Around canonical heights in arithmetic dynamics Shu Kawaguchi - PowerPoint PPT Presentation

Around canonical heights in arithmetic dynamics Shu Kawaguchi Arithmetic 2015 - Silvermania August 14, 2015 1 / 39

Plan of the talk (2005 Email? 2008 AIM) 1 Canonical heights for polarized dynamical systems 2 Canonical heights on affine space 3 Canonical heights for surface automorphisms 4 Arithmetic degrees 2 / 39

Plan of the talk (2005 Email? 2008 AIM) 1 Canonical heights for polarized dynamical systems 2 Canonical heights on affine space 3 Canonical heights for surface automorphisms 4 Arithmetic degrees The emphasis is on canonical heights other than N´ eron-Tate heights or those on P n for morphisms. 2 / 39

Part 1 Canonical heights for polarized dynamical systems a projective variety defined over ¯ X X X : Q (for simplicity) f : X → X f : X → X f : X → X : a morphism D D D ∈ Div( X ) R := Div( X ) ⊗ R : a Cartier R -divisor on X Assume that f ∗ D ∼ d D f ∗ D ∼ d D f ∗ D ∼ d D for some d > 1. ( X, f, D ) If D is ample, then the triple ( X, f, D ) ( X, f, D ) is called a polarized dynamical system . Example (polarized dynamical systems) • X : Abelian variety, D ample with [ − 1] ∗ D ∼ D , f = [2]: twice multiplication map (= ⇒ N´ eron-Tate height ) • X = P N , f : a morphism of degree > 1, D : a hyperplane ⇒ canonical height ˆ h f : P N (¯ (= Q ) → R ) 3 / 39

Theorem (Call–Silverman 1993) ˆ ( D : not necessarily ample.) There exists a unique height function ˆ ˆ h D h D h D associated to D , ˆ h D : X (¯ h D : X (¯ ˆ ˆ h D : X (¯ Q ) → R Q ) → R Q ) → R , satisfying ˆ h D ◦ f = d ˆ ˆ h D ◦ f = d ˆ ˆ h D ◦ f = d ˆ h D h D h D . Properties of Call–Silverman canonical heights 1 Assume that D is ample. Then ˆ h D is non-negative and ˆ ˆ ˆ h D ( x ) = 0 if and only if x ∈ X (¯ h D ( x ) = 0 h D ( x ) = 0 Q ) is preperiodic . PrePer( f, ¯ In particular, the set of preperiodic points PrePer( f, ¯ PrePer( f, ¯ Q ) Q ) Q ) is a set of bounded height. 4 / 39

2 Assume that D is ample. Take x ∈ X (¯ Q ) that is not preperiodic. Then f ( x ) | h H ( y ) ≤ T } ∼ log T # { y ∈ O + as T → ∞ , log d where O + f ( x ) is the forward orbit of x under f , and H is any ample divisor. 3 Decomposition into the sum of local canonical heights Take a number field K over which f is defined. For a finite extension L/K and x ∈ X ( L ) \ | D | , one has ∑ [ L v : K v ] ˆ ˆ h D ( x ) = λ D,v ( x ) . [ L : K ] v ∈ M L 5 / 39

4 Variation of the canonical height π : V → C a family over a smooth projective curve C C ◦ a Zariski open subset of C , and set V ◦ := π − 1 ( C ◦ ) f : V ◦ → V ◦ over C ◦ and D ∈ Div( V ◦ ) R as before h C : a Weil height on C corresponding to a divisor of degree 1 P : C → V a section ˆ h D t ( P t ) = ˆ lim h D ( P ) . h T ( t ) h T ( t ) →∞ Followed by stronger results by Ingram More properties of the canonical heights . . . From talks of this conference: Equidistribution (Baker–Rumely, Chambert-Loir, Favre–Rivera-Letlier, Yuan ...), Masser–Zannier unlikely intersection (Baker–DeMarco, Ghioca–Tucker–Hsia, DeMarco–Wang–Ye, Ghioca–Krieger–Nguyen–Ye ...) . . . 6 / 39

