André's theorem and weakly bounded height

Technical Summary: André's Theorem and Weakly Bounded Height

Problem Statement
The paper addresses the effective version of André's theorem, a specific case of the André–Oort conjecture for the affine plane $\mathbb{A}^2(\mathbb{C})$. The conjecture posits that a non-special algebraic curve $V$ contains only finitely many "special points," defined as pairs $(x, y)$ where both $x$ and $y$ are singular moduli (j-invariants of elliptic curves with complex multiplication). While the finiteness of such points was established by André (unconditionally) and Pila (for higher dimensions), these proofs were ineffective, providing no computable bounds on the height or the number of such points.

Previous effective results by Kühne and by Bilu, Masser, and Zannier established bounds for the height of special points, but these bounds exhibited poor dependence on the height of the curve $V$ (specifically, an exponent of 8 or higher, or no explicit dependence). The central problem this paper tackles is to establish an effective bound for the height of special points on a specific class of non-special curves that depends linearly on the height of the curve, $h(V)$.

Methodology
The author employs a refined version of the Pila–Zannier strategy, which combines o-minimal point counting with arithmetic lower bounds for Galois orbits. The proof is structured around controlling special points based on the size of their discriminants relative to a "bad" exceptional field $K^*$ arising from Tatuzawa's effective version of the Siegel–Landau bound on class numbers.

1. Classification of Special Points: Special points $(x, y)$ are divided into two categories:

   • Non-exceptional points: Those where the discriminants $\Delta_x, \Delta_y$ are not associated with the single possible exceptional field in Tatuzawa's theorem. For these, the Galois orbit size is large, allowing the use of effective point counting (via Binyamini's results) to bound the discriminants uniformly.
   • $\epsilon$-exceptional points: Those where the discriminants correspond to the exceptional field. These points have abnormally small Galois orbits and small class numbers.

2. Handling $\epsilon$-Exceptional Points:

   • The paper utilizes a result by Habegger which provides a height bound for the intersection of a curve $V$ and a modular curve $V(\Phi_N)$, provided the degrees of the coordinate functions on $V$ are distinct ($d_1 \neq d_2$).
   • The author explicitly calculates the constants in Habegger's result to show a linear dependence on $h(V)$.
   • A contradiction argument is used: If an $\epsilon$-exceptional point has a very large height, it must lie on a modular curve with a large level $N$. However, isogeny estimates and lower bounds on the Galois orbit (derived from the small class number of exceptional points) create a conflict between the upper bound on height (from Habegger) and the lower bound on the degree of the field extension. This forces the discriminants (and thus the heights) to be bounded.

3. Conditional Uniformity:

   • The paper introduces the property WBH($\eta, \kappa$) (Weakly Bounded Height), which posits a specific height bound for points on the intersection of $V$ and modular curves.
   • The author demonstrates that if WBH($\eta, \kappa$) holds, one can derive effective bounds for all non-special curves, potentially leading to a uniform effective version of André's theorem (Theorem 1.4).

Key Contributions and Results

• Theorem 1.1 (Main Unconditional Result): For a geometrically irreducible, non-special curve $V$ defined over a number field $L$, where the degrees of the coordinate functions $X$ and $Y$ restricted to $V$ are distinct ($d_1 \neq d_2$), the paper proves the existence of an effectively computable constant $c$ such that for every special point $(x, y) \in V$:
  $$\max\{h(x), h(y)\} \leq c_1([L:\mathbb{Q}], \max\{d_1, d_2\}) + 26 \max\{d_1, d_2\}^3 h(V)$$
  Furthermore, the discriminants satisfy a bound with an exponent of $2+\delta$ on $h(V)$, a significant improvement over the exponent 8 found in previous work.

• Theorem 1.3 (Conditional Result): Assuming the property WBH($\eta, \kappa$) holds, the paper establishes effective height bounds for special points on any non-special curve (removing the $d_1 \neq d_2$ restriction). If $\kappa < 2\eta$, this implies a uniform bound on the discriminants independent of $h(V)$, effectively proving Theorem 1.4 conditionally.

• Explicit Constants: The paper provides explicit formulas for the constants involved in the bounds, particularly in the proof of Theorem 6.3, which handles the $\epsilon$-exceptional case. These constants are shown to be independent of the field of definition $L$ in certain regimes.

Significance
The paper claims to establish the first effective version of André's theorem with a linear dependence on the height of the curve $V$ for a non-trivial class of curves. This represents a substantial improvement over previous effective results (Kühne, Wüstholz) which had polynomial dependence with high exponents (e.g., 8).

The author notes that the condition $d_1 \neq d_2$ is strictly stronger than the condition that $V$ is non-special (it excludes modular curves), but it is necessary for the current proof technique which relies on Habegger's specific height bound. The paper frames its conditional result (Theorem 1.3) as a strategy to extend these linear bounds to all non-special curves, contingent on proving the WBH property. The work also offers a new method for deducing finiteness results directly from height bounds without relying on Masser–Wüstholz endomorphism estimates, offering a potentially more direct path to uniform effective results.