Some effectivity results for primitive divisors of elliptic divisibility sequences

Let $P$ be a non-torsion point on an elliptic curve defined over a number field $K$ and consider the sequence $\{B_n\}_{n\in \mathbb{N}}$ of the denominators of $x(nP)$. We prove that every term of the sequence of the $B_n$ has a primitive divisor for $n$ greater than an effectively computable constant that we will explicitly compute. This constant will depend only on the model defining the curve.


Introduction
Let E be an elliptic curve defined by the equation with coefficients in a number field K .Let P ∈ E(K ) be a nontorsion point and let O K be the ring of integers of K .Let us define the fractional ideal (1) with A n and B n two relatively prime integral O K -ideals.We want to study the sequence of integral O K -ideals {B n } n∈‫ގ‬ .These are the so-called elliptic divisibility sequences.In particular, we want to study when a term B n has a primitive divisor, i.e., when there exists a prime ideal P such that Silverman [1988, Proposition 10] proved that, if E is defined over ‫,ޑ‬ then B n has a primitive divisor for n large enough.This result was generalized for every number field K in [Cheon and Hahn 1999], where the following theorem is proved.
Theorem 1.1 [Cheon and Hahn 1999, main theorem].Let E be an elliptic curve defined over a number field K and let P be a nontorsion point in E(K ).Consider the sequence {B n } n∈‫ގ‬ of integral O K -ideals as defined in (1).Then B n has a primitive divisor for all but finitely many n ∈ ‫.ގ‬ The previous theorem is not effective.Indeed, the proof relies on Siegel's ineffective theorem about integral points on elliptic curves.The aim of this paper is to make the work of [Cheon and Hahn 1999] effective.Indeed, we will explicitly compute a constant C so that B n has always a primitive divisor for n > C.
Theorem 1.2.Let E be an elliptic curve defined over a number field K and let P be a nontorsion point in E(K ).Consider the sequence {B n } n∈‫ގ‬ of integral O K -ideals as defined in (1).There exists a constant C(E/K , M) > 0, effectively computable and depending only on the curve E over the field K equipped with a model M also defined over K , such that B n has a primitive divisor for n > C(E/K , M).
Remark 1.3.The dependence on the model M is necessary.Indeed, given a nontorsion point P on an elliptic curve E and a positive constant C, it is easy to show that we can find a model of E such that B n does not have a primitive divisor for all n ≤ C.
Remark 1.4.It is conjectured that, in the case when M is minimal, the constant C(E/K , M) should depend only on the field K .In [Ingram and Silverman 2012, Theorem 1] it is proved that the number of terms without a primitive divisor of an elliptic divisibility sequence can be bounded by a constant that does not depend on E and P, in the case when E is given by a minimal model, K = ‫,ޑ‬ and assuming the abc-conjecture.
Remark 1.5.We believe that the techniques used in this paper can be applied also to a generalization of elliptic divisibility sequences.Let O be the endomorphism ring of E and, given α ∈ O, define B α as the denominator of (x(α P))O K .The sequence {B α } α∈O is a sequence of ideals and one can give a definition of primitive divisors also for these sequences (see [Streng 2008, Section 1]).It has been shown in [Streng 2008, main theorem] that also in this case there are only finitely many terms that do not have a primitive divisor (see also [Verzobio 2021b]).In the case when End(E) = ‫ޚ‬ this is a trivial corollary of Theorem 1.1, but in the case End(E) ̸ = ‫ޚ‬ (i.e., when E has complex multiplication) this is far from being easy.We believe that using the techniques of this paper one can find an explicit upper bound for the degree of α such that B α does not have a primitive divisor, in the case when End(E) is a maximal order and it is a principal ideal domain.

Notation
The curve E is defined by the equation with coefficients in the number field K .
The following notation will be used during the paper: is the discriminant of the equation defining the curve.