Part 2 Canonical heights on affine space H´ enon map affine plane (over ¯ A 2 : Q ) f : A 2 → A 2 : a H´ enon map , i.e., an automorphism of the form f ( x, y ) = ( y + P ( x ) , x ) for some polynomial P ( x ) ∈ ¯ Q [ x ] with d := deg( P ) ≥ 2. Then f extends to a birational map f : P 2 ��� P 2 , ( x : y : z ) �→ ( yz d − 1 + z d P ( x/z ) : xz d − 1 : z d ). Since f has the indeterminacy set I f = { (1 : 0 : 0) } , f is not a morphism. 7 / 39

enon map f : P 2 ��� P 2 is not a polarized dynamical system, but The H´ Silverman proved the following theorem. Theorem (Silverman 1994) Let f : A 2 → A 2 be a H´ enon map of degree 2 over ¯ Q . Then Per( f, ¯ 1 the set of periodic points Per( f, ¯ Per( f, ¯ Q ) is a set of bounded height. Q ) Q ) 2 Take x ∈ A 2 (¯ Q ) that is not periodic. Then # { y ∈ O f ( x ) | h Weil ( y ) ≤ T } ∼ 2 log T as T → ∞ , log 2 where O f ( x ) = { f n ( x ) | n ∈ Z } is the f -orbit, and h Weil is the standard Weil function. Remark The proof uses blow-ups along the indeterminacy sets I f and I f − 1 . 8 / 39

Regular polynomial automorphism affine N -space (over ¯ A N : Q ) f : A N → A N : a polynomial automorphism Then f extends to a birational map f : P N ��� P N . Following Sibony, f is called a regular polynomial automorphism I f ∩ I f − 1 = ∅ , if where I f and I f − 1 is the indeterminacy sets of f and f − 1 . Example { H´ enon maps } ⊂ { regular polynomial automorphisms } Indeed, for a H´ enon map, I f = { (1 : 0 : 0) } and I f − 1 = { (0 : 1 : 0) } . 9 / 39

Silverman’s results are generalized by Denis and Marcello. Theorem (Denis, Marcello) Let f : A N → A N be a regular polynomial automorphism of degree d ≥ 2 . Then 1 the set of periodic points Per( f, ¯ Per( f, ¯ Per( f, ¯ Q ) is a set of bounded height. Q ) Q ) 2 Take x ∈ A N (¯ Q ) that is not periodic. Then # { y ∈ O f ( x ) | h Weil ( y ) ≤ T } ∼ 2 log T as T → ∞ , log d where O f ( x ) = { f n ( x ) | n ∈ Z } is the f -orbit, and h Weil is the standard Weil function. 10 / 39

Canonical heights have not appeared so far, but they exist. For a polynomial automorphism f : A 2 → A 2 on the affine plane, the (first) dynamical degree (explained more later) is defined by n →∞ (deg f n ) 1 /n , δ δ δ := δ δ δ f := lim which in this case is an integer. (For a H´ enon map, δ = deg f .) 11 / 39

Canonical heights have not appeared so far, but they exist. For a polynomial automorphism f : A 2 → A 2 on the affine plane, the (first) dynamical degree (explained more later) is defined by n →∞ (deg f n ) 1 /n , δ δ δ := δ δ δ f := lim which in this case is an integer. (For a H´ enon map, δ = deg f .) Theorem (K. (dim N = 2)) Let f : A 2 → A 2 be a polynomial automorphism with δ > 1 . Then the limits 1 1 ˆ ˆ h + ( x ) := lim δ n h Weil ( f n ( x )) , δ n h Weil ( f − n ( x )) h − ( x ) := lim n →∞ n →∞ h + + ˆ ˆ h := ˆ h + + ˆ h + + ˆ h − : A 2 (¯ exist for all x ∈ A 2 (¯ Q ) . We set ˆ h := ˆ ˆ h := ˆ h − h − Q ) → R ≥ 0 . Then ˆ h satisfy � h ≫≪ h Weil ( ) ( ( ) ) δ + 1 δ + 1 δ + 1 h ◦ f − 1 = h ◦ f − 1 = h ◦ f − 1 = and ˆ h ◦ f + ˆ h ◦ f + ˆ ˆ ˆ h ◦ f + ˆ ˆ ˆ ˆ h h h . Further (... continued) δ δ δ 11 / 39

