-Integral Points on a Mordell Curve

Abstract

We use an extension of quadratic Chabauty to number fields, recently developed by the author with Balakrishnan, Besser and Müller, combined with a sieving technique, to determine the integral points over on the Mordell curve $y^2 = x^3-4$.

1 Introduction

Let E be an elliptic curve over a number field K, described by a Weierstrass equation

$$\begin{aligned} E:y^2 + a_1xy+a_3y = x^3+a_2x^2+a_4x+a_6, \qquad a_i\in \mathcal {O}_K, \end{aligned}$$

(1)

where $\mathcal {O}_K$ is the ring of integers of K. By the set of (K-)integral points on E we mean the subset of solutions (x, y) to (1) with $x,y\in \mathcal {O}_K$. Such a set is finite by Siegel’s Theorem [19].

Assume for now that . Different solutions to the problem of effectively determining the set of integral points for a fixed elliptic curve have been given. The most notable include reducing the problem to solving some Thue equations, and using elliptic logarithms. See [24] for an overview.

An alternative more recent approach to an effective version of Siegel’s Theorem for elliptic curves comes from a very special instance of Kim’s non-abelian Chabauty programme. At present, explicit versions of this are known only for elliptic curves of Mordell–Weil rank at most 1 [5, 6, 11, 16] and can be understood in terms of the theory of p-adic heights and formal group logarithms, where p is some fixed prime. Algorithmically, this method, often referred to as “quadratic Chabauty” in the literature, outputs a finite set of (approximations of) p-adic points on E, containing the integral points. When the rank is equal to one, this set typically also contains some points that we cannot recognise as rational, or even as algebraic. In order to determine the integral points, we need to be able to prove that such points are not p-adic approximations of rational points.

Unfortunately, there is no known method to achieve this, which might place this approach at a disadvantage compared to the previously mentioned ones. However, the idea of using p-adic heights and linear -valued functionals on and to study the integral points turned out to be amenable to extensions to some curves of higher genus [2, 3], where the Mordell–Weil sieve is an effective tool used to address the problem of eliminating spurious p-adic points.

Generalisations to higher genus curves are very interesting from a Diophantine point of view. Nevertheless, we are left with the somewhat unsatisfactory impression that most quadratic Chabauty computations for integral points on elliptic curves appearing in the literature are carried out more as a “proof of concept” than as an actual way to determine integral points (but see [3, Appendix A] and [10, Appendix 4.A] for some examples of full computations of integral points).

In recent work with Balakrishnan, Besser and Müller [1], using restriction of scalars, we extended these explicit quadratic Chabauty techniques for integral points on elliptic curves, and, more generally, odd degree hyperelliptic curves, to arbitrary number fields. Combined with the Mordell–Weil sieve, the method successfully determined the integral points over an imaginary quadratic field on a genus 2 curve, as well as the rational points over some quadratic fields on some genus 2 bielliptic curves. Once again, though, examples of quadratic Chabauty computations for elliptic curves over number fields were also presented, but not turned into a provable determination of the set of integral points.

In this note we use the techniques of [1] to determine the set of -integral points on the Mordell curve

$$\begin{aligned} E:y^2 = x^3-4. \end{aligned}$$

(2)

The goal is twofold. First, we want to show that quadratic Chabauty can indeed be a tool to determine the set of integral points on an elliptic curve (even over number fields), and that it is not only a test case for more general techniques that apply to higher genus curves. In order to achieve this, we need to substitute the Mordell–Weil sieve step with an analogous sieve for elliptic curves: we do this by extending to number fields the technique of [3, Appendix A]. Secondly, presenting this particular computation allows us to overcome a lot of the technicalities and notational complexity of [1], while still conveying the general strategy.

