Technical Summary: Exceptional Sets for Certain 2F1 Hypergeometric Functions

1. Problem Statement

The paper addresses the determination of exceptional sets for Gauss hypergeometric functions ${}_2F_1(a, b, c; z)$ with rational parameters $a, b, c \in \mathbb{Q}$. The exceptional set is defined as:
$$E(a, b, c) := \{z \in \mathbb{Q} \mid {}_2F_1(a, b, c; z) \in \mathbb{Q}\}.$$

While the finiteness of the monodromy group implies the hypergeometric function is algebraic (a result by Beukers and Heckman), the general case involves non-algebraic functions. Wolfart conjectured that the exceptional set is infinite if and only if the monodromy group is arithmetic. This paper focuses on explicitly determining these sets for data $(a, b, c)$ where the associated monodromy group is an arithmetic triangle group belonging to Takeuchi's Class I. Specifically, it targets the nine non-compact arithmetic triangle groups commensurable with $\text{PSL}_2(\mathbb{Z})$.

2. Methodology

The author employs a dual approach combining transcendence theory with explicit modular form identities:

A. Geometric and Transcendence Framework

1. Abelian Varieties: Using Euler's integral representation of the hypergeometric function, the paper constructs associated algebraic curves $C(N, z)$ and their desingularizations $X(N, z)$. The periods of these curves are related to the hypergeometric function values.
2. Jacobian Varieties: The author considers the Jacobian variety $\text{Jac}(X(N, z))$ and its new part $\text{Jac}^{\text{new}}(X(N, z))$, which has dimension $\phi(N)$.
3. Transcendence Results: The paper applies deep transcendence results by Wüstholz and Wolfart. These results provide necessary conditions for the algebraicity of periods of abelian varieties with complex multiplication (CM). Specifically, they link the rationality of the hypergeometric value to the CM nature of the underlying period lattice.

B. Hypergeometric-Modular Identities

1. Schwarz Triangle Groups: The hypergeometric differential equation is analyzed via the Schwarz map, which uniformizes the upper half-plane $\mathbb{H}$ onto a hyperbolic triangle. The inverse of this map is a Hauptmodul (a generator of the function field) for the corresponding triangle group $\Gamma$.
2. Explicit Identities: The paper derives identities expressing ${}_2F_1(a, b, c; t(\tau))$ in terms of modular forms (specifically products of Dedekind eta functions). Theorem 2 establishes that for a Hauptmodul $t$ and a specific branch, the hypergeometric function is proportional to a linear combination of $\tau$ and a square root of a Wronskian factor $W(\tau)$.
3. CM Points: By evaluating these modular identities at CM points (points $\tau \in \mathbb{H} \cap \mathbb{Q}(\sqrt{-d})$), the paper identifies specific algebraic values of the Hauptmodul $t(\tau)$ that yield rational hypergeometric values.

3. Key Contributions and Results

Main Theorem (Theorem 1)

For data $(a, b, c)$ where the monodromy group is an arithmetic triangle group in Takeuchi's Class I (with $0 < a, b, c < 1$), the exceptional set is characterized as:
$$E(a, b, c) = \{ z \in \mathbb{Q} \mid \exists \tau_d \in \mathbb{Q}(\sqrt{-d}) \cap \mathbb{H} \text{ such that } z = t(\tau_d) \},$$
where:

• $t$ is a specific Hauptmodul chosen for the triangle group.
• $d \in \{1, 2, 3\}$ is a positive integer determined by the specific group.
• If $c \ge 1$, the exceptional set is trivial: $E(a, b, c) = \{0\}$.

Explicit Determination

The paper explicitly computes the exceptional sets for all such cases in Takeuchi's Class I.

• Table 1 lists the triangle groups, their corresponding Hauptmoduln (e.g., $S_2, S_3, J_3, R_3$), and their relations to standard modular functions.
• Table 2 provides the explicit hypergeometric-modular identities, including the constant $\alpha$ (a root of unity) and the Wronskian factor $W(\tau)^{1/2}$ for each case.
• Table 4 summarizes the integers $d$ and the specific modular functions $t$ for each datum.
• Table 5 lists explicit examples of algebraic values. For instance, for the datum $(1/4, 1/4, 1/2)$ corresponding to the group $(2, \infty, \infty)$ (identified with $\Gamma_0(2)$), the exceptional set is generated by $S_2(\tau)$ where $\tau \in \mathbb{Q}(i) \cap \mathbb{H}$. A specific evaluation yields $S_2(i) = 9$, leading to:
  $${}_2F_1\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{2}; 9\right) = 2 - \frac{i}{2\sqrt{2}}.$$

4. Significance and Claims

The paper claims to provide a complete and explicit description of the exceptional sets for the specific class of hypergeometric functions associated with Takeuchi's Class I arithmetic triangle groups.

• Synthesis of Methods: The work demonstrates how combining transcendence theory (Wolfart/Wüstholz) with explicit modular form identities allows for the precise identification of rational points in the exceptional set, rather than just proving their existence or density.
• Connection to L-values: The author notes that the explicit CM-point evaluations of Hauptmoduln obtained in this work (specifically in Table 5) can serve as algebraic inputs for computing special L-values of CM Hecke eigenforms, referencing prior work by Edixhoven, Yafaev, and others who utilize such values.
• Verification of Conjectures: The results confirm Wolfart's conjecture in this specific context by showing that the exceptional sets are indeed infinite and Zariski dense in $\mathbb{C}$, generated by the images of CM points under the relevant modular functions.

The paper does not propose new experimental methods or future applications beyond the immediate context of determining these sets and their utility in L-value computations as previously established in the literature.