A Criterion for Weak Convergence on Berkovich Projective Space
aa r X i v : . [ m a t h . AG ] J a n A CRITERION FOR WEAK CONVERGENCE ONBERKOVICH PROJECTIVE SPACE
CLAYTON PETSCHE
Abstract.
We give a criterion for the weak convergence of unit Borelmeasures on the N -dimensional Berkovich projective space P NK over acomplete non-archimedean field K . As an application, we give a suf-ficient condition for a certain type of equidistribution on P NK in termsof a weak Zariski-density property on the scheme-theoretic projectivespace P N ˜ K over the residue field ˜ K . As a second application, in the caseof residue characteristic zero we give an ergodic-theoretic equidistribu-tion result for the powers of a point a in the N -dimensional unit torus T NK over K . This is a non-archimedean analogue of a well-known re-sult of Weyl over C , and its proof makes essential use of a theorem ofMordell-Lang type for G Nm due to Laurent. Introduction K be a field which is complete with respect to a nontrivial, non-archimedean absolute value. Given an integer N ≥
1, the N -dimensionalprojective space P N ( K ) is compact (with respect to its Hausdorff analytictopology) if and only if the field K is locally compact, and this occurs onlywhen K has both a discrete value group and a finite residue field. On theother hand, the N -dimensional Berkovich projective space P NK over K isa Hausdorff space which contains the ordinary projective space P N ( K ) asa subspace, and P NK is always compact, regardless of whether or not K islocally compact. Moreover, the Hausdorff topology on P NK is closely relatednot only to the analytic topology on P N ( K ), but also to the Zariski topologyon the scheme-theoretic projective space P N ˜ K over the residue field ˜ K . Forthese and other reasons, there are many situations in which it is preferableto work on the larger space P NK rather than P N ( K ) itself.An example of an analytic notion which is best studied on compact spacesis that of equidistribution. For each integer ℓ ≥
1, let Z ℓ be a finite multisetof points in P NK (a multiset is a set whose points occur with multiplicities),and let µ be a unit Borel measure on P NK . The sequence h Z ℓ i + ∞ ℓ =1 is said tobe µ -equidistributed if the limit(1) lim ℓ → + ∞ | Z ℓ | X z ∈ Z ℓ ϕ ( z ) = Z ϕdµ The author is partially supported by NSF Grant DMS-0901147. holds for all continuous functions ϕ : P NK → R . Many researchers have beenworking recently to establish equidistribution results on Berkovich analyticspaces, often for sequences h Z ℓ i + ∞ ℓ =1 arising naturally from questions in arith-metic geometry and dynamical systems; we mention [1], [3], [10], [12], [13],and [17] to name a few examples of such work.The main result of this paper implies that, in order to establish theequidistribution of a sequence h Z ℓ i + ∞ ℓ =1 of multisets in P NK , it suffices to check(1) for a special class of continuous functions ϕ arising naturally from thegeometric structure of P NK . Since it requires no extra work to do so, we for-mulate our main result using the more general notion of weak convergence ofmeasures on P NK , although our motivating concern is with equidistribution.Moreover, for reasons which we will discuss below, our main result will bestated using nets, rather than sequences; again this requires no extra effort.We give two applications of our main result, both of which establishequidistribution theorems with respect to the Dirac measure δ γ supportedat the Gauss point γ of P NK . Letting P N ˜ K denote the scheme-theoretic pro-jective space over the residue field ˜ K , there exists a natural reduction map r : P NK → P N ˜ K , and the Gauss point γ can be described as the unique point of P NK reducing to the generic point of P N ˜ K . Our first application gives a usefulnecessary and sufficient condition for δ γ -equidistribution, and uses this toestablish the δ γ -equidistribution of nets whose reduction satisfy a certainweak Zariski-density property.Our second application is an ergodic-theoretic equidistribution result forthe sequence formed by taking the powers of a point in the N -dimensionalunit torus T NK = { ( a : a : · · · : a N ) ∈ P N ( K ) | | a | = | a | = · · · = | a N | = 1 } over K . Identifying the group variety G Nm over the residue field ˜ K with thesubvariety of P N defined by x x . . . x N = 0, the reduction map r : P NK → P N ˜ K restricts to a map r : T NK → G Nm ( ˜ K ). A point ˜ a in G Nm ( ˜ K ) is said tobe non-degenerate if it is not contained in any proper algebraic subgroup of G Nm . Theorem 1.
