A Quotient Of Elliptic Curves - Weak Neron Models ´ For Lattes Maps

A QUOTIENT OF ELLIPTIC CURVES - WEAK NÉRON MODELS FOR LATTÈS MAPS 1. Introduction Let R be a Dedekind domain with quotient field F , and let A be an abelian variety defined over F. In studying the canonical height function on A, A. Néron [13] introduced the notion of a Néron model A over R. He showed that A has a Néron model over R and that for each non-Archimedean place v of F , the canonical local height over the local field Fv can be interpreted as an intersection multiplicity on the special fiber of A ⊗R Rv over the ring Rv of integers of Fv . Call and Silverman [8] extended Néron’s theory by introducing the notion of a weak Néron model V for a smooth variety V over non-Archimedean local field k = Fv together with a k-morphism φ : V → V. Definition 1. (Weak Néron Model) Let V be a smooth variety over k and let φ : V → V be a finite morphism over k. A smooth scheme V over the ring of integers Ok is called a weak Néron model for (V /k, φ) if the following conditions hold: (i) Vk (the generic fiber) ≃ V over k, (ii) V(Ok ) ≃ V (k), and (iii) there exists an Ok -morphism Φ : V → V such that Φ |Vk = φ. Note that the Néron model of an abelian variety A over k is a weak Néron model for (A/k, [m]), where [m] : A → A is the multiplication-by-m map. In [8], Call and Silverman showed if a weak Néron model V exists for (V /k, φ), then the canonical local height associated to φ [8, Sect. 2] can also be interpreted as an intersection multiplicity on the special fiber of V. Thus, weak Néron models play an important role in the study of the canonical heights associated to a given morphism φ, just as Néron models do in the arithmetic theory of abelian varieties. It is natural to ask whether a weak Néron model always exists for the pair (V /k, φ) if V is not an abelian variety. In fact, as shown in [10], if V is not an abelian variety over k, one cannot expect that a weak Néron model always exists An obstruction to the existence of a weak Néron model is closely related to the existence of an “unstable subset” of V (k nr ) (where k nr is the maximal unramified extension field of k) under the action of φ on V (k nr ) via iterations of φ. In the case that V = P1 over k, the unstable subset is called the (non-Archimedean) Julia set Jφ (k nr ) of φ, which is the non-Archimedean analogue of the classical Julia set in the theory of complex dynamical systems in one variable. For background on complex dynamics, we refer the reader to any of [1, 9, 12]; papers on non-Archimedean dynamics and Julia sets include [2, 3, 5, 6, 10, 11, 14, 15]. In the case V = P1 over k, if φ has good reduction, then the projective line P1Ok over Ok is a weak Néron model for (P1 /k, φ). If φ does not have good reduction, then either there is a non-empty Julia set Jφ (k nr ), so that no weak Néron model exists for (P1 /k, φ), 1 or else Jφ (k nr ) is empty. If Jφ (k nr ) is empty, it could happen that Jφ (L) is non-empty after some (ramified) extension L over k nr , and we could conclude that no weak Néron model exists for φ over L. Otherwise, Jφ (L) is empty for any extension L of k nr . In many such examples, φ turns out to have good reduction over some extension L over k nr (see Example 1 below). It is an interesting problem to characterize morphisms φ : P1 → P1 which do not have good reduction over any extension L of k nr and yet have weak Néron models over OL . The aim of this note is to exhibit a particular family of such morphisms; our examples are special cases of so-called Lattès maps on P1 . They might not have good reduction, but they have weak Néron models over any discretely valued field that is an algebraic extension of k. Throughout this note, we fix the notation k to denote a non-Archimedean field that is complete with respect to a discrete valuation v. The ring of integers of k is denoted by Ok , and the maximal ideal of Ok is denoted by Mk . Also, the residue field of k is denoted by e k := Ok /Mk . In our discussion of weak Néron models, it is convenient to work with a non-Archimedean field with algebraically closed residue field. We let k nr be the maximal nr unramified extension of k, so that the maximal ideal of Oknr is Mnkr = Mk Oknr , and kf is an algebraic closure of e k. 2. Weak Néron models for rational maps on P1 We consider the case that V = P1 , and that φ : P1 → P1 is a rational map defined over k. Fixing an affine coordinate z on P1 , we have φ(z) = f (z)/g(z), where f, g are polynomials in z. It is clear that f, g can be chosen to be relatively prime polynomials with coefficients in