Density of orbits of endomorphisms of abelian varieties
aa r X i v : . [ m a t h . N T ] D ec DENSITY OF ORBITS OF ENDOMORPHISMS OFABELIAN VARIETIES
DRAGOS GHIOCA AND THOMAS SCANLON
Abstract.
Let A be an abelian variety defined over ¯ Q , and let ϕ bea dominant endomorphism of A as an algebraic variety. We prove thateither there exists a non-constant rational fibration preserved by ϕ , orthere exists a point x ∈ A ( ¯ Q ) whose ϕ -orbit is Zariski dense in A . Thisprovides a positive answer for abelian varieties of a question raised byMedvedev and the second author in [MS14]. We prove also a strongerstatement of this result in which ϕ is replaced by any commutativefinitely generated monoid of dominant endomorphisms of A . Introduction
The following conjecture was raised in [MS14, Conjecture 7.14] (motivatedby a conjecture of Zhang [Zha10] for polarizable endomorphisms of projectivevarieties).
Conjecture 1.1.
Let K be a field of characteristic , let K be an algebraicclosure of K , let X be an irreducible algebraic variety defined over K , andlet ϕ : X −→ X be a dominant rational self-map. We suppose there existsno positive dimensional algebraic variety Y and dominant rational map f : X −→ Y such that f ◦ ϕ = f . Then there exists x ∈ X ( K ) whose forward ϕ -orbit is Zariski dense in X . We denote by O ϕ ( x ) the forward ϕ -orbit, i.e. the set of all ϕ n ( x ) for n ≥ ϕ n we denote the n -th compositional power of ϕ . Conjecture 1.1was proven in [MS14, Theorem 7.16] in the special case X = A m , and ϕ := ( f , . . . , f m ) is given by the coordinatewise action of m one-variablepolynomials f i . In this paper we prove Conjecture 1.1 when X is an abelianvariety. This is the fourth known case of Conjecture 1.1 (besides the caseproven by Medvedev and the second author in [MS14], Amerik, Bogomolovand Rovinsky [ABR11] exploited the local dynamical behaviour of the map ϕ to prove a special case of Conjecture 1.1 assuming there is a good p -adicanalytic parametrization for the orbit O ϕ ( x ), and recently, the case when X is a surface was proven in [BGT] using also p -adic methods). D.G. is partially supported by an NSERC grant. T.S. is partially supported by NSFGrant DMS-1363372. This material is based upon work supported by the National ScienceFoundation under Grant No. 0932078 000 while the authors were in residence at theMathematical Sciences Research Institute in Berkeley, California, during the Spring 2014semester.
Theorem 1.2.
Let K be a field of characteristic , and let K be a fixedalgebraic closure of K . Let A be an abelian variety defined over K , andlet σ : A −→ A be a finite map of agebraic varieties. Then the followingstatements are equivalent: (1) there exists x ∈ A ( K ) such that O σ ( x ) is Zariski dense in A . (2) there exists no non-constant rational map f : A −→ P such that f ◦ σ = f . The motivation for Conjecture 1.1 comes from two different directions.First, Zhang [Zha10, Conjecture 4.1.6] proposed a variant of Conjecture 1.1for polarizable endomorphisms ϕ of projective varietieties X defined over ¯ Q (we say that ϕ is polarizable if there exists an ample line bundle L on X so that ϕ ∗ ( L ) ∼ → L ⊗ d for some integer d > ϕ preserves no non-constant fibration of X . The motivation for thestronger hypothesis appearing in [Zha10, Conjecture 4.1.6] lies in the factthat in his seminal paper [Zha10], Zhang was interested in the arithmeticproperties exhibited by the dynamics of endomorphisms of projective vari-eties. In particular, Zhang was interested in formulating good dynamicalanalogues of the classical Manin-Mumford and Bogomolov Conjectures, andthus he wanted to use the canonical heights associated to polarizable en-domorphisms (previously introduced by Call and Silverman [CS93]). Thesecond motivation for Conjecture 1.1 comes from the fact that its conclusionis known assuming K is an uncountable field of characteristic 0 (see [AC08]).More precisely, in [AC08], Amerik and Campana proved that if ϕ preservesno non-constant rational fibration, then there exist countably many propersubvarieties Y i of X so that for each x ∈ X ( K ) \ ∪ i Y i ( K ), the orbit O ϕ ( x )is Zariski dense in X . However, if K is countable, then the result of Amerikand Campana leaves open the possibility that each algebraic point of X isalso an algebraic point of some subvariety Y i for some positive integer i .Hence, Conjecture 1.1 raises a deeper arithmetical question.We are able to extend Theorem 1.2 to the action of any commutativefinitely generated monoid of dominant endomorphisms of an abelian variety.For a monoid S of endomorphisms of an abelian variety A , and for any point x ∈ A , we let O S ( x ) be the S -orbit of x , i.e. the set of all ψ ( x ), where ψ ∈ S . Theorem 1.3.
Let K be a field of characteristic , let K be an algebraicclosure of K , and let S be a finitely generated, commutative monoid of dom-inant endomorphisms of an abelian variety A defined over K . Then eitherthere exists x ∈ A ( K ) such that O S ( x ) is Zariski dense in A or there exists anon-constant rational map f : A −→ P such that f ◦ σ = f for each σ ∈ S . It is reasonable to formulate an extension of Conjecture 1.1 to the settingof a monoid action of rational self-maps on an algebraic variety X . However,there are several additional complications arising from such a generalization ENSITY OF ORBITS OF ENDOMORPHISMS OF ABELIAN VARIETIES 3 even in the case of the dynamics of endomorphisms of an abelian variety A ,such as:(i) Should we impose any restriction on the monoid S ? Theorem 1.3is valid only for finitely generated, commutative monoids, and ourmethod of proof does not seem to extend beyond this case (at leastnot in the case of arbitrary endomorphisms of an abelian variety A ; if S is an arbitrary commuting monoid of dominant group endo-morphisms of A , then the conclusion of Theorem 1.3 holds easily).(ii) Assuming there is no non-constant fibration preserved by the entiremonoid S , is it true that there exists some σ ∈ S and there exists x ∈ A ( K ) such that O σ ( x ) is Zariski dense in A ? We have examples ofnon-commuting monoids S generated by two group homomorphismsof A such that there is no non-constant fibration preserved by S ,even though for each σ ∈ S there exists a non-constant fibrationpreserved by σ . On the other hand, if S is a commutative monoidof group homomorphisms of A , then it is easy to see that the abovequestion has a positive answer.Finally, we note that Amerik-Campana’s result [AC08] was extended in [BGZ]for arbitrary monoids S acting on an algebraic variety X through dominantrational endomorphisms, i.e. if there is no non-constant rational fibrationpreserved by S , then there exist countably many proper subvarieties Y i ⊂ X such that for each x ∈ X ( K ) \ ∪ i Y i ( K ), the orbit O S ( x ) is Zariski dense in X . Again, similar to [AC08], the result of [BGZ] leaves open the possiblitythat if K is countable, then X ( K ) may be covered by ∪ i Y i ( K ).Here is the strategy for our proof. By the classical theory of abelian va-rieties, we know that each endomorphism ϕ of an abelian variety A is ofthe form T y ◦ τ (for a translation map T y , with y ∈ A ) and some (algebraic)group homomorphism τ . Since the endomorphisms ϕ from the given monoid S commute with each other, we obtain that also the corresponding grouphomomorphisms τ commute with each other. This gives us a lot of controlon the action of the corresponding group homomorphisms τ ; in particular, ifall endomorphisms from S would also be group homomorphisms, then The-orem 1.3 would follow easily. Essentially, in that special case, the problemwould reduce to the following dichotomy: either there exists a positive di-mensional algebraic subgroup of A which is fixed by a finite index submonoidof S , or there exists a single element σ of S , and there exists an algebraicpoint x of A whose σ -orbit is Zariski dense in A (essentially, such a point x has the property that the cyclic subgroup generated by x is Zariski densein A ). So, if S consists only of group homomorphisms, the conclusion ofTheorem 1.3 holds even in a stronger form. However, if the endomorphismsfrom S are not all group endomorphisms of A , then the proof is much morecomplicated. One can still find a necessary and sufficient condition underwhich there exists a non-constant rational fibration preserved by all elementsin S , but that condition is very technical. Finally, we note that the exact DRAGOS GHIOCA AND THOMAS SCANLON same proof works to prove a variant of Theorem 1.3 with the abelian variety A replaced by a power of the torus. On the other hand, our proof does notseem to generalize to the case of semiabelian varieties due to the failure ofthe Poincar´e Reducibility Theorem (see Fact 3.2) for semiabelian varietieswhich are not isogenuous to split semiabelian varieties.The plan of the paper is as follows. In Section 2 we note several easystatements regarding monoids. We continue by stating some basic factsabout abelian varieties in Section 3. Then, in Section 4 and then in Sec-tion 5 we prove various reductions of the Theorem 1.3, respectively someauxilliary results needed later. In Section 6 we prove Theorem 1.2 as a wayto introduce the reader to the more elaborate argument needed for the proofof Theorem 1.3 (which is completed in Section 7). While Theorem 1.2 is aspecial case of Theorem 1.3, we have chosen to prove them separately be-cause we believe it is easier for the reader to read first the argument donefor a cyclic monoid (Theorem 1.2), which avoids some of the technicalitiesappearing in the proof of Theorem 1.3. Acknowledgments.