A few words about our choices of curve and field. The techniques for the computation of integral points of elliptic curves over were extended to arbitrary number fields by Smart and Stephens [23]. However, it appears that they are currently implemented in Magma [12] only over totally real fields; hence the choice of an imaginary quadratic field. In order to get rid of presumed non-integral points in the quadratic Chabauty computation, it is convenient to compare quadratic Chabauty outputs at different primes. For our choice of curve, there are two prime numbers, namely $p_1 = 7$ and $p_2 =13$, which satisfy for $i\ne j$, making the comparison of the respective quadratic Chabauty outputs easier. In fact, this was used in [3, Appendix A] to compute the integral points over , since E has rank 1 over and hence “classical” quadratic Chabauty is applicable. Finally, E attains rank 2 over , both $p_1$ and $p_2$ split in , and the latter is also the field of complex multiplication of E, making it a nice example to which our techniques can be applied.

The code to perform quadratic Chabauty for an elliptic curve over base-changed to an imaginary quadratic field, as well as for the sieving routines of this example, is available at [9].

2 Quadratic Chabauty for

Let K be a number field of degree d and let $\mathbb {A}_{K,f}^{\times }$ be the group of finite ideles of K; let p be an odd prime, unramified in K. For ease of exposition, we assume that K has class number 1 and for every prime $\mathfrak {q}$ of K we fix a generator $\xi _{\mathfrak {q}}$ for $\mathfrak {q}$. Furthermore, we let $|\cdot |_{\mathfrak {q}}$ be the absolute value on $K_{\mathfrak {q}}$, normalised so that $|q|_{\mathfrak {q}} = q^{-1}$, if q is the rational prime below $\mathfrak {q}$. Thus $|\xi _{\mathfrak {q}}| = q^{-1/r_{\mathfrak {q}}}$, for some positive integer $r_{\mathfrak {q}}$ which divides the degree $n_{\mathfrak {q}}$ of the extension .

An idele class character is a continuous homomorphism

In principle, the idele class character $\chi$ is determined by the value, at each prime $\mathfrak {q}$, of $\chi _{\mathfrak {q}}$ on $\xi _{\mathfrak {q}}$ and on the units $\mathcal {O}_{\mathfrak {q}}^{\times }$ of the ring of integers of $K_{\mathfrak {q}}$. However, the incompatibility between the $\mathfrak {q}$-adic and p-adic topology for $\mathfrak {q}\ne p$ and the required vanishing of $\chi$ on $K^{\times }$ have two strong consequences:

• The restrictions of $\chi _{\mathfrak {p}}$ to $\mathcal {O}_{\mathfrak {p}}^{\times }$ at the primes $\mathfrak {p}\mid p$ uniquely determine $\chi$. Indeed, $\chi _{\mathfrak {q}}$ vanishes on $\mathcal {O}_{\mathfrak {q}}^{\times }$ if $\mathfrak {q}\not \mid p$, and for every prime $\mathfrak {q}$ we have the formula

  $$\begin{aligned} \chi _{\mathfrak {q}}(\xi _{\mathfrak {q}}) = -\sum _{\begin{array}{c} \mathfrak {p}\mid p\\ \mathfrak {p}\ne \mathfrak {q} \end{array}} \chi _{\mathfrak {p}}(\xi _{\mathfrak {q}}). \end{aligned}$$

  (3)

• Each fundamental unit for K imposes an additional constraint on the characters $\chi _{\mathfrak {p}}$. In particular, the idele class characters for K form a -vector space V of dimension greater than or equal to $d-rank (\mathcal {O}_K^{\times })$.

Computing a basis for V amounts to doing some linear algebra.

Example 1

In our situation of interest, K is an imaginary quadratic field, in which p splits, say $p = \mathfrak {p}_1 \mathfrak {p}_2$, and we may fix isomorphisms . Then V is spanned by two characters: the cyclotomic character $\chi ^{cyc }$ and the anticyclotomic character $\chi ^{anti }$. The cyclotomic character is the unique character satisfying

$$\begin{aligned} \chi ^{cyc }_{\mathfrak {p}_i}(x) = \log (x)\qquad \text {for all } x\in \mathcal {O}_{\mathfrak {p}_i}^{\times }, \end{aligned}$$

where is the p-adic logarithm, and we view x as an element of via our fixed isomorphism. The anticyclotomic character (depending on our choice of ordering of the primes above p) is determined by

$$\begin{aligned} \chi _{\mathfrak {p}_1}^{anti }(x) = \log (x)\quad \text {for all } x\in \mathcal {O}_{\mathfrak {p}_1}^{\times };\qquad \chi _{\mathfrak {p}_2}^{anti }(x) = -\log (x)\quad \text {for all } x\in \mathcal {O}_{\mathfrak {p}_2}^{\times }. \end{aligned}$$