E/K is the minimal discriminant of the elliptic curve.j (E) is the j-invariant of the curve.
where ‫ގ‬ K ‫ޑ/‬ is the norm of the field extension, is the Szpiro quotient; if E/K has everywhere good reduction (and then f E/K = 1), we put If n ∈ ‫ގ‬ is nonzero, then ω(n) is the number of rational prime divisors of n.
If x ∈ K * , define m(x) = max P {ord P (x)}, where the maximum runs over all primes in O K .

Preliminaries
Let M K be the set of all places of K , take ν ∈ M K , and let | • | ν be the absolute value associated with ν.Let n ν be the degree of the local extension K ν ‫ޑ/‬ ν .We normalize the absolute values as in [Silverman 2009, Section VIII.5, after Example VIII.5.1].If ν is finite, then | p| ν = p −1 , where p is the rational prime associated to ν.If ν is infinite, then |x| ν = max{x, −x} for every x ∈ ‫.ޑ‬ Thanks to this choice, we have the usual product formula, i.e., for every x ∈ K * .Define M ∞ K as the set of infinite places of K and M 0 K as the set of finite places.Now, we define the height of a point on the curve; more details can be found in [Silverman 2009, Chapter VIII].Given x ∈ K * , define and the height of the point as So, for every R ∈ E(K ) \ {O}, Finally, put where O is the identity of the curve.
Given a point R in E(K ), define the canonical height as in [Silverman 2009, Proposition VIII.9.1], i.e., We recall the properties of the height and of the canonical height that will be necessary for this paper.
• It is known that the difference between the height and the canonical height can be bounded by an explicit constant.In particular, we will use the following result. Let If E is defined by a Weierstrass equation in short form and with integer coefficients, then, for every R ∈ E(K ), This is proved in [Silverman 1990, equation 3].
There exists a positive constant J E , effectively computable and depending only on E and K , such that J E ≤ ĥ(P) for every nontorsion point P ∈ E(K ).Thanks to [Petsche 2006, Theorem 2], we can take 2009, beginning of Section VIII.11].
In order to prove that B n has a primitive divisor for all but finitely many terms, Silverman [1988] and Cheon and Hahn [1999] used a theorem of Siegel that says for every ν ∈ M K , as is proved in [Silverman 2009, Theorem IX.3.1].This result is not effective and hence their results are not effective.We will use some results that tell us effectively how this limit goes to 0. As we will show later, for the finite places we will use some results on the formal group of the elliptic curve, and for the infinite places we will use the work in [David 1995].The idea of using the result of David to study primitive divisors of elliptic divisibility sequences was introduced, as far as we know, by Streng [2008, Section 3].We conclude this section by showing that we can focus only on the case when E is defined by a Weierstrass equation in short form and with integer coefficients.We will do that in Lemma 3.2.In order to prove that lemma, we need the following.
Lemma 3.1.Let E be an elliptic curve defined over K by a Weierstrass equation with integer coefficients and let P ∈ E(K ).Let ν ∈ M 0 K , P be the associated prime, and p be the associated rational prime.There exists ‫ގ2(‬ K ‫ޑ/‬ (P) + 1) max{4, ord P ( j (E) −1 )} such that ν(x(k P)) < 0.
Proof.Let E P be a minimal model for the elliptic curve over K P and let P P be the image of P under the change of variables from E to E P .So, x(P) = u 2 P x(P P )+r P for some u P , r P ∈ K P .By [Silverman 2009, Proposition VII.1.3.d],ν(u P ) ≥ 0 and ν(r P ) ≥ 0. Note that 12ν(u P ) = ν( (E)) − ν( (E P )) ≤ ν( (E)) and so ν(u P ) ≤ ν( (E))/12.There exists a multiple k P P with k ≤max{4, ord P ( j (E P ) −1 )} such that k P P is not a singular point in E P ‫ކ(‬ P ) (see [Silverman 2009, Corollary C.15.2.1]).Observe that ‫ކ‬ P has ‫ގ‬ K ‫ޑ/‬ (P) elements and then the group of nonsingular points in E P ‫ކ(‬ P ) has at most ‫ގ2‬ K ‫ޑ/‬ (P) + 1 elements.So, the order of k P P in the group of nonsingular points modulo P is at most ‫ގ2‬ K ‫ޑ/‬ (P) + 1.Hence, there exists n P (P P ) ≤ ‫ގ2(‬ K ‫ޑ/‬ (P) + 1) max{4, ord P ( j (E P ) −1 )} such that n P (P P )P P reduces to the identity modulo P. Given a point Q in E(K ), it is easy to show that Q reduces to the identity modulo P if and only if ν(x(Q)) < 0. Therefore, ν(x(n P (P P )P P )) < 0.