Assume that the residue field ˜ K has characteristic zero. Let a ∈ T NK , and for each integer ℓ ≥ , define Z ℓ = { a, a , . . . , a ℓ } , consid-ered as a multiset in T NK ⊂ P N ( K ) ⊂ P NK of cardinality | Z ℓ | = ℓ . Thesequence h Z ℓ i ∞ ℓ =1 is δ γ -equidistributed in P NK if and only if the point ˜ a isnon-degenerate in G Nm ( ˜ K ) . This theorem is a non-archimedean analogue of a well-known archimedeanequidistribution result of Weyl [22]. Given a point a in the compact unittorus T N C over C , Weyl’s result gives necessary and sufficient conditions forthe Haar-equidistribution of the sets Z ℓ = { a, a , . . . , a ℓ } . Usually stated inits additive (rather than multiplicative) form, this theorem is often presented EAK CONVERGENCE ON BERKOVICH PROJECTIVE SPACE 3 as the first nontrivial example of “uniform distribution modulo 1”. We willgive a more detailed discussion of Weyl’s theorem and related results in § G Nm , due to Laurent [20].This paper is organized as follows: • In § • In § A N +1 K and P NK , and we establish the needed topologicalproperties of these spaces. • In § P NK . • In § § K denotes a field which is complete withrespect to a nontrivial, non-archimedean absolute value | · | . Denote by K ◦ = { a ∈ K | | a | ≤ } the valuation ring of K , by K ◦◦ = { a ∈ K | | a | < } its maximal ideal, and by ˜ K = K ◦ /K ◦◦ its residue field. Given an element a ∈ K ◦ , we denote by ˜ a the image of a under the quotient map K ◦ → ˜ K .Let K be the completion of an algebraic closure of K with respect to theunique extension of | · | ; thus K is both complete and algebraically closed([7] § K ◦ , K ◦◦ , and ˜ K analogously.Let N ≥ K [ X ] = K [ X , X , . . . , X N ] be thepolynomial ring over K in the N + 1 variables X = ( X , X , . . . , X N ). By amultiplicative seminorm on K [ X ] extending | · | we mean a nonnegative real-valued function [ · ] on K [ X ] satisfying [ a ] = | a | for all constants a ∈ K , andsatisfying [ f + g ] ≤ max { [ f ] , [ g ] } and [ f g ] = [ f ][ g ] for all pairs f, g ∈ K [ X ].Given an arbitrary polynomial f ∈ K [ X ], denote by H ( f ) the maximumabsolute value of the coefficients of f . Thus H ( f ) ≤ f isdefined over the valuation ring K ◦ ; in this case we denote by ˜ f ∈ ˜ K [ X ] thereduction of f . If H ( f ) = 1, we say that f is normalized.If the non-archimedean field K has a countable dense subset, then it ispossible to show using the Urysohn metrization theorem that the space P NK is metrizable; see [2] § P NK is not homeomorphic toa metric space. Consequently, notions of convergence in P NK are best studiedusing nets, rather than sequences.Briefly, a net in a set X is a function α x α from a directed set I into X ;it is usually denoted by h x α i , suppressing the dependence on I . A sequencein X is simply a net in X indexed by the directed set N = { , , , . . . } ofpositive integers. In order to distinguish them from arbitrary nets, we willgenerally refer to sequences using the notation h x ℓ i + ∞ ℓ =1 . If the set X is ametric space, then many familiar topological concepts can be reformulated CLAYTON PETSCHE in terms of convergence properties of sequences in X . These results continueto hold when X is an arbitrary Hausdorff space, but only if one takes care toproperly interpret the statements using nets in place of sequences. We referthe reader to Folland [14] § Berkovich Affine and Projective Space A N +1 K over K is defined to be the set ofmultiplicative seminorms on K [ X ] extending | · | . As a matter of notation,we will refer to a point ζ ∈ A N +1 K , and denote by [ · ] ζ its correspondingseminorm. The topology on A N +1 K is defined to be the weakest topologywith respect to which the real-valued functions ζ [ f ] ζ are continuous forall f ∈ K [ X ]. Equivalently, define a family of subsets of A N +1 K by U s,t ( f ) = { ζ ∈ A N +1 K | s < [ f ] ζ < t } for f ∈ K [ X ] and s, t ∈ R . By definition, the subsets U s,t ( f ) generate a baseof open sets for the topology on A N +1 K .To see the relation between A N +1 K and the classical affine space K N +1 ,consider a point a ∈ K N +1 . Letting [ · ] a denote the multiplicative semi-norm defined by evaluation [ f ] a = | f ( a ) | , we obtain a continuous embedding K N +1 ֒ → A N +1 K given by a [ · ] a . We regard this embedding as an inclu-sion K N +1 ⊂ A N +1 K by identifying K N +1 with its image in A N +1 K . Moregenerally, evaluation a [ · ] a defines a map K N +1 → A N +1 K whose image ishomeomorphic to the quotient of K N +1 by the action of Gal( K /K ).The classical affine space K N +1 is always a proper subset of A N +1 K . Forexample, an important class of points in A N +1 K arises by considering poly-discs D ( c, r ) = { a ∈ K N +1 | | c n − a n | ≤ r n for all 0 ≤ n ≤ N } with center c ∈ K N +1 and polyradius r ∈ | K | N +1 . The function [ · ] ζ c,r : K [ X ] → R defined by the supremum [ f ] ζ c,r := sup {| f ( a ) | | a ∈ D ( c, r ) } is amultiplicative seminorm on K [ X ] extending | · | , and therefore it defines apoint of A N +1 K ; it is convenient to denote this point by ζ c,r .2.2. Define a function k · k : A N +1 K → R by k ζ k = max { [ X ] ζ , [ X ] ζ , . . . , [ X N ] ζ } . Observe that k · k is continuous, k ζ k ≥ ζ ∈ A N +1 K , and k ζ k = 0if and only if ζ is the point 0 = (0 , , . . . ,
0) corresponding to the origin in K N +1 ⊂ A N +1 K . For each real number r >
0, define two subsets of A N +1 K by E r = { ζ ∈ A N +1 K | k ζ k ≤ r } U r = { ζ ∈ A N +1 K | k ζ k < r } .U r is clearly open, and in proving the following proposition we will see that E r is compact. EAK CONVERGENCE ON BERKOVICH PROJECTIVE SPACE 5
