1.1 Notation and setup for our problem

For an elliptic curve E defined over a field L of characteristic $0$ , we denote by ${\textrm {End}}(E)$ the ring of group endomorphisms of E (defined over an algebraic closure ${\overline {L}}$ of L). It is known (see [Reference Silverman33, Ch. 2]) that ${\textrm {End}}(E)$ is isomorphic either with ${\mathbb Z}$ (when E has no complex multiplication) or with an order in an imaginary quadratic number field (when E has complex multiplication, also referred to as CM). So, we identify ${\textrm {End}}(E)$ with either ${\mathbb Z}$ (in the non-CM case) or with an order $\mathcal {O}$ in an imaginary quadratic number field (in the CM case); more precisely, for each $n\in {\mathbb Z}$ (respectively, for each $\alpha \in \mathcal {O}$ ), we denote by $[n]$ (respectively, $[\alpha ]$ ) the corresponding endomorphism of E.

Throughout our paper, C is a smooth, projective (irreducible) curve defined over ${\overline {\mathbb Q}}$ . Also, we consider $\pi :\mathcal {E}\longrightarrow C$ as an elliptic surface, where the generic fibre of $\pi $ is an elliptic curve E defined over ${\overline {\mathbb Q}}(C)$ , while for all but finitely many points $\lambda \in C({\overline {\mathbb Q}})$ , the fibre $E_\lambda :=\pi ^{-1}(\{\lambda \})$ is an elliptic curve defined over ${\overline {\mathbb Q}}$ . We often refer to $E_\lambda $ as the $\lambda $ -fibre of $\pi $ . Each section $\sigma $ of $\pi $ (that is, a map $\sigma :C\to \mathcal {E}$ such that $\pi \circ \sigma ={\textrm {id}}|_C$ ) gives rise to a ${\overline {\mathbb Q}}(C)$ -rational point of E. Conversely, a point $P\in E({\overline {\mathbb Q}}(C))$ corresponds to a section of $\pi $ , denoted $\sigma _P$ . For all but finitely many $\lambda \in C({\overline {\mathbb Q}})$ , the intersection of the image of $\sigma _P$ in $\mathcal {E}$ with the smooth fibre $E_\lambda $ above $\lambda $ is a point $P_\lambda := \sigma _P(\lambda )$ on the elliptic curve $E_\lambda $ ; we refer to $P_\lambda $ as the $\lambda $ -specialisation of the point P and we drop the mention of the corresponding section $\sigma _P$ . Furthermore, when working with a point $P_i\in E({\overline {\mathbb Q}}(C))$ (for some index i), we use the notation $P_{i,\lambda }$ for the corresponding $\lambda $ -specialisation of $P_i$ .

We say that $\mathcal {E}$ is isotrivial if the j-invariant (see [Reference Silverman33, Ch. 3]) of the generic fibre E is a constant function (on C); for isotrivial elliptic surfaces $\mathcal {E}$ , all smooth fibres of $\pi $ are isomorphic to E. If $\mathcal {E}$ is isotrivial, then at the expense of replacing C by a finite cover $C'$ (defined over ${\overline {\mathbb Q}}$ ) and, also, replacing $\mathcal {E}$ by $\mathcal {E}\times _CC'$ , we may assume $\mathcal {E}$ is a constant elliptic surface. So, when E is isotrivial, since our question is unchanged when replacing C by a finite cover, we always assume there exists an elliptic curve $E_0$ defined over ${\overline {\mathbb Q}}$ such that $\mathcal {E}=E_0\times _{{\textrm {Spec}}({\overline {\mathbb Q}})}C$ . In this case, we say that a section $\sigma _P$ of $\mathcal {E}$ is constant if $P\in E_0({\overline {\mathbb Q}})$ .

We study the following question for elliptic surfaces $\mathcal {E}\to C$ defined over ${\overline {\mathbb Q}}$ ; as before, we denote by E the generic fibre of $\mathcal {E}$ . So, with the above notation, given two starting points $P_1,P_2\in E({\overline {\mathbb Q}}(C))$ and one target point $Q\in E({\overline {\mathbb Q}}(C))$ (all three of them nontorsion), we obtain necessary and sufficient conditions (in terms of the dynamics of $P_1,P_2,Q$ under the action of ${\textrm {End}}(E)$ ) so that $Q_\lambda $ is contained in both the cyclic group generated by $P_{1,\lambda }$ and in the cyclic group generated by $P_{2,\lambda }$ inside $E_\lambda $ .

1.2 Our main result

We prove the following statement.

Our Theorem 1.1 strengthens [Reference Ghioca, Hsia and Tucker23, Theorem 1.1], which we state next.

