Elsevier

Journal of Number Theory

Volume 213, August 2020, Pages 152-186
Journal of Number Theory

General Section
Primitive divisors of elliptic divisibility sequences over function fields with constant j-invariant

https://doi.org/10.1016/j.jnt.2019.12.002Get rights and content

Abstract

We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the j-invariant of the elliptic curve is constant.

In more detail, given an elliptic curve E with a point P of infinite order over a global field, the sequence D1, D2, of denominators of multiples P, 2P, of P is a strong divisibility sequence in the sense that gcd(Dm,Dn)=Dgcd(m,n). This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences.

A number N is called a Zsigmondy bound of the sequence if each term Dn with n>N presents a new prime factor. The optimal uniform Zsigmondy bound for the genus-zero sequences over Q is 30 by Bilu-Hanrot-Voutier [2], but finding such a bound remains an open problem in genus one, both over Q and over function fields.

We prove that the optimal Zsigmondy bound for ordinary elliptic divisibility sequences over function fields is 2 if the j-invariant is constant. In the supersingular case, we give a complete classification of which terms can and cannot have a new prime factor.

Introduction

An elliptic divisibility sequence (EDS) over Q is a sequence D1, D2, D3, of positive integers defined as follows. Given an elliptic curve E over Q and a point PE(Q) of infinite order, choose a globally minimal Weierstrass equation for E and write for every QE(Q):Q=(AQDQ2,CQDQ3), where the fractions are in lowest terms. Then set Dn=DnP.

A result of Silverman [26] shows that all but finitely many terms Dn have a primitive divisor, that is, a prime divisor p|Dn such that pDm for all 1m<n. Equivalently, this says that all but finitely many positive integers n occur as the order of (Pmodp) for some prime p. The question whether there is a uniform bound N such that Dn has a primitive divisor for all pairs (E,P) and all n>N remains open, see [2], [4], [8], [14].

The definition of DQ of (1.1) is equivalent tov(DQ)=max{12v(xv(Q)),0} for all non-archimedean valuations v and xv the x-coordinate function for a v-minimal Weierstrass equation. If E and P are defined over a number field F, then we define the EDS of the pair (E,P) to be the sequence of ideals Dn=DnP of OF defined by (1.2).

Similarly, if E and P are defined over the function field F=K(C) of a smooth, projective, geometrically irreducible curve C over a field K, then we define the EDS of the pair (E,P) to be the sequence of divisors Dn=DnP on C defined by (1.2). See Section 1.2 for an equivalent definition in the case of perfect K. Elliptic divisibility sequences over function fields are studied in [6], [9], [16], [28].

From now on, we will speak of primitive valuations instead of primitive divisors, so as not to confuse with the terms themselves, which are divisors in the function field case. A positive integer N is a Zsigmondy bound of the sequence (Dn)n if for every n>N the term Dn has a primitive valuation.

Silverman's result and proof are also valid in the number field case [15]. In the case of function fields of characteristic zero, the same result is true, as shown by Ingram, Mahé, Silverman, Stange and Streng [16, Theorems 1.7 and 5.5].

This was extended to ordinary elliptic curves E over function fields of characteristic 2,3 by Naskręcki [21]. Conditionally Naskręcki makes the result uniform in E. The special case of the results of [21] where j(E) is constant gives a Zsigmondy bound N as follows.

  • For fields K(C) of characteristic 0 we have N72 (see [12, p. 437] and [21, Lemma 7.1]).

  • For fields K(C) with p=charK(C)5 and field of constants K=Fq, q=ps we have N<10100(15+20g(C))p84 for ‘tame’ elliptic curves (cf. [21, Definition 8.3]) and a bound N=N(g(C),p,χ,s) for ‘wild’ ordinary elliptic curves where χ is the Euler characteristic of the elliptic surface attached to E over K(C).

All previous Zsigmondy bound estimates exclude the case of supersingular curves. In this paper, we consider the case of function fields F=K(C) and assume j(E)K, which includes the case of supersingular E. In a companion paper we will deal with the case j(E)FK, where we extend the results of Naskręcki [21] to arbitrary characteristic and improve the bound N.

In the ordinary case, we prove a bound N=2 and show that it is optimal. In the supersingular case in characteristic p, we show that the terms Dn for n>8p have a primitive divisor if and only if pn, and we give a sharp version for every characteristic.

In more detail, the main results are as follows.

Theorem A Theorem 8.1

Let F be the function field of a smooth, projective, geometrically irreducible curve over a field K.

Let E be an ordinary elliptic curve over F and let PE(F) be a point of infinite order such that j(E)K, but the pair (E,P) is not constant, cf. Definition 2.1. Then for all integers n>2, the term Dn has a primitive valuation.

