Article

Zariski dense orbits for regular self-maps of split semiabelian varieties in positive characteristic

Authors:
DRAGOS GHIOCA
DRAGOS GHIOCA
DRAGOS GHIOCA
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Sina Saleh at University of British Columbia
Sina Saleh
Download citation
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

We prove the Zariski dense orbit conjecture in positive characteristic for regular self-maps of split semiabelian varieties.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... The picture in positive characteristic. If K has characteristic p > 0, then Conjecture 1.1 does not hold due to the presence of the Frobenius endomorphism (see [GS21b,Remark 1.2]). In particular, if K = F p (the algebraic closure of the finite field F p ), then each orbit of a point α ∈ X(K) is finite under a rational self-map Φ : X −→ X defined over K = F p ; furthermore, Φ does not have to preserve a non-constant rational function. ...
... In particular, if K = F p (the algebraic closure of the finite field F p ), then each orbit of a point α ∈ X(K) is finite under a rational self-map Φ : X −→ X defined over K = F p ; furthermore, Φ does not have to preserve a non-constant rational function. So, the authors proposed the following conjecture as a variant of conjecture 1.1 in positive characteristic (see also [GS21b]). ...
... Discussion of our proof. Our proof of Theorem 1.3 follows the general strategy we employed in [GS21b] to treat the case of algebraic tori; however, there are significant complications due to the more complex structure of the endomorphism ring of a semiabelian variety compared with the power of the multiplicative group G N m . In particular, Sections 3 and 6 contain technical difficulties which are significantly more delicate than any of the arguments necessary for the case of tori. ...
Zariski dense orbits for regular self-maps of split semiabelian varieties in positive characteristic
Preprint
Full-text available
  • Aug 2021
We prove the Zariski dense orbit conjecture in positive characteristic for regular self-maps of split semiabelian varieties.
View
... As an application, they reduced the Zariski dense orbit conjecture for f to a terminal threefold with only f -equivariant Fano contractions. Now we consider the variants of the Zariski dense orbit conjecture in positive characteristic proposed in [17,Conjecture 1.3] and [40,Section 1.6]. Ghoica and Saleh [17,18,19] proved the conjecture dense orbit conjecture in positive characteristic for regular selfmaps of the tori G N m , the split semiabelian varieties, and the additive group scheme G N a . ...
... Now we consider the variants of the Zariski dense orbit conjecture in positive characteristic proposed in [17,Conjecture 1.3] and [40,Section 1.6]. Ghoica and Saleh [17,18,19] proved the conjecture dense orbit conjecture in positive characteristic for regular selfmaps of the tori G N m , the split semiabelian varieties, and the additive group scheme G N a . Now let X be a projective variety over an algebraically closed field k of positive characteristic over X and H an ample divisor on X . ...
A note on Zariski dense orbit conjecture
Preprint
Full-text available
  • Aug 2022
In this paper we first note a result of birational automorphisms with bounded degree of projective varieties related with the Zariski dense orbit conjecture (ZDO) and the Zariski density of periodic points. Next, we give a reduced result of ZDO for automorphisms of projective threefolds, and show ZDO for automorphisms of projective varieties X with the irregularity q(X)≥dim⁡X−1q(X)\ge\dim X-1.
View
... We first note that Conjecture 1.3.2 fits into a recent series of papers searching for analog statements over fields of positive characteristic for some of the most important results from the past 20 years in arithmetic dynamics over fields of characteristic 0 (see [GS23,Xie23,XY], for example). Two of the most active areas of recent research in arithmetic dynamics have been the unlikely intersection principle and the dynamical Mordell-Lang conjecture. ...
Intersection of orbits for polynomials in characteristic $p
Preprint
Full-text available
  • Aug 2024
In [GTZ08, GTZ12], the following result was established: given polynomials f,g∈C[x]f,g\in\mathbb{C}[x] of degrees larger than 1, if there exist α,β∈C\alpha,\beta\in\mathbb{C} such that their corresponding orbits Of(α)\mathcal{O}_f(\alpha) and Og(β)\mathcal{O}_g(\beta) (under the action of f, respectively of g) intersect in infinitely many points, then f and g must share a common iterate, i.e., fm=gnf^m=g^n for some m,n∈Nm,n\in\mathbb{N}. If one replaces C\mathbb{C} with a field K of characteristic p, then the conclusion fails; we provide numerous examples showing the complexity of the problem over a field of positive characteristic. We advance a modified conjecture regarding polynomials f and g which admit two orbits with infinite intersection over a field of characteristic p. Then we present various partial results, along with connections with another deep conjecture in the area, the dynamical Mordell-Lang conjecture.