Proposition 2. A N +1 K is a locally compact Hausdorff space.Proof. To show that A N +1 K is Hausdorff, let ζ, ζ ′ ∈ A N +1 K be distinct points.Then [ f ] ζ = [ f ] ζ ′ for some f ∈ K [ X ]. Selecting disjoint open intervals ( s, t )and ( s ′ , t ′ ) containing [ f ] ζ and [ f ] ζ ′ , respectively, it follows that U s,t ( f ) and U s ′ ,t ′ ( f ) are disjoint open neighborhoods of ζ and ζ ′ , respectively.We will now show that the sets E r are compact; since A N +1 K = ∪ r> U r , itwill follow at once that A N +1 K is locally compact. In order to show that E r iscompact it suffices to show that every net h ζ α i in E r has a subnet convergingto a limit in E r . For each f ∈ K [ X ], there exists a constant C f,r > f ] ζ ≤ C f,r for all ζ ∈ E r . (Each seminorm [ · ] ζ satisfies the ultrametricinequality, so one could take C f,r = H ( f ) r deg( f ) .) Therefore the association ζ [ f ] ζ defines a continuous map E r → [0 , C f,r ]. Letting Π denote theproduct Q f ∈ K [ X ] [0 , C f,r ], we obtain a continuous map ι : E r → Π, whichis injective by the definition of A N +1 K . Since Π is compact (Tychonoff’stheorem, [14] § h ι ( ζ α ) i has a subnet h ι ( ζ β ) i converging to somepoint ( ξ f ) f ∈ K [ X ] ∈ Π. Define a function [ · ] ξ : K [ X ] → R by [ f ] ξ = ξ f . Then[ · ] ξ is a multiplicative seminorm on K [ X ] restricting to | · | on K , and thus itcorresponds to a point ξ ∈ A N +1 K . Moreover, ξ ∈ E r and ζ β → ξ , as desired,completing the proof that E r is compact. (cid:3) N ≥
1. Define an equivalence relation ∼ on A N +1 K \{ } by declaring that ζ ∼ ξ if and only if there exists a constant λ > f ] ζ = λ deg( f ) [ f ] ξ for all homogeneous polynomials f ∈ K [ X ]. TheBerkovich projective space P NK is defined to be the quotient of A N +1 K \ { } modulo ∼ , endowed with the quotient topology; denote by π : A N +1 K \{ } → P NK the quotient map.The embedding K N +1 ֒ → A N +1 K discussed in § K N +1 \{ } → A N +1 K \ { } , which descends modulo ∼ to an embedding P N ( K ) ֒ → P NK . We again regard this map as an inclusion P N ( K ) ⊂ P NK by identifying P N ( K ) with its image in P NK . Similarly, the map K N +1 → A N +1 K descendsmodulo ∼ to a map P N ( K ) ֒ → P NK whose image is homeomorphic to thequotient of P N ( K ) by the action of Gal( K /K ).Consider the subset of A N +1 K defined by S NK = { ζ ∈ A N +1 K | k ζ k = 1 } . Note that S NK is compact, since S NK = E \ U for the compact set E and theopen set U defined in § π : A N +1 K \ { } → P NK remains surjective when restricted to S NK ; wewill still use the notation π : S NK → P NK to refer to this restricted map. Lemma 3.
Given a point z ∈ P NK , there exists a point ζ z ∈ S NK such that π ( ζ z ) = z . CLAYTON PETSCHE
Proof.
Let ζ be an arbitrary point in A N +1 K \ { } such that π ( ζ ) = z . Bysymmetry, we may assume without loss of generality that X N is the coor-dinate at which the maximum k ζ k = [ X N ] ζ is attained; thus [ X N ] ζ = 0.Given f ∈ K [ X ], observe that X ℓN f ( X /X N , X /X N , . . . , X N − /X N ,
1) is apolynomial for all sufficiently large integers ℓ ≥
0. Given such an integer ℓ ,define [ f ] ζ z := [ X N ] − ℓζ [ X ℓN f ( X /X N , X /X N , . . . , X N − /X N , ζ . The value of [ f ] ζ z does not depend on ℓ since [ · ] ζ is multiplicative. Moreover,[ · ] ζ z inherits the axioms of a multiplicative seminorm from [ · ] ζ , and if f ishomogeneous then [ f ] ζ = [ X N ] deg( f ) ζ [ f ] ζ z ; therefore [ · ] ζ z defines an element ζ z ∈ A N +1 K \ { } with π ( ζ z ) = π ( ζ ) = z . Finally, we have [ X N ] ζ z = [1] ζ = 1and [ X n ] ζ z = [ X N ] − ζ [ X n ] ζ ≤ ≤ n ≤ N −
1, whereby k ζ z k = 1,and thus ζ z ∈ S NK as desired. (cid:3) Proposition 4. P NK is a compact Hausdorff space.Proof. The following is a standard result of general topology ([8] § S is a compact Hausdorff space and f : S → S ′ is a surjective quotientmap onto a topological space S ′ , then S ′ is Hausdorff if and only if the set { ( x, y ) | f ( x ) = f ( y ) } is closed in S × S .Applying this result to the map π : S NK → P NK , in order to show that P NK is Hausdorff it suffices to show that the set R = { ( ζ, ξ ) ∈ S NK × S NK | ζ ∼ ξ } is closed in S NK × S NK . To show that R is closed, consider a convergent net h ( ζ α , ξ α ) i in S NK × S NK with ζ α ∼ ξ α for all α ∈ A , and with ( ζ α , ξ α ) → ( ζ, ξ ) ∈ S NK × S NK ; we must show that ζ ∼ ξ . By the definition of ∼ , thereexists a net of positive real numbers h λ α i such that [ f ] ζ α = λ deg( f ) α [ f ] ξ α forall homogeneous f ∈ K [ X ] and all α ∈ A . Since ζ α → ζ and ξ α → ξ , andsince the maps ζ [ f ] ζ are continuous, we deduce that [ f ] ζ α → [ f ] ζ and[ f ] ξ α → [ f ] ξ for all f ∈ K [ X ]. It follows that the net h λ α i converges tothe number λ := ([ f ] ζ / [ f ] ξ ) / deg( f ) for all homogeneous f ∈ K [ X ]. Since R is Hausdorff, the limit λ is unique and therefore independent of f . Since[ f ] ζ = λ deg( f ) [ f ] ξ , we deduce that ζ ∼ ξ . This concludes the proof that R is closed in S NK × S NK , and therefore that P NK is Hausdorff. Since S NK iscompact and π : S NK → P NK is continuous and surjective, P NK must also becompact. (cid:3) § A N +1 K . Theconstruction of P NK via the equivalence relation ∼ on A N +1 K , which is similarto the scheme-theoretic Proj construction, is due to Berkovich himself [5].Baker-Rumely ([2] § N = 1 at length, but forgeneral N ≥ P NK donot seem to have been written out in detail before now. EAK CONVERGENCE ON BERKOVICH PROJECTIVE SPACE 7
The fundamental compactness argument, used here in the proof of Propo-sition 2, is due to Baker-Rumely ([2] Thm. C.3); it is slightly different thanBerkovich’s original argument ([4] Thm. 1.2.1). Naturally, both proofs useTychonoff’s theorem.3.
