aa r X i v : . [ m a t h . N T ] M a r CANONICAL METRICS OF COMMUTING MAPS
J. PINEIRO
Abstract.
Let ϕ : X → X be a map on an projective variety.It is known that whenever ϕ ∗ : Pic( X ) ⊗ R → Pic( X ) ⊗ R hasan eigenvalue α >
1, we can build a canonical measure, a canoni-cal height and a canonical metric associated to ϕ . In the presentwork, we establish the following fact: if two commuting maps ϕ, ψ : X → X satisfy these conditions, for eigenvalues α and β andthe same eigenvector L , then the canonical metric, the canonicalmeasure, and the canonical height associated to both maps, areidentical. Introduction
Let X be a projective variety defined over a number field K . Supposethat ϕ : X → X is a map on X , also defined over K . Assume that wecan find an ample line bundle L on X and a number α >
1, such that L α ∼ = ϕ ∗ L . Under this conditions, we can build the canonical heightˆ h ϕ ([9] theorem 1.1) and the canonical measure dµ ϕ ([20] proposition3.1.4) associated to ϕ and L . They satisfy nice properties with respectto the map ϕ , for example we have ˆ h ϕ ◦ ϕ = deg( ϕ )ˆ h ϕ and ϕ ∗ µ ϕ = µ ϕ . Sometimes it happens that a whole set of maps are associated to thesame canonical height function and measure. As our first exampleconsider the collection of maps φ k : P K → P K on the Riemann Sphere,where φ k is defined as φ k ( t ) = t k . The line bundle L = O (1) on P satisfies the isomorphism φ ∗ k L ∼ = L k . If one builds the canonical heightand measure associated to φ k and O (1), one obtains:(i) All φ k have the same canonical height namely, the naive height h nv on P Q . The naive height h nv ( P ) is a refined idea of the func-tion sup {| a | , | a |} , measuring the computational complexity ofthe projective point P = ( a : a ). For a precise definition seelater definition 2.10. Mathematics Subject Classification.
Primary: 14G40; Secondary: 28C10,14K22, 14H52, 16W22.I would like to express my gratitude to Professor Lucien Szpiro for the valuablediscussions on the subject of this paper.The author was partially supported by the PSC-CUNY Award 60029-3637. (ii) All φ k have the same canonical measure, that is, the Haarmeasure dθ on the unit circle S .Similar properties are fulfilled by the collection of maps [ n ] : E → E ,representing multiplication by n on an elliptic curve E defined over K .If L is an ample symmetric line bundle on E , we have the isomorphism[ n ] ∗ L ∼ = L n , along with the properties:(i) All maps [ n ] share the same canonical height, that is, theNeron-Tate height ˆ h E on E . In fact this will be our defini-tion (2.11) of the Neron-Tate height on E . For many otherinteresting properties we refer to B-4 in [8].(ii) All maps [ n ] have the same canonical measure, that is, theHaar measure i/ Im ( τ ) dz ∧ d ¯ z on E = C / Z + τ Z .We observe that any two maps in each collection commute for thecomposition of maps. Besides, the line bundle L ∈
Pic( X ) ⊗ R , suitableto make everything work, is the same within each collection. Thepresent work establish the general fact: Proposition 1.1.
