An archimedean approach to singular moduli on Shimura curves

This paper provides a new Archimedean proof of the Giampietro-Darmon conjecture regarding singular moduli on genus 0 Shimura curves by evaluating Green's functions at CM points, offering an alternative to Daas's previous $p$-adic approach while highlighting the parallels and distinctions between the two methods.

The Gist

Here is an explanation of the paper "An archimedean approach to singular moduli on Shimura curves," translated into everyday language with creative analogies.

The Big Picture: Solving a Mathematical Puzzle with a New Lens

Imagine you are a detective trying to solve a mystery about numbers. Specifically, you are looking at a special type of number called a "singular modulus." These aren't just random numbers; they are like the "DNA" of certain geometric shapes (curves) that appear in advanced math.

In 1985, two famous mathematicians, Gross and Zagier, discovered a beautiful pattern. They found that if you take two of these special numbers and subtract them, the result isn't just a messy decimal. It's a number that can be broken down into a very specific, clean list of prime numbers (like 2, 3, 5, 7...). It's as if the universe whispered a secret code: "If you subtract these two, you get exactly $2^{15} \times 3^7 \times 5^3$..."

This paper is about a new proof of a modern version of that secret code.

The Setting: From Flat Maps to 3D Worlds

To understand the paper, we need to change our scenery:

1. The Old Map (The Original Problem): Gross and Zagier worked on a flat, 2D map called the "upper half-plane." Think of this as a standard sheet of graph paper where you draw curves.
2. The New World (Shimura Curves): This paper moves the action to a "Shimura curve." Imagine this not as a flat sheet, but as a twisted, 3D sculpture or a complex maze. It's much harder to navigate.
3. The Goal: Mathematicians Giampietro and Darmon guessed that the same "clean prime number code" exists in this 3D maze. A mathematician named Daas proved it recently, but he used a very specific, high-tech tool: $p$-adic numbers.

What are $p$-adic numbers?
Think of them as looking at the world through a microscope that only sees one specific color (a specific prime number). It's a powerful lens, but it's very specialized. Daas used this microscope to prove the code exists.

The Author's Innovation: The "Archimedean" Approach

The author of this paper, Mateo Crabit Nicolau, says: "Wait a minute. We don't need that specific microscope. Let's look at the whole picture with our naked eyes."

In math terms, he uses an "Archimedean" approach. Instead of zooming in on one prime number, he looks at the entire shape using standard calculus and geometry (Green's functions).

The Analogy:

• Daas's Proof: Like trying to understand a symphony by listening to just the violin section, one note at a time, using a super-sensitive ear.
• This Paper: Like standing in the concert hall and listening to the entire orchestra at once, feeling the vibrations in the air.

How the Proof Works (The Detective Story)

The author uses a strategy inspired by the original 1985 proof, which involves three main steps:

1. Building a "Ghost" Machine (Eisenstein Series)

The author builds a complex mathematical machine (a non-holomorphic modular form) that acts like a ghost. This ghost is designed to be silent (zero) in the specific 3D maze (Shimura curve) they are studying.

• Why? Because if the machine is silent, it means two different ways of calculating the same thing must cancel each other out perfectly.

2. Splitting the Ghost (Holomorphic Projection)

The author takes this ghost machine and splits it into two parts:

• Part A (The Right Side): This part calculates the "clean prime number code" (the product of primes).
• Part B (The Left Side): This part calculates the "distance" between points on the 3D maze using a concept called Green's functions.

What is a Green's function?
Imagine you drop a pebble in a pond. The ripples spreading out are the Green's function. In this math world, it measures the "energy" or "distance" between two points on the twisted 3D curve.

3. The "Aha!" Moment

The author proves that Part A and Part B are actually the same thing, just written in different languages.

• Part A says: "The answer is a product of primes."
• Part B says: "The answer is the distance between these special points."

Because the "ghost machine" must be silent (zero), Part A must equal Part B. Therefore, the distance between the points is exactly equal to the product of the primes.

Why Does This Matter?

1. It's a New Perspective: It shows that you don't need the specialized "microscope" ($p$-adic numbers) to solve this problem. You can solve it by looking at the geometry of the whole shape.
2. It Connects Two Worlds: It bridges the gap between analysis (calculus, distances, waves) and number theory (primes, factors). It shows that the "shape" of the universe dictates the "arithmetic" of numbers.
3. It's a Generalization: It proves that the beautiful patterns Gross and Zagier found in 1985 hold true even in these more complex, twisted 3D worlds.

Summary in One Sentence

The author proves that the secret code of prime numbers hidden in complex 3D mathematical shapes can be unlocked not by zooming in on a single number, but by measuring the "ripples" (distances) between points on the shape, revealing a deep harmony between geometry and arithmetic.

Technical Summary

Here is a detailed technical summary of the paper "An archimedean approach to singular moduli on Shimura curves" by Mateo Crabit Nicolau.

1. Problem Statement and Context

The paper addresses the arithmetic factorization of differences of singular moduli (values of modular functions at Complex Multiplication (CM) points) on Shimura curves.

• Background: In their seminal 1985 work, Gross and Zagier proved a formula for the factorization of the difference $j(\tau_1) - j(\tau_2)$ of singular moduli on the classical modular curve (genus 0, $N=1$). Their result expresses the norm of this difference as a product of prime powers determined by the discriminants of the CM fields.
• The Generalization: Giampietro and Darmon conjectured, and Daas (2025) proved, a generalization of this result to Shimura curves $X_N$ of genus 0 (where $N \in \{6, 10, 22\}$). These curves are associated with indefinite quaternion algebras $B_N$ ramified at two primes $p, q$.
• The Gap: Daas's proof relied on $p$-adic methods, specifically using $p$-adic $\Theta$-functions and the Čerednik-Drinfeld isomorphism to relate the cross-ratio of singular moduli to $p$-adic values.
• The Goal: Nicolau provides a new archimedean proof of this conjecture. Instead of working $p$-adically, the author evaluates Green's functions at CM points on the Shimura curve, mirroring the original analytic strategy of Gross and Zagier.