A Criterion for Weak Convergence C ( P NK ) denote the space of continuous functions P NK → R . Thus C ( P NK ) is a Banach algebra (with respect to the supremum norm). By aBorel measure µ on P NK we mean a positive measure on the Borel σ -algebraof P NK ; we say µ is a unit Borel measure if µ ( P NK ) = 1. Given a net h µ α i of Borel measures on P NK , and and another Borel measure µ on P NK , we saythat µ α → µ weakly if R ϕdµ α → R ϕdµ for all ϕ ∈ C ( P NK ).We will now state and prove the main result of this paper. Given ahomogeneous polynomial f ∈ K [ X ], it follows from the definition of theequivalence relation ∼ that the real-valued function ζ [ f ] ζ / k ζ k deg( f ) on A N +1 K is constant on ∼ -equivalence classes. We may therefore define thefunction λ f : P NK → R λ f ( π ( ζ )) = [ f ] ζ k ζ k deg( f ) . Theorem 5.
Let h µ α i be a net of unit Borel measures on P NK , and let µ be another unit Borel measure on P NK . Then µ α → µ weakly if and only if R λ f dµ α → R λ f dµ for all normalized homogeneous polynomials f ∈ K [ X ] .Proof. The “only if” direction is trivial since each function λ f is continuous.To prove the “if” direction, assume that R λ f dµ α → R λ f dµ for all nor-malized homogeneous polynomials f ∈ K [ X ]. Then in fact this limit musthold for arbitrary nonzero homogeneous f ∈ K [ X ], which is easy to see byscaling f and using the fact that λ cf = | c | λ f for all nonzero c ∈ K .Denote by A ( P NK ) the subspace of C ( P NK ) generated over R by the func-tions of the form λ f : P NK → R for homogeneous f ∈ K [ X ]. Then A ( P NK )is a dense subalgebra of C ( P NK ). To see this, note that A ( P NK ) is closedunder multiplication, since λ f λ g = λ fg , and it is therefore a subalgebra.In order to show that A ( P NK ) is dense in C ( P NK ), it suffices by the Stone-Weierstrass theorem ([14] § A ( P NK ) separates the pointsof P NK . Consider two points z, w ∈ P NK such that λ f ( z ) = λ f ( w ) for allhomogeneous f ∈ K [ X ]. Taking ζ ∈ π − ( z ) and ξ ∈ π − ( w ), we have[ f ] ζ = ( k ζ k / k ξ k ) deg( f ) [ f ] ξ for all homogeneous f ∈ K [ X ], which means that ζ ∼ ξ , whereby z = w . In other words, z = w implies that λ f ( z ) = λ f ( w )for some homogeneous f ∈ K [ X ], showing that A ( P NK ) separates the pointsof P NK , and completing the proof that A ( P NK ) is dense in C ( P NK ).To show that R ϕdµ α → R ϕdµ for any ϕ ∈ C ( P NK ), it suffices by astandard approximation argument verify it for ϕ in a dense subspace of C ( P NK ). By linearity and what we have already shown, one only needs to CLAYTON PETSCHE check it when ϕ = λ f for an arbitrary normalized homogeneous polynomial f ∈ K [ X ], which holds by hypothesis. (cid:3) Remark.
The main ingredient in the proof of Theorem 5, namely the densityof the subalgebra A ( P NK ) in C ( P NK ), is similar in spirit to a density resultof Gubler ([16] Thm. 7.12). Specifically, working over an arbitrary compactBerkovich analytic space X , Gubler considers the space of functions z log k z ) k /m , for integers m ≥ k·k defined on the trivial line bundle O X . After showing that the space of allsuch functions is point-separating and closed under taking maximums andminimums, he appeals to the lattice form of the Stone-Weierstrass theoremto show that this space is dense in the space C ( X ) of continuous functions.Working only over P NK , our density result is rather simpler than Gubler’s.Given a normalized homogeneous polynomial f ∈ K [ X ] of degree d , wemay view it as a section f ∈ Γ( P N , O ( d )), and we may therefore write λ f ( z ) = k f ( z ) k sup where k·k sup is the sup-metric on O ( d ). Taking advantageof the identity λ f λ g = λ fg , we use the multiplicative form of the Stone-Weierstrass theorem to obtain the density of the algebra A ( P NK ) generatedby the functions λ f .3.2. Theorem 5 was stated in terms of the weak convergence of nets ofarbitrary unit Borel measures on P NK , but our principal concern is with themore specific notion of equidistribution. Given a finite multiset Z of pointsin P NK , define a unit Borel measure δ Z on P NK by δ Z = 1 | Z | X z ∈ Z δ z . Here δ z is the unit Dirac measure at z , characterized by the formula R ϕdδ z = ϕ ( z ) for all ϕ ∈ C ( P NK ). Since Z is a multiset, we understand the cardinality | Z | and the sum over z ∈ Z to be computed according to multiplicity. Givena net h Z α i of finite multisets in P NK , and a unit Borel measure on P NK , wesay that the net h Z α i is µ -equidistributed if δ Z α → µ weakly. Corollary 6.