From a classic result on formal groups, ν x( p ν(u P ) n P (P P )P P ) < −2ν(u P ).
Then, there exists an elliptic curve E ′ defined over K by a short Weierstrass model M ′ with integer coefficients that is isomorphic over K to E, and a positive rational integer s for E, M. The constant s(E/K , M) is effectively computable and will be defined during the proof (see ( 2)).It depends only on E and M.
Proof.Recall that E is defined by the equation Let u be the smallest positive rational integer such that, after the change of variables, If r = 0, we take q such that ν(q) = |ν(u 2 )| for every ν ∈ M 0 K .Note that q depends only on E and M. Let B ′ n be the elliptic divisibility sequence associated with E ′ and P ′ .
Let ν be the absolute value associated with a prime P coprime with q.We have ν(u 2 ) = 0 and ν(r ) ≥ 0. If P divides B n , then ν(x(nP)) < 0 and In the same way, if So, if P is coprime with q, then P divides B n if and only if it divides B ′ n .Let (2) s = max P|q p (12ν P (u)+ν P ( (E)))/12 ‫ގ2(‬ K ‫ޑ/‬ (P) + 1)(max{4, ord P ( j where p is the rational prime associated with P.
Assume n > s.We will show that, if P is a primitive divisor of B ′ n , then P is a primitive divisor also for B n .
Let P be a primitive divisor of B ′ n .Suppose that P divides q.By Lemma 3.1, there exists But, since P is a primitive divisor of B ′ n we know that k ≥ n.Hence, n ≤ s and this is absurd since we assumed n > s.So, P does not divide q.Since P is a primitive divisor of B ′ n and P is coprime with q, then P divides B n and does not divide B k for k < n.Therefore, it is a primitive divisor for B n .
In conclusion, if n > max{C(E ′ /K , M ′ ), s}, then B ′ n has a primitive divisor P. As we showed, P is also a primitive divisor for B n .Therefore, B n has a primitive divisor for all n > max{C(E ′ /K , M ′ ), s}.
Observe that s depends on j (E ′ ), (E), u, r , and q.It is easy to show these five values depend only on E and the model defining the curve.So, we are done.□ From now on, we will assume that E is defined by a short Weierstrass equation with coefficients in O K of the form Once we prove Theorem 1.2 under this assumption, then we can prove it in general using Lemma 3.2.It is useful to have E in this form in order to apply the work in [David 1995], as we will do in Section 6.

Structure of the proof
We start by recalling the structure of the proof of Cheon and Hahn of Theorem 1.1.
(1) If P is a nonprimitive divisor of B n , then P divides B n/q for q a prime divisor of n.Moreover, if ν is the place associated to P, then h ν (nP) and h ν n q P are roughly the same.
(5) For every n we have q|n q −2 < 1 and then the inequality of (4) does not hold for n large enough.So, B n does not have a primitive divisor only for finitely many n ∈ ‫.ގ‬ In order to make this proof effective, we need to make Steps (1) and (3) effective.

