended these results over a much broader family of elliptic curves.

We are interested in the product of the denominators; in particular, we study how often such products are perfect powers. Hajdu, Laishram and Szikszai in [23] have studied this over denominators of multiples of a single fixed point. That is, the terms are from a fixed elliptic divisibility sequence. More precisely, they showed that there are finitely many perfect \(\ell\)th powers among the products \[d_{nP}d_{(n+h)P} \cdots d_{(n+(s-1)h)P},\quad n,s,h \in \mathbb{N},\] and in Theorem 22, we record an analogue for arbitrary shifts.

However, the arguments of Hajdu, Laishram and Szikszai, do not work when the points are of the form \(nP+Q\), where \(Q,P\) are fixed rational points, and \(n\) is coming from an interval. One of key the obstacles is that, we no longer have the required \(p\)-adic properties (as in [23]) of these points. Similarly, the same arguments also do not work when we consider the products over all \(s\)-tuples of rational points. Recently, the products of denominators of rational points of such flavours have been considered by Bérczes, Hajdu, Ostafe, Shparlinski and the present author in [24], and they have shown that the denominators of such points are rarely multiplicatively dependent. Building upon [24] and [23], we study the frequency of the tuples of rational points, whose product of denominators is multiplicatively related to a perfect \(\ell\)th power. For any prime \(\ell\), we denote \[{\mathcal{N}}_{\ell}=\{n^{\ell}:n\in { \mathbb{N}}\}\quad \mathrm{and}\quad {\mathcal{Q}}_{\ell}=\{x^{\ell}:x\in {\mathbb{Q}}\}.\]

Assume that we are given \(s\) not necessarily distinct elliptic curves \(E_1, \ldots, E_s\) as in 1 over \({\mathbb{Q}}\) of positive ranks, and a prime \(\ell\). For fixed \(s\)-tuples \[\label{eq:32P32Q} \mathsf P=(P_1, \ldots, P_s) \qquadand\qquad\mathsf Q=(Q_1, \ldots, Q_s),\tag{3}\] of non-torsion points \(P_i\in E_i({\mathbb{Q}})\) and arbitrary points \(Q_i \in E_i({\mathbb{Q}})\), \(i =1, \ldots, s\), we estimate the size of the following set \[\begin{align} &{\mathcal{D}}_{\ell,\mathsf P,\mathsf Q}(M,N)\cr &\quad=\{ (n_1, \ldots, n_s)\in (M,M+N]^s:\cr &\qquad\quad\quad d_{n_1P_1 + Q_1}^{k_1} \cdots d_{n_sP_s+Q_s}^{k_s} \in {\mathcal{N}}_{\ell},~\text{for some }(k_1,\ldots,k_s)\in [1,\ell-1]^s \}. \end{align}\]

Furthermore, we denote by \(\widehat h(P)\) the canonical height of a rational point \(P\in E({\mathbb{Q}})\), see for instance [18]. Then, we also estimate the size of the following set, for some \((k_1,\ldots,k_s)\in [1,\ell-1]^s\) \[\begin{align} &{\mathcal{D}}_{\ell}(H)\cr &\quad= \{ (P_1, \ldots, P_s):P_i \in E_{i}({\mathbb{Q}}),\;\widehat h(P_i) \le H, \;i =1, \ldots, s,\;\text{and}\cr & \qquad\qquad\quad\quad d_{P_1}^{k_1} \cdots d_{P_s}^{k_s} \in {\mathcal{N}}_{\ell},~\text{for some }(k_1,\ldots,k_s)\in [1,\ell-1]^s\}. \end{align}\]

1.2 Main results↩︎

Furthermore, we have a better estimate over the products of the terms of an elliptic divisibility sequences, that is, when \(Q_i=O_i\) for each \(i=1,\ldots,s\), where we denote \(O_i\) to be the point at infinity for \(E_i\).

Certainly, a more interesting question is whether one can obtain a power saving. Unfortunately, we do not currently have an affirmative answer. See Section 7.3 for some conditional perspectives, and the second paragraph of Section 7.2 for further remarks, which provide a power saving over the integers in \((0,N]\) with at most a fixed number of prime factors, when counted with multipilcity.

Let us also arrange the rational points with respect to their canonical heights. If \(E({\mathbb{Q}})\) is of rank \(r\), then by [25], we have \[H^{r/2} \ll \sharp\, \{P \in E({\mathbb{Q}}):~\widehat h(P) \le H\} \ll H^{r/2}.\]

Using the same arguments as in the proof of Theorem 1, we also have the following result.

Furthermore, we obtain analogues of all these results for products of the \(x\)-coordinates (see Section 6.2). In fact, denoting \[\begin{align} X_{\ell,\mathsf P}(N) =\sharp\, \left\{ (n_1, \ldots, n_s)\in (0,N]^s :\, x(n_1P_1)^{k_1} \cdots x(n_sP_s)^{k_s} \in {\mathcal{Q}}_{\ell}, \right.\cr \left. \qquad\qquad\qquad\qquad\qquad\qquad \text{for some } (k_1,\ldots,k_s)\in [1,\ell-1]^s \right\}, \end{align}\] we prove the following estimate.

Note that [26] also imposes ?? , but deriving an estimate for a significantly denser set. Thus, it is natural to ask the following question.

Furthermore, the techniques used to prove these results also apply to sequences that are significantly denser than the set of \(\ell\)th powers, such as the integers or rationals representable as sums of two squares; see Section 6.3.

1.3 Some further motivations↩︎

Consider the toy model of estimating the number of pairs \((n_1,n_2)\in (M,M+N]^2\), for which \(d_{n_1P}\cdot d_{n_2P}\) is a perfect square. Equivalently, this amounts to counting pairs \((n_1,n_2)\in (M+M+N]^2\) such that \(u_{n_1}=u_{n_2}\), where \(u_n\) denotes the squarefree part of \(d_{nP}\), that is, the product of all distinct primes dividing \(d_{nP}\) with an odd valuation. Following the treatment in [24], it is useful to know a quadratic exponential growth of \(u_n\), that is, a Siegel-type property as in [24]. However, to the best of our knowledge, only the \(\mathrm{abc}\) conjecture implies such a growth of \(u_n\); see the proof of [27] and some related discussions in Section 7.3. In particular, the \(\mathrm{abc}\) conjecture implies that the number of pairs \((n_1,n_2)\in (M,M+N]^2\) with \(u_{n_1}=u_{n_2}\) is about \(N\). Hence, by the results in Section 1.2, we obtain an unconditional nontrivial upper bound, which in turn also provides a lower bound for the number of distinct values \(u_n\) with \(n\in (M,M+N]\). See also Remark 17 for some unconditional lower bounds on \(u_n\).