Let h Z α i be a net of finite multisets in P NK , and let µ bea unit Borel measure on P NK . Then h Z α i is µ -equidistributed if and onlyif | Z α | P z ∈ Z α λ f ( z ) → R λ f dµ for all normalized homogeneous polynomials f ∈ K [ X ] . Equidistribution and Reduction r : P N ( K ) → P N ( ˜ K ) be the usual reduction map on the ordinaryprojective space; thus r ( a : a : · · · : a N ) = (˜ a : ˜ a : · · · : ˜ a N ), where homo-geneous coordinates have been chosen so that max {| a | , | a | , . . . , | a N |} = 1.In § r : P NK → P N ˜ K from the Berkovich projective space P NK onto the scheme-theoretic projective space P N ˜ K over the residue field ˜ K . EAK CONVERGENCE ON BERKOVICH PROJECTIVE SPACE 9
Recall that ζ , denotes the point of A N +1 K corresponding to the supre-mum norm on the polydisc D (0 ,
1) in K N +1 with center 0 = (0 , . . . ,
0) andpolyradius 1 = (1 , . . . , § γ be the point of P NK defined by γ = π ( ζ , ), and let δ γ be the unit Dirac measure supported at γ . In this section we will give a useful necessary and sufficient condition for δ γ -equidistribution in terms of the functions λ f . In § δ γ -equidistribution of a net h Z α i in P NK , provided theimage of h Z α i under the reduction map r : P NK → P N ˜ K satisfies a certainweak Zariski-density property.We begin with a well-known lemma which records a basic property of theseminorm [ · ] ζ , . Given a polynomial f ∈ K [ X ], recall that H ( f ) denotesthe maximum absolute value of the coefficients of f . Lemma 7.
The identity [ f ] ζ , = H ( f ) holds for all f ∈ K [ X ] .Proof. This is trivial if f = 0, so we may assume f = 0. Scaling f by anappropriate element of K , we may assume without loss of generality that f is normalized, and thus we must show that [ f ] ζ , = 1. Plainly | f ( a ) | ≤ a ∈ D (0 ,
1) by the ultrametric inequality, whereby [ f ] ζ , ≤
1. Since f hascoefficients in K ◦ , and thus in K ◦ , it reduces to a polynomial ˜ f ( X ) ∈ ˜ K [ X ].Since H ( f ) = 1, the reduced polynomial ˜ f ( X ) is nonzero and therefore isnonvanishing on a nonempty Zariski-open subset of ˜ K N +1 (note that ˜ K isalgebraically closed). Select some ˜ a ∈ ˜ K N +1 such that ˜ f (˜ a ) = 0, and let a ∈ D (0 ,
1) be a point which reduces to ˜ a . Then 1 = | f ( a ) | ≤ [ f ] ζ , ,completing the proof that [ f ] ζ , = 1. (cid:3) Remark.
It follows from Lemma 7 and the multiplicativity of the seminorm[ · ] ζ , that H ( f g ) = H ( f ) H ( g ) for any two polynomials f, g ∈ K [ X ]; thisfact is essentially equivalent to Gauss’s lemma from algebraic number theory.Consequently, ζ , is commonly called the Gauss point of A N +1 K , and likewise γ = π ( ζ , ) the Gauss point of P NK . Theorem 8.
Let h Z α i be a net of nonempty finite multisets in P NK . Then h Z α i is δ γ -equidistributed if and only if the limit (2) lim α |{ z ∈ Z α | λ f ( z ) < t }|| Z α | = 0 holds for each normalized homogeneous polynomial f ∈ K [ X ] and each realnumber < t < .Proof. Given a normalized homogeneous polynomial f ∈ K [ X ], we have 0 ≤ λ f ( z ) ≤ z ∈ P NK (the upper bound following from the ultrametricinequality), and λ f ( γ ) = [ f ] ζ , / k ζ , k deg( f ) = 1 (by Lemma 7). For eachmultiset Z α , define the sum S α ( f ) = 1 | Z α | X z ∈ Z α λ f ( z ) . Plainly 0 ≤ S α ( f ) ≤
1, and it follows from the above observations andCorollary 6 that h Z α i is δ γ -equidistributed if and only if S α ( f ) → f ∈ K [ X ]. In order to prove thetheorem, it therefore suffices to show that S α ( f ) → < t < < t <
1, we have S α ( f ) ≥ t |{ z ∈ Z α | λ f ( z ) ≥ t }|| Z α | = t (cid:18) − |{ z ∈ Z α | λ f ( z ) < t }|| Z α | (cid:19) → t. As 0 < t < S α ( f ) ≤
1, we deduce that S α ( f ) → α |{ z ∈ Z α | λ f ( z ) < t }|| Z α | = ǫ > . for some 0 < t <
1. Passing to a subnet, we may assume that this limsup isactually a limit. Using the fact that 0 ≤ λ f ( z ) ≤ z ∈ P NK , we have S α ( f ) = 1 | Z α | X z ∈ Zα λ f ( z ) Here the affine reduction map S NK → A N +1˜ K \ { p } is given by ζ ˜ ℘ ζ , and tosee that there exists a unique map r : P NK → P N ˜ K completing the commutativediagram (4), it suffices to show that π sch ( ˜ ℘ ζ ) = π sch ( ˜ ℘ ξ ) whenever π ( ζ ) = π ( ξ ); this is straightforward to check using the definitions of π and π sch . Itis also a standard exercise in the definitions to show that the reduction map r is surjective (see [4] § γ = π ( ζ , ) can be characterized as the unique pointof P NK which reduces to the generic point of P N ˜ K . To see this, note that˜ ℘ ζ , is the zero ideal of ˜ K [ X ] by Lemma 7; thus r ( γ ) = π sch ( ˜ ℘ ζ , ) is thegeneric point of P N ˜ K . Conversely, if ζ ∈ S NK and π sch ( ˜ ℘ ζ ) is the generic pointof P N ˜ K , then the ideal ˜ ℘ ζ contains no nonzero homogeneous polynomials in˜ K [ X ]. In other