View
... Here one needs to modify the statement to take into account the possible action of the Frobenius on varieties defined over finite fields. In fact, the characteristic p analogue of the Zariski dense orbit conjecture stated in [GS21] involves a condition similar to the one appearing in part (b) of our Theorem 1.1. Only sporadic results are known in positive characteristic (e.g., the case when K is uncountable is handled in [BGR17] and the case of an endomorphisms of G N a defined over a finite field in [GS]). ...
Rational self-maps with a regular iterate on a semiabelian variety
Preprint
Full-text available
  • Aug 2022
Let G be a semiabelian variety defined over an algebraically closed field K of characteristic 0. Let Φ ⁣:G⇢G\Phi\colon G\dashrightarrow G be a dominant rational self-map. Assume that an iterate Φm ⁣:G→G\Phi^m \colon G \to G is regular for some m⩾1m \geqslant 1 and that there exists no non-constant homomorphism τ:G→G0\tau: G\to G_0 of semiabelian varieties such that τ∘Φmk=τ\tau\circ \Phi^{m k}=\tau for some k⩾1k \geqslant 1. We show that under these assumptions Φ\Phi itself must be a regular. We also prove a variant of this assertion in prime characteristic and present examples showing that our results are sharp.
View
... Obtaining the precise description of the return set N from (2) as a finite union of sets of the form (1) is beyond the known results available in the literature. The fact that the problem in positive characteristic turns out to be more subtle than the corresponding question in characteristic 0 is encountered in many similar questions in arithmetic geometry (such as the classical Mordell-Lang conjecture, see [Hru96,MS04] for the corresponding results in characteristic p) or arithmetic dynamics (such as the Zariski dense orbit conjecture in positive characteristic, see [GS21]). So, we find it interesting that the results of [BCH21], which involve a language akin to both the Dynamical Mordell-Lang Conjecture and to the classical Mordell-Lang conjecture, hold with suitable modifications over fields of positive characteristic. ...
A fusion variant of the classical and dynamical Mordell-Lang conjectures in positive characteristic
Preprint
Full-text available
  • May 2022
We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let K be an algebraically closed field of positive characteristic, let G be a finitely generated subgroup of the multiplicative group of K, and let X be a (irreducible) quasiprojective variety defined over K. We consider K-valued sequences of the form an:=f(φn(x0))a_n:=f(\varphi^n(x_0)), where φ ⁣:X→X\varphi\colon X\rightarrow X and f ⁣:X→P1f\colon X\rightarrow\mathbb{P}^1 are rational maps defined over K and x0∈Xx_0\in X is a point whose forward orbit avoids the indeterminacy loci of φ\varphi and f. We show that the set of n for which an∈Ga_n\in G is a finite union of arithmetic progressions along with a set of upper Banach density zero. In addition, we show that if an∈Ga_n\in G for every n and the φ\varphi orbit of x is Zariski dense in X then {there is} a multiplicative torus Gmd\mathbb{G}_m^d and maps Ψ:Gmd→Gmd\Psi:\mathbb{G}_m^d \to \mathbb{G}_m^d and g:Gmd→Gmg:\mathbb{G}_m^d \to \mathbb{G}_m such that an=g∘Ψn(y)a_n = g\circ \Psi^n(y) for some y∈Gmdy\in \mathbb{G}_m^d. We then describe various applications of our results.
View
... Conjecture 1.2 has been proven in the case of algebraic tori in [GS21] and more generally in the case of all split semiabelian varieties defined over F p in [GSb]. For an illustration of the trichotomy in the conclusion of Conjecture 1.2, we refer the reader to [GS21, Example 1.6]. ...