We thank Jason Bell, Thomas Tucker and ZinovyReichstein for several useful discussions while writing this paper.2.
General results regarding monoids
We need some basic facts about finitely generated, commutative monoids.First we need a definition.
Definition 2.1.
Let S be any finitely generated, commutative monoid. Foreach submonoid T ⊆ S , we denote by ¯ T the submonoid containing all x ∈ S with the property that there exist y, z ∈ T such that xy = z . We also recall that a monoid S is called cancellative if whenever xy = xz for x, y, z ∈ S , then y = z . We note that a monoid of dominant endomor-phisms of a given algebraic variety is a cancellative monoid. Lemma 2.2.
Let S be a cancellative, commutative monoid generated by theelements γ , . . . , γ s , and let T be a submonoid of S such that ¯ T = S . Thenthere exists a finitely generated submonoid T ⊂ T and there exists a positiveinteger n such that γ ni ∈ ¯ T for each i = 1 , . . . , s .Proof. Let f : N s −→ S be the homomorphism of monoids given by f ( e i ) = γ i , where e i ∈ N s is the s -tuple consisting only of zeros with the exceptionof the i -th entry which equals 1. Let U be the set of all a ∈ N s such that f ( a ) ∈ T , and let H be the subgroup of Z s generated by U . Since ¯ T = S ,then H = Z s . Therefore there exist s linearly independent tuples in U ; callthem u , . . . , u s . We claim that the monoid T spanned by f ( u ) , . . . , f ( u s )satisfies the conclusion of our Lemma.Indeed, we first show that ¯ T = f ( H ∩ N s ) where H is the subgroupof Z s generated by u , . . . , u s . To see this, on one hand, it is clear that¯ T ⊆ f ( H ∩ N s ). Now, to see the reverse inclusion, note that ¯ T satisfies ENSITY OF ORBITS OF ENDOMORPHISMS OF ABELIAN VARIETIES 5 ¯¯ T = ¯ T . Indeed, if x , x ∈ ¯ T and x ∈ S such that xx = x , we showthat x ∈ ¯ T . We have that there exist y i , z i ∈ T such that x i y i = z i for i = 1 ,
2. Then we claim that x ( y z ) = y z which would indeed showthat x ∈ ¯ T because y z , y z ∈ T (note that T is a submonoid). To seethe above equality in the cancellative monoid S , it suffices to prove that x xy z = x y z . Using that x x = x , x y = z and x y = z , and that S is commutative, we obtain the desired equality; hence ¯¯ T = ¯ T and thus¯ T = f ( H ∩ N s ).Now, since u , . . . , u s are linearly independent over Z (as elements of Z s ),then H has finite index in Z s . So, there exists a positive integer n suchthat ne i ∈ H for each i = 1 , . . . , s , and therefore f ( ne i ) = γ ni ∈ ¯ T . (cid:3) We also need some simple results from linear algebra. The first is aconsequence of the Lie-Kolchin triangularization theorem [Ko48].
Fact 2.3.
Let S be a finitely generated, commuting monoid of matrices withentries in ¯ Q . Then there exists an invertible matrix C (with entries in ¯ Q )such that for each A ∈ S , the matrix C − AC is upper triangular. Fact 2.3 will be used repeatedly throughout our proof. An importantconsequence of it is that the eigenvalues of each matrix in a commutingmonoid S are simply the entries on the diagonal (after a suitable change ofcoordinates). In particular, this has the following easy corollaries. Lemma 2.4.
Let S be a commuting monoid of matrices with entries in ¯ Q , generated by matrices A , . . . , A s . Then there exists a positive integer n such that for each matrix A contained in the submonoid of S generated by A n , . . . , A ns , if λ is an eigenvalue of A which is also a root of unity, then λ = 1 .Proof. The conclusion holds with n being the cardinality of the group ofroots of unity contained in the number field L which is generated by all theeigenvalues of the matrices A i . (cid:3) Lemma 2.5.
Let S be a finitely generated, commuting monoid of matriceswith the property that for each matrix A in S , if λ is an eigenvalue of A which is a root of unity, then λ = 1 . Let U be the set of matrices in S withthe property that the eigenspace corresponding to the eigenvalue has thesmallest dimension among all the matrices in S . Let U be the submonoidgenerated by U . Then ¯ U = S .Proof. Using Fact 2.3, we can choose a basis so that each matrix in S isrepresented by an upper triangular matrix. Furthermore, we may assumeeach matrix in U has the first r entries on the diagonal equal to 1, and noneof the other entries on the diagonal are equal to 1 (or to a root of unity).Indeed, we know each matrix in U has r entries on the diagonal equal to 1;if these entries equal to 1 would not be in the same places of the diagonalfor two distinct matrices A and B in U , then for some positive integers m and n we would have that A m B n has fewer than r entries equal to 1 on the DRAGOS GHIOCA AND THOMAS SCANLON diagonal. So, indeed the r entries equal to 1 appear in the same positionon the diagonal for each matrix in U ; so we may assume they are the first r entries, while the remaining ℓ − r entries on the diagonal of each matrix in U is not a root of unity.Let A ∈ U . Now, for each matrix B ∈ S , even if there exist entries in thepositions i = r + 1 , . . . , ℓ on the diagonal which are equal to 1, there existsa positive integer n such that the entries on the diagonal of A n B in thepositions i = r + 1 , . . . , ℓ are not equal to 1. This completes our proof. (cid:3) Abelian varieties
First we recall several results regarding abelian varieties (see [Mil] for moredetails). The setup will be as follows: A is an abelian variety defined over afield K of characteristic 0; since one needs only finitely many parameters inorder to define A , then we may assume K is a finitely generated extensionof Q . We let K be a fixed algebraic closure of K . At the expense ofreplacing K by a finite extension we may assume that all algebraic groupendomorphisms of A are defined over K ; we denote by End( A ) the ring ofall these endomorphisms. Since the torsion subgroup C tor of any algebraicsubgroup C ⊆ A is Zariski dense in C , we conclude that any algebraicsubgroup of A is defined over K ( A tor ). Frequently we will use the followingfacts. Fact 3.1.