The proper algebraic subgroups of \(\mathbb{G}_m^s\) are defined by finitely many equations of the form \(x_1^{k_1}\cdots x_s^{k_s} = 1\), where \(k_1,\ldots,k_s \in \mathbb{Z}\) are not all zero. In [28], Bombieri, Masser, and Zannier have initiated the study of intersections of geometrically irreducible algebraic curves \({\mathcal{C}}\) in \(\mathbb{G}_m^s\) (for \(s \ge 2\)) defined over a number field \(K\), with proper algebraic subgroups of \(\mathbb{G}_m^s\). More precisely, if \({\mathcal{C}}\) is not contained in any translate of a proper algebraic subgroup of \(\mathbb{G}_m^s\), then [28] shows that the set of points in \({\mathcal{C}}(\overline{{\mathbb{Q}}})\) with multiplicatively dependent coordinates, is a set of a bounded Weil height. Later in [29], the authors have proved finiteness of such points over the maximal abelian extension of \(K\), when \({\mathcal{C}}\) has a positive genus. This shows finiteness of complex points with multiplicatively dependent coordinates on elliptic curves with complex multiplication (CM). Later Barroero and Sha [30], has removed the CM condition.

Here we consider the product of elliptic curves \(\mathbb{E}:=E_1 \times \cdots \times E_s\), and the map \[\Phi_X : \mathbb{E}^{\circ} \longrightarrow \mathbb{G}_m^{s}, \qquad (P_1,\ldots,P_s)\mapsto \bigl(x(P_1),\ldots,x(P_s)\bigr),\] where \(\mathbb{E}^{\circ}\) be the subset of \(\mathbb{E}\) given by \[\mathbb{E}^{\circ} := \left\{(P_1,\ldots,P_s)\in \mathbb{E} : P_i \neq O_i,\; x(P_i)\neq 0, \; \text{for } i=1,\ldots,s \right\}.\]

Recently, a different viewpoint has been considered in [24], where the authors study the intersections of \(\Phi_{X}(\mathbb{E}^{\circ}({\mathbb{Q}}))\) with proper algebraic subgroups of \(\mathbb{G}_m^{s}\). In this paper, we instead consider the intersections of \(\Phi_{X}(\mathbb{E}^{\circ}({\mathbb{Q}}))\times \mathbb{G}_m\) with typical (see Section 7.1) proper algebraic subgroups of \(\mathbb{G}_m^{s+1}\). Moreover, the arguments used to prove the results in Section 1.2 are applied to study the analogous problems for the maps defined by non-constant rational functions \(f_i \in \mathbb{Q}(E_i)\), \(i=1,\ldots,s\), provided that at least one of the \(f_i\) has either a zero or a pole at \(O_i\). For further details, we refer the reader to Section 6.2.

1.4 Overview and the strategies of the proofs↩︎

In our setting, we do not always (e.g., in Theorem 1 and Theorem 3) restrict to the terms of an elliptic divisibility sequences. Therefore, the convenient \(p\)-adic properties from [23] are no longer available. To address this, we borrow the general counting strategy from [24], [31], suitably adapted to our context. The main distinction is that, here we need to count the denominators that are perfect \(\ell\)th powers up to a \(\mathcal{W}\)-unit, for some appropriately chosen finite set of primes \(\mathcal{W}\). While in [24], it has been sufficed to count (Siegel-type estimates) only the terms that are \(\mathcal{W}\)-units.

The counting is carried out in Lemma 8, using a factorization argument similar to the proof of Siegel’s lemma [18], and closely following the argument in [19]. This problem is, in essence, concerned with estimating the number of rational points on a family of generalized Fermat curves with some \(\mathcal{W}\)-unit coefficients. To estimate this, we use Lemma 6 of Dimitrov, Gao, and Habegger. Furthermore, Lemma 8 allows us to remove the assumption Hypothesis 4.4 imposed in the recent preprint of Kym [32]; see Section 7.2 for more details.

Furthermore in Section 4.1, we also study these questions for products of terms of elliptic divisibility sequences via a version of the elliptic sieve due to Loughran, Myerson, Nakahara, and the present author [26]. This is essentially a sieving with respect to the reductions \(E({\mathbb{Q}}) \to E({\mathbb{Z}}/p{\mathbb{Z}})\), for which the order of the reduction of the base point \(P\) modulo \(p\) is a prime, in the sense of Kowalski [25] (it is also worth noting that a sieve for strong divisibility sequences has recently appeared by Browning and Verzobio in [33]).

In fact as in [26], we also need to sieve with respect to the reductions \(E({\mathbb{Q}}) \to E({\mathbb{Z}}/p^{m\ell}{\mathbb{Z}}),~m \ge 1\), in analogy with the case of integers over intervals; see, for example, [34]. This is also the same sieve setup used in [26], and we follow that in the proof of Lemma 14. One of the key difference is that, since we consider powers up to a \(\mathcal{W}\)-unit, we need to restrict to primes \(p\) exceeding a threshold (as in 13 ) that depends on \(\mathcal{W}\).

However, our situation is considerably simpler than in [26], where the primes \(p\) are required to avoid certain arithmetic progressions, whereas no such restriction is present for us. This relaxation simplifies the argument, and allowing an additional logarithmic saving. In fact, it turns out that over the long intervals, the main contribution comes only from the smooth numbers (Lemma 16), leading to a subpolynomial saving in ?? . In particular, both Lemma 14 and Lemma 16 (see Remark 15 for example) provide a non-trivial estimates over some higher ranges of \(\sharp\, {\mathcal{W}}\), where Lemma 8 holds trivially. As a consequence, we prove Theorem 4, which gives an elliptic analogue of [17] for the \(x\)-coordinates.

We prove Theorem 3 in Section 5. The key ingredient is in Section 5.1, where we estimate the number of points of bounded canonical height that reduce to \(O\) modulo a large prime. This is obtained by counting lattice points in boxes over a family of sublattices of \({\mathbb{Z}}^r\) of large index.

In Section 6, we collect several applications of our methods. First we prove Theorem 22, which serves as a version of [23] for arbitrary shifts. Then in Section 6.2, we discuss analogues of the results in Section 1.2 for \(x\)-coordinates and more general rational functions \(f \in {\mathbb{Q}}(E)\). Lastly in Section 6.3, we present an application of the elliptic sieve in estimating the \(s\)-tuples of rational points such that the product of their \(y\)-coordinates is a sum of two squares. Analogous to Lemma 14, the main ingredient here is in 23 , which proves Theorem 23 and recovers [26] for \(s=1\).

