General SectionPrimitive divisors of elliptic divisibility sequences over function fields with constant j-invariant
Introduction
An elliptic divisibility sequence (EDS) over Q is a sequence , , of positive integers defined as follows. Given an elliptic curve E over Q and a point of infinite order, choose a globally minimal Weierstrass equation for E and write for every : where the fractions are in lowest terms. Then set .
A result of Silverman [26] shows that all but finitely many terms have a primitive divisor, that is, a prime divisor such that for all . Equivalently, this says that all but finitely many positive integers n occur as the order of for some prime p. The question whether there is a uniform bound N such that has a primitive divisor for all pairs and all remains open, see [2], [4], [8], [14].
The definition of of (1.1) is equivalent to for all non-archimedean valuations v and 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 to be the sequence of ideals of defined by (1.2).
Similarly, if E and P are defined over the function field of a smooth, projective, geometrically irreducible curve C over a field K, then we define the EDS of the pair to be the sequence of divisors 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 if for every the term 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 by Naskręcki [21]. Conditionally Naskręcki makes the result uniform in E. The special case of the results of [21] where is constant gives a Zsigmondy bound N as follows.
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For fields of characteristic 0 we have (see [12, p. 437] and [21, Lemma 7.1]).
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For fields with and field of constants , we have for ‘tame’ elliptic curves (cf. [21, Definition 8.3]) and a bound for ‘wild’ ordinary elliptic curves where χ is the Euler characteristic of the elliptic surface attached to E over .
All previous Zsigmondy bound estimates exclude the case of supersingular curves. In this paper, we consider the case of function fields and assume , which includes the case of supersingular E. In a companion paper we will deal with the case , where we extend the results of Naskręcki [21] to arbitrary characteristic and improve the bound N.
In the ordinary case, we prove a bound and show that it is optimal. In the supersingular case in characteristic p, we show that the terms for have a primitive divisor if and only if , 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 be a point of infinite order such that , but the pair is not constant, cf. Definition 2.1. Then for all integers , the term has a primitive valuation. Conversely, for all ordinary j-invariants there exist an elliptic curve with and a point of infinite order such that the terms and do not have a primitive valuation and there exist an elliptic curve with and a point of infinite order such that all terms for 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 . 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 with non-constant and P of infinite order, the term has a (respectively no) primitive valuation. If the entry is ‘⁎’, then there exist E and P as in the previous paragraph such that has a primitive valuation and there exist E and P such that 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 . Let E be an elliptic curve over K and a point of infinite order. Let n be a positive integer, if or E is ordinary, then has a primitive valuation, if and E is supersingular, then 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 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 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 with generic fibre E and a section , cf. [24, §1], [27, III, §3]. For example, if the curve E is constant (that is, defined over K), then we can take with the natural projection .
Let P be a point of infinite order in the Mordell–Weil group . We define a family of effective divisors parametrised by natural numbers n. For each the divisor is the pull-back of the image of the section O through the morphism induced by the point nP, that is,
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 and related sequences were discussed in several places. We collect some known results here.
First of all, they satisfy the strong divisibility property for all positive integers , where . Indeed, the proof in e.g. [30, Lemma 3.3] carries over.
Theorem 1.5 of [16] shows that in case with K a number field (and again ) the set of prime numbers n such that is irreducible has positive lower Dirichlet density.
Cornelissen and Reynolds [6] study perfect power terms in the case for global function fields F of characteristic . Everest, Ingram, Mahé, and Stevens study primality of terms of elliptic divisibility sequences for 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 , 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 , 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 . In that case can be viewed as a dominant morphism over K. The primitive valuations of 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 over K with the same j-invariant and an isomorphism over . Then Theorem 2.3 applies to the sequence obtained from . See Section 3.
At that point, we know exactly which terms of have primitive valuations, and the goal is to conclude which terms of have primitive valuations.
For this, we look at the rank of apparition of a valuation v of F in the sequence , which is the positive integer or ∞ if the set is empty. A valuation v is primitive in the term if and only if .
The key to our proof is to see how much the rank of apparition of a valuation v of can vary between the sequences and . 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 for sequences in characteristic always has a primitive valuation if . 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 be its function field. Let be an elliptic curve and a point. For a field extension and an elliptic curve over L, let be the base change of to M. Definition 2.1 We say that E is constant if there exists an elliptic curve and an isomorphism defined over F. We say that the pair is constant if there exist such and ϕ that also satisfy . We say that the j-invariant
Definitions and example
Let E be an elliptic curve over and let be a point of infinite order. Now suppose . 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 obtained from P with constant j-invariant to an EDS obtained from a point on a constant elliptic curve and then to apply Theorem 2.3 to .
Lemma 3.1 Let K be a field, let 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 (Lemma 4.1(2–3)), while those of good reduction appear in the same place as in the corresponding constant sequence (Lemma 4.2).
Recall that the rank of apparition of a valuation v of F is the smallest positive integer n such that (with 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 of a point in the component group.
By a local function field, we mean a completion of the function field of a smooth, projective, geometrically irreducible curve C over a field K at a discrete valuation v with and . 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
In this section we give a separate result, with an elementary proof, for the terms and in the case , 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 . If , then E is ordinary. If , then E is supersingular. If , 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 for a field K, that is, the examples have . 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 be a point of infinite order such that , but the pair is not constant, cf. Definition 2.1. Then for all integers , the term has a primitive valuation. Conversely, for all ordinary j-invariants there exist an elliptic curve with
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