Let B and C be algebraic subgroups of the abelian variety A . Then ( B + C ) = (cid:0) B + C (cid:1) , where for any algebraic subgroup H ⊆ A , we denoteby H the connected component of H (containing ∈ A ).Proof. The algebraic group B + C is the image of the connected group B × C under the sum map and is therefore connected. As B × C hasfinite index in B × C , its image under the sum map has finite index in B + C .Hence, B + C = ( B + C ) . (cid:3) The following result is proven in [Mil, Proposition 10.1].
Fact 3.2 (Poincar´e’s Reducibility Theorem) . If B ⊆ A is an abelian subvari-ety of A , then there exists an abelian subvariety C ⊆ A such that A = B + C and B ∩ C is finite; in particular A/B and C are isogenous. Poincar´e’s Reducibility Theorem yields that any abelian variety is isoge-nous with a direct product of finitely many simple abelian varieties, i.e. A ∼ → A := Q ri =1 C k i i , where each C i is simple. Then End( A ) ∼ → End( A )(see also [Mil, Section 1.10]), and moreover End( A ) ∼ → Q ri =1 M k i ( R i ),where M k i ( R i ) is the ring of all k i -by- k i matrices with entries in the ring R i := End( C i ). For any simple abelian variety C , the ring R := End( C ) isa finite integral extension of Z . Therefore we have the following fact. Fact 3.3.
Let A be an abelian variety defined over a field of characteristic .For each algebraic group endomorphism φ : A −→ A there exists a minimalmonic polynomial f ∈ Z [ t ] of degree at most A ) such that f ( φ ) = 0 . ENSITY OF ORBITS OF ENDOMORPHISMS OF ABELIAN VARIETIES 7
The following result is proven in [Mil, Corollary 1.2].
Fact 3.4 (Rigidity Theorem) . Each endomorphism ψ : A −→ A is of theform T y ◦ φ for some y ∈ A , where T y : A → A is the translation map x x + y , and φ ∈ End( A ) is an algebraic group endomorphism. Inparticular, if ψ is a finite map, then φ : A −→ A is an isogeny. Furthermore,the pair ( T y , φ ) is uniquely determined by ψ . As a simple consequence of Fact 3.4, we obtain.
Lemma 3.5.
Let ψ , ψ : A −→ A be endomorphisms of the form ψ i := T y i ◦ ϕ i (for i = 1 , ) where ϕ i : A −→ A are group endomorphisms. If ψ ◦ ψ = ψ ◦ ψ , then ϕ ◦ ϕ = ϕ ◦ ϕ . The following result is an immediate application of the structure theoremfor the ring of group endomorphisms of an abelian variety.
Fact 3.6.
Let S be a finitely generated commutative monoid of endomor-phisms of an abelian variety A as an algebraic variety. Then for each point x ∈ A , there exists a finitely generated subgroup Γ ⊂ A containing O S ( x ) .Proof. Let { σ , . . . , σ s } be a set of generators for S . For each i = 1 , . . . , s ,we let γ i := T y i ◦ τ i for some translations T y i (where y i ∈ A ) and somegroup endomorphisms τ i . Let d := dim( A ). Then, by Fact 3.3, for each i = 1 , . . . , s , there exist integers c i,j such that τ di + c i, d − τ d − i + · · · + c i, τ i + c i, · id = 0 , where id always represents the identity map. Then O S ( x ) is contained inthe subgroup Γ ⊂ A generated by γ ( x ) , γ ( y ) , . . . , γ ( y s ), where γ variesamong the finitely many elements of S of the form γ := γ m ◦ · · · γ m s s , with0 ≤ m i < d , for each i = 1 , . . . , s . (cid:3) The next result is a relatively simple application of Fact 3.2.
Lemma 3.7.
Let B ⊆ A be an algebraic subgroup of the abelian variety A .Then B = A if and only there exists a nonzero algebraic group endomor-phism ψ : A −→ A such that ψ ( B ) = { } .Proof. Clearly, if B = A , then there exists no nonzero endomorphism ψ of A such that ψ ( B ) = { } . Now, assume B = A . We note that it suffices toprove the existence of ψ ∈ End( A ) such that B ⊆ ker( ψ ), where B is theconnected component of B containing 0 for if B ⊆ ker( ψ ) and N := [ B : B ]is the index of B in B , then B ⊆ ker( φ ) where φ = [ N ] · ψ . So, from now onassume B is an abelian subvariety of A . We consider π : A −→ A/B be thecanonical quotient map. By Fact 3.2, we obtain that there exists an abeliansubvariety C ⊆ A and an isogeny τ : A/B −→ C . So, letting ι : C −→ A be the canonical injection map, we get that ψ := ι ◦ τ ◦ π : A −→ A is anendomorphism with the property that ψ ( B ) = { } . We claim that ψ = 0.Indeed, by construction, the image of ψ is C which is a positive dimensionalvariety (since B = A ). (cid:3) DRAGOS GHIOCA AND THOMAS SCANLON
The following result is the famous consequence of Mordell-Lang Conjec-ture proven by Faltings [Fal94].
Fact 3.8 (Faltings’ theorem; Mordell-Lang Conjecture) . Let V ⊂ A be anirreducible subvariety with the property that there exists a finitely generatedsubgroup Γ ⊆ A ( K ) such that V ( K ) ∩ Γ is Zariski dense in V . Then V is acoset of an abelian subvariety of A . We will also employ the following easy result.
Lemma 3.9.
Let A be an abelian variety. If x ∈ A is a point generatinga cyclic group which is Zariski dense in A , then for each positive integer ℓ ,the cyclic group generated by ℓx is Zariski dense in A .Proof. Let H be the Zariski closure of the cyclic group generated by ℓx ;then H is an algebraic subgroup of A . Furthermore, because the cyclicgroup generated by x is Zariski dense in A , then A = ℓ − [ i =0 ( ix + H ) . Since A is connected, we conclude that H = A , as desired. (cid:3) Finally, for any simple abelian variety A defined over a field K of charac-teristic 0, the action of Gal( K/K ) on A tor yields the following result. Fact 3.10.
The group
Gal( K ( A tor ) /K ) embeds into GL d (ˆ Z ) , where d =dim( A ) and ˆ Z is the ring of finite ad´eles. Reductions
Next we proceed with several preliminary results used later in the proofof Theorem 1.2. The following result was proven in the case of a cyclic group S of automorphisms in [BRS10]; we thank Jason Bell for pointing out howto extend the result from [BRS10] to our setting. Lemma 4.1.