2. Methodology

The proof follows the analytic framework of the original Gross-Zagier theorem but adapts it to the setting of Shimura curves and indefinite quaternion algebras.

A. Construction of a Modular Form

The author constructs a specific non-holomorphic modular form to apply the holomorphic projection technique:

1. Hilbert Eisenstein Series: A family of Hilbert Eisenstein series $E_{s, \lambda}$ is defined over the real quadratic field $F = \mathbb{Q}(\sqrt{D_1 D_2})$.
2. Derivative at $s=0$: A specific linear combination of these series is differentiated with respect to $s$ at $s=0$. This yields a non-holomorphic cusp form of weight 2, denoted $E_N(z)$.
3. Holomorphic Projection: The author applies the holomorphic projection operator to $E_N(z)$ to obtain a holomorphic cusp form $f \in S_2(\Gamma_0(N))$.
4. Vanishing Condition: For the specific cases $N \in \{6, 10, 22\}$, the space of cusp forms $S_2(\Gamma_0(N))$ is trivial (or the form is forced to be zero by Atkin-Lehner involution symmetry). Consequently, the Fourier coefficients of $f$ must vanish. Specifically, the first coefficient $a_1 = b_1 - c_1 = 0$, implying $b_1 = c_1$.

B. Interpretation of Coefficients

The core of the proof lies in identifying the two terms $b_1$ and $c_1$:

• The Term $b_1$ (Arithmetic Side): This term arises from the holomorphic projection of the Eisenstein series. It is explicitly calculated to be the logarithm of the right-hand side of the Giampietro-Darmon conjecture (the product of prime powers involving the function $F(m)$).
• The Term $c_1$ (Geometric Side): This term arises from the non-holomorphic part of the Eisenstein series. The author rewrites this sum as an evaluation of Green's functions on the Shimura curve $X_N$.
  • A quadratic form $\det_{F, \alpha}$ is defined on the quaternion algebra $B_N$ associated with embeddings of the CM fields.
  • The sum over elements in the quaternion order is transformed into a sum over values $\nu \in F$ represented by this quadratic form.
  • The author proves that the logarithm of the cross-ratio of singular moduli (the left-hand side of the conjecture) is exactly equal to the sum of these Green's function evaluations.

C. The Archimedean Analogy

The paper highlights a striking parallel between the two approaches:

• $p$-adic proof (Daas): Relates the cross-ratio to a limit of products involving $p$-adic $\Theta$-functions.
• Archimedean proof (Nicolau): Relates the cross-ratio to a limit of products involving the Legendre function of the second kind (via Green's functions), which serves as the archimedean analogue to the $p$-adic $\Theta$-function.

3. Key Contributions

1. New Proof of the Giampietro-Darmon Conjecture: The paper provides an independent, archimedean proof of the factorization formula for singular moduli on Shimura curves of genus 0, confirming the result previously established by Daas using $p$-adic methods.
2. Green's Function Evaluation: It establishes a precise link between the cross-ratio of singular moduli on Shimura curves and the values of Green's functions at CM points, extending the Gross-Zagier analytic philosophy to the quaternionic setting.
3. Structural Comparison: The work explicitly details the similarities and differences between the $p$-adic and archimedean proofs, offering a unified perspective on how singular moduli factorization arises from the vanishing of modular forms.
4. Generalization to Hecke Operators: The paper extends the analysis to higher Fourier coefficients $a_m$. It shows that for general $N$, these coefficients correspond to Néron height pairings of CM points on the Shimura curve, analogous to the classical Gross-Zagier formula relating derivatives of $L$-functions to heights.

4. Main Results

• Theorem 2 (Daas, proved here): For $N \in \{6, 10, 22\}$ and coprime fundamental discriminants $D_1, D_2$, the norm of the cross-ratio of singular moduli $J_N(D_1, D_2)$ satisfies:
  $$J_N(D_1, D_2)^{\pm 1} = \pm \prod_{\substack{x^2 < D \\ x^2 \equiv D \pmod{4N}}} F\left(\frac{D-x^2}{4N}\right)^{\delta(x)}$$
  where $F(m)$ is a specific product of prime powers and $\delta(x)$ depends on the choice of square roots of $D$ modulo $2N$.
• Theorem 26: For general $N$, the Fourier coefficients $a_m$ of the constructed modular form $f$ are equal to the Néron height pairing $\langle P_1, T_m P_2 \rangle_N$ of CM divisors on the Shimura curve. This unifies the arithmetic factorization (via $b_m$) and the geometric height (via $c_m$).

5. Significance

• Methodological Shift: By avoiding $p$-adic deformation theory and Čerednik-Drinfeld theory, this proof makes the result accessible via classical analytic number theory and the theory of automorphic forms on the upper half-plane.
• Conceptual Clarity: It clarifies that the factorization of singular moduli is a consequence of the vanishing of a specific modular form, driven by the geometry of the Shimura curve (Green's functions) and the arithmetic of the real quadratic field (Eisenstein series).
• Bridge to Heights: The interpretation of higher coefficients as height pairings suggests a deeper connection between the arithmetic of singular moduli and the geometry of the Jacobian of Shimura curves, potentially paving the way for generalizations to higher genus curves or different levels.

In summary, Nicolau's work successfully transplants the analytic machinery of Gross and Zagier to the realm of Shimura curves, providing a robust archimedean alternative to recent $p$-adic proofs and deepening the understanding of the arithmetic of singular moduli.