On Dense Orbit Transversality for Endomorphisms of Abelian Varieties

Abstract: Let $X/K$ be a smooth projective variety defined over a number field and $f:X\to X$ be a morphism defined over $K$. Assuming there exists a point in $X(K)$ whose $f$-orbit is Zariski dense in $X$ and up to replacing $K$ by a finite extension, Pasten and Silverman studied the distribution of grand $(f,K)$-orbits and proved that many sets of representatives of grand $(f,K)$-orbits on various classes of varieties are Zariski dense. In particular, they showed that if $X$ is a geometrically simple abelian variety, then all such sets of representatives are Zariski dense. We demonstrate the existence of a dense set of representatives for maps on all abelian varieties.

• The paper establishes that if an abelian variety admits an endomorphism with a Zariski dense orbit, then a Zariski dense transversal of grand orbits exists over some finite extension.
• The paper employs isogeny invariance, Poincaré reducibility, and algebraic reductions to decompose the variety and analyze orbit dynamics across simple components.
• The paper demonstrates that both semisimple and unipotent actions lead to distinct, dense grand orbits, bridging the gap between known results on simple and general abelian varieties.

Dense Orbit Transversality for Endomorphisms of Abelian Varieties

Introduction and Motivation

The arithmetic and algebraic dynamics of endomorphisms on abelian varieties over number fields intersect deep themes in number theory, algebraic geometry, and algebraic dynamics. This paper ("On Dense Orbit Transversality for Endomorphisms of Abelian Varieties" (2606.29057)) investigates the distribution properties of grand orbits under endomorphisms of abelian varieties, extending and generalizing prior work by Pasten and Silverman on orbit propagation and Zariski density of special sets of orbit representatives.

Central to the discussion is the notion of an $(f,K)$-transversal—distinct representatives of grand $(f,K)$-orbits, collections of rational points whose orbits eventually merge under iteration of a self-map $f$ on a variety $X$ defined over a field $K$. Specifically, the question is to determine when certain sets of representatives (transversals) are Zariski dense, which has implications for the overall distribution of orbits and for dynamical analogues of classical density results.

The principal result of the paper establishes that for arbitrary (not necessarily simple) abelian varieties, given an endomorphism $f$ with a Zariski dense rational orbit, there exists a number field extension for which a Zariski dense $(f,K)$-transversal exists. This expands the scope beyond geometrically simple abelian varieties, for which Pasten and Silverman previously proved strong dense orbit transversality.

Theoretical Framework and Definitions

The paper formalizes the notion of grand $(f,K)$-orbits as the equivalence classes under the relation "eventually merge under iterates of $f$". An $(f,K)$-transversal is a set of points in $(f,K)$0 that intersects each grand orbit exactly once.

Two important density properties are defined:

• Weak Dense Orbit Transversality (W-DOT): Existence of a Zariski dense $(f,K)$1-transversal after possibly extending the ground field $(f,K)$2 finitely.
• Strong Dense Orbit Transversality (S-DOT): Every $(f,K)$3-transversal (after extending $(f,K)$4) is Zariski dense.

Prior work demonstrated S-DOT for geometrically simple abelian varieties. This paper proves W-DOT holds for all abelian varieties under the existence of a Zariski dense orbit.

Methodology and Structural Reductions

The proof strategy proceeds through a series of algebraic reductions:

• Isogeny Invariance: Key properties (existence of dense orbits and Zariski density of transversals) are preserved under isogenies between abelian varieties, allowing reduction to products of simpler building blocks.
• Decomposition via Poincaré Reducibility: Any abelian variety is isogenous to a product of powers of pairwise non-isogenous simple abelian varieties. The action of $(f,K)$5 splits accordingly, and dynamics can be studied componentwise.
• Orbit Distribution on Products: The behavior of grand orbits for maps acting diagonally on products follows from properties of the factors, modulo certain technical time synchronization issues unique to products.
• Dynamics Under Iteration: The orbit structure and density properties are stable under passing to positive powers of the self-map $(f,K)$6.