The p-adic logarithm comes in the picture because we are considering homomorphisms from the multiplicative $\mathcal {O}_{\mathfrak {p}_i}^{\times }$ to the additive .

Formula (3) gives

$$\begin{aligned} \chi _{\mathfrak {q}}^{cyc }(q)&= r_{\mathfrak {q}}\chi _{\mathfrak {q}}^{cyc }(\xi _{\mathfrak {q}}) = -n_{\mathfrak {q}}\log (q)\qquad&\text {for all } q\ne p;\end{aligned}$$

(4)

$$\begin{aligned} \chi _{\mathfrak {q}}^{anti }(q)&= r_{\mathfrak {q}}\chi _{\mathfrak {q}}^{cyc }(\xi _{\mathfrak {q}}) =0 \qquad&\text {for all } q\ne p \text { such that } n_{\mathfrak {q}} = 2. \end{aligned}$$

(5)

Let E/K now be an elliptic curve with good reduction at all primes above p, described by an equation of the form (2), which, for simplicity, we further assume to be minimal at all primes. Let S be the set of primes of bad reduction for E, and for each prime $\mathfrak {q}$ of K, let $c_{\mathfrak {q}}$ be the Tamagawa number of E at $\mathfrak {q}$ (cf. [18, Theorem 4.11]).

If none of the primes above p is in S, by work of Bernardi [7], Mazur–Tate [17], Coleman–Gross [14] and others, to every idele class character $\chi \in V$ we can attach a quadratic function

for some suitably chosen $\iota (P)\in \mathbb {A}_{K,f}^{\times }$, which is independent of $\chi$ and whose $\mathfrak {p}$-adic component at a prime $\mathfrak {p}\mid p$ depends on the choice of a subspace of $H_{dR }^1(E/K_{\mathfrak {p}})$, complementary to the space of holomorphic forms.

We call $h_p^{\chi }$ the (global) p-adic height attached to $\chi$, where we should not forget, however, that we have made some preliminary choices that are not reflected in our notation.

More generally, at every prime $\mathfrak {q}$, given $P_{\mathfrak {q}}\in E(K_{\mathfrak {q}})\setminus \{O\}$, there exists $\iota _{\mathfrak {q}}(P_{\mathfrak {q}})\in K_{\mathfrak {q}}^{\times }$ and a local p-adic height function

so that $\iota (P)_{\mathfrak {q}} = \iota _{\mathfrak {q}}(P)^{(\prod _{\mathfrak {r}\in S\setminus {\mathfrak {q}} }c_{\mathfrak {r}})}$ and hence

$$\begin{aligned} h_p^{\chi }(P) = \sum _{\mathfrak {q}}\lambda _{\mathfrak {q}}^{\chi }(P)\qquad \text {if } P\in E(K)\setminus \{O\}. \end{aligned}$$

The failure of each $\lambda _{\mathfrak {q}}^{\chi }$ to be a quadratic function is absorbed into the vanishing of $\chi$ on $K^{\times }$, i.e. we have

(6)

We do not go into the details here of how to define $\iota _{\mathfrak {q}}(P_{\mathfrak {q}})$. We just note that the definition of $h_p^{\chi }$ mimics Néron’s definition of the real canonical height (for which see, for instance, [21, Chapter VI]), and we record the following properties.