It suffices to prove Theorem 1.3 for a submonoid of S spannedby iterates of each of the generators of S .Proof. We consider a finite generating set U := { γ , . . . , γ s } for the monoid S . We assume S does not fix a non-constant fibration of A (otherwiseTheorem 1.3 holds). We let S ′ be the submonoid of S spanned by theendomorphisms in U ′ := { γ m , . . . , γ m s s } (for some positive integers m i ).We assume Theorem 1.3 holds for S ′ . If also S ′ does not fix a non-constantfibration, then there exists x ∈ A ( K ) such that the S ′ -orbit of x is Zariskidense in A ; hence also O S ( x ) is Zariski dense in A . So, it remains to provethat S ′ cannot fix a non-constant fibration if S does not fix a non-constantfibration.We assume f ◦ γ m i i = f for some non-constant map f : A −→ P (for each i ). Let S be a finite set of representatives for the cosets of S ′ in S (note ENSITY OF ORBITS OF ENDOMORPHISMS OF ABELIAN VARIETIES 9 that
S/S ′ is a finite group since it is a finite monoid in which each elementis invertible); without loss of generality we assume the identity is part of S .Let m := |S| , and let S := { σ , . . . , σ m } . Let s , . . . , s m be the elementarysymmetric functions g i : ( P ) m → P and let g i := s i ( f ◦ σ , . . . , f ◦ σ m )(for i = 1 , . . . , m ) Clearly, γ i preserves each fibration g j ; hence if one g j isnon-constant, then we are done. If each g j is a constant, then we obtain acontradiction because f = f ◦ id would be a root of the polynomial (withconstant coefficients) X m − g X m − + g X m − + · · · + ( − m g m = 0 . This completes the proof of Lemma 5.1. (cid:3)
Lemma 4.2.
With the notation as in Theorem 1.3, let T be a submonoidof S such that ¯ T = S . If the conclusion of Theorem 1.3 holds for T , then itholds for S .Proof. We assume that there exists no non-constant fibration preserved byall elements of S , and it suffices to prove that there is also no non-constantfibration preserved by the elements of T . Assume, by contradiction thatthere exists f : A −→ P such that f ◦ ψ = f for each ψ ∈ T . Now, let σ ∈ S ; then there exist ψ , ψ ∈ T such that γψ = ψ . So, using also that S is commutative, we get f ◦ γ = f ◦ ψ ◦ γ = f ◦ ψ = f. Hence f must be constant, as desired. (cid:3) Combining Lemmas 4.1 and 4.2 we obtain the following reduction of The-orem 1.3.
Lemma 4.3.
With the notation from Theorem 1.3, assume the monoid S is generated by the maps γ , . . . , γ s . Then it suffices to prove the conclusionof Theorem 1.3 for a finitely generated submonoid T of S with the propertythat γ ni ∈ ¯ T for each i = 1 , . . . , s , for some positive integer n . Auxiliary results
In this Section we present several technical results useful for our proof ofTheorems 1.2 and 1.3.
Lemma 5.1.
Let K be a finitely generated field of characteristic , and let K be an algebraic closure of K . Let ψ , . . . , ψ s : B −→ C be algebraic groupmorphisms of abelian varieties defined over K , and let y , . . . , y s ∈ C ( K ) .Then there exists x ∈ B ( K ) such that for each i = 1 , . . . , s , the Zariskiclosure of the subgroup generated by ψ i ( x )+ y i is the algebraic group generatedby ψ i ( B ) and y i .Proof. We let B = A + · · · + A m written as a sum of simple abelian varieties.Let i = 1 , . . . , s ; then ψ i ( B ) equals the sum ψ i ( A ) + · · · + ψ i ( A m ) (eachalgebraic group being either simple or trivial). We find an algebraic point x i ∈ ψ i ( B ) such that the Zariski closure of the cyclic group generated by x i + y i is the algebraic group generated by ψ i ( B ) and y i ; moreover we ensurethat ∩ si =1 ψ − i ( { x i } )is nonempty (in B ). We find x i as a sum x i, + · · · + x i,m , where each x i,j ∈ ψ i ( A j ). If for some j we have ψ i ( A j ) = { } , we simply pick x i,j = 0. Thenour goal is to construct the sequence { x i,j } such that for each j = 1 , . . . , m ,the set(5.1.1) ∩ si =1 ( ψ i ) | − A j ( { x i,j } )is nonempty (in A j ). Obviously when ψ i ( A j ) = { } , we might as welldisregard the set ( ψ i ) | − A j ( { x i,j } ) = ( ψ i ) | − A j ( { } ) = A j from the above intersection. Let now j = 1 , . . . , m such that ψ i ( A j ) isnontrivial. We will show that there exists x i,j ∈ ψ i ( A j ) such that for anypositive integer n we have(5.1.2) nx i,j / ∈ ( ψ i ( A j )) ( K ( C tor , x i, , . . . , x i,j − )) . Claim . If the above condition (5.1.2) holds for each j = 1 , . . . , m suchthat ψ i ( A j ) = { } , then the Zariski closure of the cyclic group generated by x i + y i is the algebraic subgroup B i generated by ψ i ( B ) and y i . Proof of Claim 5.2.
Indeed, assume there exists some algebraic subgroup D ⊆ C (not necessarily connected) such that x i + y i ∈ D ( K ). Let j ≤ m bethe largest integer such that x i,j = 0; then we have x i,j ∈ (( − y i − x i, − · · · − x i,j − ) + D ) ∩ ψ i ( A j ) . Assume first that ψ i ( A j ) ∩ D is a proper algebraic subgroup of ψ i ( A j ). Since ψ i ( A j ) is a simple abelian variety, then D ∩ ψ i ( A j ) is a 0-dimensional al-gebraic subgroup of C ; hence there exists a nonzero integer n such that n · ( D ∩ ψ i ( A j )) = { } . Then nx i,j is the only (geometric) point of the sub-variety n · ((( − y i − x i, − · · · − x i,j − ) + D ) ∩ ψ i ( A j )) which is thus rationalover K ( C tor , x i, , . . . , x i,j − ). But by our construction, nx i,j / ∈ ψ i ( A j )( K ( C tor , x i, , . . . , x i,j − ))which is a contradiction. Therefore ψ i ( A j ) ⊆ D if j is the largest index ≤ m such that x i,j = 0 (or equivalently, such that ψ i ( A j ) = 0). So, x i + y i ∈ D yields now x ′ i + y i ∈ D , where x ′ i := x i, + · · · + x i,j − . Repeating theexact same argument as above for the next positive integer j < j for which ψ i ( A j ) = { } , and then arguing inductively we obtain that each ψ i ( A j ) iscontained in D , and therefore ψ i ( B ) ⊆ D . But then x i ∈ ψ i ( B ) ⊆ D and so, y i ∈ D as well, which yields that the Zariski closure of the group generatedby x i + y i is the algebraic subgroup B i of C generated by ψ i ( B ) and y i . (cid:3) ENSITY OF ORBITS OF ENDOMORPHISMS OF ABELIAN VARIETIES 11
We just have to show that we can choose x i,j both satisfying (5.1.2) andalso such that the above intersection (5.1.1) is nonempty. So, the problemreduces to the following: L is a finitely generated field of characteristic 0, ϕ , . . . , ϕ ℓ are algebraic group homomorphisms (of finite kernel) between asimple abelian variety A and another abelian variety C all defined over L ,and we want to find z ∈ A ( K ) such that for each positive integer n , and foreach i = 1 , . . . , ℓ , we have(5.2.1) nϕ i ( z ) / ∈ ϕ i ( A ) ( L ( C tor )) . Indeed, with the above notation, A := A j , L is the extension of K gener-ated by x i,k (for i = 1 , . . . , s , and k = 1 , . . . , j − ϕ i ’s are thehomomorphisms ψ i ’s (restricted on A = A j ) for which ψ i ( A j ) is nontrivial.Let d be the maximum of the degree of the isogenies ϕ ′ i : A −→ ϕ i ( A ) ⊂ C .In particular, this means that for each w ∈ C ( K ), and for each z ∈ A ( K )such that φ i ( z ) = w we have(5.2.2) [ L ( z ) : L ] ≤ d · [ L ( w ) : L ] . For any subfield M ⊆ K , we let M ( d ) be the compositum of all extensionsof M of degree at most equal to d . Claim . Let L be a finitely generated field of characteristic 0, let C be anabelian variety defined over L , let L tor := L ( C tor ), and let d be a positiveinteger. Then there exists a normal extension of L ( d )tor whose Galois group isnot abelian. Proof of Claim 5.3.