Let X be a projective variety defined over a numberfield K . Suppose that two maps ϕ, ψ : X → X commute ( ϕ ◦ ψ = ψ ◦ ϕ ) and satisfy the following property: For some ample line bundle L ∈
Pic( X ) ⊗ R and real numbers α, β > , we have ϕ ∗ L ∼ −→ L α and ψ ∗ L ∼ −→ L β , then we have ˆ h ϕ = ˆ h ψ = ˆ h ϕ ◦ ψ and dµ ϕ = dµ ψ = dµ ϕ ◦ ψ . This result is known in dimension one, a proof can found for examplein [6]. Also it is a well known fact [8], that commuting maps in aprojective variety must share the same canonical height. The mainfeature of the present work it is to obtain all this results from theequality of the canonical metrics. Given a ample line bundle L on X ,it was an original idea of Arakelov [1] to put metrics on L σ = L ⊗ σ C over all places σ of K at infinity. This gave rise to heights as intersectionnumbers and curvature forms at infinity. In was then an idea of Zhang[19] to look for suitable metrics at all places v of K . In presence ofthe dynamics ϕ : X → X , the line bundle L on X can be endowed[19] with very special metrics k . k ϕ,v on L v that satisfy the functionalequation k . k ϕ,v = ( φ ∗ ϕ ∗ k . k ϕ,v ) /α , whenever we have an isomorphism φ : L α ∼ −→ ϕ ∗ L . The canonicalheight and the canonical metric will be defined (definitions 2.6 and 2.9)depending only on the metric k . k ϕ . The equality of canonical heightsand measure for commuting maps is a consequence of the followingproposition: ANONICAL METRICS OF COMMUTING MAPS 3
Proposition 1.2.
Suppose that two maps ϕ, ψ : X → X commute,and for some ample line bundle L ∈
Pic( X ) ⊗ R we have ϕ ∗ L ∼ −→ L α and ψ ∗ L ∼ −→ L β for some numbers α, β > , then k . k ϕ = k . k ψ . Towards the end of the paper we discuss maps on P arising as pro-jections of maps on elliptic curves with complex multiplication. Westudy ramification points and present examples of commuting maps onthe Riemann sphere.2. Canonical heights and canonical measures
Canonical metrics.
Consider the projective variety X definedover a number field K , a map ϕ : X → X defined over K , and anample line bundle L ∈
Pic( X ) ⊗ R such that φ : L α ∼ −→ ϕ ∗ L for some α >
1. This situation will be called [20] a polarized dynamical system(
X, ϕ, L , α ) on X defined over K .Assume that for every place v of K we have chosen a continuous andbounded metric k . k v on each fibre of L v = L ⊗ K K v . The followingtheorem is proposition 2.2 in [19]: Theorem 2.1.
The sequence defined recurrently by k . k v, = k . k v and k . k v,n = ( φ ∗ ϕ ∗ k . k v,n − ) /α for n > , converge uniformly on X ( ¯ K v ) to ametric k . k v,ϕ (independent of the choice of k . k v, ) on L v which satisfiesthe equation k . k ϕ,v = ( φ ∗ ϕ ∗ k . k ϕ,v ) /α .Proof. Denote by h the continuous function log k . k k . k on X ( ¯ K v ). Thenlog k . k n = log k . k + n − X k =0 ( 1 α φ ∗ ϕ ∗ ) k h. Since k ( α φ ∗ ϕ ∗ ) k h k sup ≤ ( α ) k k h k sup , it follows that the series givenby the expression P ∞ k =0 ( α φ ∗ ϕ ∗ ) k h , converges absolutely to a boundedand continuous function h v on X ( ¯ K v ). Let k . k ϕ,v = k . k exp( h v ), then k . k n converges uniformly to k . k ϕ,v and its not hard to check that k . k ϕ,v satisfies k . k ϕ,v = ( φ ∗ ϕ ∗ k . k ϕ,v ) /α , which was the result we wanted to prove. (cid:3) Definition 2.2.
The metric k . k v,ϕ is called the canonical metric on L v relative to the map ϕ . Example 2.3.
Consider the line bundle L = O P n (1) on P n ¯ Q and therational map φ k : P n Q → P n Q given by the expression φ ( T : ... : T n ) = J. PINEIRO ( T k : ... : T kn ) . The Fubini-Study metric k ( λ T + ... + λ n T n )( a : ... : a n ) k F S = | P λ i a i | pP i a i is a smooth metric on L C . If we take k . k = k . k F S as our metric atinfinity, the limit metric we obtain is k ( λ T + ... + λ n T n )( a : ... : a n ) k nv = | P λ i a i | sup i ( | a i | ) . Example 2.4.