Proposition 1

• 1. At a prime $\mathfrak {p}\mid p$, the function $\lambda _{\mathfrak {p}}^{\chi }$ restricts to a locally analytic function on $E(\mathcal {O}_{\mathfrak {p}})$, whose power series expansion in a local coordinate t in each residue disc¹ can be computed up to any desired $\mathfrak {p}$-adic and t-adic precision.

• 2. At a prime $\mathfrak {q}\not \mid p$, $\iota _{\mathfrak {q}}(P)$ is independent of p. We have

  • If P reduces to a non-singular point modulo $\mathfrak {q}$, then

    $$\begin{aligned} \iota _{\mathfrak {q}}(P) = \max \{1, |x(P)|_{\mathfrak {q}}\}^{-c_{\mathfrak {q}}}. \end{aligned}$$

  • If P reduces to a singular point, then $\iota _{\mathfrak {q}}(P)^{r_{\mathfrak {q}}}\in W_{\mathfrak {q}}$, where is a finite set which can be deduced from knowledge of the Tamagawa number and Kodaira symbol of E at $\mathfrak {q}$.

Corollary 1

Let S be the set of primes of K at which E has bad reduction. For every $\mathfrak {q}\in S$, there exists a finite set such that for every $P\in E(\mathcal {O}_K)$ there exists $t_{\mathfrak {q}}\in T_{\mathfrak {q}}$ with

$$\begin{aligned} h_p^{\chi }(P) = \sum _{\mathfrak {p}\mid p}\lambda _{\mathfrak {p}}^{\chi }(P) +\sum _{\mathfrak {q}\in S}c_{\mathfrak {q}}^{-1}r_{\mathfrak {q}}^{-1}\chi _{\mathfrak {q}}(t_{\mathfrak {q}}). \end{aligned}$$

Moreover, $T_{\mathfrak {q}}$ and $t_{\mathfrak {q}}$ are independent of p and $\chi$.

Corollary 1 and Eq. (6) are the key ingredients for our method. In fact, a point $P\in E(\mathcal {O}_K)$ carries global information by its being a point in E(K). This is encoded in (6), but alone would not suffice to give it a finite characterisation inside $\prod _{\mathfrak {p}\mid p} E(K_{\mathfrak {p}})$, since E(K) could be infinite. Thus, we also need to exploit the local information that it carries by its being integral at every prime and this is where Corollary 1 comes into play.

In order to use (6) effectively, we need to express the p-adic height $h_p^{\chi }$ as the restriction of a locally analytic function . This is where we are forced to impose some conditions on the rank of E(K). We will focus now on our example of interest; see [1] for the general treatment of the quadratic Chabauty steps. The sieving method that we describe is an extension to number fields of [3, Appendix A]; while for the moment we restrict the exposition to this example, the method would apply more generally: see Sect. 3.

Example 2

Let E be the curve over defined by (2). Let

$$\begin{aligned} E^{-3}:y^2 = x^3+108 \end{aligned}$$

(7)

be the quadratic twist of E by $-3$. By computing generators for and with SageMath [25], we deduce that

where

$$\begin{aligned} Q = (2,2),\qquad R = (-\sqrt{-3} + 1, 2\sqrt{-3}). \end{aligned}$$

The set of primes at which E/K has bad reduction is

$$\begin{aligned} S = \{(2), (\sqrt{-3})\},\qquad \text {with } c_{(2)} = 4,\quad c_{(\sqrt{-3})} = 3. \end{aligned}$$

Let $p\in \{7,13\}$. The prime p splits completely in K and we are in the situation of Example 1, so we have two global p-adic heights

where we have chosen the local heights at $\mathfrak {p}_i$ to be the ones of Mazur–Tate (which coincide with Bernardi’s in this case). For $i\in \{1,2\}$, we may also consider

where $Log _i$ is the unique homomorphism of abelian groups which restricts to the formal group logarithm (composed with ) on the formal group. By construction, the functions $Log _1$ and $Log _2$ are locally analytic, and Lemma 6.4 of [1] shows that $f_1$ and $f_2$ (or, more precisely, their extensions to ) are linearly independent. Since the rank of E(K) is 2 by (), it follows that any -valued quadratic function on it must be a -linear combination of

