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Arithmetic Dynamics, Arithmetic Geometry, and Number Theory Joseph H Arithmetic Dynamics, Arithmetic Geometry, and Number Theory Joseph H. Silverman Brown University MAGNTS (Midwest Arithmetic Geometry and Number Theory Series) October 12{13, 2019 0 Arithmetic Dynamics and Arithmetic Geometry 1 What is Arithmetic Dynamics? Arithmetic Geometry: Study solutions to polyno- mial equations (points on algebraic varieties) over non- algebraically closed fields. (Discrete) Dynamical Systems: Study orbits of points under iteration of a function. Arithmetic Dynamics: Study number theoretic prop- erties of orbits of points on algebraic varieties. A lot of arithmetic dynamics comes by analogy from arithmetic geometry. Sometimes the analogy is quite di- rect, sometimes less so, and there are parts of arithmetic geometry that still lack dynamical analogues. Today's talk will be a survey of what arithmetic dynamics is all about, with details on its connections to the arithmetic geometry that we all know and love. Then in tomorrow's talk I'll delve more deeply into some specific topics. Arithmetic Dynamics and Arithmetic Geometry 2 In Arithmetic Geometry We Study . Elliptic curves / higher dim'l abelian varieties Torsion points Torsion points defined over a fixed K. Fields generated by torsion points. Image of Galois G(K=K¯ ) ! Aut(Ators). Torsion points on subvarieties (dim A ≥ 2). Mordell{Weil groups Rank of K-rational points . for fixed A and varying K; for fixed K and varying A. Intersection with subvarieties (dim A ≥ 2). Moduli spaces of elliptic curve and abelian varieties Geometry of moduli spaces, e.g., X0(N) and Ag. Distribution of \special" points (CM moduli). Modular forms, L-series, Hecke operators, . Arithmetic Dynamics and Arithmetic Geometry 3 In Discrete Dynamics We Study . A Space X, a Self-Map f : X ! X, and Iteration f◦n = f ◦ f ◦ · · · ◦ f : | {z } nth iterate of f Orbit of x 2 X ◦2 ◦3 Of (x) := x; f(x); f (x); f (x);::: : Preperiodic Point x 2 X with finite orbit Of (x). Periodic Point x 2 X with f◦n(x) = x for some n ≥ 1. Moduli spaces of dynamical systems Classify (X; f) up to \dynamical equivalence." Post-critically finite (PCF) maps Arithmetic Dynamics and Arithmetic Geometry 4 In Discrete Dynamics We Study . A Space X, a Self-Map f : X ! X, and Iteration f◦n = f ◦ f ◦ · · · ◦ f : | {z } nth iterate of f Orbit of x 2 X ! \Mordell{Weil group" ◦2 ◦3 Of (x) := x; f(x); f (x); f (x);::: : Preperiodic Point ! \torsion point" x 2 X with finite orbit Of (x). Periodic Point ! \torsion point" x 2 X with f◦n(x) = x for some n ≥ 1. Moduli spaces of dynamical systems Classify (X; f) up to \dynamical equivalence." Post-critically finite (PCF) maps ! \CM points" Arithmetic Dynamics and Arithmetic Geometry 5 Two Examples Let A be an abelian variety and P 2 A. Then P is a torsion point () P is preperiodic for doubling. In other words, Ators = PrePer A; [2] . Let T (Q) = Q + P . Then P = O (0) [O (0). P Z TP T−P Let f(x) 2 Q(x) be the polynomial 29 f(x) = x2 + : 16 Then f has 8 preperiodic points in Q: 3 1 4 4 7 −4 5 1 5 −4 −4 3 4 7 −4 4 That's a lot of rational preperiodic points!! Arithmetic Dynamics and Arithmetic Geometry 6 Uniform Boundedness in Arithmetic Geometry Let E=Q be an elliptic curve. Theorem. (Mazur) E(Q) contains at most 16 torsion points. More generally: Uniform Boundedness Theorem. (Mazur, Kami- enny, Merel) Let K=Q be a number field and E=K an elliptic curve. Then #E(K)tors ≤ C [K : Q] : And even more generally: Uniform Boundedness Conjecture. Let K=Q be a number field and A=K an abelian variety. Then #A(K)tors ≤ C [K : Q]; dim(A) : Arithmetic Dynamics and Arithmetic Geometry 7 Uniform Boundedness in Arithmetic Dynamics Recall our dictionary: Mordell{Weil group ! orbit of a point torsion point ! preperiodic point Dynamical Uniform Boundedness Conjec- ture. (Morton{Silverman) Let K=Q be a number field, N N and f : P ! P a morphism of degree ≥ 2. Then N # PrePer f; P (K) ≤ C [K : Q]; deg f; N : N • Northcott: PrePer f; P (Q) is a set of bounded ht. Here's a very special case that is still far from resolved: Conjecture. (Poonen) Let c 2 Q. Then 2 # PrePer(x + c; Q) ≤ 8: N • Fakhruddin: Dynamical UBC for P =) UBC for A Arithmetic Dynamics and Arithmetic Geometry 8 Arithmetic Geometry: Elliptic Modular Curves To study points of order N on elliptic curves, one looks at the set of pairs E is an elliptic curve, and Y1(N) := (E; P ): P 2 Etors is a point of order N As is well known, Y1(N) has a natural structure as an affine curve, and (E; P ) 2 Y1(N)(K) () E=K and P 2 E(K): As usual, we let X1(N) = smooth completion of Y1(N) = Y1(N) [ fcuspsg: Then Mazur's theorem (and proof) say that N > 16 () X1(N)(Q) = fcuspsg: (True more generally if genus X1(N) ≥ 1.) Arithmetic Dynamics and Arithmetic Geometry 9 Dynamical Modular Curves (for x2 + c) 2 Let fc(x) = x + c We want to classify pairs dyn Y1 (N) := (α; c): α has (formal) period N for fc dyn Y1 (N) has a natural structure as an affine curve. Let dyn dyn X1 (N) = smooth completion = Y1 (N) [ fcuspsg: dyn Conjecture. N ≥ 4 =) X1 (N)(Q) = fcuspsg. 