Zariski dense orbits for endomorphisms of a power of the additive group scheme defined over finite fields
Preprint
Full-text available
  • Feb 2022
We prove the Zariski dense orbit conjecture in positive characteristic for endo-morphisms of a power of the additive group scheme defined over Fp.
View
... Then employing [RRZ06, Theorem 7.2] (along with [PR04, Theorem 3.1]) allows us to finish the proof of Theorem 1.6; in the language of [RRZ06], the automorphism φ is wild (see Section 2.3) and so, it does not have proper φ-invariant subvarieties. We also note that one could obtain the desired conclusion from Theorem 1.6 by using alternatively more combinatorial arguments akin to the ones employed in the proof from [GS21]. Finally, in order to prove Theorem 1.8 (whose conclusion is once again unchanged if one replaces φ = T •Ψ by a suitable iterate of it), we analyze the action of Ψ on X according to the roots of its minimal polynomial P (x); for this part our arguments are somewhat similar to the ones employed in [GS17,GS19]. ...
A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one
Preprint
Full-text available
  • Feb 2022
We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety X defined over an algebraically closed field K of characteristic 0, endowed with a birational self-map ϕ\phi of dynamical degree 1, we expect that either there exists a non-constant rational function f:X⇢P1f:X\dashrightarrow \mathbb{P}^1 such that f∘ϕ=ff\circ \phi=f, or there exists a proper subvariety Y⊂XY\subset X with the property that for any invariant proper subvariety Z⊂XZ\subset X, we have that Z⊆YZ\subseteq Y. We prove our conjecture for automorphisms ϕ\phi of dynamical degree 1 of semiabelian varieties X. Also, we prove a related result for regular dominant self-maps ϕ\phi of semiabelian varieties X: assuming ϕ\phi does not preserve a non-constant rational function, we have that the dynamical degree of ϕ\phi is larger than 1 if and only if the union of all ϕ\phi-invariant proper subvarieties of X is Zariski dense. We give applications of our results to representation theoretic questions about twisted homogeneous coordinate rings associated to abelian varieties.
View
... On the other hand, the known counter-examples often involve some Frobenius actions. See[27, Theorem 1.5, Question 1.7] for this phenomenon. We suspect that when tr.d. ...
Remarks on algebraic dynamics in positive characteristic
Preprint
  • Jul 2021
In this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. We generalise some basic facts on arithmetic degree and canonical height in positive characteristic. As applications, we prove the dynamical Mordell-Lang conjecture for automorphisms of projective surfaces of positive entropy, the Zariski dense orbit conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell-Lang and prove a counting result for backward orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces.
View
View
Remarks on algebraic dynamics in positive characteristic
Article
  • Jan 2023
In this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. Applying the arithmetic degree and canonical height in positive characteristic, we prove the Dynamical Mordell–Lang Conjecture for automorphisms of projective surfaces of positive entropy, the Zariski Dense Orbit Conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell–Lang and prove a counting result for backward orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces.
View
A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one
Article
  • Jul 2022
We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety X defined over an algebraically closed field K of characteristic 0 , endowed with a birational self-map ϕ\phi of dynamical degree 1 , we expect that either there exists a nonconstant rational function f:X⇢P1f:X\dashrightarrow \mathbb {P} ^1 such that f∘ϕ=ff\circ \phi =f , or there exists a proper subvariety Y⊂XY\subset X with the property that, for any invariant proper subvariety Z⊂XZ\subset X , we have that Z⊆YZ\subseteq Y . We prove our conjecture for automorphisms ϕ\phi of dynamical degree 1 of semiabelian varieties X . Moreover, we prove a related result for regular dominant self-maps ϕ\phi of semiabelian varieties X : assuming that ϕ\phi does not preserve a nonconstant rational function, we have that the dynamical degree of ϕ\phi is larger than 1 if and only if the union of all ϕ\phi -invariant proper subvarieties of X is Zariski dense. We give applications of our results to representation-theoretic questions about twisted homogeneous coordinate rings associated with abelian varieties.