Properties of canonical heights ˆ ˆ h ( x ) = 0 if and only if x ∈ A 2 (¯ ˆ h ( x ) = 0 h ( x ) = 0 Q ) is periodic . 1 Per( f, ¯ In particular, the set of periodic points Per( f, ¯ Per( f, ¯ Q ) Q ) Q ) is a set of bounded height (Silverman). 2 Take x ∈ X (¯ Q ) that is not preperiodic. Then, as T → ∞ , # { y ∈ O f ( x ) | h Weil ( y ) ≤ T } = 2log T log δ − � h ( O f ( x )) + O (1) , where � h ( O f ( x )) is a quantity defined by the orbit O f ( x ) and the O (1) bound does not depend on x . Remark The construction of � h uses blow-ups along I f and I f − 1 . The above estimate ⃝ 2 is similar to Silverman’s canonical heights on Wehler K3 surfaces (explained later). 12 / 39

These results are generalized to higher dimensional case by Lee and K. Theorem (Lee, K) Let f : A N → A N be a regular polynomial automorphism of degree d ≥ 2 . Then there exists a canonical height function ˆ h : A N (¯ ˆ h : A N (¯ ˆ h : A N (¯ Q ) → R Q ) → R Q ) → R defined in a similar way which satisfies � h ≫≪ h Weil and ( ( ( ) ) ) d + 1 d + 1 d + 1 h ◦ f − 1 = h ◦ f − 1 = h ◦ f − 1 = h ◦ f + ˆ ˆ ˆ h ◦ f + ˆ h ◦ f + ˆ ˆ ˆ ˆ ˆ h h h . d d d Further, ˆ h enjoys the same properties ⃝ 1 ⃝ 2 as before. Remark To construct ˆ h , Lee uses blow-ups along the indeterminacy sets I f and I f − 1 . My construction is to introduce local canonical heights and to sum up (as explained in the next page). 13 / 39

3 Decomposition into the sum of local canonical heights f : A N → A N a regular polynomial automorphism of degree d := deg( f ) ≥ 2 defined over a number field K . Set d − := deg( f − 1 ). x ∈ A N ( L ) for a finite extension L/K , v ∈ M L : a place 1 G + d n log + ∥ f n ( x ) ∥ v , v ( x ) := lim n →∞ 1 log + ∥ f − n ( x ) ∥ v . G − v ( x ) := lim d n n →∞ − Then ∑ ( ) [ L v : M v ] ˆ G + v ( x ) + G − h ( x ) := v ( x ) . [ L : K ] v ∈ M L Remark The most difficult part is to show h Weil ≫≪ ˆ h . 14 / 39

4 Variation of the canonical height for H´ enon maps Theorem (Ingram) C a smooth projective curve over a number field K f : A 2 → A 2 enon map of degree d ≥ 2 defined over K ( C ) a H´ h C : a Weil height on C corresponding to a divisor of degree 1 P ∈ A 2 ( K ( C )) Then ˆ h t ( P t ) = ˆ h ( P ) h C ( t ) + ε ( t ) , (√ ) where ε ( t ) = O (1) if C = P 1 and ε ( t ) = O h C ( t ) in general. Thus canonical heights for H´ enon maps (and regular polynomial automorphisms) enjoy various nice properties. 15 / 39