2 In other words, for all c 2 Q, the map x + c has no Q-rational points of period 4 or larger. This is true for: N = 4 (Morton);N = 5 (Flynn{Poonen{Schaefer); ∗ N = 6 (Stoll) ∗assuming B-SwD for a certain abelian 4-fold. dyn Remark: Jac X1 (N) tends to be irreducible and have positive MW rank over Q, so no Eisenstein quotient! Arithmetic Dynamics and Arithmetic Geometry 10 Dynamical Modular Curves (for x2 + c) dyn The curve Y1 (N) is given explicitly by the equation dyn Y ◦d µ(N=d) ΦN (x; y) := fy (x) − x = 0: djN The inclusion/exclusion is needed to get rid of points of period smaller than N, just as in the formula for the cyclotomic polynomials. Two complications: dyn (1) One needs to prove that ΦN (x; y) is a polynomial. dyn (2) Even if ΦN (α; c) = 0, it may not be true that α 3 has exact period N for fc. For example, c = −4 gives dyn 3 dyn 3 2 Φ1 (x; −4) = (2x + 1)(2x − 3); Φ2 (x; −4) = (2x + 1) : Following Milnor, one says that α has formal period N. dyn Theorem. (Bousch) X1 (N) is geometrically irre- ducible, and there is an explicit formula for its genus. Arithmetic Dynamics and Arithmetic Geometry 11 Unlikely Intersections in Arithmetic Geometry Many theorems and conjectures in arithmetic geometry fall into the \unlikely intersection" paradigm: If an algebraic subvariety has a Zariski dense set of special points, then the subvariety is itself special, i.e., there is a geometric reason that it contains so many special points. Let A be an abelian variety and Y ⊆ A a subvariety. Theorem. (Faltings, n´eeMordell{Lang Conjecture) Let Γ ⊂ A be a finitely generated subgroup. If Γ \ Y = Y , then Y is a finite union of translates of abelian subvarieties of A. Theorem. (Raynaud, n´eeManin{Mumford Conj.) If Ators \ Y = Y , then Y is a finite union of translates of abelian subvarieties of A. Arithmetic Dynamics and Arithmetic Geometry 12 Unlikely Intersections in Arithmetic Dynamics 1 Our dictionary says that: finitely generated subgroup Γ ! orbit Of (x): Dynamical Mordell{Lang Conjecture. (Bell, Denis, Ghioca, Tucker) Let X=C be a smooth projective variety, let Y ⊆ X, let f : X ! X be a morphism, and let P 2 X. Then Of (P ) \ Y = Y =) Y is periodic for f. The extensive literature proving special cases includes: Theorem. Dynamical Mordell{Lang is true if: 2 2 (a) (Xie): f : A ! A is a morphism defined over Q. (b) (Bell-Ghioca-Tucker) f : X ! X is ´etaleand de- N N fined over Q, e.g., f : A ! A is an polynomial auto- morphism. [Proof uses Skolem{Mahler{Lech method.] Arithmetic Dynamics and Arithmetic Geometry 13 Unlikely Intersections in Arithmetic Dynamics 2 Our dictionary also says that: torsion subgp Ators ! preperiodic pts PrePer(f; X): Dynamical Manin{Mumford Conjecture. Let X=C be a smooth projective variety, let Y ⊆ X, and let f : X ! X be a polarized morphism (9 ample L with f∗L ∼= L⊗d with d > 1). Then PrePer(f) \ Y = Y =) Y is periodic for f. Ghioca{Tucker: This statement is FALSE! • There is a partial fix due to Ghioca{Tucker{Zhang. Arithmetic Dynamics and Arithmetic Geometry 14 Arithmetic Geometry: The Image of Galois m Classical Case: Let fm(x) = x . We consider cyclo- tomic fields and their Galois groups, −1 Qm := Q fm (1) = Q(µm);Gm := Gal(Qm=Q): A fundamental theorem on cyclotomic fields says that ∗ ρm : Gm −! Aut(µm) = (Z=mZ) is surjective. Iwasawa theory gives information about the ideal class groups in the tower of fields Q ⊂ Qp ⊂ Qp2 ⊂ Qp3 ::: . Elliptic Curves: Let E=K and [m]: E ! E. Let −1 Km := K [m] (0) = K E[m] ;Gm := Gal(Km=K): Theorem. (Serre) If E does not have CM, then Image of ρE;m : Gm −! Aut E[m] = GL2(Z=mZ) has index that is bounded independent of m. Arithmetic Dynamics and Arithmetic Geometry 15 Arithmetic Dynamics: Arboreal Representations Let K=Q be a number field, let N N f : PK ! PK N be a map of degree d ≥ 2, and let P 2 P (K).
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