View
Density of orbits of endomorphisms of commutative linear algebraic groups
Article
Full-text available
  • Mar 2018
We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected commutative linear algebraic groups G defined over an algebraically closed field k\mathbb{k} of characteristic 0. That is, if Φ ⁣:G⟶G\Phi\colon G\longrightarrow G is a dominant endomorphism, we prove that one of the following holds: either there exists a non-constant rational function f∈k(G)f\in \mathbb{k}(G) preserved by Φ\Phi (i.e., f∘Φ=ff\circ \Phi = f), or there exists a point x∈G(k)x\in G(\mathbb{k}) whose Φ\Phi-orbit is Zariski dense in G.
View
The Dynamical Mordell-Lang Conjecture for endomorphisms of semiabelian varieties defined over fields of positive characteristic
Article
Full-text available
  • Feb 2018
Let K be an algebraically closed field of prime characteristic p, let X be a semiabelian variety defined over a finite subfield of K, let f be a regular self-map on X defined over K, let V be a subvariety of X defined over K, and let x be a K-point of X. The Dynamical Mordell-Lang Conjecture in characteristic p predicts that the set S consisting of all positive integers n for which f^n(x) lies on V is a union of finitely many arithmetic progressions, along with finitely many p-sets, which are sets consisting of all integers of the form c_1 p^{k_1 n_1} + ... + c_m p^{k_m n_m} (as we vary n_1,..,n_m among all positive integers), for some given positive integer m, some rational numbers c_i and some non-negative integers k_i. We prove that this conjecture is equivalent with some difficult Diophantine problem in characteristic 0. In the case X is an algebraic torus, we can prove the conjecture in two cases: either when the dimension of V is at most 2, or when no iterate of f is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of X.
View
On the Medvedev-Scanlon Conjecture for Minimal Threefolds of Non-Negative Kodaira Dimension
Article
Full-text available
  • Oct 2016
Motivated by work of Zhang from the early `90s, Medvedev and Scanlon formulated the following conjecture. Let K be an algebraically closed field of characteristic 0 and let X be a quasiprojective variety defined over K endowed with a dominant rational self-map Φ\Phi. Then there exists a point α∈X(K)\alpha\in X(K) with Zariski dense orbit under Φ\Phi if and only if Φ\Phi preserves no nontrivial rational fibration, i.e., there exists no non-constant rational function f∈K(X)f\in K(X) such that Φ∗(f)=f\Phi^*(f)=f. The Medvedev-Scanlon conjecture holds when K is uncountable. The case where K is countable (e.g., K=Q‾K=\overline{\mathbb{Q}}) is much more difficult; here the conjecture has only been proved in a small number of special cases. In this paper we show that the Medvedev-Scanlon conjecture holds for all varieties of positive Kodaira dimension, and explore the case of Kodaira dimension 0. Our results are most complete in dimension 3.
View
Density of orbits of endomorphisms of Abelian varieties
Article
Full-text available
  • Dec 2014
  • T AM MATH SOC
Let A be an abelian variety defined over Qˉ\bar{\mathbb{Q}}, and let φ\varphi be a dominant endomorphism of A as an algebraic variety. We prove that either there exists a non-constant rational fibration preserved by φ\varphi, or there exists a point x∈A(Qˉ)x\in A(\bar{\mathbb{Q}}) whose φ\varphi-orbit is Zariski dense in A. This provides a positive answer for abelian varieties of a question raised by Medvedev and the second author ("nvariant varieties for polynomial dynamical systems", Ann. of Math. (2) 179 (2014), no. 1, 81-177). We prove also a stronger statement of this result in which φ\varphi is replaced by any commutative finitely generated monoid of dominant endomorphisms of A.
View
Distributions in Algebraic Dynamics
Article
Full-text available
  • Jan 2005
View
On a dynamical version of a theorem of Rosenlicht
Article
Full-text available
  • Aug 2014
  • ANN SCUOLA NORM-SCI
Consider the action of an algebraic group G on an irreducible algebraic variety X all defined over a field k. M. Rosenlicht showed that orbits in general position in X can be separated by rational invariants. We prove a dynamical analogue of this theorem, where G is replaced by a semigroup of dominant rational self-maps of X. Our semigroup G is not required to have the structure of an algebraic variety and can be of arbitrary cardinality.