Finally in Section 7.2, we address some challenges in obtaining power savings in our main results. Subsequently in Section 7.3, we outline approaches for obtaining power savings under variants of the abc-conjecture, indicating possible directions for further improvement.

1.5 Notations↩︎

We recall the standard convention that the notations \(U \ll V\) and \(U = O(V)\) are equivalent to \(|U| \le cV\) for some constant \(c>0\), where \(c\) may depend on \(s,\ell\), the \(s\)-tuples \(\mathsf P\) and \(\mathsf Q\) as in 3 , the curves \(E_1,\ldots,E_s\) (i.e., the coefficients of their Weierstrass equations 1 ), and, in some instances, their ranks. Furthermore we write \(U^{o(1)}\) for any quantity \(V\), which for any \(\delta > 0\) satisfies \(|V| \le c(\delta) U^{\delta}\), with some constant \(c(\delta)\) that may depend on \(\delta\) and all the invariants mentioned earlier. In particular, \(U^{o(1)} \cdot U^{o(1)} = U^{o(1)}\).

For any finite set \(\mathcal{S}\), we denote by \(\sharp\, \mathcal{S}\) its cardinality. Given a prime \(p\) and a nonzero rational number \(x\), we denote by \(\nu_p(x)\) the \(p\)-adic valuation of \(x\). Finally, as usual, \({\mathbb{Q}}\), \({\mathbb{Z}}\), and \({ \mathbb{N}}\) denote the sets of rational numbers, integers, and natural numbers, respectively.

2 Perfect powers among denominators↩︎

Let \(K\) be a number field. We denote by \(M_K\) the set of all places of \(K\), both archimedean and non-archimedean, and let \(M_{K,\infty}\subset M_K\) be the set of all archimedean places.

Let \(\mathcal{S} \subset M_K\) be a finite set of places containing \(M_{K,\infty}\). The ring \(\mathcal{O}_{K,{\mathcal{S}}}\) of \({\mathcal{S}}\)-integers in \(K\) is given by \[\mathcal{O}_{K,{\mathcal{S}}}=\{x \in K : \nu(x) \ge 0,~\mathrm{for~all}~\nu \in M_K\setminus {\mathcal{S}}\},\] and the unit group \(\mathcal{O}^{*}_{K,{\mathcal{S}}}\) of \(\mathcal{O}_{K,{\mathcal{S}}}\) is given by \[\mathcal{O}^{*}_{K,{\mathcal{S}}} = \{x \in K : \nu(x) = 0,~\mathrm{for~all}~\nu \in M_K\setminus {\mathcal{S}}\}.\]

Let \(C\) be a smooth, geometrically irreducible, projective curve of genus \(g \ge 2\) defined over \({\mathbb{Q}}\), and \({\mathrm{Jac}}(C)\) be the Jacobian of \(C\). We denote \({\mathrm{rank}}\bigl({\mathrm{Jac}}(C)(K)\bigr)\) be the Mordell-Weil rank of \({\mathrm{Jac}}(C)\) over \(K\).

We shall use the following estimate due to Dimitrov, Gao and Habegger [35].

2.1 A family of generalised Fermat curves↩︎

Let \(\mathcal{W}_K\) be a finite set of places of \(K\) containing \(M_{K,\infty}\) and at least one non-archimedean place, and let \(\ell\) be as in Theorem 1. Consider the following family of curves \[\label{eqn:Cuvell} C_{u,v,\ell}: ux^{2\ell} + vy^{2\ell} = 1, \quad u,v \in \mathcal{O}^{*}_{K,{\mathcal{W}}_K}/(\mathcal{O}^{*}_{K,{\mathcal{W}}_K})^{2\ell}.\tag{4}\]

We need to estimate the following sum \[\label{eqn:sumofsizes} \sum_{u,v \in \mathcal{O}^{*}_{K,{\mathcal{W}}_K}/(\mathcal{O}^{*}_{K,{\mathcal{W}}_K})^{2\ell}} \sharp\, C_{u,v,\ell}(K).\tag{5}\]

Since \(u,v \neq 0\), each such affine curve \(C_{u,v,\ell}\) is smooth. Let \(\widetilde{C}_{u,v,\ell}\) denote its projective model, given by \(\widetilde{C}_{u,v,\ell}: ux^{2\ell} + vy^{2\ell} = z^{2\ell}\). This is a smooth, geometrically irreducible projective curve. It is well-known that its genus is \((2\ell-1)(\ell-1)\); see, e.g., [18]. Moreover, each of them is geometrically irreducible, since over \(\overline{K}\) one can rescale \(x\) and \(y\) to reduce the defining equation to \(x^{2\ell} + y^{2\ell}-z^{2\ell}=0\), which is irreducible over \(\overline{K}\) (for instance, by Eisenstein’s criterion).

Since \(C_{u,v,\ell}\) is a affine piece of \(\widetilde{C}_{u,v,\ell}\), clearly we have \(\sharp\, C_{u,v,\ell}(K) \le \sharp\, \widetilde{C}_{u,v,\ell}(K)\), and hence, we can apply Lemma 6 to \(\widetilde{C}_{u,v,\ell}\).

To estimate 5 we argue as in the proof of [36], except that we use Lemma 6, instead of [36] and the bound at the top of page 760 [36].

2.2 Perfect powers up to a \({\mathcal{W}}\)-unit↩︎

It is known, due to [19], that there are only \(O(1)\) many \(P\in E({\mathbb{Q}})\) such that \(d_{P}\) is a perfect \(\ell\)th power. We need to estimate the following slightly modified quantity, which will be needed to prove Theorem 1.

For any prime \(\ell\), and any set of rational primes \({\mathcal{W}}\), we denote \[\label{eqn:dlw} {\mathcal{D}}_{\ell,{\mathcal{W}}}=\{P\in E({\mathbb{Q}}):d_{P}\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}} \cdot {\mathcal{N}}_{\ell}\}.\tag{6}\]

We now prove the following estimate. The proof of Lemma 8 below, follows almost verbatim the argument in [19]. For completeness, we include a brief self-contained proof.

3 Products over shifted points↩︎

3.1 Preliminary↩︎

Let us first recall the following ingredients from [24], which be used throughout.

We now estimate the number of terms with a large prime factor. The following lemma is taken from [24].

For any positive real number \(R\), let \({\mathcal{W}}(R)\) be the set of all primes \(p\) with \(\rho_{i,p} \le R\) for at least one \(i =1, \ldots, s\). Here we denote \(\rho_{i,p}\) be the index of appearance of \(p\) with respect to \(P_i\), for each \(i=1,\ldots, s\), where \(P_1,\ldots,P_s\) be the rational points as in Theorem 1.

We shall also appeal to the following lemma from [31], which provides a saving that grows with \(s\).