Conversely, for all ordinary j-invariants jK there exist an elliptic curve E/F with j(E)=j and a point PE(F) of infinite order such that the terms D1 and D2 do not have a primitive valuation and there exist an elliptic curve E/F with j(E)=j and a point PE(F) of infinite order such that all terms Dn for n1 have a primitive valuation.

Theorem B Theorem 8.2

Let F be the function field of a smooth, projective, geometrically irreducible curve over a field of characteristic p>0. Let n be a positive integer.

If the entry corresponding to n and p in Table 1 is ‘yes’ (respectively ‘no’), then for every supersingular elliptic curve E over F, and every PE(F) with (E,P) non-constant and P of infinite order, the term Dn has a (respectively no) primitive valuation.

If the entry is ‘’, then there exist E and P as in the previous paragraph such that Dn has a primitive valuation and there exist E and P such that Dn has no primitive valuation.

In the case where E itself is defined over K (and not just its j-invariant), the result is much stronger, as follows.

Theorem C Theorem 2.3

Let F be the function field of a smooth, projective, geometrically irreducible curve over a field K of characteristic p0. Let E be an elliptic curve over K and PE(F)E(K) a point of infinite order. Let n be a positive integer,

  • 1.

    if pn or E is ordinary, then Dn has a primitive valuation,

  • 2.

    if p|n and E is supersingular, then Dn=p2Dn/p has no primitive valuation.

We now give a more standard, but more technical, definition of elliptic divisibility sequences over function fields in the case of perfect base fields K. It is proven in [16, Lemma 5.2] that this defines the same sequence (DnP)n in the case of number fields K; and the proof at [16] extends to perfect fields K.

Let E be an elliptic curve over the function field K(C) of a smooth, projective, geometrically irreducible curve C over a perfect field K. Let S be the Kodaira–Néron model of E, i.e., a smooth, projective surface with a relatively minimal elliptic fibration π:SC with generic fibre E and a section O:CS, cf. [24, §1], [27, III, §3]. For example, if the curve E is constant (that is, defined over K), then we can take S=E×C with the natural projection π:E×CC.

Let P be a point of infinite order in the Mordell–Weil group E(K(C)). We define a family of effective divisors DnPDiv(C) parametrised by natural numbers n. For each nN the divisor DnP is the pull-back of the image O of the section O through the morphism σnP:CS induced by the point nP, that is,DnP=σnP(O).

The delicate issues with non-perfect coefficient fields K are discussed in detail in Section 7 and Example 7.6.

Elliptic divisibility sequences over function fields F=K(C) and related sequences were discussed in several places. We collect some known results here.

First of all, they satisfy the strong divisibility propertygcd(Bm,Bn)=Bgcd(m,n) for all positive integers m,n, where gcd(Bm,Bn):=vmin{v(Bm),v(Bn)}[v]. Indeed, the proof in e.g. [30, Lemma 3.3] carries over.

Theorem 1.5 of [16] shows that in case E/K with K a number field (and again PE(F)E(K)) the set of prime numbers n such that DnPD1P is irreducible has positive lower Dirichlet density.

Cornelissen and Reynolds [6] study perfect power terms in the case j(E)FFp for global function fields F of characteristic p5. Everest, Ingram, Mahé, and Stevens study primality of terms of elliptic divisibility sequences for K(C)=Q(t) in the context of magnified sequences, see [9, Theorem 1.5]. Silverman [28] and Ghioca-Hsia-Tucker [11] study the common subdivisor for two simultaneous divisibility sequences on elliptic curves over K(t), where K is a field of characteristic 0.

In a broad context, Flatters and Ward [10] prove an analogue of Theorem 8.1 for divisibility sequences of Lucas type for polynomials and Akbar-Yazdani [1] study the greatest degree of the prime factors of certain Lucas polynomial divisibility sequences.

Hone and Swart [13] study examples of Somos 4 sequences over K(t), which are constructed from specific elliptic divisibility sequences. They construct a certain elliptic surface and show that the corresponding sequence is a sequence of polynomials.

The main idea behind the proof is to reduce to the case where E is defined over the base field K of F=K(C). In that case PE(F) can be viewed as a dominant morphism CE over K. The primitive valuations of Dn then are exactly the pull-backs of points of order n on E, which gives Theorem C. For details, see Section 2.

For elliptic curves over F where only the j-invariant is in K, we find an elliptic curve E˜ over K with the same j-invariant and an isomorphism ϕ:EE˜ over F. Then Theorem 2.3 applies to the sequence (DnP)n obtained from (E˜,ϕ(P)). See Section 3.

At that point, we know exactly which terms of (DnP)n have primitive valuations, and the goal is to conclude which terms of (DnP)n have primitive valuations.