View
Invariant varieties for polynomial dynamical systems
Article
Full-text available
  • Jan 2009
  • ANN MATH
We study algebraic dynamical systems (and, more generally, σ\sigma-varieties) Φ:ACn→ACn\Phi:{\mathbb A}^n_{\mathbb C} \to {\mathbb A}^n_{\mathbb C} given by coordinatewise univariate polynomials by refining a theorem of Ritt. More precisely, we find a nearly canonical way to write a polynomial as a composition of "clusters". Our main result is an explicit description of the (weakly) skew-invariant varieties. As a special case, we show that if f(x)∈C[x]f(x) \in {\mathbb C}[x] is a polynomial of degree at least two which is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and X⊆AC2X \subseteq {\mathbb A}^2_{\mathbb C} is an irreducible curve which is invariant under the action of (x,y)↦(f(x),f(y))(x,y) \mapsto (f(x),f(y)) and projects dominantly in both directions, then X must be the graph of a polynomial which commutes with f under composition. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius. We also show that in models of ACFA0_0, a disintegrated set defined by σ(x)=f(x)\sigma(x) = f(x) for a polynomial f has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew-conjugacy class of f is defined over a fixed field of a power of σ\sigma, and that nonorthogonality between two such sets is definable in families if the skew-conjugacy class of f is defined over a fixed field of a power of σ\sigma.
View
F-structures and integral points on semiabelian varieties over finite fields
Article
Full-text available
  • Jun 2002
  • AM J MATH
Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination and stability result for finitely generated modules over certain finite simple extensions of the integers given together with predicates for cycles of the distinguished generator of the ring. 1. introduction The Mordell-Lang conjecture asserts that for G a semiabelian variety over the complex numbers, X ⊂ G a subvariety, and Γ ≤ G(C) a finitely generated subgroup of the complex points, the set of points X(C) ∩ Γ is a finite union of cosets of subgroups of Γ. This fails when C is replaced by a field of positive characteristic. For example, suppose that X and G are defined over a finite field Fq and that F: G → G is the corresponding Frobenius morphism. Let K: = Fq(X) and let Γ ≤ G(K) be the Z[F]-submodule generated by γ: = idX: X → X thought of as an element of X(K). Then X(K) ∩ Γ contains the infinite set {F n γ: n ∈ N}. If X
View
Remarks on algebraic dynamics in positive characteristic
Article
  • Jan 2023
In this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. Applying the arithmetic degree and canonical height in positive characteristic, we prove the Dynamical Mordell–Lang Conjecture for automorphisms of projective surfaces of positive entropy, the Zariski Dense Orbit Conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell–Lang and prove a counting result for backward orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces.
View
View
A SPARSITY RESULT FOR THE DYNAMICAL MORDELL–LANG CONJECTURE IN POSITIVE CHARACTERISTIC
Article
  • Feb 2021
We prove a quantitative partial result in support of the dynamical Mordell–Lang conjecture (also known as the DML conjecture ) in positive characteristic. More precisely, we show the following: given a field K of characteristic p , a semiabelian variety X defined over a finite subfield of K and endowed with a regular self-map Φ:X⟶X\Phi :X{\longrightarrow } X defined over K , a point α∈X(K)\alpha \in X(K) and a subvariety V⊆XV\subseteq X , then the set of all nonnegative integers n such that Φn(α)∈V(K)\Phi ^n(\alpha )\in V(K) is a union of finitely many arithmetic progressions along with a subset S with the property that there exists a positive real number A (depending only on X , Φ\Phi , α\alpha and V ) such that for each positive integer M , \begin{align*}\scriptsize\#\{n\in S\colon n\le M\}\le A\cdot (1+\log M)^{\dim V}.\end{align*}
View
Zariski dense orbits for regular self-maps on split semiabelian varieties
Article
  • Mar 2021
We provide a direct proof of the Medvedev–Scanlon’s conjecture from Medvedev and Scanlon (Ann. Math. Second Series 179 (2014), 81–177) regarding Zariski dense orbits under the action of regular self-maps on split semiabelian varieties defined over a field of characteristic 0 . Besides obtaining significantly easier proofs than the ones previously obtained in Ghioca and Scanlon (Trans. Am. Math. Soc. 369 (2017), 447–466; for the case of abelian varieties) and Ghioca and Satriano (Trans. Am. Math. Soc. 371 (2019), 6341–6358; for the case of semiabelian varieties), our method allows us to exhibit numerous starting points with Zariski dense orbits, which the methods from Ghioca and Scanlon (Trans. Am. Math. Soc. 369 (2017), 447–466) and Ghioca and Satriano (Trans. Am. Math. Soc. 371 (2019), 6341–6358) could not provide.