1.3 Unlikely intersections in arithmetic dynamics

Our Theorem 1.1 fits into the larger theme of unlikely intersections in arithmetic dynamics. The following result was proved by Masser and Zannier (see [Reference Masser and Zannier28, Reference Masser and Zannier29]); an alternative proof was given by DeMarco and Mavraki (see [Reference DeMarco and Mavraki16]).

Theorem 1.4 shows that the infinite occurrence of the unlikely event that both $P_{1,\lambda }$ and $P_{2,\lambda }$ are torsion on $E_\lambda $ can only be explained by a global dynamical relation between the points $P_1$ and $P_2$ ; such a conclusion fits the pattern behind several other unlikely intersection problems, going back to classical questions in arithmetic geometry (for more details, see [Reference Zannier36]). We also note that several extensions of the results from [Reference Masser and Zannier28, Reference Masser and Zannier29] were obtained both in the case of elliptic surfaces (see [Reference Barroero and Capuano6]) and also in the general case of semi-abelian schemes (see [Reference Barroero and Capuano7, Reference Barroero and Capuano8]).

Our Theorem 1.1 can be viewed as an extension of Theorem 1.4 since we replace the target $0\in E$ from Theorem 1.4 with an arbitrary point $Q\in E({\overline {\mathbb Q}}(C))$ . Also, we note that for one starting point $P_1$ and a target point Q, it is known, as proved in [Reference Corvaja and Zannier15, Theorem A2] (see also Lemma 4.2), that there exist infinitely many integers n such that $[n](P_{1,\lambda })=Q_\lambda $ . However, the fact that for some point $\lambda \in C({\overline {\mathbb Q}})$ , the point $Q_\lambda $ is a multiple both of $P_{1,\lambda }$ and of $P_{2,\lambda }$ is an unlikely event, which can only be explained by a global dynamical relation between the starting points $P_1,P_2$ and the target point Q (as in conclusions (A)–(C) from Theorem 1.1).

The results of Masser and Zannier [Reference Masser and Zannier28, Reference Masser and Zannier29] (see Theorem 1.4) were the catalysts for an intense study in arithmetic dynamics of unlikely intersection questions (see the survey article [Reference Benedetto, Ingram, Jones, Manes, Silverman and Tucker10] for a more in-depth discussion of it as one of the central questions in arithmetic dynamics, and also the book [Reference Zannier36] for a thorough study of the origins of unlikely intersection questions in arithmetic geometry). In particular, the so-called question of simultaneously preperiodic points (see also the survey [Reference Ghioca18]) generated a great deal of research in recent years (see [Reference Baker and DeMarco4, Reference Baker and DeMarco5, Reference Ghioca20, Reference Ghioca, Hsia and Tucker21] and the references therein). Consider a family $F_\lambda (z)$ of polynomials parametrised by the ${\mathbb C}$ -points $\lambda $ on some projective smooth curve C; also, let $\mathbf {a},\mathbf {b}:C\to \mathbb {P}^1$ . Then, one obtains (see [Reference Favre and Gauthier17]) a complete characterisation (based on the dynamics of $\mathbf {a},\mathbf {b}$ under the action of the family of maps $\{F_\lambda \}_{\lambda \in C}$ ) for the following unlikely scenario to happen: there exist infinitely many $\lambda \in C({\mathbb C})$ such that both $\mathbf {a}(\lambda )$ and $\mathbf {b}(\lambda )$ are preperiodic under the action of $F_\lambda (z)$ . The general principle behind this question involving simultaneously preperiodic points is that the aforementioned infinite occurrence of the unlikely event that both the orbits of $\mathbf {a}(\lambda )$ and of $\mathbf {b}(\lambda )$ are finite under the action of $F_\lambda (z)$ is explained only due to a global dynamical relation between the two starting points $\mathbf {a}$ and $\mathbf {b}$ under the action of $F_\eta (z)$ , where $\eta $ is the generic point on C.