words, λ f ( π ( ζ )) = [ f ] ζ = 1 for all normalized homogeneouspolynomials f ∈ K [ X ]. Since λ f ( γ ) = 1 for all such f as well, and since thefunctions λ f separate the points of P NK , we conclude that π ( ζ ) = γ .4.3. Given a finite multiset Z of points in P NK , define its reduction ˜ Z to bethe finite multiset in P N ˜ K where the multiplicity of a point ˜ z in ˜ Z is the sumof the multiplicities of the points z ∈ r − (˜ z ) in Z . Thus | Z | = | ˜ Z | .Let h ˜ Z α i be a net of nonempty finite multisets in P N ˜ K . We say that thenet h ˜ Z α i is generic if, given any subnet h ˜ Z β i of h ˜ Z α i and any proper Zariski-closed subset W ⊂ P N ˜ K , there exists β such that ˜ Z β ∩ W = ∅ for all β ≥ β .We say that the net h ˜ Z α i is weakly generic if the limit(5) lim α | ˜ Z α ∩ W || ˜ Z α | = 0holds for all proper Zariski-closed subsets W ⊂ P N ˜ K . Note that a generic netis weakly generic: if h ˜ Z α i is generic, then | ˜ Z α ∩ W | = 0 for all sufficientlylarge α , whereby | ˜ Z α ∩ W | / | ˜ Z α | → Theorem 9. Let h Z α i be a net of nonempty finite multisets in P NK . If thereduction h ˜ Z α i is weakly generic in P N ˜ K , then h Z α i is δ γ -equidistributed.Proof. Given a normalized homogeneous polynomials f ∈ K [ X ], let ˜ f ∈ ˜ K [ X ] denote its reduction, and define V ( ˜ f ) = { p ∈ P N ˜ K | ( ˜ f ) ⊆ p } to be the hypersurface in P N ˜ K associated to ˜ f . Given a multiset Z α and apoint z ∈ Z α , by Lemma 3 we may select ζ z ∈ A N +1 K such that π ( ζ z ) = z and k ζ z k = 1. It follows from the assumptions H ( f ) = 1 and k ζ z k = 1, alongwith the ultrametric inequality, that λ f ( z ) = [ f ] ζ z ≤ 1. Moreover, it is easyto check using the definition of the reduction map that λ f ( z ) = [ f ] ζ z < and only if ˜ z = r ( z ) is contained in the hypersurface V ( ˜ f ). Thus(6) |{ z ∈ Z α | λ f ( z ) < }|| Z α | = | V ( ˜ f ) ∩ ˜ Z α || ˜ Z α | . Since h ˜ Z α i is weakly generic in P N ˜ K , the right-hand-side of (6) goes to zero inthe limit. It follows that the limit (2) holds for all normalized homogeneouspolynomials f ∈ K [ X ] and each real number 0 < t < 1, and we concludethat h Z α i is δ γ -equidistributed using Theorem 8. (cid:3) Remark. The converse of Theorem 9 is false. For example, let h x α i be a netin K such that | x α | < α , but such that | x α | → 1. Theorem 8 impliesthat the net h Z α i of singleton sets Z α = { (1 : x α ) } is δ γ -equidistributed. Buteach point (1 : x α ) reduces to the same point (1 : 0) in P ( ˜ K ), so h ˜ Z α i isnot weakly generic in P K .In practice, however, the converse of Theorem 9 may hold for certainclasses of nets h Z α i of particular interest. We will see an example of this inthe proof of Theorem 11.5. A Ergodic Equidistribution Theorem T NK in P N ( K ) by T NK = { ( a : a : · · · : a N ) ∈ P N ( K ) | | a | = | a | = · · · = | a N | = 1 } . Note that T NK is a group under coordinate multiplication, with neutral el-ement 1 = (1 : · · · : 1). In this section we will prove an equidistributionresult, in the case of residue characteristic zero, for the sequence h a ℓ i + ∞ ℓ =1 formed by taking the powers of a point a ∈ T NK .5.2. We begin with some algebraic preliminaries. Let k be an arbitrary fieldof characteristic zero, and fix homogeneous coordinates ( x : x : · · · : x N )on P N over k . We identify the group variety G Nm over k with the subvarietyof P N defined by x x . . . x N = 0; the group law on G Nm ( k ) is given bycoordinate multiplication, with neutral element 1 = (1 : · · · : 1).Given an arbitrary point a ∈ G Nm ( k ), we denote by a , . . . , a N the uniqueelements of k × such that a = (1 : a : · · · : a N ). We say that a is degenerateif the elements a , . . . , a N are multiplicatively dependent in k × ; otherwisewe say that a is non-degenerate.Consider a subgroup Λ of the integer lattice Z N . The group Λ gives riseto an algebraic subgroup G Λ of G Nm by G Λ ( k ) = { a ∈ G Nm ( k ) | a ℓ . . . a ℓ N N = 1 for all ( ℓ , . . . , ℓ N ) ∈ Λ } . Conversely, all algebraic subgroups of G Nm arise in this way ([6] § a ∈ G Nm ( k ) is degenerate if andonly if a is contained in some proper algebraic subgroup of G Nm . Moreover,it is clear that a is degenerate if and only if a ℓ is degenerate for all nonzero ℓ ∈ Z . EAK CONVERGENCE ON BERKOVICH PROJECTIVE SPACE 13 Proposition 10. A point a ∈ G Nm ( k ) is non-degenerate if and only if, foreach proper Zariski-closed subset W of G Nm ( k ) , there exist only finitely manyintegers ℓ such that a ℓ ∈ W .Proof. The “if” direction is trivial. For if a is degenerate then a ∈ G ( k ) forsome proper algebraic subgroup G of G Nm , and therefore a ℓ ∈ G ( k ) for all ℓ ∈ Z .To prove the “only if” direction, assume that a is non-degenerate, andconsider a proper Zariski-closed