As proven in [Tho13], the field L tor is Hilbertian (notethat L itself is Hilbertian since it is a finitely generated field of characteristic0). For each positive integer n , according to [FJ08, Corollary 16.2.7 (a)],there exists a Galois extension L n of L tor such that Gal( L n /L tor ) ∼ → S n (thesymmetric group on n letters). Assume there exists a abelian extension L of L ( d )tor containing L n . If n > max { , d ! } , we will derive a contradiction fromour assumption.We let G := Gal (cid:16) L /L ( d )tor (cid:17) and G := Gal( L /L tor ). Then there existsa surjective group homomorphism f : G −→ S n . Because G is a normalsubgroup of G (and f is a surjective group homomorphism), we get that f ( G ) is a normal subgroup of S n , and moreover, it is abelian since G isabelian. Because n ≥
5, the only proper normal subgroup of S n is A n , whichis not abelian. Hence, G ⊆ ker( f ), and therefore, f induces a surjectivegroup homomorphism (also denoted by f ) from G /G to S n ; more precisely,we have a surjective group homomorphism f : G ( d ) −→ S n , where G ( d ) :=Gal (cid:16) L ( d )tor /L tor (cid:17) . But G ( d ) is a group of exponent d !, and so, S n = f ( G ( d ) )is also a group of exponent d !, which is a contradiction with the fact that n > d !. (cid:3) Claim 5.3 yields that there exists a point z ∈ A ( K ) which is not definedover a abelian extension of L ( C tor ) ( d ) ; i.e., nz / ∈ A (cid:16) L ( C tor ) ( d ) (cid:17) for allpositive integers n . Hence, nφ i ( z ) / ∈ φ i ( A ) ( L ( C tor )) (see (5.2.2)), whichconcludes the proof of Lemma 5.1. (cid:3) The next result will be used (only) in the proof of Theorem 1.2.
Lemma 5.4.
It suffices to prove Theorem 1.2 for a conjugate γ − ◦ σ ◦ γ ofthe automorphism σ under some automorphism γ .Proof. Since O γ − σγ ( γ − ( x )) = γ − ( O σ ( x )), we obtain that there exists aZariski dense orbit of an algebraic point under the action of σ if and only ifthere exists a Zariski dense orbit of an algebraic point under the action of γ − ◦ σ ◦ γ . Also, σ preserves a non-constant fibration f : A −→ P if andonly if γ − σγ preserves the non-constant fibration f ◦ γ . (cid:3) The conclusion of the next result shares the same philosophy as the con-clusion of Lemma 5.1: one can find an algebraic point in an abelian varietyso that it is sufficiently generic with respect to any given set of finitely manypoints.
Lemma 5.5.
Let Γ ⊆ A ( K ) be a subgroup such that End( A ) ⊗ Z Γ is afinitely generated End( A ) -module, and let B ⊆ A be a nontrivial abeliansubvariety. Then there exists x ∈ B ( K ) such that for each ψ ∈ End( A ) satisfying ψ ( x ) ∈ Γ , we must have that B ⊆ ker( ψ ) .Proof. Each abelian variety is isogenous to a product of simple abelian va-rieties; so let π : A −→ A := Q ri =1 C k i i be such an isogeny, where each C i is a simple abelian variety. Then it suffices to find an algebraic point y ∈ C := π ( B ) such that for each φ ∈ End( A ), if φ ( y ) ∈ π (Γ), then C ⊆ ker( φ ).At the expense of replacing C with an isogenous abelian variety, we mayassume that C := Q ri =1 C m i i with 0 ≤ m i ≤ k i . Each endomorphism φ ∈ End( A ) is of the form ( J , . . . , J r ) where each J i ∈ M k i ( R i ), where M k i ( R i ) is the k i -by- k i matrices with entries in the ring R i of endomor-phisms of C i (note that R i is a finite integral extension of Z ). We let Γ i be the finitely generated R i -module generated by the projections of π (Γ) oneach of the k i copies of C i contained in the presentation of A = Q ri =1 C k i i .We let y i, , . . . , y i,ℓ i be generators of the free part of Γ i as an R i -module.Without loss of generality, we may assume the points y i, , . . . , y i,ℓ i are lin-early independent over R i .Then it suffices to pick x ∈ C of the form( x , , . . . , x ,m , x , , . . . , x ,m , . . . , x r, , . . . , x r,m r ) , where each x i,j ∈ C i such that for each i , the points x i, , . . . , x i,m i , y i, , . . . , y i,ℓ i are linearly independent over R i . The existence of such points x i,j followsfrom the fact that each C i ( K ) ⊗ R i Frac( R i ) has the structure of a Frac( R i )-vector space of infinite dimension. (cid:3) ENSITY OF ORBITS OF ENDOMORPHISMS OF ABELIAN VARIETIES 13
The next result is an application of Fact 3.8.
Lemma 5.6.
Let y , . . . , y r ∈ A ( K ) , and let P , · · · , P r ∈ Q [ z ] such that P i ( n ) ∈ Z for each n ≥ and for each i = 1 , . . . , r , while deg( P r ) > · · · > deg( P ) > . Then for each infinite subset S ⊆ N , there exist nonzerointegers ℓ , . . . , ℓ r such that the Zariski closure V of the set { P ( n ) y + · · · + P r ( n ) y r : n ∈ S } contains a coset of the subgroup Γ generated by ℓ y , · · · , ℓ r y r .Proof. Because V ( K ) ∩ Γ is Zariski dense in V , then by Fact 3.8 we obtainthat V is a finite union of cosets of algebraic subgroups of A . So, at theexpense of replacing S by an infinite subset, we may assume V = z + C ,for some z ∈ A ( K ) and some irreducible algebraic subgroup C of A . Thisis equivalent with the existence of an endomorphism ψ : A −→ A suchthat ker( ψ ) = C (the construction of ψ is identical with the one givenin the proof of Lemma 3.7); hence ψ is constant on the set { P ( n ) y + · · · P r ( n ) y r } n ∈ S . We will show there exist nonzero integers ℓ i such that ℓ i y i ∈ ker( ψ ) for each i = 1 , . . . , r ; since ker( ψ ) = C , then we obtain thedesired conclusion.We proceed by induction on r . The case r = 1 is obvious since then { P ( n ) } n ∈ S takes infinitely many distinct integer values (note that deg( P ) ≥ ψ is constant on the set { P ( n ) y } n ∈ S , then ψ ( ℓy ) = 0 forsome nonzero ℓ := P ( n ) − P ( n ) with distinct n , n ∈ S . Next we assumethe statement holds for all r < s (where s ≥ r = s .Let n ∈ S . At the expense of replacing each P i ( n ) by P i ( n ) − P i ( n ), wemay assume from now on that the set { P ( n ) y + · · · P s ( n ) y s } n ∈ S lies in thekernel of ψ . Let n ∈ S such that P ( n ) = 0 (note that deg( P ) ≥ i = 2 , . . . , s we let Q i ( z ) := P ( n ) · P i ( n ) − P ( n ) · P i ( n ). Thenthe set { P si =2 Q i ( n ) y i } n ∈ S is in the kernel of ψ . Because deg( Q i ) = deg( P i )for each i = 2 , . . . , s , we can use the induction hypothesis and concludethat there exist nonzero integers ℓ , . . . , ℓ s such that ℓ i y i ∈ ker( ψ ) for each i . Since ψ ( P ( n ) y + · · · + P s ( n ) y s ) = 0 and P ( n ) = 0, then also( P ( n ) · Q si =2 ℓ i ) y ∈ ker( ψ ). This concludes our proof. (cid:3) The above Lemma has the following important consequence for us.