The process further leverages group-theoretic and algebraic results such as:

• Goursat's Lemma in the Isogeny Category: Description of surjective subgroups (or subvarieties) in products, crucial for density arguments and understanding subgroup structure.
• Faltings' Theorem on Subgroups: Structure of the Zariski closure of finitely generated subgroups in abelian varieties supports reductions based on projections.

Construction of Dense Transversals

Two major cases are addressed for the action of $(f,K)$7 on each isotypic component:

• Case 1: Existence of Nontrivial Semisimple Part. When, after possible iterates, the action on some simple factor is not unipotent, explicit construction of points with Zariski dense orbits and pairwise disjoint grand orbits is facilitated by linear independence in the $(f,K)$8-module structure and orbit calculations.
• Case 2: Entirely Unipotent Action. When $(f,K)$9 acts by a unipotent endomorphism plus translation, the dynamics are more intricate, but it is shown that if the translation part is nontorsion, infinitely many points in a dense subgroup (e.g., generated by a rational point with appropriately independent components) correspond to distinct grand orbits.

Key technical results utilized include:

• Explicit bounds (at most two) for the number of multiples in the same grand orbit in the unipotent case (see Proposition \ref{unipq} in the paper).
• Application of lemmas reducing the density check to the density of projections and using cyclic subgroups with prescribed linear independence properties in the relevant modules.

Main Theorem and Proof Summary

The main theorem asserts W-DOT for all abelian varieties:

Let $f$0 be an abelian variety and $f$1 an endomorphism with a Zariski dense orbit. Then, over some finite extension of the ground number field $f$2, there exists a Zariski dense set of rational points with pairwise distinct grand orbits.

The proof combines the aforementioned reductions, taking a point $f$3 whose orbit is dense by construction, and exploiting the structure of the abelian variety and module-theoretic properties of the action to ensure that infinitely many multiples of $f$4 are in distinct grand orbits and that this set is Zariski dense.

The argument extends to show that in the case of uncountable fields and dominant maps, every transversal is automatically Zariski dense, reflecting the inability for a countable union of proper algebraic subsets to cover the variety.

Numerical and Structural Highlights

The paper establishes that dense transversals always exist under mild dynamical assumptions for abelian varieties. The proof is completely constructive, making detailed use of the algebraic group structure, isogeny decompositions, and properties of unipotent affine maps—showing, for instance, that in the unipotent case a pigeonhole argument and module theory restrict the possible collisions in grand orbits.

The work clarifies a previously ambiguous logical gap between simple and general abelian varieties, and places the orbit structure in a comprehensive algebraic context.

Implications and Future Directions

The results have substantial implications for arithmetic dynamics and the broader study of algebraic dynamics on group varieties:

• They provide complete information about the possible distribution of grand orbits under endomorphisms with a dense orbit, which in turn bears on questions of dynamical uniformity, orbit growth, and the interplay between group structure and dynamics.
• The equivalence between dense transversals and other notions of density sheds light on connections between dynamical properties and Diophantine or geometric density (parallel to work in unlikely intersections and the dynamical Mordell-Lang conjecture).
• The methods employed could inform future work on dynamical generalizations of Manin–Mumford and Pink–Zilber problems, where a refined understanding of orbit distribution is essential.
• While the methods are specific to abelian varieties, the reduction techniques and group-theoretic approach may extend to semiabelian varieties, or more generalized algebraic groups, posing natural questions for subsequent research.

Conclusion

This paper demonstrates that every endomorphism with a Zariski dense orbit on an abelian variety over a number field (after possible extension) admits a Zariski dense transversal of grand orbits. The work unifies and generalizes earlier special cases, deploying a variety of tools from the structure theory of abelian varieties, algebraic dynamics, and group theory. Both the theoretical insight and the structural, constructive techniques are likely to inform ongoing work on the dynamics of algebraic group endomorphisms and their arithmetic properties.

Reference: "On Dense Orbit Transversality for Endomorphisms of Abelian Varieties" (2606.29057).