3.2 Proof of Theorem 1↩︎

First, by the Siegel’s theorem, except for at most \(O(N^{s-1})\) many tuples we may assume that \(d_{n_iP_i+Q_i}>1\), for each \(i=1,\ldots, s\).

Now, let \(1<R< N\) be a real number to be chosen suitably later. We first count the \(s\)-tuples of first type, that is, the \(s\)-tuples \((n_1, \ldots, n_s)\in {\mathcal{D}}_{\ell,\mathsf P,\mathsf Q}(M,N)\), for which \[\nu_p(d_{n_iP_i+Q_i})\not\equiv 0 \pmod \ell \implies p \in {\mathcal{W}}(R),~\mathrm{for~some }~i=1,\ldots, s.\]

Lemma 8 and Lemma 12 show that, the number of such \(s\)-tuples of first type is bounded by \[N^{s-1} \exp\left(O\left(R^3\right)\right).\]

For each remaining \(s\)-tuple \((n_1,\ldots, n_s)\in {\mathcal{D}}_{\ell,\mathsf P,\mathsf Q}(M,N)\) of second type, and for any \(i=1,\ldots, s\), we have some prime \(p\not\in {\mathcal{W}}(R)\) with \[\nu_p(d_{n_iP_i+Q_i})\not\equiv 0 \pmod \ell.\]

In particular, for any such tuple \((n_1,\ldots, n_s)\) of second type, and for any \(i=1,\ldots,s\), there exists some \(1 \le j\le s\) with \(j\ne i\), satisfying \[\label{eqn:edge} \nu_p(d_{n_iP_i+Q_i})\not\equiv 0 \pmod \ell \quad \mathrm{and}\quad \nu_p(d_{n_jP_j+Q_j})\not\equiv 0 \pmod \ell,\tag{11}\] for some prime \(p\not\in {\mathcal{W}}(R)\).

We claim that the number of such \(s\)-tuples of second type is \[\label{eqn:largeones} O\left(N^{s}R^{-\left\lceil s/2\right\rceil}\right).\tag{12}\]

To prove this claim, we consider the graph \({\mathcal{G}}\) on \(s\) vertices \(n_1, \ldots, n_s\), and the edges are defined by the condition at 11 . By the observation above, \({\mathcal{G}}\) has no isolated vertex. By Lemma 13, we can fix a subset \({\mathcal{I}}\) of \(\{1,\ldots, s\}\) with \(m=\sharp\, {\mathcal{I}}\leq \left\lfloor s/2\right\rfloor\), such that for any \(j\in \{1,\ldots,s\}\setminus {\mathcal{I}}\), there exists some \(i\in {\mathcal{I}}\), such that \(n_i\) and \(n_j\) satisfy 11 .

Trivially, we can fix such \(m\)-tuples by \(O(N^m)\) ways. Without loss of generality, we may assume that the fixed ones are \((n_1,\ldots, n_m)\). For any such fixed \(m\)-tuple, applying Lemma 11, the number of remaining \((s-m)\)-tuples \((n_{m+1},\ldots, n_s)\in (M,M+N]^{s-m}\) (such that \((n_1,\ldots,n_{m},n_{m+1},\ldots,n_s)\) is of second type) is \[O\left((N/R+1)^{s-m}\right)= O\left(N^{s-m}R^{-\left\lceil s/2\right\rceil}\right),\] since \(R< N\) and \(m\le \lceil s/2 \rceil\). This proves the claim at 12 . For a more detailed argument, we also refer the reader to the estimations of \(K_2\) and \(K_3\) in the proof of [24].

Therefore, we finally derive \[\sharp\, {\mathcal{D}}_{\ell,\mathsf P,\mathsf Q}(M,N)\le N^{s-1} \exp\left(O\left(R^3\right)\right)+O\left(N^{s}R^{-\left\lceil s/2\right\rceil}\right).\]

The proof concludes by taking \(R=c(\log N)^{1/3}\), for some sufficiently small constant \(c>0\). \(\sqcup\)

4 Products over terms of elliptic divisibility sequences↩︎

Let \({\mathcal{W}}(R)\) be the same set of primes as in the proof of Theorem 1. For any fixed \(\ell\), we need to estimate the size of the following set \[{\mathcal{D}}_{\ell,P,{\mathcal{W}}(R)}(M,N)=\{M<n\le M+N: d_{nP}\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}(R)}\cdot {\mathcal{N}}_{\ell}\}.\]

Let \({\mathcal{W}}_P = \{p \;\text{prime} : p \mid d_P\}\), and include \(2\) if it is not already contained in \({\mathcal{W}}_P\). Clearly we have \(L({\mathcal{W}}_P)\ll \log d_P\). Therefore Lemma 8 shows that there exists a constant \(C_{E,P,\ell}>0\), such that for any prime \(q>C_{E,P,\ell}\) we have \(qP\not \in {\mathcal{D}}_{\ell,{\mathcal{W}}_P}\), where \({\mathcal{D}}_{\ell,{\mathcal{W}}_P}\) be as in 6 .

4.1 Elliptic sieve and powers up to a \({\mathcal{W}}(R)\)-unit↩︎

In this section we follow the sieving argument as in the proof of [26], and prove stronger versions of Lemma 8, over the intervals whose lengths are much larger than \(\sharp\, {\mathcal{W}}(R)\).

Let us denote \(\Psi(N,y)\) be the number of \(y\)-smooth positive integers up to \(N\), and as usual, we say that a positive integer \(n\) is \(y\)-smooth if any prime factor of \(n\) is at most \(y\). Now, we also have the following version of Lemma 14 for \(M=0\).

Therefore, Lemma 16 suggests that we need a comparison between \(\Psi(N,R)\) and \(N/R\). Writing \(u=\frac{\log N}{\log R}\), we deduce from [39] that \[\label{eqn:smooth} \Psi(N,R) \le N \exp\big((-1+o(1))\, u \log u\big),\tag{15}\] provided that \(u \to \infty\) and \(R > (\log N)^2\). Therefore, taking any \(R\) satisfying \[\label{eqn:newR} \frac{\log N}{\log R}\left(\log \log N -\log \log R\right)=\log R,\tag{16}\] we obtain \[\label{eqn:psinrbound} \Psi(N,R)\le N/R^{1+o(1)}.\tag{17}\]

4.2 Proof of Theorem 2↩︎

To prove ?? , we follow a similar argument as in the proof of Theorem 1. The only difference is that we split the \(s\)-tuples \((n_1,\ldots,n_s)\in {\mathcal{D}}_{\ell,\mathsf P,\mathsf O}(M,N)\) into two types.