Lemma 5.7.
Let K be a field of characteristic , let K be an algebraicclosure of K , let A be an abelian variety defined over K , let τ ∈ End( A ) with the property that there exists a positive integer r such that ( τ − id) r = 0 ,let y ∈ A ( K ) , let σ : A −→ A be an endomorphism as algebraic varietiessuch that σ = T y ◦ τ , and let x ∈ A ( K ) . Let γ ∈ End( A ) with the propertythat there exists an infinite set S of positive integers such that γ is constanton the set { σ n ( x ) : n ∈ S } . Then there exists a positive integer ℓ such that ℓ · ( β ( x ) + y ) ∈ ker( γ ) , where β := τ − id . Proof.
We compute σ n ( x ) for any n ∈ N ; first of all, we have(5.7.1) σ n ( x ) = τ n ( x ) + n − X i =0 τ i ( y ) . Then (since β = τ − id and also) noting that β r = 0 we have σ n ( x )(5.7.2) = n X i =0 (cid:18) ni (cid:19) β i ( x ) + n − X i =0 τ i ( y )(5.7.3) = r − X i =0 (cid:18) ni (cid:19) β i ( x ) + n − X i =0 i X j =0 (cid:18) ij (cid:19) β j ( y )(5.7.4) = r − X j =0 (cid:18) nj (cid:19) β j ( x ) + r − X j =0 n − X i = j (cid:18) ij (cid:19) β j ( y )(5.7.5) = r − X j =0 (cid:18) nj (cid:19) β j ( x ) + r − X j =0 (cid:18) nj + 1 (cid:19) β j ( y )(5.7.6) = x + r X j =1 (cid:18) nj (cid:19) β j ( x ) + r X j =1 (cid:18) nj (cid:19) β j − ( y )(5.7.7) = x + r X j =1 (cid:18) nj (cid:19) β j − ( β ( x ) + y ) . (5.7.8)Since γ is constant on the set { σ n ( x ) : n ∈ S } , then letting n ∈ S we havethat for each n ∈ S ,(5.7.9) r X j =1 (cid:18)(cid:18) nj (cid:19) − (cid:18) n j (cid:19)(cid:19) β j − ( β ( x ) + y ) ∈ ker( γ ) . Using Lemma 5.6 and (5.7.9), we obtain the desired conclusion. (cid:3)
Then the following result is an immediate consequence of Lemma 5.7 andof Lemma 3.9.
Corollary 5.8.
With the notation as in Lemma 5.7, if the cyclic groupgenerated by β ( x ) + y is Zariski dense in A , then γ = 0 . Moreover, the set { σ n ( x ) : n ∈ S } is Zariski dense in A .Proof. Indeed, Lemmas 3.9 and 5.7 yield that any group homomorphism γ which is constant on the set U := { σ n ( x ) : n ∈ S } must be trivial.Now, for the ‘moreover’ part of Corollary 5.8, Fact 3.6 yields that U (along with O σ ( x )) is contained in a finitely generated subgroup of A , andso, Fact 3.8 yields that the Zariski closure of U is a finite union of cosets ofalgebraic subgroups of A . Pick such a coset w + H which contains infinitely ENSITY OF ORBITS OF ENDOMORPHISMS OF ABELIAN VARIETIES 15 many σ n ( x ). Then another application of Lemma 5.7 (coupled with Lem-mas 3.7 and 3.9) yields that H = A , thus completing our proof that U isZariski dense in A . (cid:3) The cyclic case
Now we are ready to prove Theorem 1.3 for cyclic monoids.
Proof of Theorem 1.2.
By Fact 3.4, there exists an isogeny τ : A −→ A , andthere exists y ∈ A ( K ), such that σ ( x ) = τ ( x ) + y for all x ∈ A . At theexpense of replacing σ by an iterate σ n (and in particular, replacing τ by τ n ; see also (5.7.1)), we may assume dim ker( τ m − id) = dim(ker( τ − id))for all m ∈ N (see Lemma 4.1 which shows that it is sufficient to proveTheorem 1.2 for an iterate of σ ). In other words, we may assume that theonly root of unity, if any, which is a root of the minimal polynomial f (withcoefficients in Z ) of τ ∈ End( A ) is equal to 1.Let r be the order of vanishing at 1 of f , and let f ∈ Z [ t ] such that f ( t ) = f ( t ) · ( t − r . Then f is also a monic polynomial, and if r = 0, then f = f .Let A := ( τ − id) r ( A ) and let A := f ( τ )( A ), where f ( τ ) ∈ End( A ) andid is the identity map on A . If r = 0, then A = 0 and therefore A = A . Bydefinition, both A and A are connected algebraic subgroups of A , hencethey are both abelian subvarieties of A . Furthermore, by definition, therestriction of τ | A ∈ End( A ) has minimal polynomial equal to f whoseroots are not roots of unity. On the other hand, ( τ − id) r | A = 0. Lemma 6.1.
With the above notation, A = A + A .Proof of Lemma 6.1. By the definition of r and of f , we know that thepolynomials f ( t ) and ( t − r are coprime; so there exist polynomials g , g ∈ Z [ t ] and there exists a nonzero integer k (the resultant of f ( t ) and of ( t − r )such that f ( t ) · g ( t ) + ( t − r · g ( t ) = k. Let x ∈ A ( K ) and let x ∈ A ( K ) such that kx = x . Then clearly x := ( τ − id) r ( g ( τ ) x ) ∈ A and x := f ( τ ) ( g ( τ ) x ) ∈ A , and moreover, x + x = kx = x , as desired. (cid:3) Arguing similarly, one can show that A ∩ A ⊆ A [ k ] since if x ∈ A ∩ A then f ( τ ) x = 0 = ( τ − id) r x and thus kx = ( g ( τ ) f ( τ ) + g ( τ )( τ − id) r ) x = 0 . Let y ∈ A and y ∈ A such that y = y + y ; furthermore, we mayassume that if y ∈ A then y = 0. We note that τ restricts to an endo-morphism to each A and A ; we denote by τ i the action of τ on each A i .Let y ∈ A ( K ) such that (id − τ )( y ) = y (note that (id − τ ) : A −→ A is an isogeny because the minimal polynomial f of τ ∈ End( A ) does nothave the root 1). Using Lemma 5.4, it suffices to prove Theorem 1.2 for T − y ◦ σ ◦ T y ; so, we may and do assume that y = 0. Let σ i : A i −→ A i be given by σ ( x ) = τ ( x ) and σ ( x ) = τ ( x ) + y .Then for each x ∈ A , we let x ∈ A and x ∈ A such that x = x + x ; wehave: σ ( x ) = σ ( x + x ) = τ ( x + x ) + y = τ ( x ) + τ ( x ) + y = σ ( x ) + σ ( x ) . Moreover, σ n ( x + x ) = σ n ( x ) + σ n ( x ) for all n ∈ N .We let β := ( τ − id) | A ∈ End( A ); then β r = 0. Let B be the Zariskiclosure of the subgroup of A generated by β ( A ) and y . Then B is analgebraic subgroup of A . Lemma 6.2. If B = A , then σ preserves a nonconstant fibration.Proof of Lemma 6.2. If B = A , then dim( B ) < dim( A ) (note that A is connected) and since A ∩ A is finite, we conclude that the algebraicsubgroup C := A + B is a proper abelian subvariety of A . We let f : A −→ A/C be the quotient map; we claim that f ◦ σ = f . Indeed, for each x ∈ A ,we let x ∈ A and x ∈ A such that x = x + x and then f ( σ ( x )) = f ( σ ( x + x )) = f ( σ ( x ) + σ ( x )) = f ( σ ( x )) = f ( x ) = f ( x ) . Since