View
View
Algebraic dynamics of skew-linear self-maps
Article
  • Jan 2018
Let X be a variety defined over an algebraically closed field k of characteristic 0, let N∈NN\in\mathbb{N}, let g:X⇢Xg:X\dashrightarrow X be a dominant rational self-map, and let A:AN→ANA:\mathbb{A}^N\to \mathbb{A}^N be a linear transformation defined over k(X), i.e., for a Zariski open dense subset U⊂XU\subset X, we have that for x∈U(k)x\in U(k), the specialization A(x) is an N-by-N matrix with entries in k. We let f:X×AN⇢X×ANf:X\times\mathbb{A}^N\dashrightarrow X\times \mathbb{A}^N be the rational endomorphism given by (x,y)↦(g(x),A(x)y)(x,y)\mapsto (g(x), A(x)y). We prove that if the determinant of A is nonzero and if there exists x∈X(k)x\in X(k) such that its orbit Og(x)\mathcal{O}_g(x) is Zariski dense in X, then either there exists a point z∈(X×AN)(k)z\in (X\times \mathbb{A}^N)(k) such that its orbit Of(z)\mathcal{O}_f(z) is Zariski dense in X×ANX\times \mathbb{A}^N or there exists a nonconstant rational function ψ∈k(X×AN)\psi\in k(X\times \mathbb{A}^N) such that ψ∘f=ψ\psi\circ f=\psi. Our result provides additional evidence to a conjecture of Medvedev and Scanlon.
View
Density of orbits of dominant regular self-maps of semiabelian varieties
Article
  • Aug 2017
We prove a conjecture of Medvedev and Scanlon in the case of regular morphisms of semiabelian varieties. That is, if G is a semiabelian variety defined over an algebraically closed field K of characteristic 0, and φ ⁣:G→G\varphi\colon G\to G is a dominant regular self-map of G which is not necessarily a group homomorphism, we prove that one of the following holds: either there exists a non-constant rational fibration preserved by φ\varphi, or there exists a point x∈G(K)x\in G(K) whose φ\varphi-orbit is Zariski dense in G.
View
View
Integral Points on Subvarieties of Semiabelian Varieties, II
Article
  • Sep 1996
This paper proves a finiteness result for families of integral points on a semiabelian variety minus a divisor, generalizing the corresponding result of Faltings for abelian varieties. Combined with the main theorem of the first part of this paper, this gives a finiteness statement for integral points on a closed subvariety of a semiabelian variety, minus a divisor. In addition, the last two sections generalize some standard results on closed subva- rieties of semiabelian varieties to the context of closed subvarieties minus divisors.
View
View
The isotrivial case in the Mordell-Lang Theorem
Article
  • Jul 2008
We determine the structure of the intersection of a finitely generated subgroup of a semiabelian variety G defined over a finite field with a closed subvariety X ⊂ G. We also study a related question in the context of a power of the additive group scheme.
View
View
Abelian varieties. lecture notes available online
  • J Milne
J. MILNE. Abelian varieties. lecture notes available online.
Zariski dense orbits for regular self-maps of tori in positive characteristic
  • Jan 2021
  • 1274
  • Ghioca
The general case of S. Lang's conjecture. Barsotti Symposium in Algebraic Geometry
  • Jan 1991
  • 175-182
  • G Faltings
G. FALTINGS. The general case of S. Lang's conjecture. Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991). Perspect. Math., vol. 15 (Academic Press, San Diego, CA, 1994), pp. 175-182.
  • Jan 1984
  • 299-327
  • M Laurent
M. LAURENT. Équations diophantiennes exponentielles. Invent. Math. 78(2) (1984), 299-327.