Antisymmetry of real quadratic singular moduli

This paper confirms the Darmon-Vonk conjecture on the antisymmetry of real quadratic singular moduli by analyzing rigid meromorphic cocycles for the split orthogonal group on four variables and establishing the modularity of a generating series of Kudla-Millson divisors.

The Gist

Here is an explanation of Sören Sprehe's paper, "Antisymmetry of Real Quadratic Singular Moduli," translated into everyday language using analogies.

The Big Picture: Solving a Mathematical Riddle

Imagine you have a magical map called the Complex Upper Half-Plane. On this map, there are special "treasure spots" called CM points (Complex Multiplication points). If you drop a magical function (the $j$-function) onto these spots, you get special numbers called Singular Moduli.

Mathematicians have known for a long time that these numbers are like perfect, algebraic jewels. They have a very neat property: if you take two different treasure spots, $\tau$ and $\omega$, and calculate the difference between their values, the result is "antisymmetric." This means if you swap the spots, the answer flips its sign (or becomes its inverse). It's like a seesaw: if $\tau$ is up, $\omega$ is down.

The Problem:
For a long time, mathematicians tried to find a similar map for Real Quadratic Fields (a different kind of number system). But there was a catch: the real quadratic "treasure spots" (called RM points) don't live on the magical map where the $j$-function works. They live in a different, "p-adic" world (a world based on prime numbers rather than infinity).

In 2019, Darmon and Vonk proposed a way to build a new map for this p-adic world. They created a new kind of function that acts like the $j$-function but for these real spots. They calculated values for pairs of spots and noticed something amazing: The values seemed to be antisymmetric too. If you swapped the spots, the value flipped.

The Conjecture:
Darmon and Vonk guessed this was true for all pairs of spots. They called this Conjecture A. But guessing isn't proving. The roles of the two spots in their construction were very different (like a chef and a sous-chef), so it wasn't obvious why the result should be perfectly symmetrical.

The Solution:
Sören Sprehe's paper proves that Conjecture A is true. He does this by building a new, symmetric map where both spots are treated equally, like two dancers on a stage rather than a chef and a sous-chef.

---

The Key Concepts (The Metaphors)

1. The "Rigid Meromorphic Cocycle" (The Magical Recipe)

Think of a Cocycle not as a math term, but as a recipe or a rulebook.

• In the old world (complex numbers), the rulebook tells you how to get a number from a spot.
• In this new p-adic world, the rulebook is called a Rigid Meromorphic Cocycle. It's a bit more abstract; it's a "rule" that assigns values to points, but it's defined by how it behaves when you move around the map.

2. The "Divisor" (The Footprint)

Every recipe leaves a footprint (a divisor).

• If you have a recipe for spot $\omega$, its footprint is a specific pattern of lines on the map.
• Darmon and Vonk had a recipe for a single spot, but its footprint was messy and asymmetrical.
• Sprehe's breakthrough was realizing that if you look at the product of two maps (a 4-dimensional space), you can create a symmetrical footprint.

3. The "Twisted Diagonal" (The Dance Floor)

Imagine a dance floor where two dancers, $\tau$ and $\omega$, are moving.

• In the old approach, you looked at the dance from the side (one dancer leading, one following).
• Sprehe looked at the dance from above. He realized that the "footprint" of the rulebook lies on a twisted diagonal.
• Think of a diagonal line on a checkerboard. If you swap the coordinates (swap the dancers), the diagonal flips.
• Because the underlying "footprint" is perfectly symmetric (it looks the same if you swap the dancers, just flipped), the resulting numbers must be antisymmetric.

4. The "Cup Product" (The Magic Glue)

The paper uses a tool from topology called the Cup Product.

• Imagine you have two separate streams of water (two mathematical objects).
• The Cup Product is a magical glue that combines them into a single, larger object.
• A fundamental rule of this glue is that it is graded commutative. In plain English: If you swap the order of the two streams you are gluing, the result flips its sign.
• Sprehe showed that his new function is essentially the result of gluing two things together. Because of the "glue rule," the antisymmetry happens automatically. It's not a coincidence; it's a law of the universe!

5. The "Kudla-Millson Divisors" (The Modular Series)

The paper also proves a second big result: Modularity.

• Imagine you have a collection of these footprints (divisors) for every possible integer $n$.
• You line them up in a row to form a long series.
• The paper proves that this entire row of footprints behaves like a Modular Form.
• Analogy: Think of a kaleidoscope. If you turn the dial (apply a transformation), the pattern inside shifts, but the overall symmetry remains perfect. The series of footprints is like that kaleidoscope pattern; it has a hidden, rigid structure that mathematicians call "modularity." This connects the p-adic world to the classical world of modular forms.