A/C is a positive dimensional algebraic group and f : A −→ A/C isthe quotient map, then we conclude that σ preserves a nonconstant fibration. (cid:3) From now on, assume B = A . We will prove that there exists x ∈ A ( K )such that O σ ( x ) is Zariski dense in A . First we prove there exists x ∈ A ( K )such that O σ ( x ) is Zariski dense in A .Because we assumed that the group generated by β ( A ) and y is Zariskidense in A , then Lemma 5.1 yields the existence of x ∈ A ( K ) such thatthe group generated by β ( x ) + y is Zariski dense in A . Then Corollary 5.8yields that any infinite subset of O σ ( x ) is Zariski dense in A . If A istrivial, then A = A and σ = σ and Theorem 1.2 is proven. So, from nowon, assume that A is positive dimensional.Let Γ be the subgroup of A ( K ) generated by all φ ( x ) and φ ( y ) as wevary φ ∈ End( A ). Then Γ is a finitely generated End( A )-module. UsingLemma 5.5, we may find x ∈ A ( K ) with the property that if ψ ∈ End( A )has the property that ψ ( x ) ∈ Γ, then A ⊆ ker( ψ ). Let x := x + x ; wewill prove that O σ ( x ) is Zariski dense in A .Let V be the Zariski closure of O σ ( x ). The orbit O σ ( x ) is contained ina finitely generated group (see Fact 3.6). Then Fact 3.8 yields that V is afinite union of cosets of algebraic subgroups of A . So, if V = A , then thereexists a coset c + C of a proper algebraic subgroup C ⊂ A which contains { σ n ( x ) } n ∈ S for some infinite subset S ⊆ N . By Lemma 3.7, there exists anonzero ψ ∈ End( A ) such that ψ ( σ n ( x )) = ψ ( c ) for each n ∈ S , i.e. ψ isconstant on the set { σ n ( x ) : n ∈ S } .Let n > m be two elements of S . Then ψ ( σ n ( x ) − σ m ( x )) = 0, and so, ψ ( τ n − τ m )( x ) = ψ ( σ m − σ n )( x ) ∈ Γ . ENSITY OF ORBITS OF ENDOMORPHISMS OF ABELIAN VARIETIES 17
Using the fact that x ∈ A was chosen to satisfy the conclusion of Lemma 5.5with respect to Γ and the fact that τ n − τ m = τ m ( τ n − m − id) is an isogenyon A , we obtain that ψ ( A ) = 0. Thus ψ is constant on { σ n ( x ) } n ∈ S . ThenCorollary 5.8 yields that A ⊆ ker( ψ ). Hence A + A = A ⊆ ker( ψ ) whichcontradicts the fact that ψ = 0. This concludes our proof. (cid:3) The general case
The proof of Theorem 1.3 follows the same strategy as the proof of The-orem 1.2.
Proof of Theorem 1.3.
We let γ , . . . , γ s be a set of generators for S . Welet S be the monoid of group endomorphisms of A consisting of all τ : A −→ A such that there exists some y ∈ A such that T y ◦ τ ∈ S . Welet U := { γ , . . . , γ s } , and also let U be a finite set of generators for S corresponding to the elements in U (i.e., for each ϕ ∈ U , there exists y ∈ A such that T y ◦ ϕ ∈ U ).By Fact 3.2, A is isogenuous with a product of simple abelian varieties Q i A r i i and so, End( A ) (the ring of group endomorphisms of A ) is isomorphicto Q i M r i (End( A i )). We let R i := End( A i ) and F i := Frac( R i ). Then eachelement in S is represented by a tuple of matrices in Q i M r i ( R i ); from nowon, we use freely this identification of the group endomorphisms from S withtuples of matrices in Q i M r i ( R i ). Using Lemma 2.4 and also Lemma 4.1, itsuffices to assume that for each τ ∈ S , and for each positive integer n , wehave(7.0.1) dim ker( τ − id) = dim ker( τ n − id) . Let U be the submonoid of S generated by all τ ∈ S such that(7.0.2) max n ≥ dim ker( τ − id) n is minimal as we vary τ in S . Then, by Lemma 2.5, ¯ U = S . Let U bethe submonoid of S corresponding to U , i.e. the set of all σ ∈ S such thatthere exists some τ ∈ U and there exists a translation T y on A for which σ = T y ◦ τ . Because ¯ U = S , then also ¯ U = S . Using Lemma 2.2, thereexists a finitely generated submonoid U ′ of U (and therefore of S ) and thereexists a positive integer n such that for each i = 1 , . . . , s , we have γ ni ∈ ¯ U ′ .By Lemma 4.3, it suffices to prove Theorem 1.3 for U ′ . So, from now on, weassume U ′ = S . In particular, this means that S is generated (as a monoid)by finitely many endomorphisms τ satisfying (7.0.2); we denote this set by U (as before). Finally, we recall our notation that U = { γ , . . . , γ s } is afinite set of generators of S , and that for each generator τ ∈ U of S thereexists some translation T y and some i = 1 , . . . , s such that T y ◦ τ = γ i .Let τ , τ in U . Assume r is the order of the root 1 of the minimalpolynomial for τ , and let B := ker( τ − id) r . Since τ commutes with τ ,we obtain that τ acts on B . Furthermore, because both τ and τ are in U , it must be that the restriction of the action of τ on B is also unipotent (see also the proof of Lemma 2.5); otherwise for some positive integer m ,the element τ := τ m τ ∈ S would have the property thatmax n ≥ dim ker( τ − id) n is smaller than dim B (which is minimal among all elements of S ).We let B be a complementary connected algebraic subgroup of A suchthat A = B + B , and moreover, each element of S induces an endomor-phism of B . So, we reduced to the case that each element of S is of theform T y ◦ τ , where τ acts on A = B + B as follows:(i) τ restricted to B acts unipotently, i.e. there exists some positiveinteger r τ such that ( τ − id) r τ | B = 0;(ii) for each τ ∈ U , the action of τ on B (which by abuse of notation,we also denote by τ ) has the property that τ n − id is a dominantmap for each positive integer n (see (7.0.1)).We proceed similarly to the case S is cyclic. Then for each σ i ∈ U (for i = 1 , . . . , s ), we let τ i ∈ U , z i ∈ B and y i ∈ B such that σ i = T y i + z i ◦ τ i .Note that it may be that τ i = τ j for some i = j , but this is not relevantfor the proof. We let C i be the algebraic subgroup of B spanned by y i and( τ i − id)( B ) (for each i = 1 , . . . , s ). We recall that β i := ( τ i − id) | B is anilpotent endomorphism of B ; we let U be the finite set of all β i . Finally,we let C S be the algebraic subgroup of B generated by all C i .If the algebraic subgroup C S + B does not equal A , then the exact sameargument as in Lemma 6.2 yields the existence of a non-constant rationalmap fixed by each σ ∈ S . Essentially, the projection map π : A −→ A/ ( B + C S ) is a non-constant morphism with the property that π ◦ σ = π for each σ ∈ S .Next assume C S + B = A ; we will show there exists x ∈ A ( K ) whoseorbit under S is Zariski dense. The strategy is the same as in the case S is cyclic. We can find algebraic points x ∈ B and x ∈ B such that the S -orbit of x = x + x is Zariski dense in A . First we choose x ∈ B ( K ) asin Lemma 5.1 with respect to the algebraic group endomorphisms β i and thepoints y i , for i = 1 , . . . , s ; hence the Zariski closure of the group generatedby β i ( x ) + y i is C i for each i .Let Γ be the End( A )-module spanned by x , y , . . . , y s , z , . . . , z s , which isa finitely generated subgroup of A ( K ). Then (using Lemma 5.5) we choose x ∈ B ( K ) such that if ψ ∈ End( A ) has the property that ψ ( x ) ∈ Γ, then B ⊆ ker( ψ ). Let x := x + x ; we will prove that O S ( x ) is Zariski dense in A .Using Facts 3.6 and 3.8, the Zariski closure of O S ( x ) is a union of finitelymany cosets w j + H j of algebraic subgroups of A . Lemma 7.1.