Suppose that X = E is an elliptic curve and assumethat [ n ] : E → E is denoting the multiplication by n on E . As aconsequence of the theorem of the cube, the ample symmetric line bundle L on E satisfies φ : [ n ] ∗ L ∼ −→ L n . The canonical metric is the metricof the cube discussed in [11] and suitable to make φ and isomorphismof metrized line bundles. The following proposition relates the canonical metrics associated tocommuting maps. It represents the main result of this paper.
Proposition 2.5.
Let ( X, ϕ, L , α ) and ( X, ψ, L , β ) be two polarizedsystems on X defined over K . Suppose that the maps ϕ and ψ satisfy ϕ ◦ ψ = ψ ◦ ϕ , then k . k ϕ = k . k ψ .Proof. The key idea is that the canonical metric associate to a mor-phism does not depend on the metric we start the iteration with.Let s ∈ Γ( X, L ) be a non-zero section of L . We are going to con-sider two metrics k . k v, = k . k ϕ and k . k ′ v, = k . k ψ on the line bundle L . By our definition of canonical metric for ϕ , we can start with k . k ′ v, and obtain k s ( x ) k ϕ = lim k k s ( ϕ k ( x ) k /α k ψ , but also by our def-inition of canonical metric for ψ starting with k . k v, = k . k ϕ we get k s ( x ) k ψ = lim l k s ( ϕ l ( x )) k /β l v, . So using the uniform convergence andthe commutativity of the maps, k s ( x ) k ϕ = lim k,l k s ( ϕ k ◦ ψ l ( x )) k /α k β l v, = lim l,k k s ( ψ l ◦ ϕ k ( x )) k /β l α k v, = k s ( x ) k ψ , which was the result we wanted to prove. (cid:3) Canonical measures.
Let X be a n-dimensional projective va-riety defined over a number field K and suppose that ( X, ϕ, L , α ) isa polarized dynamical system defined over K . let v be a place of K over infinity. We can consider the morphism ϕ ⊗ v : X v → X v on the complex variety X v . Associated to ϕ and v we also have the ANONICAL METRICS OF COMMUTING MAPS 5 canonical metric k . k ϕ,v and therefore the distribution c ( L , k . k ϕ,v ) = πi ) ∂∂ log k s ( P ) k ϕ,v analogous to the first Chern form of ( L , k . k ϕ,v ).It can be proved that c ( L , k . k ϕ,v ) is a positive current in the sense ofLelong, and following [5] we can define the n-product c ( L , k . k ϕ,v ) n = c ( L , k . k ϕ,v ) ...c ( L , k . k ϕ,v ) , which represents a measure on X v . Definition 2.6.
The measure dµ ϕ = c ( L v , k . k ϕ,v ) n /µ ( X ) , is calledthe canonical measure associated to ϕ and v . Once we have fixed L , itdepends only on the metric k . k ϕ,v . Example 2.7.
Consider the rational map φ k : P n Q → P n Q given by theformula φ k ( T : ... : T n ) = ( T k : ... : T kn ) , the canonical measure dµ φ k isthe normalize Haar measure on the n-torus S × ... × S . Example 2.8.
Let E be an elliptic curve, L a symmetric line bundleon E and the map [ n ] : E → E . The canonical measure associated to ( E, [ n ] , L , [ n ] ) can be proved to be [11] the normalized Haar measureon E . Canonical heights as intersection numbers.
