Weak Neron Models for Lattes Maps

Weak Néron Models for Lattès Maps 2013 Notation Throughout this talk, we set the following notation: I K is a field of characteristic zero, I v is a discrete valuation on K , I | · |v is an absolute value associated with v , I π = πv ∈ K is a uniformizer for K , I Ov = {x ∈ K : |x|v ≤ 1} ⊆ K is the ring of integers, I Mv = πv Ov = {x ∈ K : |x|v < 1} ⊆ K is the maximal ideal, I k = Ov /Mv is the residue field, I p = char k ≥ 0 is the residue characteristic. For example, K = Qp , Ov = Zp , k = Fp , π = p, ur or K = Qur p , Ov = Zp , k = Fp , π = p. Néron Models of Elliptic Curves: Quick Summary Definition Let E /K be an elliptic curve, and let E be a smooth, separated Spec Ov -scheme. Suppose that I The generic fiber EK of E is isomorphic to E over K . I (“Néron Mapping Property”) For any smooth Spec Ov -scheme X and rational map ψK : XK → EK , there is a unique Spec Ov -morphism ψ : X → E extending ψK . Then we say E is a Néron model for E over K . In particular, the Néron model E has the following properties: I I I E is of finite type over Spec Ov , E(Ov ) ∼ = E (K ), i.e., every point x ∈ E (K ) extends to a section sx : Spec Ov → E, For any endomorphism ψ : E → E there is a Spec Ov -morphism Ψ : E → E such that Ψ|E = ψ. Dynamics on P1 , and Good Reduction Let φ ∈ K (z) be a rational function of degree d ≥ 2. deg φ := max{deg f1 , deg f2 }, where φ = f1 /f2 in lowest terms. Setting φn := φ ◦ · · · ◦ φ, consider the dynamical system {φn : n ≥ 0} acting on P1 . Definition Given φ = f1 /f2 ∈ K (z) as above, we may assume without loss that f1 , f2 ∈ Ov [z] have no common factors, and that at least one coefficient a of f1 or f2 has |a|v = 1. Let f 1 , f 2 ∈ k[z] be the reductions of f1 and f2 mod Mv . If deg(f 1 /f 2 ) = deg(φ), then we say φ has good reduction. Motivating Weak Néron Models In scheme-theoretic language: Consider the Spec Ov -scheme P1Ov , which has generic fiber P1K . Then φ ∈ K (z) has good reduction if and only if φ : P1 (K ) → P1 (K ) extends from a morphism of the generic fiber to a morphism Φ : P1Ov → P1Ov of Spec Ov -schemes. Recall: A Néron model E for E is a finite type, smooth, separated Spec Ov -scheme with the properties that (among other things): I I I The generic fiber EK of E is isomorphic to E over K . E(Ov ) ∼ = E (K ), i.e., every point x ∈ E (K ) extends to a section sx : Spec Ov → E, Every endomorphism ψ : E → E extends to a Spec Ov -morphism Ψ : E → E Weak Néron Models Definition Let V /K be a smooth variety, and let φ : V → V be a finite morphism defined over K . Meanwhile, let V be a finite type, smooth, separated Spec Ov -scheme, and let Φ : V → V be a morphism of Spec Ov -schemes. Suppose that I The generic fiber VK of V is isomorphic to V over K . V(Ov ) ∼ = V (K ), i.e., every point x ∈ V (K ) extends to a section sx : Spec Ov → V, I Φ|V = φ. I Then we say (V, Φ) is a weak Néron model for (V , φ) over K . Example. If φ ∈ K (z) has good reduction, define Φ : P1Ov → P1Ov ( φ(x) for x ∈ P1 (K ), by Φ(x) = φ(x) for x ∈ P1 (k). Then (P1Ov , Φ) is a weak Néron model for (P1K , φ) over K . Lattès Maps Definition A morphism φ : P1 → P1 over K is a Lattès map if there exist an elliptic curve E /K , a morphism ψ : E → E over K , and a finite separable morphism α : E → P1 over K so that ψ E −−−−→  α y E  α y φ P1 −−−−→ P1 commutes. Fact. Given a Lattès map φ over a field of characteristic zero, we can always assume that I ψ ∈ End(E ), i.e., ψ(O) = O; and I there is a nontrivial finite subgroup Γ ⊆ Aut(E ) such that α is the projection map α : E → E /Γ ∼ = P1 . Examples of Lattès Maps Example. E : y 2 + a1 xy + a3 y = x 3 + a2 x 2 + a4 x + a6 has quotient map α : E → P1 by α(x, y ) = x. If ψ = [2] ∈ End(E ), then φ(x) = x 4 − b4 x 2 − 2b6 x − b8 4x 3 + b2 x 2 + 2b4 x + b6 is Lattès map for ψ, making the diagram [2] E −−−−→  α=x y E  α=x y φ P1 −−−−→ P1 commute. Example. If E has good reduction, then φ also has good reduction. Multiplicative Reduction, Type I1 Example. E : y 2 + xy = x 3 + π has type I1 reduction, i.e., the special fiber of the Néron model E is a nodal cubic with the singularity (x, y ) = (0, 0) removed. Taking the quotient by ±1, the Lattès map for multiplication-by-2 is x 4 − 8πx − π φ(x) = 3 . 4x + x 2 + 4π But E/{±1} is not a weak Néron model, because φ is bad at 0. E had no K -rational points with |x|v ≤ |π|v , but P1 does! Fixing the I1 model For φ(x) = x 4 − 8πx − π , make a model with two components: 4x 3 + x 2 + 4π x = x0 , x0 = πx1 . with X0 → X0 by x00 = φ(x0 ) = x04 − 8πx0 − π 4x03 + x02 + 4π ≡ x 20 , 4x 0 + 1 and X1 → X0 by x00 = φ(πx1 ) = π 3 x14 − 8πx1 − 1 4π 2 x13 + πx12 + 4 NOTE: This is no longer a finite morphism. ≡ −1 . 4 More General Multiplicative Reduction Example. E : y 2 + xy = x 3 + π n has type In reduction: E0 : y02 + x0 y0 = x03 + π n x0 = πx1 , y0 = πy1 E±1 : y12 + x1 y1 = πx13 + π n−2 x1 = πx2 , y1 = πy2 E±2 : y22 + x2 y2 = π 2 x23 + π n−4 .. . ( 2 + x y = πm x 3 + π E±m : ym m m m 2 + x y = πm x 3 + 1 Em : ym m m m if n = 2m + 1, if n = 2m. The Néron model E consists of the union of the n components Ej with singularities on the special fiber removed. Multiplicative Reduction: Quotient of the Néron model Multiplication-by-2 on E : y 2 + xy = x 3 + π n , x 4 − 8π n x − π n under quotient by ±1, induces φ(x) = . 4x 3 + x 2 + 4π n Taking the quotient of the Néron model by ±1 gives x0 = πx1 , x1 = πx2 , ··· , xm−1 = πxm with x 20 x 4 − 8π n x0 − π n ≡ , X0 → X0 : x00 = φ(x0 ) = 0 3 4x 0 + 1 4x0 + x02 + 4π n Xj → X2j , for 1 ≤ j ≤ n/4: xj4 − 8π n−3j xj − π n−4j 0 x2j = π −2j φ(π j xj ) = 4π j xj3 + xj2 + 4π n−2j ≡ x 2j − (0 or 1 ) x 2j Xj → Xn−2j , for n/4 < j ≤ n/2: π 4j−n xj4 − 8π j xj − 1 −1 0 xn−2j = π 2j−n φ(π j xj ) = j 3 ≡ 2 2 n−2j 4π xj + xj + 4π x j + (0 or 4) Multiplicative Reduction: The Quotient, Continued The Upshot: For type In reduction, with n = 2m or n = 2m + 1, the quotient V of the Néron model E by ±1 consists of the m + 1 components given by variables x0 , x1 , . . . , xm If m = 2n is even: Then V, with the maps on the previous slide, is a weak Néron model for φ. But if n = 2m + 1 is odd: Then V has a problem for x ∈ P1 (K ) with |x| < |π m |, as we’ll now see. . . Fixing the Odd In case x 4 − 8π n x − π n and n = 2m + 1, we have 4x 3 + x 2 + 4π n x0 = πx1 , x1 = πx2 , ··· , xm−1 = πxm For φ(x) = with Xm → X1 by x10 = π −1 φ(π m xm ) = 4 − 8π m x − 1 −1 π 2m−1 xm m ≡ 2 m 3 2 4π xm + xm + 4π xm which fails to be a morphism at x m ≡ 0, because X1 (k) has no point at ∞. So let’s blow up again at x m = 0, i.e., add a new component: Xm+1 , with variable xm+1 satisfying xm = πxm+1 , and with Xm+1 → X0 by xm+1 7→ φ(π m+1 xm+1 ), i.e., 4 π 2m+3 xm+1 − 8π m+1 xm+1 − 1 1 xm+1 7→ ≡− 3 2 4 4π m+2 xm+1 + πxm+1 +4 NOTE: Again, this is not a finite morphism. Additive Reduction: Quotient of the Néron model Example. p 6= 2, E : y 2 = x 3 − π 2 x has type I∗0 reduction, with four components y02 = x03 − π 2 x0 , y12 = π 2 x13 − x1 , y22 = π 2 x23 + 3πx22 + 2x2 , y32 = π 2 x33 − 3πx32 + 2x3 where x0 = π 2 x1 = π + π 2 x2 = −π + π 2 x3 . The quotient V of the Néron model by ±1 has the same four xi -components but is not a weak Néron model for the Lattès map φ(x) = (x 2 + π 2 )2 . 4x(x 2 − π 2 ) After all, no points of P1 (K ) with |x|v = |x − π|v = |x + π|v = |π|v extend to the special fiber of V. Fixing the type I∗0 model (x 2 + π 2 )2 , 4x(x 2 − π 2 ) make the coordinate change x = πx1 to get Instead, for E : y 2 = x 3 − π 2 x and φ(x) = ψ(x1 ) = π −1 φ(πx1 ) = (x 2 + 1)2 , 4x(x 2 − 1) which has good reduction. Equivalently, we could work over the field √ L = K ( π), where E has good reduction, take the quotient of the (type I0 ) Néron model by ±1, and then base change back to K . The Berkovich Projective Line Informal Definition. Let Cv be the completion of an algebraic closure of K with respect to | · |v . The Berkovich projective line P1Ber is a compact, path-connected topological space containing P1 (Cv ). (Most of) the extra points added correspond to closed disks D(a, r ) ⊆ Cv . More Formal Definition. A1Ber is the set of multiplicative seminorms on Cv [z] extending | · |v on Cv ⊆ Cv [z], equipped with the Gel’fand topology, i.e., the weakest topology such that for all f ∈ Cv [z], the function A1Ber → [0, ∞) by k · k 7→ kf k is continuous. P1Ber is then A1Ber together with a point at infinity, i.e., the one-point compactification of A1Ber . Berkovich Points Points of P1Ber come in four types: I Type 1: “classical points,” i.e., points of P1 (Cv ). I Type 2: There is one Type 2 point ζ(a, r ) for each rational closed disk D(a, r ) ⊆ Cv . [Here, “rational” means r ∈ |C× v |v .] I Type 3: There is one Type 3 point ζ(a, r ) for each irrational disk D(a, r ) ⊆ Cv . I Type 4: These points correspond to chains of disks D1 ⊃ D2 ⊃ · · · with empty intersection. We well not be concerned with Type 4 points. A Rough Sketch of P1Ber ζ(a,r) ∞ 1 ζ(0,1) ζ(0,|π|)=ζ(π,|π|) π a (a) 0 (b) Definition A type 2 point ζ = ζ(a, r ) is K -rational if at least three different components of P1Ber r {ζ} intersect P1 (K ). Equivalently, the disk D(a, r ) contains a point of K and has r ∈ |K × |v . Models as sets of type 2 points ζ1=ζ(0,1) W1 W2 ζ2=ζ(0,|π|) W4 W3 ζ3=ζ(0,|π|2) W5 ζ4=ζ(0,|π|3) (a) ζ5=ζ(π2,|π|3) (b) A finite-type model V becomes a finite set of type 2 points. I I Each component in the model V corresponds to a K -rational type 2 point in P1Ber . If V is to have V(Ov ) ∼ = P1 (K ), then for any two K -rational type 2 points in the Berkovich model, all K -rational type 2 points between them must also be in the Berkovich model. Berkovich weak Néron models Definition Let M ⊆ P1Ber be a finite set of K -rational type 2 points. I We say M is a Berkovich model for P1 over K if: for any two points ζ1 , ζ2 ∈ M, all K -rational type 2 points between ζ1 and ζ2 also belong to M. I Let φ ∈ K (z) be a non-constant rational function. We say M is a Berkovich weak Néron model for φ over K if: for every ζ ∈ M and every connected component U of P1Ber r {ζ}, if U ∩ M = ∅, then φ(U) ∩ M = ∅. Idea: In the second condition above, I ζ corresponds to a component Wi of the special fiber of the model V, and I U, which is a disk, corresponds to a point on Wi . Berkovich models under base change Let TK ⊆ P1Ber denote the set of K -rational type 2 points in P1Ber . Definition Let L/K be a finite extension, and let N ⊆ TL be a Berkovich model for P1 over L. Define [ M = (N ∩ TK ) ∪ ∂U, U where the union is over all connected components U of P1Ber r TK that contain points of N. Then we say M is the Berkovich model over K induced by N. Idea: Each U is an annulus; ∂U consists of two K -rational type 2 points. Berkovich weak Néron models under base change Theorem Let L be a finite extension of K , and let φ ∈ K (z) be a nonconstant rational function. Suppose that N ⊆ TL is a Berkovich weak Néron model for φ over L, and let M be the Berkovich model over K induced by N. Then M is a Berkovich weak Néron model for φ over K . Proof (sketch): (1) (M is a Berkovich model over K ): Given two points ζ1 , ζ2 ∈ M, and a third ζ ∈ TK lying between them, we can show that ζ lies between two points of N. Then ζ ∈ N ∩ TK ⊆ M. (2) (weak Néron property): Given ζ ∈ M and an open Berkovich disk U at ζ, disjoint from M, U is contained in a corresponding disk V for ζ 0 ∈ N, and hence φ(U) ∩ N = ∅. Slightly more work then shows φ(U) ∩ M = ∅. Constructing Berkovich weak Néron models Corollary Let φ be a Lattès map defined over K . Then φ has a weak Néron model defined over K . Proof (Sketch). Let E be the corresponding elliptic curve, and let L/K be a finite extension over which E has reduction of type In for n even. If n = 0, then E /L has good reduction, and so φ also has good reduction (after L-rational coordinate change) and hence a weak Néron model V 0 over L. If n ≥ 2, let E 0 be the Néron model over L, and let V 0 be E 0 /{±1}, which is a weak Néron model for φ over L. Let N be the Berkovich weak Néron model over L associated to V 0 . Let M be the Berkovich model over K induced by N, which is a Berkovich weak Néron model by the previous theorem. Then the associated model V is a weak Néron model over K . Example 2 3 ∗ Let K = Qur 2 , and E : y = x − 4x, which has type I2 reduction. Consider the Lattès map for multiplication-by-2, (x 2 + 4)2 . φ(x) = 4x(x 2 − 4) √ Warning: Wild ramification: E.g. over K 0 = K ( 2), E is isomorphic to E 0 : y 2 = x 3 − x, which has type I∗3 reduction. Back to K = Qur 2 : besides the 2-torsion points O, (0, 0), (±2, 0), all the 3-√and 4-torsion points have x-coordinates in the disk D(2 + 2 2, |4|2 ) ⊆ K . So if E has semistable reduction over L/K , the algorithm says the associated Berkovich weak Néron model over K will have two points: ζ(0, |2|2 ) and ζ(2, |4|2 ). Example, Continued 2 3 Recall: K = Qur 2 , and E : y = x − 4x, and φ(x) = (x 2 + 4)2 . 4x(x 2 − 4) Our Berkovich weak Néron model over K has two points: ζ(0, |2|2 ) and ζ(2, |4|2 ). That is, x = 2x0 and x0 = 1 + 2x1 , with X0 → X0 by 1 (x02 + 1)2 x00 = φ(2x0 ) = ≡∞ 2 4x0 (x02 − 1) and X1 → X0 by 1 (2x12 + 2x1 + 1)2 x00 = φ(4x1 + 2) = ≡∞ 2 4x1 (x1 + 1)(2x1 + 1)