Unlikely intersections with CM abelian varieties in a family and explicit bounds for canonical heights under endomorphisms

Technical Summary: Unlikely Intersections with CM Abelian Varieties in a Family and Explicit Bounds for Canonical Heights under Endomorphisms

Problem Statement
The paper addresses a specific instance of the Zilber–Pink conjecture regarding "unlikely intersections" within families of abelian varieties. Let $S$ be a smooth irreducible curve over $\mathbb{Q}$ and $\pi: A \to S$ be an abelian scheme of relative dimension $g \geq 1$. The authors investigate the intersection of an irreducible curve $C \subset A$ (defined over $\mathbb{Q}$) with the union of all proper algebraic subgroups of the fibers $A_s$ that possess Complex Multiplication (CM).

Previous work by Barroero and Capuano (2020) established that if $C$ is not contained in a proper subgroup scheme, its intersection with the union of flat subgroup schemes of codimension at least 2 is finite. This paper extends that result to the case where the intersections occur with the algebraic subgroups of the CM fibers specifically. The main theorem asserts that if $A \to S$ is not isotrivial and $C$ is not contained in a fixed fiber or a translate of a proper flat subgroup scheme by a constant section, then the set of points $P \in C(\mathbb{C})$ such that the fiber $A_{\pi(P)}$ has CM and $P$ lies in a proper algebraic subgroup of that fiber is finite.

Methodology
The proof follows the Pila–Zannier strategy, combining functional transcendence (o-minimality) with arithmetic geometry. The methodology proceeds through the following stages:

1. Reduction to the Universal Family: The problem is reduced to the case where $A$ is the universal family of principally polarized abelian varieties over a curve $S \subset \mathcal{A}_g$. This involves finite base changes and isogenies to ensure the existence of a principal polarization and a level-3 structure, allowing the use of the fine moduli space $\mathcal{A}_{g,1,3}$.
2. O-minimality and Definability: Using the uniformization of the universal family by the Siegel upper half-space $\mathbb{H}_g \times \mathbb{C}^g$, the authors consider the preimage of the curve $C$. By restricting to a Siegel fundamental domain, this preimage becomes a definable set in the o-minimal structure $\mathbb{R}_{an, \exp}$.
3. Point Counting: The authors apply a theorem of Habegger and Pila to bound the number of points on this definable set that lie on algebraic subvarieties of bounded arithmetic complexity. This requires establishing that the algebraic relations defining the intersections have controlled height.
4. Arithmetic Bounds: The core of the arithmetic argument involves deriving explicit bounds for the canonical height of points on $C$ in terms of the Faltings height of the fiber and the degree of the field of definition. Crucially, this relies on constructing a non-zero endomorphism of the fiber that vanishes at the point $P$.
5. Explicit Height Control: A significant portion of the work is dedicated to providing explicit bounds for the canonical height under endomorphisms. The authors determine constants $\gamma_1, \gamma_2$ such that $\gamma_1 \hat{h}(P) \leq \hat{h}(f(P)) \leq \gamma_2 \hat{h}(P)$, where these constants are derived from the eigenvalues of the analytic representation of $f^\dagger f$ (where $\dagger$ is the Rosati involution).

Key Contributions and Results

• Main Theorem (Theorem 1.1): Proves the finiteness of the intersection of a non-isotrivial curve $C$ in an abelian scheme with the proper algebraic subgroups of CM fibers, provided $C$ is not contained in a fixed fiber or a translate of a flat subgroup scheme. This generalizes a previous result by Barroero (2019) from fibered powers of elliptic schemes to general abelian schemes.
• Explicit Canonical Height Bounds (Theorem 1.4 / Theorem 7.3): The paper establishes a general inequality for the canonical height under endomorphisms:
  $$\alpha_D^-(f) \cdot \hat{h}_{A,D}(P) \leq \hat{h}_{A,D}(f(P)) \leq \alpha_D^+(f) \cdot \hat{h}_{A,D}(P)$$
  Here, $\alpha_D^\pm(f)$ are the minimum and maximum eigenvalues of the analytic representation of $f^\dagger f$. The authors prove these constants are optimal and provide explicit formulas for them. This result is of independent interest, generalizing the classical identity $\hat{h}([n]P) = n^2 \hat{h}(P)$.
• Arithmetic Complexity Bounds: The authors derive explicit bounds for the height of the period matrix and the endomorphism associated with a point $P$ in the intersection set. Specifically, they show that the Rosati norm of the endomorphism vanishing at $P$ is bounded by a polynomial in the degree $[k(P):k]$.
• Matrix Bounds for Endomorphisms (Section 5): The paper provides effective bounds relating the Rosati norm of an endomorphism to the sup-norm of its rational representation matrix, depending on the period matrix and the polarization type.

Significance and Claims
The paper claims to settle the Zilber–Pink conjecture for curves in non-isotrivial abelian schemes in the specific context of intersections with CM fibers. The authors note that while the full Zilber–Pink conjecture for curves in non-isotrivial abelian schemes was previously known only for fibered powers of elliptic schemes (via work of Barroero, Capuano, and others), this result extends the scope to general abelian schemes.

The significance of the work lies in two areas:

1. Generalization: It moves beyond the specific setting of elliptic schemes to arbitrary abelian schemes, requiring a more sophisticated treatment of endomorphisms and heights.
2. Explicitness: Unlike many results in the field which rely on the existence of constants without explicit values, this paper provides explicit bounds for the canonical height under endomorphisms and for the arithmetic complexity of the relevant algebraic relations. This explicit control is a key ingredient in the proof, allowing the authors to compare the arithmetic lower bound (derived from the existence of the endomorphism) with the geometric upper bound (derived from the Pila–Zannier strategy) to conclude finiteness.

The authors acknowledge that the result is a particular case of the broader Zilber–Pink conjecture and that the functional transcendence tools used (specifically the Ax-Schanuel type results) currently limit the proof to the form stated, particularly regarding sections that are not constant. The work is presented as a contribution to the understanding of unlikely intersections and the arithmetic of abelian varieties, building upon the foundational works of Masser, Zannier, Pila, and others.