---

The Step-by-Step Story of the Proof

1. The Setup: Sprehe starts by looking at a 4-dimensional space (two copies of the p-adic map).
2. The Construction: He builds a new "rulebook" (a cocycle) on this 4D space. This rulebook has a special property: its footprint is supported on "twisted diagonals."
3. The Symmetry: He proves that this new rulebook is symmetric. If you swap the two coordinates of the 4D space, the rulebook stays the same (up to a tiny, harmless factor).
4. The Evaluation: He shows that if you take this new rulebook and "evaluate" it at a pair of points $(\tau, \omega)$, you get a number.
5. The Connection: He proves that this new number is exactly the same as the old number Darmon and Vonk were calculating (with a small square factor).
6. The "Aha!" Moment: Because the new rulebook is symmetric, and the "glue" (Cup Product) flips signs when you swap inputs, the final number must be antisymmetric.
   • Old way: "Let's calculate A, then B, then swap." (Hard to prove).
   • New way: "We built a symmetric machine. The machine's output is guaranteed to flip when inputs are swapped." (Proven).

Why Does This Matter?

This paper is a bridge.

• It confirms a deep guess made by Darmon and Vonk.
• It shows that the "p-adic world" (which seems very strange and disconnected from our usual numbers) actually follows the same elegant, symmetric laws as the "complex world."
• It opens the door to understanding how these special numbers generate "abelian extensions" (a fancy way of saying they help build new number systems) for real quadratic fields, which has been a major unsolved mystery in number theory for decades.

In a nutshell: Sprehe took a messy, one-sided puzzle, built a symmetrical 4D frame around it, and showed that the solution was hiding in plain sight all along, waiting for the right perspective to reveal its perfect balance.

Technical Summary

Here is a detailed technical summary of the paper "Antisymmetry of Real Quadratic Singular Moduli" by Sören Sprehe.

1. Problem Statement and Context

The paper addresses a fundamental problem in the arithmetic of real quadratic fields, specifically concerning the construction of abelian extensions and the definition of "singular moduli" for real quadratic irrationalities.

• Background: In the theory of Complex Multiplication (CM), the classical modular $j$-function evaluated at CM points (imaginary quadratic numbers) yields algebraic integers (singular moduli) that generate abelian extensions of imaginary quadratic fields. The difference of two such values, $J_\infty(\tau_1, \tau_2) = j(\tau_1) - j(\tau_2)$, is symmetric (antisymmetric in the sense $J_\infty(\tau_1, \tau_2) = -J_\infty(\tau_2, \tau_1)$) and its norm has a known prime factorization (Gross-Zagier).
• The Challenge: Extending this to Real Quadratic Fields is obstructed because real quadratic irrationalities do not lie in the complex upper half-plane $\mathbb{H}$, where $j(z)$ is defined.
• The Darmon-Vonk Framework: Darmon and Vonk proposed a $p$-adic analogue using rigid meromorphic cocycles on Drinfeld's $p$-adic upper half-plane $\mathcal{H}_p$. They defined a $p$-adic number $J_p(\tau, \omega)$ associated with a pair of Real Quadratic (RM) points $\tau, \omega$.
• The Conjecture: Darmon and Vonk conjectured that this $p$-adic number satisfies an antisymmetry property:
  $$J_p(\tau, \omega) = J_p(\omega, \tau)^{-1}$$
  (up to roots of unity and units in the relevant quadratic fields). While numerically verified, a theoretical proof was missing because the construction of $J_p(\tau, \omega)$ treats $\tau$ and $\omega$ asymmetrically (one is the evaluation point, the other defines the divisor).

2. Methodology

Sprehe resolves the conjecture by constructing a symmetric, two-variable framework for these objects, moving from the Ihara group $\Gamma = \text{SL}_2(\mathbb{Z}[1/p])$ to the product group $\Gamma_2 = \Gamma \times \Gamma$.