Finite places
Take P a prime over a valuation ν ∈ M 0 K .Let p be the rational prime under P. Recall that E is defined by a Weierstrass equation with integer coefficients.The group of points of E(K P ) that reduce to the identity modulo P is a group that is isomorphic to a formal group, as proved in [Silverman 2009, Proposition VII.2.2].Observe that, in the hypotheses of this proposition, there is the requirement that E is in minimal form.Anyway, the proof works in the exact same way only requiring that the coefficients of E are integers in K P , that is our case.Let Q ∈ E(K P ) and, using the equation defining the elliptic curve, it is easy to show that 3ν(x(Q)) = 2ν(y(Q)) and therefore Lemma 5.1.Take ν ∈ M 0 K and let P be the associated prime.Define n P as the smallest integer such that n P P reduces to the identity modulo P.Then, k P reduces to the identity modulo P if and only if k is a multiple of n P .Moreover, ν(x(k P)) < 0 if and only if k is a multiple of n P .
Proof.Let E ns ‫ކ(‬ P ) be the group of nonsingular points of the curve E reduced (with respect to the given model) modulo P. Suppose by contradiction that k P reduces to the identity but k is not a multiple of n P .Take q and r the quotient and the remainder of the division of k by n P .Since n P does not divide k, we have that 0 < r < n P .So, and this is absurd since n P is the smallest positive integer such that n P P ≡ O mod P. Vice versa, if k = qn P , then k P ≡ q(n P P) ≡ q O ≡ O mod P. Now, we conclude by observing that a point Q reduces to the identity modulo P if and only if ν(x(Q)) < 0.
. Now, we proceed by induction.The case e = 0 is trivial.Assume that we know that ν(z( = p e−1 Q and for the observation at the beginning of the proof we know ν(z Now, we deal with the second part of the lemma.Let n = p e n ′ and, by Lemma 5.1, Proof.This follows by [Silverman 1988, Theorem IV.6.4,Proposition VII.2.2].□ Definition 5.4.Let S be the set of finite places of K such that ν | 2 or ν ramifies over ‫.ޑ‬ Observe that this set is finite.
for all n ≥ 1.
Proof.Since ν / ∈ S, we have ν( p) = 1 and p−1 ≥ 2. So, ν(z(Q)) ≥ 1 > ν( p)/( p−1) and we apply Lemma 5.3.□ Proposition 5.6.Let E be an elliptic curve defined over a number field K and let P ∈ E(K ) be a nontorsion point.Take ν ∈ M 0 K , let P be the associated prime, and p be the rational prime under P. Recall that n P is the smallest positive integer such that n P P reduces to the identity modulo P. Assume that n P | n and n P ̸ = n.Then, one of the following hold: • There exists a prime q | n such that ν(z((n/q)P)) > 0 and ν(z(nP)) = ν z n q P + ν(q).
Proof.Assume ν / ∈ S and let Q = n P P. Since n/n P is an integer greater than 1, there is a prime q that divides it.By Corollary 5.5, So, we focus on the case ν ∈ S. Assume that there exists q ̸ = p such that q | n/n P .Then, ν(z(nP)) = ν z n q P by [Silverman 1988, Corollary IV.4.4] and we are done since ν(q) = 0. Assume now that there is no q ̸ = p such that q | n/n P .So, n = p e n P with e ≥ 1 (since n ̸ = n P ) and recall that we defined Q = n P P.
Since P is a prime over a place in S and the primes that ramify divide the discriminant of the field K , we have ‫ގ‬ K ‫ޑ/‬ (P) ≤ gpf(2 K ) D .Therefore, Proposition 5.10.Let E be an elliptic curve defined over a number field K and let P ∈ E(K ) be a nontorsion point.Take ν ∈ M 0 K and let P be the associated prime.Assume that n P | n, that n P ̸ = n, and that n ≥ C 1 .Then, there exists a prime q | n such that h ν (nP) = h ν n q P + 2h ν (q −1 ).