$$\begin{aligned} f_1^2,\quad f_1f_2,\quad f_2^2. \end{aligned}$$

In particular, there exist such that

$$\begin{aligned} h_p^{cyc }= \alpha _1^{cyc } f_1^2 + \alpha _2^{cyc } f_1f_2 + \alpha _3^{cyc }f_2^2,\qquad h_p^{anti }= \alpha _1^{anti } f_1^2 + \alpha _2^{anti } f_1f_2 + \alpha _3^{anti }f_2^2. \end{aligned}$$

(8)

Properties of the cyclotomic and anticyclotomic characters show that $\alpha _1^{cyc } = \alpha _3^{cyc }$; $\alpha _1^{anti } =- \alpha _3^{anti }$ and $\alpha _2^{anti } = 0$ (cf. [1, §6.2]). Furthermore, since $h_p^{\chi }$ is invariant under any automorphism of E/K (while $f_1^2$ and $f_2^2$ are only invariant under $\pm 1$), in this case all the constants are identically zero, except for $\alpha _2^{cyc }$, which is non-zero by [8]. We can compute it up to our desired precision, by comparing the values of the p-adic height and of $f_1f_2$ on any point of infinite order of E(K).

To make (8) sensitive to the difference between K-rational and K-integral points, we use Corollary 1. By work of Cremona–Prickett–Siksek [15] (see also [11, Proposition 2.4]), we may take

$$\begin{aligned} T_{(2)} = \{1,2^{4}\}, \qquad T_{(\sqrt{-3})} =\{1, 3^2\}. \end{aligned}$$

Consider the locally analytic functions

defined by

$$\begin{aligned} \rho _p^{\star }(P_1,P_2) = \lambda _{\mathfrak {p}_1}^{\star }(P_1) +\lambda _{\mathfrak {p}_2}^{\star }(P_2)-\alpha _2^{\star }Log _1(P_1)Log _2(P_2). \end{aligned}$$

By Corollary 1 and (4)-(5), for every $P\in E(\mathcal {O}_{K})$, there exists $t_{(2)}\in T_{(2)}$ and $t_{(\sqrt{-3})}\in T_{(\sqrt{-3})}$ such that

$$\begin{aligned} \rho _p^{cyc }(\psi _p(P)) -\frac{1}{2}\log t_{(2)}-\frac{1}{3}\log t_{(\sqrt{-3})} = 0 = \rho _p^{anti }(\psi _p(P)), \end{aligned}$$

where $\psi _p:E(\mathcal {O}_K)\hookrightarrow E(\mathcal {O}_{\mathfrak {p}_1})\times E(\mathcal {O}_{\mathfrak {p}_2})$ is the map induced by the completion maps. The strategy is then as follows.

• 1. Fix a positive integer B and compute²

  $$\begin{aligned} E(\mathcal {O}_K)_{known } = \{nQ+mR: |n|,|m|\le B \quad \text {and}\quad nQ+mR\in E(\mathcal {O}_K)\}. \end{aligned}$$

  By picking B sufficiently large, we might expect that $E(\mathcal {O}_K)_{known } = E(\mathcal {O}_K)$ and the following steps try to prove the equality.

• 2. Fix $(t_{(2)},t_{(\sqrt{-3})})\in T_{(2)}\times T_{(\sqrt{-3})}$.

• 3. Let $p= 7$. Compute, modulo some fixed p-adic precision, all $(P_1,P_2)\in E(\mathcal {O}_{\mathfrak {p}_1})\times E(\mathcal {O}_{\mathfrak {p}_2})$ such that

  $$\begin{aligned} \rho _7^{cyc }(P_1,P_2) -\frac{1}{2}\log t_{(2)}-\frac{1}{3}\log t_{(\sqrt{-3})} = 0 = \rho _7^{anti }(P_1,P_2). \end{aligned}$$

  (9)

  If $\varphi \in Aut (E/K)$ with induced $\varphi _i\in Aut (E/K_{\mathfrak {p}_i})$, then $(\varphi _1(P_1),\varphi _2(P_2))$ is a solution to (9) if $(P_1,P_2)$ is. Furthermore, if $(P_1,P_2)$ is a solution, then so is $(P_2,P_1)$ where we are abusing notation, in view of the isomorphisms .