There exists a coset w + H of an algebraic subgroup appearingas a component of the Zariski closure of O S ( x ) , and there exists a positiveinteger N such that w + H is invariant under γ Ni for each i = 1 , . . . , s . ENSITY OF ORBITS OF ENDOMORPHISMS OF ABELIAN VARIETIES 19
Proof.
So, we know that the Zariski closure of O S ( x ) is the union of cosetsof (irreducible) algebraic subgroups ∪ ℓi =1 ( w i + H i ). Let γ ∈ S . Then, usingthat γ ( O S ( x )) ⊆ O S ( x ), we obtain ∪ ℓi =1 ( γ ( w i ) + γ ( H i )) ⊆ ∪ ℓi =1 ( w i + H i ) . On the other hand, each γ ∈ S is a dominant endomorphism of A , and there-fore, for each i = 1 , . . . , ℓ , we have dim( γ ( H i )) = dim( H i ). So, that means γ permutes the subgroups H i of maximal dimension appearing above. Inparticular, there exists a positive integer N such that for each i = 1 , . . . , s ,the endomorphism γ N i fixes each algebraic group H i of maximal dimension.Let S ( N ) be the submonoid of S consisting of all γ N for γ ∈ S . Now, let H be one such algebraic group of maximal dimension among the algebraicgroups H i (for i = 1 , . . . , ℓ ). Let w i + H with i = 1 , . . . , k be all the cosetsof H appearing as irreducible components of the Zariski closure of O S ( x ).Then each element γ ∈ S ( N ) induces a map f γ : { , . . . , k } −→ { , . . . , k } given by f γ ( w i + H ) = w f γ ( i ) + H ; the map is not necessarily bijective.Moreover, we get a homomorphism of monoids f : S ( N ) −→ F k given by f ( γ ) := f γ , where F k is the monoid of all functions from the set { , . . . , k } into itself. Clearly, there exists j ∈ { , . . . , k } , and there exists a positiveinteger N such that f γ N ( j ) = j for each generator γ ∈ { γ N , . . . , γ N s } of S ( N ) . Then Lemma 7.1 holds with N := N · N . (cid:3) Let w + H be one coset as in the conclusion of Lemma 7.1, and let N be the positive integer from the conclusion of Lemma 7.1 with respect tothe coset w + H . We let S ′ be the submonoid of S generated by γ Ni for i = 1 , . . . , s . Then w + H contains a set of the form O S ′ ( x ′ ), for some x ′ ∈ O S ( x ); in other words, w + H contains a set of the form { γ m + Nn · · · γ m s + Nn s s ( x ) : n , . . . , n s ≥ } , for some positive integers m , . . . , m s .Let then π : A −→ A/H be the canonical projection. Then π (cid:16) γ m + Nn · · · γ m s + Nn s s ( x ) (cid:17) = w for all n , . . . , n s ≥
0. Restricted on B , for each group endomorphism τ i (for i = 1 , . . . , s ), the action on the tangent space of B has no eigenvaluewhich is a root of unity (see (ii) above); hence ψ := (cid:16) τ m + N τ m · · · τ m s s − τ m τ m · · · τ m s s (cid:17) | B is an isogeny.So, we get that ( π ◦ ψ )( x ) ∈ Γ. Because of our choice for x and the factthat ψ is an isogeny on B , we conclude that B ⊆ ker( π ) (note also that B is connected by our assumption). Thus B ⊆ H . So, we can view π asa group homomorphism π : B −→ A/H with the property that for each n , . . . , n s ≥ π (cid:16) γ m + n N · · · γ m s + n s Ns − γ m · · · γ m s s (cid:17) ( x ) = 0 . Letting γ ′ := γ m · · · γ m s s | B , we have that π ◦ γ ′ is constant (equal to w ) oneach orbit O γ Ni ( x ). Then Corollary 5.8 yields that the connected componentof the Zariski closure C i of the cyclic group generated by ( τ i − id)( x ) + y i is contained in the kernel of π ◦ γ ′ . Since the C ′ i s generate the algebraicgroup C S (and therefore the connected components of the C i ’s generatethe connected component of C S ; see also Fact 3.1), and furthermore, theconnected component of C S contains the connected component of B , weconclude that π ◦ γ ′ is identically 0 on B . Because γ ′ is an isogeny, weconclude that B ⊆ ker( π ), and therefore H = A since H contains both B and B . This concludes our proof. (cid:3) References [ABR11] E. Amerik, F. Bogomolov, and M. Rovinsky,
Remarks on endomorphisms andrational points , Compos. Math. (2011), 1819–1842.[AC08] E. Amerik and F. Campana,
Fibrations m´eromorphes sur certaines vari´et`e a fibr´ecanonique trivial , Pure Appl. Math. Q. (2008), 509–545.[BGT] J. P. Bell, D. Ghioca, and T. J. Tucker, Applications of p -adic analysis forbounding periods of subvarieties under etale maps , to appear in Int. Math. Res.Not. (2014), 18 pages.[BGZ] J. P. Bell, D. Ghioca, and Z. Reichstein, On a dynamical version of a theoremof Rosenlicht , preprint (2014), 15 pages, arXiv:1408.4744 .[BRS10] J. P. Bell, D. Rogalski, and S. J. Sierra,
The Dixmier-Moeglin equivalence fortwisted homogeneous coordinate rings , Israel J. Math. (2010), 461–507.[CS93] G. S. Call and J. H. Silverman,
Canonical heights on varieties with morphisms ,Compositio Math. (1993), 163–205.[Fal94] G. Faltings, The general case of S. Lang’s conjecture , Barsotti Symposium inAlgebraic Geometry (Abano Terme, 1991), Perspect. Math., no. 15, AcademicPress, San Diego, CA, 1994, pp. 175–182.[FJ08] M. D. Fried and J. Moshe,
Field arithmetic . Third edition. Springer-Verlag,Berlin, 2008. xxiv+792 pp.[Ko48] E. R., Kolchin,
Algebraic matric groups and the Picard-Vessiot theory of homo-geneous linear ordinary differential equations , Ann. of Math. (2) , (1948), 1 -42.[MS14] A. Medvedev and T. Scanlon, Invariant varieties for polynomial dynamical sys-tems , Ann. of Math. (2) (2014), no. 1, 81–177.[Mil] J. Milne,
Abelian varieties , course notes available online: .[Tho13] C. Thornhill,
Abelian varieties and Galois extensions of Hilbertian fields , J. Inst.Math. Jussieu (2013), no. 2, 237–247.[Zha10] S. W. Zhang, Distributions in algebraic dynamics , a tribute to professor S. S.Chern. volume 10 of Survey in Differential Geometry, pages 381–430. Interna-tional Press, 2006.
Department of Mathematics, University of British Columbia, Vancouver,BC V6T 1Z2, Canada
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