First, those for which, for every \(i=1,\ldots,s\) we have \[\label{eqn:foreach} \nu_p(d_{n_iP_i}) \not\equiv 0 \pmod \ell \implies p \in {\mathcal{W}}(R).\tag{18}\] Second, those for which 18 fails for at least one \(1 \le i \le s\).

For any parameter \(R\) with \(\log N \le R < N\), Lemma 14 shows that the number of \(s\)-tuples satisfying 18 is \(\ll N^s \left(\frac{\log R}{\log N}\right)^s\), while Lemma 11 implies that the number of \(s\)-tuples failing 18 is \(\ll N^s / R\).

Combining these bounds, we derive \[\sharp\, {\mathcal{D}}_{\ell,\mathsf P,\mathsf O}(M,N) \ll N^s \left(\frac{\log R}{\log N}\right)^s + \frac{N^s}{R}.\] The proof of ?? concludes by taking \(R = (\log N)^s\).

To prove ?? , we take the \(R\) satisfying 16 . Then for each \(1\le t\le s\), we count the number of \(s\)-tuples \((n_1, \ldots, n_s) \in {\mathcal{D}}_{\ell,\mathsf P,\mathsf O}(0,N)\) for which 18 fails for \(t\) many \(i\) in \(\{1,\ldots,s\}\). Without loss of generality, we may assume that the corresponding integers are \(n_1,\ldots,n_t\), and \(n_{t+1},\ldots,n_s\), respectively.

By 17 , the number of such \((s-t)\)-tuples \(\left(n_{t+1},\ldots,n_s\right) \in (0,N]^{s-t}\) is bounded by \[\left(\frac{N}{R^{1+o(1)}}\right)^{s-t}.\]

As in the proof of Theorem 1, we again consider the graph \({\mathcal{G}}_{t}\), where the vertices are \((n_1,\ldots,n_t)\), and the edges are defined by the condition \[\nu_p(d_{n_iP_i})\not\equiv 0 \pmod \ell \quad \mathrm{and}\quad \nu_p(d_{n_jP_j})\not\equiv 0 \pmod \ell,\] for some prime \(p\not \in {\mathcal{W}}(R)\). Then, again as in the proof of Theorem 1, applying Lemma 11 and Lemma 13, the number of such \(t\)-tuples \((n_1,\ldots, n_t)\in (0,N]^t\) is \(O\left(\frac{N^t}{R^{\left\lceil t/2\right\rceil}}\right).\)

Therefore, summing over \(t=0, \ldots, s\), we finally derive \[\sharp\, {\mathcal{D}}_{\ell,\mathsf P,\mathsf O}(0,N) \le \, \sum_{t=0}^{s}\left(\frac{N}{R^{1+o(1)}}\right)^{s-t} \, \frac{N^t}{R^{\left\lceil t/2\right\rceil}}\le \, \frac{N^{s}}{R^{\lceil s/2 \rceil \left(1+o(1)\right)}}.\]

Also, it is clear from 16 that \[(\log N \log \log N)^{1/2} \ge \log R \ge \frac{1}{\sqrt{2}}\left(\log N \, (\log\log N - 2\log\log\log N)\right)^{1/2},\] and the proof concludes. \(\sqcup\)

4.3 Proof of Theorem 4↩︎

We argue exactly as in the proof of Theorem 2. For this, one of the ingredient is already available, due to Lemma 11. Therefore, we only need an analog of Lemma 16. This is straightforward for \(\ell>2\), as \[x(nP)\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}(R)} \cdot {\mathcal{Q}}_{\ell} \implies d_{nP}\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}(R)} \cdot {\mathcal{N}}_{\ell}.\]

For \(\ell=2\), we use the identity \[|x(nP) - x(mP)|=\frac{|\psi_{n+m}(P)\psi_{n-m}(P)|}{\psi^2_n(P)\psi^2_m(P)},\quad \mathrm{for~any }~n>m\ge 1,\] where \(\psi_n\) denotes the \(n\)th division polynomial; see [18]. Moreover, it follows from [40] that \(\nu_p(d_{nP})=\nu_p(|\psi_n(P)|)\) for any \(n\ge 1\), and for any prime \(p \nmid d_P\) at which \(E\) has a good reduction.

In particular if \(x(mP)=0\) for some \(m\ge 1\), then for any \(n>m\), we have \[x(nP)\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}(R)} \cdot {\mathcal{Q}}_{2} \implies d_{(n+m)P}\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}'(R)} \cdot {\mathcal{N}}_{2},\] where \(\mathcal{W}'(R)\) is the union of \(\mathcal{W}(R)\) with a finite set of primes depending only on \(E\) and \(P\). Thus, one can again use Lemma 16. The proof concludes. \(\sqcup\)

5 Products over all points of a bounded height↩︎

5.1 Counting points in a family of lattices↩︎

Let \(E({\mathbb{Q}})\) be of rank \(r\), and \(G_1, \dots, G_r\) generate the free part of \(E({\mathbb{Q}})\). For any prime \(p\) of good reduction, consider the lattices in \({\mathbb{Z}}^r\) given by \[\Lambda_p = \left\{\mathbf{n} \in {\mathbb{Z}}^r : n_1 G_1+\cdots +n_rG_r \equiv O \pmod p\right\},\] where again \(O\) be the point at infinity for \(E\), and \[\Lambda'_p = \left\{\mathbf{n} \in {\mathbb{Z}}^r : n_1 G_1+\cdots +n_rG_r \in E({\mathbb{Q}})_{\mathrm{tor}} \pmod p\right\}.\]

Here, we denote \(E({\mathbb{Q}})_{\mathrm{tor}}\) be the subgroup of torsion points in \(E({\mathbb{Q}})\). Moreover, for the definition of reduction modulo prime \(p\) of a rational point \(P\), or equivalently, for the meaning of the standard notation \(P \pmod p\), we refer the reader to [26].

Now consider the box \[{\mathcal{B}}(X)=\{\mathbf{x}\in {\mathbb{R}}^r: \|\mathbf{x}\|_{\infty}\le X\},\] where \(\|\cdot\|_{\infty}\) denotes the sup norm, and let \(\Lambda\) be a lattice in \({\mathbb{Z}}^r\).

For any \(\mathbf{m} \in {\mathbb{Z}}^r\), we denote \(N_{m,\Lambda}(X)\) be the size of \({\mathcal{B}}(X)\cap \left(\mathbf{m}+\Lambda\right)\), where by \(\mathbf{m}+\Lambda\) we mean the translation of the lattice \(\Lambda\) by vector \(\mathbf{m}\). Then, we record the following presumably well-known estimate.