• 4. For every $(P_1,P_2)$ computed in step 3, check if there exists $P\in E(\mathcal {O}_K)_{known }$ with $\psi _7(P) \equiv (P_1,P_2)$. If such P does not exist, suppose there exists $T\in E(\mathcal {O}_K)\setminus E(\mathcal {O}_K)_{known }$ with $\psi _7(T) = (P_1,P_2)$. Then $T = nQ+mR$ for some satisfying

  $$\begin{aligned} \left( \begin{matrix} n\\ m \end{matrix}\right) = \left( \begin{matrix} f_1(Q) &{} f_1(R)\\ f_2(Q) &{} f_2(R) \end{matrix}\right) ^{-1}\left( \begin{matrix} Log _1(P_1)\\ Log _2(P_2) \end{matrix}\right) \bmod {7}. \end{aligned}$$

  Furthermore we can compute

  $$\begin{aligned} \left( \begin{matrix} n\\ m \end{matrix}\right) \bmod {13}, \end{aligned}$$

  by considering the images of $P_1,P_2,Q,R$ in and , since . (Note that it is straightforward to verify that there can be at most one such possibility).

• 5. The solution set of

  $$\begin{aligned} \rho _{13}^{cyc }(P_1,P_2) -\frac{1}{2}\log t_{(2)}-\frac{1}{3}\log t_{(\sqrt{-3})} = 0 = \rho _{13}^{anti }(P_1,P_2) \end{aligned}$$

  must contain $\psi _{13}(T)$. Since , we can, similarly to steps 3 and 4, deduce another list of possibilities for the pair

  $$\begin{aligned} \left( \begin{matrix} n\\ m \end{matrix}\right) \bmod {13}, \qquad \left( \begin{matrix} n\\ m \end{matrix}\right) \bmod {7}. \end{aligned}$$

  If none of these pairs matches the one of step 4, then we have shown that $(P_1,P_2)$ is not in the image of $\psi _7$.

• 6. We repeat steps 3–5 for each element of $T_{(2)}\times T_{(\sqrt{-3})}$.

Remark 1

The computations of the zero sets in steps 3 and 5 are carried out locally, i.e. for every pair , we can give a two-variable parametrisation of the points in $E(K_{\mathfrak {p}_1})\times E(K_{\mathfrak {p}_2})$ reducing to $(\overline{P_1},\overline{P_2})$. The task is then reduced to computing zero sets of systems of two-variable equations in two variables. This can be done either naively, or, when the hypotheses apply, using a two-variable version of Hensel’s lemma [1, Appendix A]. For the strategy to give a provable determination of the integral points, it is necessary to verify that the approximations of the zeros corresponding to points in $E(\mathcal {O}_K)$ lift to unique zeros. It is on the other hand not necessary in general to show that the other zeros lift uniquely.

We implement this technique in SageMath and run it with $B=10$. As we vary $t_{(2)}$ and $t_{(\sqrt{-3})}$, this gives 426 solutions to (9) which do not come from points in $E(\mathcal {O}_K)_{known }$. Out of these 426, there are only 4 up to automorphisms and conjugation which survive our sieve.