subset W of G Nm ( k ). Denote by a Z thecyclic subgroup of G Nm ( k ) generated by a . Since a is non-degenerate, itis non-torsion, and therefore in order to complete the proof it is enoughto show that a Z ∩ W is finite. Replacing W with the Zariski-closure of a Z ∩ W , we may assume without loss of generality that a Z ∩ W is Zariski-dense in W . A result of Laurent ([20]; see also [6] Thm. 7.4.7) implies that,since a Z ∩ W is Zariski-dense in W , we must have W = ∪ Jj =1 y j G j for somefinite set y , . . . , y J ∈ G Nm ( k ) and some finite collection G , . . . , G J of properalgebraic subgroups of G Nm . Therefore, in order to show that a Z ∩ W isfinite, it suffices to show that a Z ∩ yG ( k ) is finite for an arbitrary y ∈ G Nm ( k )and an arbitrary proper algebraic subgroup G of G Nm . In fact, a Z ∩ yG ( k )can contain at most one point. For if a ℓ and a ℓ ′ are elements of yG ( k ) forsome distinct integers ℓ, ℓ ′ ∈ Z , then a ℓ − ℓ ′ ∈ G ( k ), implying that a ℓ − ℓ ′ isdegenerate. This contradicts the assumption that a is non-degenerate. (cid:3) Remark. The result of Laurent used in this proof is the G Nm case of what iscommonly called the Lang (sometimes Mordell-Lang) conjecture. It is nowa much more general theorem, holding for finite-rank subgroups of semi-abelian varieties, due to Laurent, Faltings, Vojta, and McQuillen; see [19] § F.1.1 for a survey.5.3. We return to our complete non-archimedean field K . Observe that,given a point a ∈ P N ( K ), we have a ∈ T NK if and only if ˜ a ∈ G Nm ( ˜ K ). Theorem 11. Assume that the residue field ˜ K has characteristic zero. Let a ∈ T NK , and for each integer ℓ ≥ , define Z ℓ = { a, a , . . . , a ℓ } , consid-ered as a multiset in T NK ⊂ P N ( K ) ⊂ P NK of cardinality | Z ℓ | = ℓ . Thesequence h Z ℓ i ∞ ℓ =1 is δ γ -equidistributed in P NK if and only if the point ˜ a isnon-degenerate in G Nm ( ˜ K ) .Proof. Assume that ˜ a is non-degenerate in G Nm ( ˜ K ). By Proposition 10, ˜ a Z ∩ W is finite for any proper Zariski-closed subset W of P N ( ˜ K ), which impliesthat the sequence h ˜ Z ℓ i + ∞ ℓ =1 is weakly generic in P N ( ˜ K ). By Theorem 9, weconclude that h Z ℓ i ∞ ℓ =1 is δ γ -equidistributed.Conversely, suppose that ˜ a is degenerate in G Nm ( ˜ K ). Writing a = (1 : a : · · · : a N ) ∈ T NK ⊂ P N ( K )the fact that ˜ a is degenerate means that ˜ a ℓ . . . ˜ a ℓ N N = 1 in ˜ K for a nonzerovector ( ℓ , . . . , ℓ N ) ∈ Z N . Therefore the element A := a ℓ . . . a ℓ N N − ∈ K satisfies | A | < 1. Select an integer r ≥ ℓ n + r ≥ ≤ n ≤ N , let ℓ = P Nn =1 ℓ n , and define f ∈ K [ X ] by f ( X ) = X ℓ + r . . . X ℓ N + rN − X ℓ ( X . . . X N ) r . Note that f is nonzero, homogeneous, and satisfies H ( f ) = 1. In particular,(7) Z λ f dδ γ = λ f ( γ ) = [ f ] δ , = H ( f ) = 1 . On the other hand, it is easy to check that | f (1 , a , . . . , a N ) | = | A | , and thatmore generally | f (1 , a j , . . . , a jN ) | ≤ | A | for all integers j ≥ 1. Therefore(8) 1 | Z ℓ | X z ∈ Z ℓ λ f ( z ) = 1 ℓ ℓ X j =1 | f (1 , a j , . . . , a jN ) | ≤ | A | < . Letting ℓ → + ∞ , (7) and (8) together show that the sequence h Z ℓ i ∞ ℓ =1 failsthe criterion for δ γ -equidistribution stated in Corollary 6. (cid:3) T N C in P N ( C ) by T N C = { ( a : a : · · · : a N ) ∈ P N ( C ) | | a | = | a | = · · · = | a N | = 1 } . Then T N C is a compact topological group, and as such it carries a uniquenormalized Haar measure. Theorem 12 (Weyl) . Let a ∈ T N C , and for each integer ℓ ≥ , define Z ℓ = { a, a , . . . , a ℓ } , considered as a multiset in T N C of cardinality | Z ℓ | = ℓ .The sequence h Z ℓ i ∞ ℓ =1 is Haar-equidistributed in T N C if and only if a is non-degenerate in G Nm ( C ) . An important difference between Theorems 11 and 12 stems from the factthat, in the non-archimedean case, the assumption that char( ˜ K ) = 0 ensuresthat ˜ K has infinitely many elements, which implies that the field K is notlocally compact. Consequently, the unit torus T NK is noncompact and thushas no Haar measure in the traditional sense. On the other hand, observethat T NK is contained in the compact Berkovich unit torus T NK = { π ( ζ ) | ζ ∈ A N +1 K and [ X ] ζ = [ X ] ζ = · · · = [ X N ] ζ = 1 } , and that the Dirac measure δ γ supported on the Gauss point γ ∈ T NK isinvariant under the translation action of the group T NK on T NK . Thus δ γ is anatural substitute for Haar measure in this setting.Analogues of Weyl’s Haar-equidistribution result have been investigatedover the locally compact non-archimedean field Q p , at least in the case N = 1; see Bryk-Silva [9] and Coelho-Parry [11].As pointed out by the referee, several things can be said in the directionof Theorem 11 when the residue field ˜ K has characteristic p = 0. First, the“only if” direction of the theorem continues