Let us now denote \[{\mathcal{E}}_{p}(H)=\{P\in E({\mathbb{Q}}) :\widehat h(P)\le H,~\nu_p\left(x(P)\right) \ne 0 \}.\]

As a consequence to Lemma 19, we deduce the following estimate.

For any positive real number \(R\), denote \(\mathcal{T}(R)\) be the set of primes at which \(E\) has a bad reduction, together with the primes \(p\) of good reduction with \(\rho_{E,p} \le R\). We have the following analogue of Lemma 12.

5.2 Proof of Theorem 3↩︎

The proof is completely analogous to that of Theorem 1, so we provide only a brief sketch.

Let \(1<R<H^{1/2}\) be a real number to be chosen suitably later, and first count the \(s\)-tuples \((P_1, \ldots, P_s)\in {\mathcal{D}}_{\ell}(H)^s\), for which we have \[\nu_p(d_{P_i})\not\equiv 0 \pmod \ell \implies p \in {\mathcal{T}}(R),~\mathrm{for~some}~1\leq i\le s.\]

Lemma 8 and Lemma 21 show that the number of such \(s\)-tuples is bounded by \[H^{\frac{r_1+\ldots+r_s-r_{ \min}}{2}} \exp\left(O\left(R^{1+\frac{2}{r_{\min}}}\right)\right).\]

To count the remaining tuples \((P_1,\ldots, P_s)\in {\mathcal{D}}_{\ell}(H)^s\), we consider the graph \(\widehat{{\mathcal{G}}}\) on such \(s\) vertices \(P_1, \ldots, P_s\), where the edges are defined by the condition \[\nu_p(d_{P_i})\quad \mathrm{and}\quad \nu_p(d_{P_j})\not\equiv 0 \pmod \ell,\quad \mathrm{for~some}~p\not \in {\mathcal{T}}(R).\]

As in the proof of Theorem 1, \(\widehat{{\mathcal{G}}}\) has no isolated vertex. Therefore, by Lemma 13, there exists a subset \({\mathcal{I}}\) of \(\{1,\ldots, s\}\) with \(m=\sharp\, {\mathcal{I}}\leq \left\lfloor s/2\right\rfloor\), such that for any \(j\in \{1,\ldots,s\}\setminus {\mathcal{I}}\), there exists some \(i\in {\mathcal{I}}\), such that \(P_i\) and \(P_j\) satisfy 11 . Then arguing as exactly in the proof of Theorem 1, applying Lemma 20 (instead of Lemma 11) and Lemma 13, the number of such \(s\)-tuples is \[O\left(H^{\frac{r_1+\ldots+r_s}{2}}R^{-\left\lceil s/2\right\rceil}\right),\quad \mathrm{as}\quad R< H^{1/2}.\]

Therefore, we finally derive \[\sharp\,{\mathcal{D}}_{\ell}(H)\le \, H^{\frac{r_1+\ldots+r_s-r_{\min}}{2}} \exp\left(O\left(R^{1+\frac{2}{r_{\min}}}\right)\right)+O\left(H^{\frac{r_1+\ldots+r_s}{2}}R^{-\left\lceil s/2\right\rceil}\right).\]

The proof concludes by taking \(R=c(\log H)^{1/\left(1+\frac{2}{r_{\min}}\right)}\), for some sufficiently small constant \(c>0\). \(\sqcup\)

6 Further applications of the underlying techniques↩︎

6.1 Perfect powers in the single-variable shifted products↩︎

Let us assume that \(E_i=E\) for each \(i=1,\ldots, s\), and fix \(P\in E({\mathbb{Q}})\) be a non-torsion point. Consider the shifted products of the form \[d_{(n+h_1)P} \cdots d_{(n+h_s)P},\quad n\in \mathbb{N}.\]

In this context, again recall that Hajdu, Laishram and Szikszai [23] have shown the same for any fixed \(\ell\ge 2\), there are finitely many perfect \(\ell\)th powers among such products, uniformly over the shifts of the form \[\mathsf h= (h, 2h, \dots, sh), \quad h,s \in \mathbb{N},\] In light of their result, for any fixed \(\ell\) and \(s\), let us consider \[\begin{align} &S_{\ell,\mathsf P}(M,N,H)\cr &\quad\quad =\max_{\mathsf h\in {\mathcal{H}}_s}\sharp\, \left\{ n\in (M,M+N]: d_{(n+h_1)P}\cdots d_{(n+h_s)P} \in {\mathcal{N}}_{\ell}\right\}, \end{align}\] where we denote \[{\mathcal{H}}_s=\{(h_1,\cdots,h_s) \in ({\mathbb{Z}}\cap [-H,H])^s: h_i\ne h_j,~\mathrm{for}~1\le i\ne j\le s\}.\] Assume that \(2H<R\). Then for any prime \(q>R\), we have \[h_i\ne h_j \pmod q,\quad \mathrm{for~any}~1\le i\ne j\le s.\]

Using the same sieving argument as in the proof of Lemma 14, one can easily derive the following estimate \[\begin{align} &\max_{\mathsf h\in {\mathcal{H}}_s}\sharp\, \left\{n\in (M,M+N]: d_{(n+h_i)P}\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}(R)}\cdot {\mathcal{N}}_{\ell},~\for~i=1,\cdots,s\right\}\cr &\qquad\qquad \qquad\qquad\quad \ll N \, \left(\frac{\log \, \max\{R,\log N\}}{\log N}\right)^s. \end{align}\]

For any remaining \(n\in (M,M+N]\) with \(d_{(n+h_1)P}\cdots d_{(n+h_s)P} \in {\mathcal{N}}_{\ell}\), there exists a prime \(p\not\in {\mathcal{W}}(R)\) such that \[p\mid d_{(n+h_i)P}\quad \mathrm{and}\quad p\mid d_{(n+h_j)P}\quad \mathrm{for~some}~i\ne j.\] In particular, we have \[\label{eqn:impcond} \rho_p\mid h_i-h_j,\quad \mathrm{for~some}~i\ne j.\tag{21}\]

However, 21 does not hold, as \(\mathbf{h} \in \mathcal{H}_s\) and \(2H < R\). Therefore, choosing \(R=3H\), we finally derive the following result.

Thus, \(S_{\ell,\mathsf P}(M,N,H)\) is \(o(N)\) over the range \(1\le H\le \exp\left(\frac{\log N}{G(N)}\right)\), for any function \(G(N)\to \infty\) as \(N\to \infty\).

6.2 Products of images of rational functions↩︎

Assume again that we are given \(s\), not necessarily distinct elliptic curves \(E_1, \ldots, E_s\) as in 1 over \({\mathbb{Q}}\) of positive ranks. Fix a \(s\)-tuples \(\mathsf P=(P_1, \ldots, P_s)\) of non-torsion points \(P_i\in E_i({\mathbb{Q}})\), for each \(i=1,\ldots, s\).