We need to show these 4 solutions do not come from points in $E(\mathcal {O}_K)$. We first note that they are p-adically isolated. Then we show that, if one of them did come from a point T in $E(\mathcal {O}_K)$, then, up to automorphism, it would come from a point in or a point in where we take the minimal model (7) (this amounts to showing that in the automorphism class of the solution, there is a point of the form $(P_1,P_1)$ or $(P_1,-P_1)$, where we are using our isomorphisms ). At this point, we could invoke existing implementations to compute and , to show that all the points in $E(\mathcal {O}_K)$ coming from points in these sets are in $E(\mathcal {O}_K)_{known }$. Alternatively, we observe that being in the image of or imposes additional contraints on the local heights away from p of P, which allow us to eliminate at once 3 of the automorphisms classes. The remaining one must come from and hence must be of the form mR. There is some information that we have not used yet: since , we also know $m\bmod {3}$. In particular, $m\equiv 0\bmod {3}$. But then mR is a multiple of 3R, which is in the formal group at $(\sqrt{-3})$: thus mR cannot be integral. This shows that

Theorem 1

$E(\mathcal {O}_K) = \{\varphi (P),\varphi (P^c) : P\in G, \varphi \in Aut (E/K)\}$, where $P^c$ is the Galois conjugate of P and

$$\begin{aligned} G = \{&(2,2),(5, 11),(-2,2\sqrt{-3}),(1,\sqrt{-3}),\\&(-122, 778\sqrt{-3} ), (3\sqrt{-3} - 5, 4\sqrt{-3} + 18) \}. \end{aligned}$$

3 A More General Approach

We end this article with an outline of the strategy that was implicitly used in Example 2, in the hope that this discussion will convince the reader that similar methods could be used to determine the integral points of other elliptic curves over number fields.

Let E be an elliptic curve over an imaginary quadratic field K. Suppose further that existing algorithms - e.g. as implemented in [12] - succeed in the computation of the rank r of E(K). If $r\ge 3$, our methods are not applicable. If $r=0$, the computation of $E(\mathcal {O}_K)$ is trivial; if $r=1$, minor modifications to the quadratic Chabauty method over are sufficient [4].

Thus, we may assume that r is exactly equal to 2, and that we can compute generators for the Mordell–Weil group E(K). For simplicity, we further assume that the torsion subgroup of E(K) is trivial. Hence,

$$\begin{aligned} E(K) = \langle Q,R\rangle \end{aligned}$$

for some points of infinite order Q and R. Our goal is the computation of the integral points $E(\mathcal {O}_K)$.

It is reasonable to first compute the integral points up to a certain height bound and we shall denote the resulting set by $E(\mathcal {O}_K)_{known }$. Indeed, for instance, if Q and R are non-integral at the same prime $\mathfrak {q}$, then $E(\mathcal {O}_K)$ is empty and no further step should be taken (see [22, Chapter VII]).

Let p be an odd prime such that E has good reduction at every $\mathfrak {p}\mid p$. Current restrictions in the implementation of some of the techniques that we need also require us to assume that p is split in K, say $p\mathcal {O}_K=\mathfrak {p}_1\mathfrak {p}_2$ for distinct primes $\mathfrak {p}_1$ and $\mathfrak {p}_2$ of $\mathcal {O}_K$. We have homomorphisms

obtained by composing the completion map $E(K)\hookrightarrow E(K_{\mathfrak {p}_i})$ with $\mathfrak {p}_i$-adic abelian integration and with the isomorphism . If $f_1$ and $f_2$ are linearly independent, then we can carry out quadratic Chabauty for the prime p on E/K. Linear independence is guaranteed by [1, Lemma 6.4] if E is the base-change of an elliptic curve over of rank 1, and we also expect it to hold if E does not descend to .

In the belief that the exposition of the quadratic Chabauty computation in Sect. 2 was, when not already in full generality, easily generalisable, we do not elaborate on that further. The upshot is that we can compute a set $U_p$ of points in $E(\mathcal {O}_{\mathfrak {p}_1})\times E(\mathcal {O}_{\mathfrak {p}_2})$ such that if $P\in E(\mathcal {O}_K)\setminus E(\mathcal {O}_K)_{known }$ then the image of P under the completion maps lies in $U_p$. By “computing”, we mean that given a large enough integer n we can find the finite set of approximations modulo³ $p^n$ of the points in $U_p$. This is provided that we can show that, modulo $p^n$, there are no points in $U_p$ that have the same reduction as any point in $E(\mathcal {O}_K)_{known }$. Based on experimental data, we can say that the latter task can generally be addressed using the multivariable Hensel’s lemma, but occasionally needs support from theoretical arguments [1, Appendix A] (see also Remark 1).