For a regularprojective variety X of dimension n, defined over a field K, the classicaltheory of intersection ([7],[14]) defines the intersection c ( L ) ...c ( L n )of the classes c ( L i ) associated to line bundles L i on X , when 0 < i ≤ n .For the purpose of defining the arithmetic intersection, we want to as-sume that X is an arithmetic variety of dimension n + 1, that is, givena number field K , there exist a map f : X → Spec( O K ), flat and offinite type over Spec( O K ). We can define (See for example [4], [3], [16],[1], [17] or [18]) the arithmetic intersection number ˜ c ( L ) ... ˜ c ( L n +1 ) ofthe classes ˜ c ( L i ) of hermitian line bundles ˜ L i = ( L i , k . k ) on X . Thefact that ˜ L i are hermitian line bundles for i = 1 ..n + 1, means thateach line bundle L i on X is equipped with a hermitian metric k . k v,i over X v = X ⊗ K Spec O K v for each place v at infinity. The numbers˜ c ( L ) ... ˜ c ( L j ) prove to be the appropriated theory of intersection inthe particular case of arithmetic varieties, adding places over infinityallows us to recover the desirable properties of the classical intersectionnumbers of varieties over fields.The last step in the theory of intersection is actually the one thatplays the more important role in our definition of the canonical heightassociated to a morphism. Suppose that X is a regular variety of di-mension n and ( L i , k . k i ) v ( i = 1 , .., p + 1) are metrized line bundleson X . Assume also that the L i are been equipped with semiposi-tive metrics over all places v (not just at infinity as before) in the J. PINEIRO sense of [19]. Such line bundles are called adelic metrized line bun-dles and following [19], we can define the adelic intersection numberˆ c ( L | Y ) ... ˆ c ( L p +1 | Y ) over a p − cycle Y ⊂ X . The adelic intersectionnumber is in fact a limit of classical numbers ˜ c ( L ) ... ˜ c ( L p +1 ) once thenotion of converge is established. The numbers ˆ c ( L | Y ) ... ˆ c ( L p +1 | Y )satisfy again nice properties, they are multilinear in each of the L i and satisfy ˆ c ( f ∗ L | Y ) ... ˆ c ( f ∗ L p +1 | Y ) = ˆ c ( L | f ( Y )) ... ˆ c ( L p +1 | f ( Y )) , whenever we have a map f : X → X . We are interested in a partic-ular case of this situation. Suppose that we are in the presence of apolarized dynamical system ( X, ϕ, L , α ), in this situation the canonicalmetric k . k ϕ of 2.1 represent a semipositive metric on L , (again we referto [19]) and we can define the canonical height associated to ( L , k . k ϕ ). Definition 2.9.
The canonical height ˆ h ϕ ( Y ) of a p − cycle Y in X isdefined as ˆ h ϕ ( Y ) = ˆ c ( L| Y ) p +1 (dim( Y ) + 1) c ( L| Y ) p . It depends only on ( L , k . k ϕ ) , where k . k ϕ is actually representing a col-lection of canonical metrics over all places of K . An important par-ticular case of canonical height will be the canonical height ˆ h ϕ ( P ) of apoint in P ∈ X . Example 2.10.
Consider the map φ k : P n ¯ Q → P n ¯ Q given by the formula φ k ( T : ... : T n ) = ( T k : ... : T kn ) , the canonical height associated to φ k is called the naive height h nv on P n . If P = [ t : ... : t n ] is a point in P n the naive height is h nv ([ t : ... : t n ]) = 1[ K : Q ] log Y places v of K sup( | t | v , ..., | t n | v ) N v , where N v = [ K v : Q w ] and w is the place of Q such that v | w . Definition 2.11.
Let E be an elliptic curve and L an ample symmetricline bundle on E . The canonical height associated to [ n ] : E → E and L is called the Neron-Tate height ˆ h E on E . The fact that this isindependent of n, will be a consequence of proposition 2.12. The collection of maps { φ k } k on P n and the collection { [ n ] } n on agiven elliptic curve E , share two important properties, the maps withineach collection commute, and share the same canonical height andcanonical measure. The following proposition establishes a general factabout canonical heights and canonical measures of commuting maps ona projective variety X . ANONICAL METRICS OF COMMUTING MAPS 7
Proposition 2.12.
Let ( X, ϕ, L , α ) and ( X, ψ, L , β ) be two polarizedsystems on X defined over K . Suppose that the maps ϕ and ψ satisfy ϕ ◦ ψ = ψ ◦ ϕ , then ˆ h ϕ = ˆ h ψ = ˆ h ϕ ◦ ψ and dµ ϕ = dµ ψ = dµ ϕ ◦ ψ . Proof.
This is a consequence of our definitions of canonical measure2.6, canonical height 2.9 and proposition 2.5. (cid:3)
Corollary 2.13.