As in the proof of Theorem 4, one can estimate the number of \(s\)-tuples of rational points whose product of the \(x\)-coordinates is a perfect \(\ell\)th power. Again note that, one of the ingredient is already available, due to Lemma 11 or Lemma 20. Therefore, we only need an analog of Lemma 8. For \(\ell>2\), we have \[x(P)\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}(R)} \cdot {\mathcal{Q}}_{\ell} \implies d_{P}\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}(R)} \cdot {\mathcal{N}}_{\ell}.\]

For \(\ell=2\), the relevant family of curves is \[C'_u:y^2=u^3x^6+aux^2+b,\quad u\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}(R)}/(\mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}(R)})^{2\ell},\] where \(a,b\) be as 1 . Now we must to assume that \(b\ne 0\), as otherwise any \(x(P)\) is a square up-to some factor of \(a\). If we have \(b\neq 0\), then each such \(C'_u\) is non-singular and of genus \(2\) (follows from [18]). We may now apply [36] directly. Consequently, in all such cases, one obtains analogues of the results of Section 1.2 for products of the \(x\)-coordinates of points of the form \(nP+Q,~n\in (M,M+N]\), as well as for products taken over rational points of bounded canonical height.

Now let \(f\in {\mathbb{Q}}(E)\) be a rational function. Mimicking the proof of Lemma 11 (or Lemma 20), we can derive corresponding analogues of these lemmas, where the implied constant depends on \(\deg(f)\), the degree of rational map \(f: E \to \mathbb{P}^1\).

On the other hand, to obtain an analogue of Lemma 8, that is, an estimate for \[\sharp\, \{P\in E({\mathbb{Q}}): f(P)\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}(R)}\cdot {\mathcal{Q}}_{\ell}\},\] we first write \(f=\frac{A(x)+yB(x)}{C(x)}\), where \(A,B,C\in \mathbb{Z}[x]\) and \(x,y\) are the Weierstrass coordinate functions on \(E\). Let us assume that \[\label{eqn:f} 2\deg A < 2\deg B + 3 \quad \text{or} \quad \deg A \ne \deg C,\tag{22}\] or in other words, we assume that \(f\) has either a zero or a pole at \(O\).

Writing \(P\) as in 2 with both \(a_P, b_P \ne 0\) (note that only finitely many points \(P\) fail to satisfy this), and expanding and clearing denominators, we obtain \(f(P) = d_P^{k_f} \, r_P\), for some nonzero integer \(k_f\) depending only on \(f\), and some rational number \(r_P\) whose numerator and denominator (in lowest terms) are both co-prime to \(d_P\). Therefore, we can again use Lemma 8, at least for any prime \(\ell\) not dividing \(k_f\).

Thus, we obtain an analogue of Theorem 4 for any function \(f\) satisfying 22 , rather than only \(x\), and for any prime \(\ell\) coprime to \(k_f\). For instance, the function \(f=y\) satisfies 22 , with \(k_y=3\).

6.3 Sum of two squares among the products↩︎

Let us denote \[\begin{align} D_{\square+\square,P,{\mathcal{W}}(R)}(M,N) &=\sharp\, \bigl\{ n\in (M,M+N]: d_{nP} \cr &\quad\quad\;\; \text{is a sum of two squares up to a}~{\mathcal{W}}(R)\text{-}\mathrm{unit} \bigr\}. \end{align}\]

To estimate \(D_{\square+\square,{\mathcal{W}}(R),P}(M,N)\) one can again argue as in Lemma 14, adapting [26]. The only difference is that one now needs to sieve out the indices of denominators, modulo the new set of primes \(\Lambda'(R)=\{q~\mathrm{prime}:2R<q\}\cap \Lambda'\), where \[\Lambda'=\{q~\mathrm{prime}:d_{qP}~\text{is not a sum of two squares}\}.\]

Thanks to [26], which ensures that \(\Lambda'\) has a positive relative density \(\omega=\omega_{E,P}>0\) in the set of all primes, assuming that \(P\) is in the connected component of \(O\) in \(E(\mathbb{R})\). Analogous to Lemma 14 and [26], one obtains \[\label{eqn:[]+[]uptoW} D_{\square+\square,P,{\mathcal{W}}(R)}(M,N) \ll N \left(\frac{\log \max\{R,\log N\}}{\log N}\right)^{\omega}.\tag{23}\]

Let us denote \[\begin{align} Y_{\square+\square,\mathsf P}(M,N) &= \sharp\, \{ (n_1, \ldots, n_s)\in (M,M+N]^s:\cr &\qquad\qquad y(n_1P_1) \cdots y(n_sP_s)~\text{is a sum of two squares} \}. \end{align}\]

Arguing as exactly in the proof of ?? , we derive the following result.

::: {#thm:y=[]+[] .thm} Theorem 23. Let \(s\ge 2\) be fixed. Then uniformly over \(M \ge 0\), we have \[Y_{\square+\square,\mathsf P}(M,N)\ll N^s\left( \frac{\log \log N}{\log N}\right)^{c(\mathsf P) \, \omega},\] where \(c(\mathsf P)\) is the number of indices \(i=1,\ldots, s\) such that \(P_i\) is in the connected component of \(O_i\) of \(E_i({\mathbb{R}})\), where \(\omega\) be as in [26]. :::

In particular, Theorem 23 extends [26] to \(s\ge 2\).

7 Concluding remarks↩︎

7.1 An illustration of the admissible exponents↩︎

Note that the conclusion of Theorem 1 also holds for equations of the form \[d_{n_1P_1 + Q_1}^{k_1} \cdots d_{n_sP_s+Q_s}^{k_s} \in {\mathcal{N}}_{m},\] provided that there exists a prime \(\ell \mid m\), which does not divide \(k_i\) for any \(i=1,\ldots,s\). It is now natural to ask how many \((s+1)\)-tuples \((k_1,\ldots,k_{s},m)\) of integers do not satisfy such a condition, with \(|k_i|\le N\) for each \(i=1,\ldots,s\) and \(|m|\le N\). For a parameter \(2\le y\le N\) to be chosen suitably, a crude estimation is given by \[\label{eqn:expsbound} N^{s}\Psi(N,y)+N^{s+1}\sum_{\substack{y< \ell\cr \ell~\mathrm{prime}}} \frac{1}{\ell^2}\ll N^{s}\Psi(N,y)+\frac{N^{s+1}}{y \log y},\tag{24}\] where again, \(\Psi(N,y)\) be the number of \(y\)-smooth positive integers up to \(N\). In particular one easily sees that the quantity in 24 is \(o(N^{s+1})\), for instance by 17 and taking \(y\) as in 16 .