Proof.Observe that we are in the hypotheses of Proposition 5.6.By Lemma 5.8 we know that, since n ≥ C 1 , we cannot be in the second case of Proposition 5.6.Therefore, there exists a q | n such that Observe that, given Q ∈ E(K ) with ν(x(Q)) < 0, then by (3), Therefore, ). □

Infinite places
We know that 2n 2 ĥ(P) is close to h(nP) and that Thanks to the previous section, we know how to bound h ν (nP) for ν finite in the case when B n does not have a primitive divisor.Now, we need to bound h ν (nP) for ν infinite.We show that, for n large enough, h ν (nP) is negligible compared to n 2 ĥ(P).
Before proceeding, we need to define some constants.Let h = max{1, h(1 : g 2 : g 3 ), h( j (E))}, where h(1 : g 2 : g 3 ) is the usual height on ‫ސ‬ 2 (for a definition see [Silverman 2009, Section VIII.5]).Let Let c 1 := 3.6 • 10 41 , that is the constant c 1 of [David 1995, Theorem 2.1] evaluated in k = 2. Define (5) Proposition 6.1.Let E be an elliptic curve defined by the equation for g 2 , g 3 ∈ K and take P ∈ E(K ).Let z ∈ ‫ރ‬ be so that φ(z) = P and suppose Since log n > C 3 , we have log n > D, log n > eh > h +1, and log n = log B. Hence, Moreover, since log n > 30, we have and then and ω(n) as the number of prime divisors of n.It is easy to prove, by direct computation, that Recall that C 1 is defined in Definition 5.9.
Lemma 7.1.Let n ≥ C 1 .If B n does not have a primitive divisor, then there exists an embedding K → ‫ރ‬ such that where with |x(nP)| we mean the absolute value in the embedding and C E is defined in Section 2.
Proof.Suppose that B n does not have a primitive divisor and take ν finite.Let P be the associated prime and assume ν(B n ) > 0. Hence, n P | n but n ̸ = n P since B n does not have a primitive divisor.So, using Proposition 5.10, there is a prime Let M 0,n K be the set of finite places ν such that h ν (nP) > 0. Therefore, Here we are using that h ν (k P) ≥ 0 for all ν ∈ M K and all k ≥ 1.Thus, K , is larger than the right-hand side.Recalling that We briefly recall the hypotheses that we are assuming.As we said in the previous section, we are assuming that E(‫)ރ‬ ∼ = ‫/ރ‬ with the lattice generated by the complex numbers ω 1 and ω 2 .Moreover, we are working with an elliptic curve defined by a Weierstrass equation with integer coefficients and in short form.Recall that C 2 is defined in (5) and define that n ≥ C 1 , and that log n ≥ C 3 , as defined in (4).If B n does not have a primitive divisor, then Proof.Fix the embedding K → ‫ރ‬ of Lemma 7.1.Since B n does not have a primitive divisor, we have This lemma is stated for K = ‫,ޑ‬ but the proof works in the exact same way for K number field.Since δ is an n-torsion point, we have for 0 ≤ m 1 , m 2 ≤ n.Using Proposition 6.1 and the assumption that log n > C 3 , we have Applying inequalities ( 8) and ( 9) we have Observe that ω(n) ≤ log 2 n and (1 − ρ(n)) > 0.5.Thus, rearranging (10), we have ĥ(P)n 2 ≤ 2 ĥ(P)n 2 (1 − ρ(n)) Here we are using that n 1/2 > log n thanks to the hypothesis log n > C 3 .Recall that we obtained this inequality assuming |x(nP Therefore, one can easily show that, both in the case |x(nP)| < C 4 and in the case |x(nP)| ≥ C 4 , we have ĥ(P)n 2 ≤ n 1/2 (2C 2 + 4 + 2C E + log C 4 ).□ We are now ready to prove our main theorem.We will show that (7) does not hold if n is large enough.
Proof of Theorem 1.2.Define ), e 30 .We want to show that B n has a primitive divisor.
Observe that, thanks to the assumption in ( 11) and the definition of C 3 in (4), we have log n > C 3 .Moreover, Therefore, equation ( 6) holds.Finally, n ≥ C 1 .Hence, we are in the hypotheses of Proposition 7.2.We assume that B n does not have a primitive divisor and we find a contradiction.Since B n does not have a primitive divisor, we know that we can apply Proposition 7.2 and (7) must hold.But .
and then (7) does not hold.Therefore, we find a contradiction and then B n must have a primitive divisor.