If $U_p$ is empty, we deduce that $E(\mathcal {O}_K)= E(\mathcal {O}_K)_{known }$. Otherwise, we can try to extrapolate more information from $U_p$, or compute $U_p$ for more than one choice of p. If the primes are chosen wisely, it is possible to compare the outputs $U_p$. We explain here a good choice; many others could be possible.

For $i\in \{1,2\}$, let $k_{\mathfrak {p}_i}$ be the greatest common divisor of the orders in of the reduction of Q and R. Assume that $(P_{\mathfrak {p}_1},P_{\mathfrak {p}_2})\in U_p$ is the localisation of a point $P\in E(\mathcal {O}_K)\setminus E(\mathcal {O}_K)_{known }$. Then there must exist such that

$$\begin{aligned} P = nQ+mR. \end{aligned}$$

Using $\mathfrak {p}_i$-adic abelian integration, we may compute a unique pair such that

$$\begin{aligned} (n,m)\equiv (n_{p},m_{p}) \bmod {p}. \end{aligned}$$

(10)

Furthermore, for every i we can compute a list of possibilities for

$$\begin{aligned} (n,m)\bmod {k_{\mathfrak {p}_i}}, \end{aligned}$$

(11)

by considering the images of Q, R and $P_{\mathfrak {p}_i}$ in .

If $p\mid k_{\mathfrak {p}_i}$ for at least one i, then (10) and (11) can sometimes be used to rule out points in $U_p$ from being localisations of points in $E(\mathcal {O}_K)$.

If this is not sufficient to prove that no point in $U_p$ can correspond to a point in $E(\mathcal {O}_K)$, or if p is coprime with each $k_{\mathfrak {p}_i}$, then we may look for a split odd prime q of good reduction such that $q\mid k_{\mathfrak {p}_i}$ and $p\mid k_{\mathfrak {q}_j}$ for at least one (i, j), if $q\mathcal {O}_K =\mathfrak {q}_1\mathfrak {q}_2$. If the element $(P_{\mathfrak {p}_1},P_{\mathfrak {p}_2})\in U_p$ corresponds to $P\in E(\mathcal {O}_K)$, then P must map to some $(P_{\mathfrak {q}_1},P_{\mathfrak {q}_2})\in U_q$ under the $(\mathfrak {q}_1,\mathfrak {q}_2)$-completion maps. By running through $U_q$, we obtain a list of possibilities for the pair

$$\begin{aligned} (n,m)\bmod {q}\qquad \text {and}\qquad (n,m)\bmod {k_{\mathfrak {q}_j}}, \end{aligned}$$

which we can compare with (10) and (11).

If this is still not sufficient to show that $E(\mathcal {O}_K) = E(\mathcal {O}_K)_{known }$, we can look for sequences of primes with similar patterns.

Elliptic curves E over admitting a pair of primes of good reduction (p, q) satisfying and were studied by Silverman–Stange [20]. Assuming that there are infinitely many primes $\ell$ for which is prime, the authors conjectured that the number of such pairs (p, q) with $p<q$ and $p\le X$ should grow like a multiple of $X/(\log X)^2$ if E has CM, and of $\sqrt{X}/(\log X)^2$ otherwise. More generally, they considered sequences of primes $(p_1,\ldots , p_{m})$ of arbitrary length m such that , where i is taken modulo m.

This study is relevant for us if our elliptic curve over K is the base-change of an elliptic curve over , as the one of Example 2. For more general elliptic curves over number fields, see [13].

Finally, the discussion of this section is readily generalisable to number fields other than imaginary quadratic ones, after deriving the analogue of Example 1 and suitably replacing the condition $r=2$ (see [1]).