Ok . We say φ has good reduction (with respect to the coordinate z) if the resultant Res(f, g) of f and g is a unit in Ok . If φ has good reduction, then φ extends to an Ok -morphism on the projective line P1Ok over Ok . By the definition of weak Néron model, if φ has good reduction, then P1Ok is a weak Néron model for (P1 /k, φ). For general φ , we cannot always expect a weak Néron model to exist for (P1 /k, φ). A necessary condition for the existence of a weak Néron model is given Proposition 2.1 below (see [10, Theorem 3.1] for a general statement). Before we can state the result, recall that the set of periodic points Perφ of φ is the set of points P of P1 such that φn (P ) = P for some n ≥ 1, where φn = φ ◦ φ ◦ · · · ◦ φ denotes the n-th iterate of φ. For any field K/k, let Perφ (K) denote the set of K-rational periodic points of φ. Proposition 2.1. Suppose that there is a weak Néron model for (P1 /k, φ) over Ok . Assume that Perφ (k nr ) is non-empty. Then for any P ∈ Perφ (k nr ) with φn (P ) = P , we have |(φn )′ (P )| ≤ 1. The number (φn )′ (P ) is called the multiplier of periodic point P. We call a periodic point P such that φn (P ) = P repelling if |(φn )′ (P )| > 1 and non-repelling otherwise. Proposition 2.1 says that if a weak Néron model exists for the pair (V /k, φ), then all the k nr -rational periodic points Perφ (k nr ) are non-repelling. As there are rational maps φ with repelling periodic points, we see from the above proposition that for such rational maps weak Néron models do not exist. It is natural to ask the following question: if all periodic points P ∈ Perφ are non-repelling, is it true that weak Néron models exist for (P1 , φ) over any finite extension L of k? Example. 1. Suppose that p 6= 2 and that k is a finite extension of Qp . Let α ∈ k ∗ with |α| < 1, and consider the following rational maps : αz 2 − z , α2 z 2 + 1 φ2 (z) = αz d + c, d ≥ 2, c ∈ Ok zp + c , c ∈ Ok∗ ψ(z) = α Note that all the maps above have bad reduction (with respect to the coordinate z). φ1 (z) = (i) For φ1 : let z = f (w) = w/α. Then w2 − w , w2 + 1 which has good reduction (with respect to the coordinate w). φf1 (w) := (f −1 ◦ φ1 ◦ f )(w) = (ii) For φ2 : choose a (d − 1)-th root δ ∈ k̄ of α and make the change of coordinate z = f (w) = w/δ. Then φf2 (w) = w d + c δ, which has good reduction with respect to the coordinate w. If α is not a (d − 1)-st power in k, then the above coordinate change is defined over the field k(δ), which is a finite ramified extension of k. However, there is no coordinate change z = f (x) over k nr such that φf2 has good reduction. Nevertheless, applying a sequence of blow-ups at closed points on the special fiber [10], it can be shown that there is a weak Néron model for (P1 /k, φ). (iii) For ψ : let s > 0 be such that |α| = |p|s , and let L = k(δ, Fix(ψ)), where Fix(ψ) = {a ∈ k̄ | ψ(a) = a}, and δ is a (p − 1)-th root of α. Case I. 0 < s ≤ 1 : Consider the coordinate change z = f (w) = λ + δw, where λ ∈ Fix(ψ) is any of the fixed points. (Note that v(λ) ≥ 0.) By a direct computation, one can show that ψ f (w) has good reduction with respect to the coordinate w. Case II. s > 1 : For λ ∈ Fix(ψ), we have |ψ ′ (λ)| = |p|1−s > 1. By Proposition 2.1, we see that (P1 , ψ) does not have a weak Néron model over any field containing k(Fix(ψ)). Remark. 1. For the map ψ in the above example, if Fix(ψ) does not contain any k nr -rational points, then it can still be shown that there is a weak Néron model X over Ok for (P1 /k, ψ). The above examples have the property that after some finite (ramified) extension of k, the maps in question either have good reduction with respect to some coordinate or have repelling periodic points defined over the extended field; in the latter case, we can conclude that there does not exist a weak Néron model. As mentioned in Section 1, we will seek rational maps φ over k that do not have good reduction (with respect to any coordinate on P1 ) over any algebraic extension L of k but admit weak Néron models over every OL . By Proposition 2.1, all periodic points of these rational maps are necessarily non-repelling. 