Special geodesics and atypical intersections

1. Problem Statement and Context

The paper addresses a question posed by Henri Darmon regarding the characterization of strongly special curves in the product of modular curves $Y(1) \times Y(1)$.

• Background: In the theory of complex multiplication, Heegner points are special algebraic points on modular curves. André's Theorem (a special case of the André–Oort conjecture) states that an irreducible complex algebraic curve $C \subset Y(1) \times Y(1)$ (which is not horizontal or vertical) is strongly special (i.e., a component of a Hecke correspondence defined by a modular polynomial $\Phi_N$) if and only if it contains infinitely many points with Heegner coordinates.
• The Analogy: Darmon and Vonk established that real quadratic geodesics in the upper half-plane $\mathbb{H}$ play a role analogous to Heegner points in the context of real quadratic fields. Specifically, the image of a real quadratic geodesic under the $j$-invariant map is a "special geodesic" in $Y(1)$.
• The Question (Question 1.2.3): Let $C \subset Y(1) \times Y(1)$ be an irreducible complex algebraic curve. If $C$ contains infinitely many real algebraic curves whose projections onto each copy of $Y(1)$ are unions of real quadratic geodesics, is $C$ necessarily a strongly special curve?

2. Methodology

Tamiozzo approaches the problem using intersection theory, Weil restriction of scalars, and the Zilber–Pink conjecture. The methodology proceeds in several key steps:

A. Translation to Real Geometry via Weil Restriction

The author interprets the complex curve $C$ as a real surface $\tilde{C}$ via the Weil restriction of scalars from $\mathbb{C}$ to $\mathbb{R}$.

• $Y(1) \cong \mathbb{A}^1_{\mathbb{C}}$ corresponds to $\mathbb{A}^2_{\mathbb{R}}$ via the map $(x, y) \mapsto x+iy$.
• Consequently, $Y(1) \times Y(1)$ corresponds to $\mathbb{A}^2_{\mathbb{R}} \times \mathbb{A}^2_{\mathbb{R}} \cong \mathbb{A}^4_{\mathbb{R}}$.
• The condition that $C$ contains real curves with "special geodesic projections" translates to the existence of real algebraic curves in $\tilde{C}(\mathbb{R})$ that lie within unions of products of special geodesics.

B. Identification of "Atypical" Intersections

The core insight is that the intersection between the real surface $\tilde{C}(\mathbb{R})$ and the union of special geodesic products is "atypically large."

• In $\mathbb{R}^4$, the intersection of two 2-dimensional surfaces is expected to be 0-dimensional (points).
• The presence of 1-dimensional curves (real algebraic curves) in this intersection implies an "atypical" intersection.
• This setup is linked to the Zilber–Pink conjecture, which predicts the finiteness of atypical components in intersections of subvarieties with special subvarieties in Shimura varieties.

C. Base Change and Reduction to $Y(1)^4$

To apply known results, the author performs a base change to $\mathbb{C}$ and utilizes a specific morphism $f: \mathbb{A}^2_{\mathbb{C}} \to \mathbb{A}^2_{\mathbb{C}}$ defined by $(X, Y) \mapsto (X+iY, X-iY)$.

• This maps the real surface $\tilde{C}$ to a complex surface $V$ in $\mathbb{A}^4_{\mathbb{C}}$.
• The real curves with special geodesic projections map to curves $A$ in $V$.
• These curves $A$ are shown to be strongly atypical subvarieties of $V$ (i.e., their dimension is larger than expected given the dimensions of $V$ and the special subvariety $S$ they intersect).

D. Application of Pila–Tsimerman's Theorem

The author invokes Theorem 2.1.2 (Pila–Tsimerman, 2016), which proves a weak version of the Zilber–Pink conjecture for $Y(1)^n$: Every subvariety contains finitely many maximal strongly atypical subvarieties.

• By showing that the assumption of "infinitely many such curves" leads to infinitely many distinct strongly atypical curves, the author deduces that the ambient surface $V$ itself must be contained in a special subvariety.
• This containment forces the original curve $C$ to be strongly special.

3. Key Contributions

1. Resolution of Darmon's Question: The paper provides a positive answer to Question 1.2.3. It proves that the existence of infinitely many real algebraic curves with special geodesic projections is a sufficient condition for a curve in $Y(1) \times Y(1)$ to be strongly special.
2. Theorem 2.3.3 (Main Result): The author establishes a characterization analogous to André's theorem but using real geodesics instead of Heegner points:

   An irreducible complex algebraic curve $C \subset Y(1) \times Y(1)$ (non-horizontal, non-vertical) is strongly special if and only if $C(\mathbb{C})$ contains infinitely many curves with special geodesic projections.

3. Bridging Real and Complex Arithmetic Geometry: The work successfully connects the arithmetic of real quadratic fields (via real quadratic geodesics) with the geometry of modular curves and the Zilber–Pink conjecture, extending the scope of "special" objects beyond complex points.
4. Distinction from Heegner Point Characterization: The author notes (Remark 2.3.4) that curves with special geodesic projections may contain few Heegner points, meaning this result cannot be trivially deduced from the classical André–Oort theorem; it requires the specific machinery of atypical intersections in the real setting.

4. Results

• Theorem 2.3.3: Proven as stated above.
• Proposition 2.4.1: Demonstrates the converse direction: if $C$ is strongly special, it indeed contains infinitely many such curves (constructed by intersecting the Weil restriction $\tilde{C}$ with specific real curves $Z_M$).
• Finiteness of Atypical Components: The proof relies on the fact that if $C$ were not strongly special, the associated surface $V$ in $\mathbb{A}^4_{\mathbb{C}}$ would contain only finitely many maximal strongly atypical curves. The assumption of infinitely many input curves leads to a contradiction unless $V$ is special.

5. Significance

• Advancement of the Zilber–Pink Conjecture: The paper applies the Zilber–Pink conjecture (specifically the Pila–Tsimerman theorem) to a novel setting involving real algebraic geometry and Weil restrictions, demonstrating the versatility of o-minimality and model-theoretic methods in number theory.
• Explicit Class Field Theory: By validating the analogy between Heegner points and real quadratic geodesics in the context of curve detection, the work supports the broader program of explicit class field theory for real quadratic fields initiated by Darmon and Vonk.
• New Characterization of Modular Curves: It provides a new geometric criterion for identifying modular curves (strongly special curves) based on the behavior of their real points and geodesic projections, rather than just their complex algebraic points.

In summary, Tamiozzo's paper successfully translates a question about real quadratic geodesics into the language of atypical intersections in Shimura varieties, using the Pila–Tsimerman theorem to prove that such geometric configurations uniquely characterize modular curves.