Theorem 1 also applies to the \(s\)-tuple of exponents \((k_1,\ldots,k_s,m)\), for which there exists a prime \(\ell \mid m\) such that \(\ell \nmid k_i\) for at least one \(1\le i \le s\). Indeed, let \(t\) denotes the number of such indices \(1\le i\le s\) such that \(\ell \nmid k_i\). Then by trivially fixing \(n_j\in (M,M+N]\) for each of remaining \(s-t\) indices \(j\), we save at least \((\log N)^{\lceil t/2 \rceil/3}\) for \(t\ge 2\), and save \(N\) when \(t=1\), compared to the trivial bound of order \(N^s\).

7.2 Difficulties in obtaining a power saving↩︎

An interesting open problem is whether the savings obtained in Theorem 1 can be substantially improved. Of course, when \(s=1\), there are only \(O(1)\) such values of \(n_1\). This indicates that one should hope for a power saving. The bottleneck arises from the bound for \(D_{\ell,{\mathcal{W}}}\) in Lemma 8. Perhaps one cannot generally do better than \(\exp(O(\sharp\, {\mathcal{W}}))\), because there are \(\mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}}/(\mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}})^{2\ell}\) equations to consider in 10 . Moreover, since \(1\) is trivially an \(\ell\)th power, the Siegel-type bounds (see [18]) suggests that such an estimate is essentially best possible.

However, even assuming a bound of order \(\exp(O(\sharp\,{\mathcal{W}}))\) in Lemma 8, we only save an additional power of \(\log\log N\) in Theorem 1 and a power of \(\log \log H\) in Theorem 3. Nevertheless, it may be of independent interest in its own right to study the family of curves \(C_{u,v,\ell}\) defined in 4 . Motivated by this, we pose the following question.

It is clear that our argument gives a power saving in ?? , when Lemma 16 saves a power with \(R=N^{\varepsilon}\), for some fixed \(\varepsilon>0\). This requires a better understanding of \(d_{nP}\) when \(n\) is \(N^{\varepsilon}\)-smooth, but having a moderate prime factor; for example, in the range \(\big((\log N)^K,N^{\varepsilon}\big)\) for some fixed and large \(K\). On the other hand, by Silverman’s primitive divisor theorem, for any integer \(1\le \Omega<1/3\varepsilon\), we have \[\sharp\, \{n\in\mathbb{N}: d_{nP}\in \mathcal{O}^{*}_{{\mathbb{Q}},{\mathcal{W}}\left(N^{\Omega\varepsilon}\right)}\}\ll N^{3\Omega \varepsilon}.\]

So for any \(n\) not as above, we get a divisor \(r_p>N^{\Omega\varepsilon}\) of \(n\). Then \(n\) is not \(N^{\varepsilon}\)-smooth, when \(n\) has at most \(\Omega\) prime factors, counted with multiplicity. Thus, Lemma 16 is considerably simpler in such cases.

A recent preprint of Kym [32] studies \(s\)-tuples of integers \((n_1,\ldots,n_s)\) for which \[d_{n_1P}\cdots d_{n_sP}\notin \mathcal{N}_{\ell},\] for a fixed prime \(\ell\); see, for example, Theorem 5.5 and the subsequent results, including Corollary 5.6, Proposition 5.7, and Corollary 5.8 therein. These results are proved under some additional conditions, such as the divisibility of \(d_P\) by \(2\) or \(3\), or under the weaker Hypothesis 4.4 of [32]. As already mentioned in Section 1.4, our Lemma 8 removes the need for this additional hypothesis, since it gives an unconditional version of [32].

7.3 A power saving under abc-conjecture↩︎

Recall the well-known abc-conjecture (over \({\mathbb{Q}}\)) of Masser and Oesterlé, as mentioned in [27]. Then the proof of [27] gives the following.

Now assume the hypotheses of Lemma 25. Let \(s,\ell \ge 2\) be fixed Then as a consequence to Lemma 25, uniformly over \(M\ge 0\), we have \[\label{eqn:Nsaving} \sharp\, {\mathcal{D}}_{\ell,\mathsf P,\mathsf O}(M,N) \ll N^{s-1}.\tag{25}\]

The proof of 25 is straightforward. Just note that for any fixed tuple \((n_2,\ldots, n_{s-1})\in (M,M+N]^{s-1}\), we have \[\sharp\left\{n_1\in { \mathbb{N}}:d_{n_1P_1}^{k_1}\in d_{n_2P_2}^{-k_2} \cdots d_{n_sP_s}^{-k_s}\cdot {\mathcal{N}}_{\ell}\right\}\le C(E,P)+1.\]

Furthermore, a generalisation of Lemma 25 dropping the condition on the \(j\)-invariant is possible, using Kühn and Müller’s [45], under the assumption of the uniform abc-conjecture [45]. Note that the abc-conjecture over \({\mathbb{Q}}\) is a special case of the uniform abc-conjecture, as mentioned in [45].

It is now natural to ask whether one can obtain a power saving for \(\sharp\, {\mathcal{D}}_{\ell,\mathsf P,\mathsf Q}(M,N)\), for any arbitrary \(s\)-tuple of rational points \(\mathsf Q\). In this setting, the primitive divisor theorem for the sequence \(d_{nP+Q}\) is currently known only when \(Q\) is a torsion point, thanks to Verzobio [46]–[48]. However, to obtain a power saving for \(\sharp\, {\mathcal{D}}_{\ell,\mathsf P,\mathsf Q}(M,N)\), it suffices to have a good lower bound on the powerless part of \(d_{nP+Q}\).

Assuming the \(abc\) conjecture over \({\mathbb{Q}}\), such a lower bound can be extracted from the proof of [27] in the case where the \(j\)-invariant is \(0\) or \(1728\), with [24] playing a crucial role in the argument. To treat the cases of arbitrary \(j\)-invariants, one may again use the uniform \(abc\) conjecture.

Acknowledgements↩︎

The author sincerely thanks Philip Habegger for suggesting useful tools for bounding the rank of Jacobians, and Alina Ostafe for bringing [35] to the author’s attention, which led to a simplification and improvement of a preliminary bound. He also is very grateful to Igor E. Shparlinski for reading the manuscript, providing many helpful remarks, and drawing attention to [17], which is one of the key motivations for this work. The author also sincerely thanks Simon L. Rydin Myerson for many helpful suggestions. Further thanks go to Lajos Hajdu and Shanta Laishram for related discussions. Some parts of this work have been carried out during a visit to ISI Delhi, and the author is grateful to Shanta Laishram for the hospitality.

The author has been supported by the Australian Research Council Grant DP230100530.

References↩︎