3. Lattès Maps We start by recalling the definition of Lattès maps. Definition 2. A morphism φ : P1 → P1 over k is called a Lattès map if there exist an elliptic curve E/k, a morphism ψ : E → E, and a finite separable morphism α : E → P1 over k so that the following diagram is commutative: ψ E −−−→  α y φ E  α y P1 −−−→ P1 For a discussion on properties of Lattès maps, we refer the reader to [18, Chap. 6]. For simplicity, we’ll only consider the case where α : E → P1 is given by the quotient E → E/{±1} ≃ P1 . Moreover, we assume that the residue characteristic of e k is odd. As an explicit example of a Lattès map, suppose that E is given by a Weierstrass equation y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 , and the quotient map to P1k is simply α = x : (x, y) 7→ x. If ψ = [2], the multiplicationby-2 map, then the map φ is given by the formula φ(x) = x4 − b4 x2 − 2b6 x − b8 , 4x3 + b2 x2 + 2b4 x + b6 where the {bi } are the usual functions of the {ai }. (See [16, III.2.3].) The following proposition [18, Prop. 6.52] gives a description of periodic points of φ and computes the multiplier of periodic points of Lattès map φ. For the proof (and the observation that part (b) really does cover all possible cases), we refer the reader to [18]. Proposition 3.1. Let φ : P1 → P1 be a Lattès map and assume that ψ(P ) = [m]P . (a) The set of n-periodic points of φ (i.e., the set of periodic points P with φn (P ) = P ) is Pern (φ) = α (E[mn − 1]) ∪ α (E[mn + 1]) , where E[ℓ] denotes the group ℓ-torsion points of E. (b) Let ζ be a periodic point of φ of exact period n (i.e. φn (P ) = P but φd (P ) 6= P for 0 < d < n). Then  n  if ζ ∈ α(E[mn − 1]) and ζ 6∈ α(E[2]), m (φn )′ (ζ) = −mn if ζ ∈ α(E[mn + 1]) and ζ 6∈ α(E[2]),  m2n if ζ ∈ α(E[mn + 1]) ∩ α(E[2]). As the valuation on k is non-Archimedean, we see that all the periodic points of φ are non-repelling. In fact, in [4] we can show the following result. Theorem 3.2. Let φ : P1 → P1 be a Lattès map over k. Then, for any discretely valued field extension L of k, there is a weak Néron model over OL for (P1 /L, φ). In the next section, we will sketch the construction of weak Néron models for Lattès maps. 4. A construction of weak Néron models for Lattès maps Fix a Lattès map φ : P1 → P1 as defined in Section 3, and let the corresponding elliptic curve E be defined by the Weierstrass equation y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 , with ai ∈ Ok . We’ll assume that the Weierstrass model of E is minimal. By this, we mean that the valuation v(∆) of the discriminant ∆ of E is as small as possible. (Of course, v(∆) ≥ 0, since ai ∈ Ok .) Let ψ : E → E be the endomorphism on E so that the diagram in Definition 2 is commutative (with α(x, y) = x), and write Γ = {±1} ⊂ Aut(E). Let E be the Néron model of E over Ok . (See [7] for an extensive treatment of Néron models, or [17], Chapter IV for a more accessible exposition for Néron models of elliptic curves.) As Néron models are invariant under faithfully flat base change (see [7, Section 7.2]; see also [17, IV.4.5] for the case of elliptic curves and finite unramified extensions), we’ll assume that k = k nr , and thus that the residue field e k is algebraically closed. This assumption will not affect our final results. It is tempting to take the quotient of E directly, just as P1 is a quotient of E. By the Néron mapping property, the map ψ extends to a morphism Ψ : E → E over Ok , and the group Γ extends to a group, which we still denote by Γ, of automorphisms of E over Ok . Let Y := E/Γ. Note that Y is a smooth model over Ok for P1k . It’s not hard to see that φ extends to an Ok -morphism Φ : Y → Y. However, there may be a point P ∈ P1 (k) that does not extend to an Ok -valued point on Y. This is possible because α−1 (P ) = {Q, −Q} ⊂ E(ℓ) for some ramified quadratic extension ℓ of k, and if Q ∈ E(ℓ) \ E(k), then Q does not extend to Ok -valued point on E. Hence, in general one cannot expect that Y is a weak Néron model for (P1 /k, φ). Instead of considering Y over Ok , we consider the quadratic ramified extension of √ k. Let ℓ = k( πk ), where πk is a uniformizer of k. Recall that by assumption, the characteristic of e k is different from 2; hence ℓ is the only ramified quadratic extension over k. Let Eℓ := E ⊗k ℓ, and let E ′ denote the Néron model over Oℓ for Eℓ . Again by the Néron mapping property, ψ and Γ extend to Ψ′ and Γ′ on E ′ over Oℓ . Let Y ′ := E ′ /Γ′ . Then Y ′ is a smooth model over Oℓ for P1ℓ . Moreover, every k-rational point on P1 extends to an Oℓ -valued point on Y ′ . We claim that a smooth Ok -model X for P1k can then be constructed from Y ′ so that every k-rational point extends to an Ok -valued point on X, and φ extends to an Ok -morphism on X. To prove the claim, we introduce the following notation. Let Ye′ denote the special fiber of Y ′ , and write [ Ye′ = Zi , i where the Zi ’s are the irreducible components of Ye′ . For each P ∈ P1 (k), let P ∈ Y ′ (Oℓ ) e for the corresponding point on the special fiber. be the closure of P in Y ′ , and write P e }. The construction of the desired