to hold, with the same proof. If EAK CONVERGENCE ON BERKOVICH PROJECTIVE SPACE 15 ˜ K is algebraic over its prime field F p , then the statement of Theorem 11 holdstrivially, since all points of G Nm ( F p ) are torsion and therefore degenerate.Finally, the statement of Theorem 11 continues to hold for an arbitrary field K of residue characteristic p = 0 in the one-dimensional case. For observethat an element ˜ a in G m ( ˜ K ) is degenerate if and only if it is torsion. Usingthe trivial fact that the cyclic subgroup ˜ a Z of G m ( ˜ K ) is either finite orZariski-dense, there is no need for Laurent’s theorem, and therefore no needto assume that char( ˜ K ) = 0. We do not know whether the statement ofTheorem 11 holds for residue characteristic p = 0 and N ≥ X be a compact Hausdorff space, let T : X → X be an automor-phism, let µ be a T -invariant unit Borel measure on X , and let x ∈ X be apoint. One of the basic goals of ergodic theory is to establish conditions un-der which the sequence h Z ℓ i + ∞ ℓ =1 of multisets Z ℓ = { T ( x ) , T ( x ) , . . . , T ℓ ( x ) } is µ -equidistributed; see for example Furstenberg [15] and Lindenstrauss[21] for discussions of such results with a particular eye toward arithmeticapplications.Weyl’s original proof of Theorem 12 uses Fourier analysis, but there existsan alternate, ergodic-theoretic proof, see Furstenberg [15] Ch. 3. It wouldbe interesting to pursue non-archimedean equidistribution results such asTheorem 11 from an ergodic-theoretic angle. In view of Proposition 10 andit’s reliance on Laurent’s theorem [20], it is especially intriguing to considerthe possibility of deeper connections between ergodic theory and questionsof Mordell-Lang type. References [1] M. Baker and C. Petsche. Global discrepancy and small points on elliptic curves. Int.Math. Res. Not. 61 (2005) 3791-3834.[2] M. Baker and R. Rumely. Potential Theory and Dynamics on the Berkovich ProjectiveLine . AMS Surveys and Mathematical Monographs series, to appear.[3] M. Baker and R. Rumely, Equidistribution of small points, rational dynamics, andpotential theory, Ann. Inst. Fourier (Grenoble) 56 no. 3 (2006) 625–688.[4] V. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields ,AMS Mathematical Surveys and Monographs 33 (AMS, Providence, 1990).[5] V. Berkovich, The automorphism group of the Drinfeld half-plane, C. R. Acad. Sci.Paris S´er. I Math. 321 (1995), no. 9, 1127-1132.[6] E. Bombieri and W. Gubler, Heights in Diophantine Geometry , Cambridge UniversityPress, New York, 2006.[7] S. Bosch, U. G¨unzter, and R. Remmert. Non-Archimedean Analysis , Springer-Verlag,Berlin, 1984.[8] N. Bourbaki. Elements of Mathematics: General Topology Part I. Hermann, Paris,1966.[9] J. Bryk and C. Silva, Measurable dynamics of simple p-adic polynomials, Amer. Math.Monthly Vol. 112 (2005), no. 3, 212-232.[10] A. Chambert-Loir, Mesures et ´equidistribution sur les espaces de Berkovich, J. f¨urdie reine und angewandte Mathematik 595 (2006) 215-235. [11] Z. Coelho and W. Parry, Ergodicity of p-adic multiplications and the distributionof Fibonacci numbers, in Topology, Ergodic Theory, Real Algebraic Geometry, Amer.Math. Soc. Transl. Ser. 2, no. 202, Providence (2001) pp. 51-70.[12] X.W.C. Faber. Equidistribution of dynamically small subvarieties over the functionfield of a curve. Acta Arith. 137 (2009) 4, 345-389.[13] C. Favre and J. Rivera-Letelier, ´Equidistribution quantitative des points de petitehauteur sur la droite projective, Math. Ann. (2) 335 (2006) 311361.[14] G. Folland. Real Analysis: Modern Techniques and their Applications (2nd ed.) JohnWiley, 1999.[15] H. Furstenberg. Recurrence in Ergodic Theory and Combinatorial Number Theory Vol. 201 of Graduate Texts in Mathematics . Princeton University Press, 1981.[16] W. Gubler. Local heights of subvarieties over non-archimedean fields. J. reine agnew.Math. 498 (1998) 61-113.[17] W. Gubler. Equidistribution over function fields. Manuscripta Math. 127 (2008) 4,485-510.[18] R. Hartshorne. Algebraic Geometry. Springer-Verlag, New York, 1977. GraduateTexts in Mathematics, No. 52.[19] M. Hindry and J. H. Silverman. Diophantine Geometry: an Introduction Vol. 201 of Graduate Texts in Mathematics . Springer-Verlag, New York, 2000.[20] M. Laurent, ´Equations diophantiennes exponentielles, Invent. Math. 78 (1984), no.2, 299-327.[21] E. Lindenstrauss. Some examples how to use measure classification in number theory,in Equidistribution in number theory, an introduction , NATO Sci. Ser. II Math. Phys.Chem., v. 237, pp. 261-303. Springer, Dordrecht, 2007.[22] H. Weyl, ¨Uber die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916)313-352. Clayton Petsche; Department of Mathematics and Statistics; Hunter Col-lege; 695 Park Avenue; New York, NY 10065 U.S.A. E-mail address ::