Suppose that two maps ϕ, ψ : P → P , satisfy thehypothesis of the previous proposition, then the two maps have the sameJulia set.Proof. The Julia set of a map ϕ : P → P is nothing but the closurein P of the set of repelling periodic points. For details we refer todefinition 2.2 in [12]. Now, the corollary is a consequence of proposition2.12 and proposition 7.2 in [12]. (cid:3) Elliptic Curves and examples
This section illustrates examples of commuting maps on P . Theyall share one thing in common: being induced in some sense by endo-morphisms on elliptic curves. Proposition 3.1.
Consider an elliptic curve E = C / Z + τ Z given byWeierstrass equation y = G ( x ) . Suppose that E admits multiplicationby the algebraic number λ , then multiplication by λ in E induces, asquotient by the action of [ − , a map ϕ λ : P → P . Besides, we have: (i) ˆ h E ( x, y ) = ˆ h λ ( x ) for any point P = ( x, y ) on E . (ii) The canonical measure on P associated to ϕ λ is dµ ϕ λ = idz ∧ d ¯ z Im ( τ ) | G ( z ) | . Proof.
The first part is a classical fact of the theory of elliptic functionsand complex multiplication. There exist polynomials P ( z ) and Q ( z )where deg( P ) = deg( Q )+1 = N ( λ ) such that ℘ ( λz ) = P ( ℘ ( z )) /Q ( ℘ ( z )),where ℘ is denoting the Weierstrass ℘ − function. Suppose that we call π the quotient map from E → P , we have a commutative diagram: E λ −−−→ E π y π y P ϕ λ −−−→ P Now, consider the line bundle L = O (1) on P , we have ϕ ∗ λ L ∼ −→ L N ( λ ) and equally for the ample symmetric line bundle π ∗ L on E . Therefore,it make sense to talk about canonical heights associated to ϕ λ : P → P J. PINEIRO and λ : E → E . The number λ lies in an imaginary quadratic extensionof Q , so we also have a commutative diagram: E λ −−−→ E ¯ λ −−−→ E π y π y π y P ϕ λ −−−→ P ϕ ¯ λ −−−→ P So, the two maps ϕ λ and ϕ ¯ λ commute. After 2.12 the canonical heightassociated to multiplication by λ on E is the same as the canonicalheight associated to multiplication by N ( λ ), that is the Neron-Tateheight on E . Take L = O (1) on P and P a point on E , the intersectionnumbers satisfy a projection formulaˆ c ( π ∗ L| P ) = ˆ c ( L| π ( P )) c ( π ∗ L| P ) = c ( L| π ( P )) . This gives (i) after definition 2.9. For (ii) consider the Haar measure i/ dz ∧ d ¯ z on E , normalized by Im ( τ ). If ℘ denote the Weierstrassfunction and ω = ℘ ( z ), we have idω ∧ d ¯ ω Im ( τ ) = idz ∧ d ¯ z | ℘ ′ ( z ) | Im ( τ ) = idz ∧ d ¯ z | y | Im ( τ ) = idz ∧ d ¯ z | G | Im ( τ ) . which gives the result we wanted to prove. (cid:3) Remark 3.2.
If the elliptic curve E admits multiplication by the num-bers λ and δ , then ϕ λ ◦ ϕ δ = ϕ δ ◦ ϕ λ . Example 3.3.
Consider an elliptic curve E given by Weierstrass equa-tion E : y = G ( x ) . For λ = 2 we have ϕ ( z ) = ( G ′ ( z )) − zG ( z )4 G ( z ) . Example 3.4.