A. Geometric and Cohomological Setup

• Quaternion Algebras: The author works with an indefinite quaternion algebra $B/\mathbb{Q}$ split at $p$. This allows the identification of the $p$-adic symmetric space with $\mathcal{H}_p \times \mathcal{H}_p$ (in the split case $B \cong M_2(\mathbb{Q})$).
• Divisor-Valued Cocycles: The paper utilizes the theory of Kudla-Millson divisors introduced by Darmon, Gehrmann, and Lipnowski. These are cohomology classes in $H^2(\Gamma_2, \text{Div}^\dagger_{rq}(\mathcal{H}_p^2))$ supported on "twisted diagonals" $\Delta_{v,p} = \{(v \cdot z, z) \mid z \in \mathcal{H}_p\}$.
• Symmetry Analysis: The author defines an involution $\varsigma_p$ on $\mathcal{H}_p^2$ that swaps coordinates. Crucially, it is shown that the cohomology class $[D_1]$ (associated with the identity matrix) is invariant under this involution.

B. Lifting Obstructions and Modularity

• Lifting to Functions: A major technical hurdle is lifting divisor-valued cocycles to actual rigid meromorphic functions (cocycles). The paper analyzes the lifting obstructions in the cohomology sequence:
  $$H^2(\Gamma_2, M^\times_{rq}) \to H^2(\Gamma_2, \text{Div}^\dagger_{rq}) \to H^3(\Gamma_2, A^\times_2)$$
  Using Hecke operators and the Universal Coefficient Theorem, the author identifies a specific ideal in the Hecke algebra that annihilates the obstruction groups, ensuring that certain Hecke translates of the divisor classes can be lifted to rigid meromorphic cocycles.
• Modularity Theorem: The paper proves that the generating series of Kudla-Millson divisors (with values in a $p$-adic Chow group analogue) is a modular form of weight 2 and level $\Gamma_0(p)$. This relies on Eichler-Shimura theory and the structure of the cohomology of $p$-arithmetic groups.

C. Intersection Theory

• Eilenberg-Zilber Map: To relate the two-variable function to the one-variable evaluation, the author uses the Eilenberg-Zilber map to compute the intersection of the two-variable cocycle with a "slice" $\mathcal{H}_p \times \{\omega\}$.
• Main Calculation: It is proven that intersecting the symmetric two-variable class $[D_1]$ with an RM point $\omega$ yields a class proportional to the square of the one-variable class $[D_\omega]$.

3. Key Contributions and Results

Theorem B (Comparison of Functions)

The author constructs a symmetric two-variable function $\hat{J}_p(\tau, \omega)$ defined as the evaluation of a rigid meromorphic cocycle $J_1$ on $\mathcal{H}_p^2$ at the pair $(\tau, \omega)$. The main technical result establishes the relationship between the original Darmon-Vonk function $J_p$ and this new symmetric function:
$$J_p(\tau, \omega)^{-2} = \hat{J}_p(\tau, \omega)$$
(up to roots of unity and units).

Proposition C (Antisymmetry)

Because the class $[D_1]$ is invariant under the coordinate swap $\varsigma_p$, the function $\hat{J}_p$ satisfies:
$$\hat{J}_p(\tau, \omega) = \hat{J}_p(\omega, \tau)^{-1}$$
Combining this with Theorem B immediately yields the Darmon-Vonk Conjecture:
$$J_p(\tau, \omega) = J_p(\omega, \tau)^{-1}$$
(up to roots of unity and units).

Theorem D (Modularity)

The paper proves that the generating series of Kudla-Millson divisors $D_n$ is modular of weight 2. This extends the Kudla program to the $p$-adic setting for the specific case of the split orthogonal group in four variables, filling a gap left by previous work of Darmon-Gehrmann-Lipnowski.

4. Significance

• Resolution of a Major Conjecture: The paper provides the first rigorous proof of the antisymmetry conjecture for real quadratic singular moduli, a central open problem in the $p$-adic theory of real quadratic fields.
• Symmetrization of the Theory: By shifting the perspective from a one-variable evaluation to a two-variable symmetric cocycle on $\mathcal{H}_p^2$, the author reveals the underlying geometric symmetry (graded commutativity of the cup product) that was previously obscured.
• Advancement of the $p$-adic Kudla Program: The proof of modularity for the generating series of Kudla-Millson divisors in the four-dimensional split case provides a new instance of the $p$-adic Kudla program, linking arithmetic cycles on $p$-adic symmetric spaces to modular forms.
• Methodological Innovation: The work demonstrates the power of combining rigid analytic geometry, group cohomology of $p$-arithmetic groups, and intersection theory on singular chain complexes to solve arithmetic problems.

In summary, Sprehe confirms that the $p$-adic "singular moduli" for real quadratic fields possess the same fundamental antisymmetry as their classical complex counterparts, achieved by constructing a symmetric two-variable framework and overcoming significant cohomological lifting obstructions.