For this, we look at the rank of apparition m(v) of a valuation v of F in the sequence (DnP)n, which is the positive integerm(v)=m(P,v):=min{n1:ordv(DnP)1}, or ∞ if the set is empty. A valuation v is primitive in the term DnP if and only if n=m(v).

The key to our proof is to see how much the rank of apparition m(v) of a valuation v of F can vary between the sequences (DnP)n and (DnP)n. Section 4 shows that this does not vary much, and bounds the variation in terms of the component group of the special fibre of the Néron model.

This is already enough to get a weaker version of the main results, which is not sharp, but is already uniform (Theorem 4.7, Theorem 4.9).

In Section 5 we prove two auxiliary results about the order of a point P in the component group at v. This is needed in the proof of the main theorems to obtain a sharp result.

In Section 6 we show that the term D3P for sequences in characteristic 2,3 always has a primitive valuation if j(E)=0. This is also needed in order to obtain a sharp result.

Section 7 contains examples which we use to show that our main theorems are optimal, that is, to prove the converse statement in Theorem A and the ⁎-entries in Theorem B.

Finally, in Section 8 we combine all of the above into a proof of Theorems A and B.

  • Acknowledgements

  • The authors would like to thank Peter Bruin and Hendrik Lenstra for helpful discussions and the anonymous referee for comments that improved the exposition.

Section snippets

Constant curves

Let C be a smooth, projective, geometrically irreducible curve over a field K and let F=K(C) be its function field. Let E/F be an elliptic curve and PE(F) a point. For a field extension ML and an elliptic curve E over L, let EM be the base change of E to M.

Definition 2.1

We say that E is constant if there exists an elliptic curve E˜/K and an isomorphism ϕ:EE˜F defined over F.

We say that the pair (E,P) is constant if there exist such E˜ and ϕ that also satisfy ϕ(P)E˜(K).

We say that the j-invariant j(E)

Definitions and example

Let E be an elliptic curve over F=K(C) and let PE(F) be a point of infinite order. Now suppose j(E)K. Note that this includes the case where E is supersingular by [29, V.3.1(a)(iii)].

The idea behind the proof of our main results is to relate the EDS (DnP)n obtained from P with constant j-invariant to an EDS (DnP)n obtained from a point on a constant elliptic curve and then to apply Theorem 2.3 to (DnP)n.

Lemma 3.1

Let K be a field, let C/K be a smooth, projective, geometrically irreducible curve and

Reduction modulo primes of curves with constant j

Elliptic curves with constant j-invariant admit only places of good or additive reduction. We show that the valuations v of additive reduction appear early on in the sequence DnP (Lemma 4.1(2–3)), while those of good reduction appear in the same place as in the corresponding constant sequence DnP (Lemma 4.2).

Recall that the rank of apparition m(v)=m(P,v) of a valuation v of F is the smallest positive integer n such that v(DnP)>0 (with m(v)= if it does not exist).

With the notation as in

Component groups

In order to sharpen Theorem 4.7, Theorem 4.9 further, we need to look at the component group. In this section we derive extra restrictions on the order dv of a point in the component group.

By a local function field, we mean a completion K(C)v of the function field K(C) of a smooth, projective, geometrically irreducible curve C over a field K at a discrete valuation v with v(K)={0} and v(F){0}.

Proposition 5.1

Let F be a local function field of characteristic 2 with valuation v and constant field K. Let E be

The third term when j=0

In this section we give a separate result, with an elementary proof, for the terms D3 and D3p in the case j=0, because the local considerations of Section 5 do not apply to that case.

We first collect some well-known results about elliptic curves with j-invariant 0 in the following lemma, of which we give a proof for completeness.

Lemma 6.1

Let E be an elliptic curve with j-invariant 0 over a field L of characteristic p>0.

  • 1.

    If p1mod3, then E is ordinary.

  • 2.

    If p1mod3, then E is supersingular.

  • 3.

    If p>3, then E

Additional examples

In this section we gather examples that are crucial for the proof of optimality in the main theorems. In our examples, the function field F is always F=K(t) for a field K, that is, the examples have C=P1. The following result shows that this suffices, in the sense that the existence of such examples implies the existence of examples over all function fields that we consider.

Theorem 7.1

Let K be a field and let F be the function field of a smooth, projective, geometrically irreducible curve over K. Let E be

Proof of the main theorems

We now have all the ingredients required for proving the following two main theorems.

Theorem 8.1

Let F be the function field of a smooth, projective, geometrically irreducible curve over a field K.

Let E be an ordinary elliptic curve over F and let PE(F) be a point of infinite order such that j(E)K, but the pair (E,P) is not constant, cf. Definition 2.1. Then for all integers n>2, the term Dn has a primitive valuation.

Conversely, for all ordinary j-invariants jK there exist an elliptic curve E/F with j(E

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