weak Néron model now comes That is, P = {P, P from following observations about Ye′ , proven in [4]. Proposition 4.1. For every irreducible component Zi of Ye′ , (1) Zi is isomorphic to a Zariski open subset of P1e . k e e e with (2) Zi (k) contains either infinitely many P with P ∈ P1 (k) or at most one P 1 P ∈ P (k). e with P ∈ P1 (k), then Zi is the special fiber of (3) If Zi (e k) contains infinitely many P Xi ⊗Ok Oℓ , for some Ok -scheme Xi . e with P ∈ P1 (k), let Ei be the blowing-up of P e . Then (4) If Zi (e k) contains only one P 1 e with P ∈ P (k). Moreover, Ei is the special fiber of Ej (k) contains infinitely many P some Xi ⊗Ok Oℓ , for some Ok -scheme Xi . e for P ∈ Now blow up all components Zi of Ye′ with exactly one point of the form P e . Using Proposition 4.1, P1 (k), and then remove all components Zi with no such points P it is not difficult to complete the proof that the resulting scheme X is a weak Néron model for φ over Ok . We close with an example of the above construction of a weak Néron model in the case that E is a Tate curve. Example. 2. Assume that the elliptic curve E has multiplicative reduction over k, and that E is defined by a minimal Weierstrass equation. Let E 0 denote that identity component of the Néron model E of E. The group of automorphisms Γ = {±1} acts on f0 (e ee E(Ok )/E 0 (Ok ) ≃ E( k)/E k). Hence, the action of Γ induces an action on the irreducible e components of E, which form a cyclic group of order n = v(∆). i Identify the irreducible components of Ee with Z/nZ, and denote by Ee the component of Ee corresponding to i ∈ Z/nZ (fig. 1). Then the action of Γ is just multiplication by 0 ±1 on the cyclic group Z/nZ. Note that Ee is simply the special fiber of the identity e 0 and Y e n/2 ; component. If n is even, then there are two components fixed by Γ, namely Y if n is odd, then only the identity component is fixed by Γ. In either case, the special e of Y can be read off easily from Ee by identifying Ee i with Een−i (see fig. 2). Denote fiber Y e ∗ the component by Y ( n/2 e ∗ if n is even, e = E /Γ Y (n−1)/2 (n+1)/2 {Ee , Ee }/Γ if n is odd, e 0. which is the component that is, in a sense, furthest from Y e It follows Note that Γ identifies smooth closed points with smooth closed points on E. e e that the double points of both E and Y are not actually part of the models, as E and Y are smooth over Ok . Moreover, if n is even, then all k-rational points reduce to smooth closed points, so that the missing double points don’t cause any violations to property (ii) in the definition of a weak Néron model. However, if n is odd, then there is a missing e ∗ , corresponding to the double point where Ee(n−1)/2 and Ee (n+1)/2 intersected. point ω e on Y Unfortunately, there are points P ∈ P1 (k) that reduce to the missing point ω e . This is the phenomenon noted earlier in this section; such points P correspond to points (x, y + ) and (x, y − ) on E with x ∈ k but y ± 6∈ k. e fig. 2 Y fig. 1 Ee n−1 Ee .. .. .. . 0 Ee j Ee .. 1 .. e .. E .. . - ... ... ... ... .. ω e ∗ e Y Instead, our algorithm stipulates that we consider the Néron model Ee′ over the ramified √ quadratic extension ℓ = k( πk ), with component group isomorphic to Z/2nZ. As before, 0 ∗ n we take the quotient Ye′ , which consists of n + 1 components: Ye′ and Ye′ = Ye′ are i i each quotients of a single component Ee′ fixed by Γ, and every other component Ye′ is i 2n−i the glueing of the pair Ee′ and Ee′ , for 1 ≤ i ≤ n − 1. We now need to turn Ye′ into a model defined over k. i It can be shown that none of the components Ye′ for odd 1 ≤ i ≤ n − 1 contain any e with P ∈ P1 (k). If n is even, all the remaining components have infinitely points P e with P ∈ P1 (k); by statement (3) of Proposition 4.1, we have our many closed points P desired weak Néron model X, up to base change. If n is odd, only the last component, n e with P ∈ P1 (k). Following Ye′ , is the exception to this rule: it contains a single point P e , which has the effect of replacing P e by a new our algorithm, we blow up (over ℓ) at P n e that is a copy of P1 . Removing the old Ye′ (which has now had the point component Z e removed), and descending back to k, we are left with a weak Néron model X. P In this special case of multiplicative reduction, we can actually construct the model X from the original quotient Y, as follows. If n is even, then the scheme Y ′ with the oddnumbered components removed is simply Y ⊗Ok Oℓ . If n is odd, then add the missing point ω e into Y and blow up (over k) at ω e to obtain a scheme Y ⊛. 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