Let’s consider some examples of elliptic curves withcomplex multiplication:
The elliptic curve E : y = x + x admits multiplication by Z [ i ].The multiplication by i morphism can be written in x, y coordinates as[ i ]( x, y ) = ( − x, iy ). The two maps ϕ i ( z ) = 1(1 + i ) z + 1 z ϕ − i ( z ) = − i ) z + 1 z commute, and their composition satisfies ϕ i ( ϕ − i ( z )) = ϕ − i ( ϕ i ( z )) = ϕ ( z ) = z − z + 14( z + z ) . ANONICAL METRICS OF COMMUTING MAPS 9
The canonical height and measure are:ˆ h ( z ) = h E ( z, ±√ z + z ) dµ ( z ) = idz ∧ d ¯ z | z + z | Other examples of maps attached to E are ϕ i ( z ) = ( − − i ) z ( z + 1 + 2 i ) (5 z + 1 − i ) ϕ − i ( z ) = (3 + 4 i ) z ( z + 1 + 2 i ) (5 z + 1 − i ) ϕ i ( z ) = (3 − i ) z ( z + 1 − i ) (5 z + 1 + 2 i ) ϕ − i ( z ) = ( − i ) z ( z + 1 − i ) (5 z + 1 + 2 i ) . The curve E : y = x + 1 admits multiplication by the ring Z [ ρ ]where ρ = ( √− /
2. The multiplication by ρ can be expressed in x, y coordinates as [ ρ ]( x, y ) = ( ρx, y ). An example of commuting mapscoming from E is ϕ √− ( z ) = − ( z + 4)3 z ϕ √− ρ ( z ) = − ρ ( z + 4)3 z ϕ √− ◦ ϕ √− ρ ( z ) = ϕ ε ( z ) = ( z − z + 48 z + 64)9 ρz ( z + 4) , where ε = ( − √− /
2. The canonical measure associated to thethree maps is dµ E ( z ) = √ idz ∧ d ¯ z | z + 1 | . To have an idea of the ramification points and indexes of the maps ϕ λ , we proof the following lemma: Lemma 3.5.
A ramification point for ϕ λ belongs to the image by π ofthe 2-torsion points on E .Proof. To see this, suppose that ϕ − λ ( π ( P )) = { π ( Q ) | λQ = P } has car-dinal strictly smaller than N ( λ ). Then there exist two points π ( Q ) = π ( − Q ) inside the set ϕ − λ ( P ), such that λQ = − λQ and consequently2 λQ = 2 P = 0 . (cid:3) Let’s see some examples of the different ramifications that a map ϕ λ may have. Let d be a positive square free integer. Assume that theelliptic curve C /Z + √− dZ , admits multiplication by λ = a + b √− d .Suppose that P = 0, P = 1 / P = 1 / √− d/ P = √− d/ E and that r j denotes the amount of pre-images of the point π ( P j ), that is, the cardinality of the set ϕ − λ ( π ( P j )).Under the conditions previously described, we can observe for examplethat for λ = 2, the points in ϕ − ( π ( P )) are not ramification points of ϕ . On the other hand for the multiplication by λ = 1 + 2 i on E ,all points in ϕ − i ( π ( P )) ∪ ϕ − i ( π ( P )) ∪ ϕ − i ( π ( P )) ∪ ϕ − i ( π ( P ))are ramification points of ϕ i . The following table summarize theresults: λ = a + b √− d , N ( λ ) r j , j = 0 , r j , j = 1 , a + bd ≡ mod (2) r = ( N ( λ ) + 1) / r = ( N ( λ ) + 1) / r = ( N ( λ ) + 1) / r = ( N ( λ ) + 1) / a ≡ b ≡ mod (2), N ( λ ) > r = N ( λ ) / r = N ( λ ) / r = N ( λ ) / r = N ( λ ) / a ≡ b ≡ mod (2), N ( λ ) = 4 r = 4 r = N ( λ ) / r = N ( λ ) / r = N ( λ ) / a ≡ bd ≡ mod (2), N ( λ ) = 2 r = 1 r = N ( λ ) r = 1 r = N ( λ ) a ≡ bd ≡ mod (2), N ( λ ) > r = N ( λ ) / r = N ( λ ) / r = N ( λ ) / r = N ( λ ) / References [1] S. Arakelov,
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Distributions in algebraic dynamics ∼ szhang/papers/dynamics.pdf Department of Mathematics and Computer Science, Bronx Commu-nity College of CUNY, University Ave. and West 181 Street, Bronx,NY 10453
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