Orbit Transversality for Abelian Varieties

Abstract: Let $X/K$ be an abelian variety defined over a number field and let $f:X\to X$ be a dominant morphism defined over $K$. We show that $(f,K)$ is strongly dense orbit transversal. That is, up to replacing $K$ with a finite extension, every set of representatives for grand $(f,K)$-orbits is Zariski dense in $X$.

• The paper establishes that for every dominant endomorphism of an abelian variety, every (f,L)-transversal is Zariski dense after a finite extension.
• It uses an orbit avoidance criterion and the Mordell–Weil structure to construct points whose orbits avoid any given proper subvariety.
• The result generalizes prior work by proving strong dense orbit transversality holds even over uncountable fields and for all abelian varieties.

Strong Orbit Transversality for Dominant Endomorphisms of Abelian Varieties

Introduction

This work presents a comprehensive and definitive result in the arithmetic dynamics of abelian varieties, focusing on the distribution and transversality properties of orbits under dominant endomorphisms. For a given abelian variety $X$ defined over a number field $K$ and a dominant morphism $f: X \rightarrow X$ also defined over $K$, the study investigates the nature of grand $(f, K)$-orbits, with an emphasis on the strong density properties of transversals representing these orbits.

The prior literature, particularly Pasten and Silverman’s framework for orbit propagation principles, introduced dense orbit transversality (DOT) as a robust measure of arithmetic and geometric richness of orbit representatives. The fundamental question is whether Zariski dense transversals exist for arbitrary dominant self-maps on arbitrary abelian varieties and under what circumstances these transversals are necessarily dense.

Main Contributions

The principal result establishes that for any abelian variety $X$ over a number field and any dominant morphism $f: X \rightarrow X$, there exists a finite extension $L/K$ such that every $(f, L)$-transversal is Zariski dense in $X$. This is encapsulated as strong dense orbit transversality (S-DOT). Notably, this result extends prior knowledge, which only addressed S-DOT for geometrically simple abelian varieties or under the existence of a single dense orbit. Furthermore, the proof avoids case distinctions based on the geometry of $K$0—it applies uniformly to all abelian varieties and all dominant self-maps.

The paper also establishes that over uncountable fields, S-DOT always holds for dominant self-maps of irreducible varieties, highlighting the interplay between field cardinality and orbit distribution phenomena.

Technical Approach

The central technical tool is the reduction of the S-DOT property to an orbit avoidance criterion: for each proper subvariety $K$1, one must construct a global point whose orbit avoids $K$2. To formalize this equivalence, the paper presents a precise lemma: S-DOT holds if and only if, after a finite extension, for every proper subvariety $K$3, there exists a point whose $K$4-grand orbit is disjoint from $K$5.

The key ingredient enabling this avoidance is a detailed analysis of the $K$6-module structure of $K$7 provided by the Mordell–Weil theorem. By Poincaré's reducibility and suitable field extension, the corank of any proper abelian subvariety in the Mordell–Weil group is made arbitrarily large, ensuring significant room to separate orbits from any finite collection of proper subvarieties. The argument then relies on constructing points in the free part of $K$8 whose orbits, under affine-linear actions induced by $K$9, avoid prescribed unions of translates of submodules, leveraging constraints on the corank and rank following Faltings’ description of irreducible subvarieties.

The technical passage to finite fields via reduction modulo primes not dividing the determinant of the linear part of the map, and the counting of orbit sizes and representatives, is handled via explicit control over $f: X \rightarrow X$0-actions and combinatorial bounds. Lifting arguments are then used to conclude the existence of desired points over number fields.

The case of uncountable fields is treated separately, with the proof there appealing to basic properties of irreducibility and countability to rule out the existence of non-dense transversals.

Numerical and Conceptual Strengths

The core assertion is for every dominant morphism $f: X \rightarrow X$1 of an abelian variety $f: X \rightarrow X$2 defined over a number field, and after finite base change, every $f: X \rightarrow X$3-transversal is Zariski dense. There are no additional geometric or arithmetic assumptions required beyond dominance of $f: X \rightarrow X$4 and the abelian variety structure.

By comparison with earlier work, this claim is stronger and more general. Prior results, such as those of Pasten and Silverman, only addressed the strongly dense transversality in the context of simple abelian varieties or required the existence of a single Zariski dense orbit. Here, the result covers the composite case and all dominant morphisms, regardless of the complexity of $f: X \rightarrow X$5's isogeny decomposition.

Implications and Future Directions

The result has significant implications for arithmetic dynamics on abelian varieties. It provides a rigorous foundation for understanding how orbits under dominant endomorphisms can be globally separated from arbitrary proper subvarieties, reinforcing the geometric uniformity and unpredictability of such orbits in the generic case.

From a practical perspective, this strengthens the methods available for constructing dense sets of arithmetic representatives for dynamical systems on abelian varieties. It also clarifies the limitations and sharpness of transversality concepts in settings beyond abelian varieties—particularly regarding the arithmetic of algebraic dynamical systems and the distribution of rational points.

Theoretically, this result points toward a refined orbit classification for other classes of varieties (e.g., semiabelian or mixed), and suggests that field-theoretic properties (notably countability) fundamentally constrain the existence of non-dense transversals. This connection may inform further exploration of orbit closure, arithmetic ergodicity, and the interplay between algebraic dynamics and Diophantine geometry.

Future work may seek analogous orbit-propagation dichotomies for endomorphisms with nontrivial ramification, non-abelian group varieties, or over global fields of positive characteristic, as well as effective bounds on orbit separation in specific arithmetic applications.

Conclusion

This paper establishes that strong dense orbit transversality universally holds for dominant endomorphisms of abelian varieties over number fields, after finite extension. The proof leverages the $f: X \rightarrow X$6-module structure of rational points and orbit avoidance via combinatorial group theory, providing both a reinforcement and generalization of prior results and setting the stage for further advances in the understanding of arithmetic dynamics on higher-dimensional varieties.