A note on Zilber-Pink in $Y(1)^n$

Abstract: Building on \cite{daworrpap,dawpap}, we prove two Zilber-Pink-type statements in $Y(1)^n$, assuming a weak form of the Lang-Trotter conjecture for pairs of elliptic curves.

• The paper refines Zilber-Pink finiteness by reducing intersection problems in Y(1)^n to those with strongly special subvarieties.
• It employs G-functions and explicit height bounds to translate geometric intersection issues into analytic estimates tied to supersingular reductions.
• Conditional on a weak Lang-Trotter assumption for non-isogenous, non-CM elliptic curves, the work yields effective finiteness and degree bounds for atypical intersections.

Zilber-Pink-Type Finiteness Results in Products of Modular Curves

Introduction and Context

The Zilber-Pink conjecture encodes finiteness expectations for intersections between subvarieties and special (i.e., Shimura-theoretically defined) loci of high codimension in ambient Shimura varieties. For moduli spaces of elliptic curves, the modular curve $Y(1)$ and its powers $Y(1)^n$ serve as principal test cases for unlikely intersection problems. The paper "A note on Zilber-Pink in $Y(1)^n$" (2605.00766) advances the conditional state of Zilber-Pink-type results for curves in $Y(1)^n$ by leveraging novel consequences of the Lang-Trotter conjecture for pairs of elliptic curves.

The author builds on strategies developed by Habegger, Pila, Daw, and Orr, further refining the reduction of the Zilber-Pink conjecture to finiteness assertions about intersections with "strongly special" subvarieties. The principal results supply explicit finiteness statements for certain atypical intersection loci, contingent on a weak form of Lang-Trotter for supersingular primes for two non-isogenous, non-CM elliptic curves.

Main Theorems and Conditional Statements

Geometric Setup and Atypical Intersections

Let $S \subset Y(1)^n$ ($n\geq 4$) be a smooth, irreducible curve defined over $\bar{\mathbb{Q}}$ and not contained in any proper special subvariety. For $I \subset \{1, \dots, n\}$, $|I|=3$, and positive integers $N, N'$, the subvariety $Y(1)^n$0 is cut out by two modular polynomial relations among coordinates specified by $Y(1)^n$1. The primary objects of interest are loci of points $Y(1)^n$2 that belong to special subvarieties defined by multiple, pairwise disjoint sets of these modular relations.

Theorem 1: Finiteness for Quadruple Special Intersections

Statement: Assume $Y(1)^n$3. If $Y(1)^n$4 meets a subvariety of the form $Y(1)^n$5 at a Hodge generic point and a weak Lang–Trotter–type conjecture for pairs of elliptic curves (see below) holds, then the set

$Y(1)^n$6

is finite.

Theorem 2: Finiteness for Higher-Order Intersections

Statement: For $Y(1)^n$7, if $Y(1)^n$8 are disjoint, $Y(1)^n$9, $Y(1)^n$0 meets $Y(1)^n$1 at a Hodge generic point, and the same weak Lang-Trotter assumption holds, then the set

$Y(1)^n$2

is finite.

Significant Point: The proofs show that, under the stated assumptions, not only is the set of such points finite, but there are explicit (albeit ineffective) height bounds on their size in terms of degrees over $Y(1)^n$3, subject to the Lang-Trotter hypothesis.

Weak Lang–Trotter Assumption and Its Role

The finiteness results rest crucially on the following conjectural input:

Let $Y(1)^n$4 be non-CM, non-isogenous elliptic curves over a number field $Y(1)^n$5. Then the number of primes (of norm $Y(1)^n$6) at which both $Y(1)^n$7 and $Y(1)^n$8 are supersingular is $Y(1)^n$9. This is far weaker than the heuristically expected $Y(1)^n$0 density but is sufficient to deduce polynomial-type bounds on contributions to the degree of the global relation constructed in the argument.

Technical Methods: G-functions, Height Bounds, and Period Relations

The main novelty is the use of the G-functions method, originally developed by André and applied to unlikely intersection contexts by Daw, Orr, and Papas. The technique proceeds as follows:

1. Reformulation in Terms of Families: The intersection problem is translated into height bounds for $Y(1)^n$1 parameterizing families of elliptic curves. Special intersection conditions are expressed in terms of isogeny relations among fibers over points of $Y(1)^n$2.
2. Construction of Period Relations: For each place of "proximity" (i.e., $Y(1)^n$3 being $Y(1)^n$4-adically close to a basepoint $Y(1)^n$5), one constructs polynomial relations among periods of the G-functions associated to the relevant family of elliptic curves.
3. Global Nontrivial Relation and Degree Bounds: By multiplying the local relations across all such places, a global nontrivial relation of controlled degree is obtained. The degree analysis is reduced to bounding the number of places where the supersingular condition holds simultaneously for two relevant fibers — precisely captured by the weak Lang-Trotter conjecture above.
4. Height Bound via André-Bombieri: The André-Bombieri “Hasse principle” for G-functions then provides explicit height bounds for the points of intersection in terms of the degree of the global relation.

Comparison with Prior Results and Conditionality

Previous results of Habegger-Pila [habeggerpila1], Daw-Orr [daworr4], and Papas [papaszpy1] established aspects of the Zilber-Pink conjecture for $Y(1)^n$6, often under asymmetric or boundary-intersecting hypotheses, or for smaller $Y(1)^n$7. The present work removes reliance on such asymmetry and deals directly with higher codimension intersections, at the price of depending on the Lang-Trotter-type conjecture for pairs. In this sense, it complements and advances the program initiated in the aforementioned works, especially in the application of the G-functions method to highly degenerate intersections.

Period-Relation Independence and the Supersingular Locus

A key observation in the proof is the independence (from $Y(1)^n$8) of the local period relation factors at ordinary places, with the supersingular places generating the essential obstruction to obtaining “too many” points of intersection. This is conceptually analogous to the detection of unlikely intersections via the distribution of "atypical" or highly degenerate points, explicitly linked here to supersingular reduction phenomena.

The paper also speculates on the extension of this method to other Shimura varieties (e.g., $Y(1)^n$9) and on the broader significance of period-relation independence except at rare places.

Implications and Future Prospects

Theoretical Implications: These results clarify the mechanism by which the sparsity of supersingular places — tied to fundamental number-theoretic distribution phenomena (Lang-Trotter/Sato-Tate) — constrains the frequency of unlikely intersections for curves in $S \subset Y(1)^n$0. They provide a conditional template for similar future results in higher rank Shimura varieties, especially under conjectures about distributions of reduction types.

Practical Consequences: The methods offer an effective strategy to reduce certain Zilber-Pink statements to established or conjectural analytic number theory, bringing the prospect of further progress closer, assuming advances on the relevant equidistribution or sparsity conjectures.

Directions for Future Work: Major avenues include making the arguments unconditional by proving the required sparsity estimates for supersingular primes for pairs of non-CM, non-isogenous elliptic curves, and adapting the G-functions framework to moduli of higher-dimensional abelian varieties and mixed Shimura settings.

Conclusion

This paper succeeds in advancing conditional finiteness statements of Zilber-Pink type for smooth irreducible curves in $S \subset Y(1)^n$1 intersecting families of higher-codimension special subvarieties. The crux is the translation of the geometry of unlikely intersections into analytic bounds (height and counting) that are realized via explicit period relations and the sparsity of simultaneous supersingular reduction, under a weak form of the Lang-Trotter conjecture. The approach illuminates how arithmetic properties of underlying moduli drive the structure of atypical intersections and offers new strategies for further progress in unlikely intersection theory.