It is natural to extend this study of unlikely intersections to the following scenario. With the above notation for the family of maps $F_\lambda $ and for the two starting points $\mathbf {a},\mathbf {b}$ , one considers also a target point $\mathbf {c}:C\to \mathbb {P}^1$ and then asks for a precise characterisation of when the following unlikely scenario occurs: there exist infinitely many points $\lambda \in C({\mathbb C})$ such that $\mathbf {c}(\lambda )$ lives in the forward orbits of both $\mathbf {a}(\lambda )$ and of $\mathbf {b}(\lambda )$ under the action of $F_\lambda (z)$ (that is, there exist positive integers m and n such that $F_\lambda ^m(\mathbf {a}(\lambda ))=\mathbf {c}(\lambda )$ and $F_\lambda ^n(\mathbf {b}(\lambda ))=\mathbf {c}(\lambda )$ ). This problem is often coined the collision of orbits question (see [Reference Asgarli and Ghioca3, Reference Ghioca19]). Using the same strategy patented in [Reference Baker and DeMarco4] (and further refined in the subsequent papers [Reference Baker and DeMarco5, Reference Ghioca, Hsia and Tucker21]), one can obtain global conditions that are necessary for this collision of orbits question; however, obtaining precise necessary and sufficient conditions is much more delicate compared with the previous question of simultaneously preperiodic points (see the discussion from both [Reference Asgarli and Ghioca3, Remark 2.8] and [Reference Ghioca19, Section 2]). Our Theorem 1.1 provides a complete answer to a variant of this collision of orbits question in the context of algebraic families of elliptic curves $E_\lambda $ , this time replacing the orbit of a point under a polynomial (or rational) map with the orbit of the corresponding point on $E_\lambda $ under the action of ${\mathbb Z}$ , seen as a subring of ${\textrm {End}}(E_\lambda )$ .

1.4 Previous results and further connections to our work

A variant of the problem of collision of orbits is the following question studied by Hsia and Tucker [Reference Hsia and Tucker24]. Given polynomials $\mathbf {a},\mathbf {b},\mathbf {c}\in {\mathbb C}[z]$ (whose compositional powers $\mathbf {a}^m,\mathbf {b}^n,\mathbf {c}^k$ for any positive integers $m,n,k$ are all distinct), then [Reference Hsia and Tucker24, Theorem 1] establishes that there exist finitely many $\lambda \in {\mathbb C}$ such that

$$ \begin{align*} (z-\lambda)\mid \gcd(\mathbf{a}^m(z)-\mathbf{c}(z),\mathbf{b}^n(z)-\mathbf{c}(z)) \quad\text{for some } m,n\in{\mathbb N}. \end{align*} $$

So, at its heart, the problem of collision of orbits is a question regarding the greatest common divisors of certain polynomials (see also the discussion from [Reference Asgarli and Ghioca3, Section 9]). This has its origins in the problem studied by Ailon and Rudnick [Reference Ailon and Rudnick1], who proved that if $\mathbf {a},\mathbf {b}\in {\mathbb C}[z]$ are multiplicatively independent, then there exists a nonzero polynomial $\mathbf {d}\in {\mathbb C}[z]$ such that

$$ \begin{align*}\gcd(\mathbf{a}(z)^k-1,\mathbf{b}(z)^k-1) \mid \mathbf{d}(z) \quad\text{for all } k\in{\mathbb N},\end{align*} $$

where $\mathbf {a}(z)^k,\mathbf {b}(z)^k$ are the (usual) kth powers of the polynomials $\mathbf {a}(z),\mathbf {b}(z)$ .

In turn, the result of Ailon and Rudnick was motivated by the work of Bugeaud, Corvaja and Zannier [Reference Bugeaud, Corvaja and Zannier11] who established an upper bound for $\gcd (a^k - 1,b^k - 1)$ (as k varies in ${\mathbb N}$ ) for given $a,b\in {\mathbb Q}$ . This problem of bounding the greatest common divisor has been studied in several other directions: for elements close to S-units [Reference Corvaja and Zannier14, Reference Luca27], for polynomials defined over fields of positive characteristic [Reference Ghioca, Hsia and Tucker22] and also for elliptic divisibility sequences [Reference Silverman32]. One of the open questions raised in [Reference Silverman32] was answered in the affirmative in [Reference Ghioca, Hsia and Tucker23, Theorem 1.1] (stated above as Theorem 1.2). Extensions of the results from [Reference Ghioca, Hsia and Tucker23] were also obtained in the more general case of semi-abelian schemes (see [Reference Barroero, Capuano and Turchet9]).

1.5 Plan for our paper

Most of the work for obtaining the direct implication in Theorem 1.1 was done in [Reference Ghioca, Hsia and Tucker23, Theorem 1.1] (see Theorem 1.2). However, in the case when the elliptic surface admits complex multiplication, we need to sharpen the statement of Theorem 1.2(ii) to deliver conclusion (C) in our Theorem 1.1. This part is done in Section 3 (see Proposition 3.1).

We prove the converse implication from Theorem 1.1 by considering separately each of the three conditions (A), (B) and (C) (see Lemmas 4.4, 4.5 and Proposition 4.6) and show that, in each case, there exist infinitely many points $\lambda \in C({\overline {\mathbb Q}})$ such that $Q_\lambda $ is (an integer) multiple of both $P_{1,\lambda }$ and of $P_{2,\lambda }$ . When the elliptic surface admits complex multiplication (that is, condition (C) holds), we need a more careful analysis; in particular, we need to employ some basic facts regarding the arithmetic of orders in an imaginary quadratic number field (for more details, see Section 2).