In conclusion, define and B n has a primitive divisor for n > C 6 (E/K , M). Observe that every constant involved in the definition of C 6 (E/K , M) does not depend on P and it is effectively computable (we will give more details in the next section).So, we are done.
Recall that we are working under the assumption that E is defined by a short Weierstrass equation with integer coefficients.In order to conclude for the general case, one has to use Lemma 3.2.□

Explicit computation
Now, we explicitly write a constant C(E/K , M) such that Theorem 1.2 holds.We assume that E is defined by a short Weierstrass equation with integer coefficients, the general case can be done using Lemma 3.2.Recall that we defined many constants in Section 2.
First of all, we show how to bound |τ |, as defined at the beginning of Section 6. Recall that we are working under the assumption that τ is in the fundamental domain.Hence, we know |ℜτ | ≤ 1/2 and then we study ℑτ , the imaginary part of τ .Put q = e 2πiτ and then We obtain By the definitions of V 1 , V 2 , and C 2 given at the beginning of Section 6, we have .

Examples
We apply our main theorem to a couple of examples.With our methods, even if we optimize all the estimates in the proof, we cannot hope to find a constant for Theorem 1.2 much smaller than the one of Example 9.1.Indeed, in the definition of c 1 and of J E appear constants that are very large (namely 10 41 and 10 15 ) and so, even if the other constants involved are small, we cannot find a constant much smaller than 10 38 .In order to find better constants, one would need to have better constants in the bound of canonical height and in logarithmic approximation.Now, we present another example where we show the techniques that one can use to find the terms without a primitive divisor.
Example 9.2.We focus on the elliptic curve y 2 = x 3 − 2x and P = (2, 2) ∈ E(‫.)ޑ‬The first terms of the sequence are B 1 = 1, B 2 = 2 2 , B 3 = 1, B 4 = (2 4 )(3 2 )(7 2 ), and B 5 = (17) 2 (19) 2 .Hence, B 1 and B 3 do not have a primitive divisor.For the terms that have very large indexes, we can use Theorem 1.2.So, we apply Theorem 1.2 with C(E/K , M) as defined in (13).In the definition of C(E/K , M) we substitute J E with 0.3.Indeed, for every rational nontorsion point of E, we have ĥ(P) > 0.3 and J E is a constant such that J E < ĥ(P).The minimum of the canonical height of the rational nontorsion points of E is computed in [LMFDB], where the canonical height is defined as the double of our canonical height.By Theorem 1.2 we have that, for n ≥ 2 • 10 31 , B n has a primitive divisor.
To deal with the terms with indexes smaller than 2•10 31 , we can use the following techniques.By [Voutier and Yabuta 2012, Theorem 1.3] and [Verzobio 2021a], B n has a primitive divisor for n even.So, we focus on the terms with odd indexes.As an easy corollary of [Voutier and Yabuta 2012, Lemma 3.4], we have that if B n does not have a primitive divisor, then log B n ≤ 0.18n 2 .So, we can compute the values of B n and check if the inequality holds (this is much faster than computing the factorization of the terms).As far as we know, the faster way to compute B n is to use [Verzobio 2022, Theorem 1.9], where it is proved that, for k ≥ 9, B k for an appropriate choice of the sign (for more details, see [Verzobio 2022, Definition B]).One can check that log B n > 0.18n 2 for 4 ≤ n ≤ 10 5 using PARI/GP [2018] and then B n has a primitive divisor for 4 ≤ n ≤ 10 5 .So, our bound is too large to be computationally useful and then new methods are needed to bridge the gap.
It remains the case e − 1 < ν( p)/( p − 1).In this case, n = n P p e < n P p To explicitly compute ν(z(nP)) in the second case of the previous proposition one can use [Stange 2016, Lemma 5.1].