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Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$ -functions

This paper establishes a relationship between the central derivative values of twisted Rankin-Selberg $L$ -functions and arithmetic heights of Hirzebruch-Zagier divisors on $X_0(N) \times X_0(N)$ , thereby refining higher Gross-Zagier formulae for Shimura varieties and providing new proofs and connections to the Birch-Swinnerton-Dyer conjecture for both imaginary and real quadratic fields.

Original authors: Jeanine Van Order

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jeanine Van Order

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a massive, cosmic puzzle. On one side of the table, you have a mathematical mystery involving a special kind of curve called an "elliptic curve" (think of it as a very specific, wavy shape that holds secrets about numbers). On the other side, you have a geometric landscape, a vast, multi-dimensional terrain made of shapes and dividers.

For decades, mathematicians have known that these two sides are connected, but the bridge between them was built using a very specific, narrow path. This paper, by Jeanine van Order, builds a new, wider bridge and shows us that the two sides are actually connected in a much more direct and surprising way.

Here is the story of the paper, broken down into simple concepts:

1. The Two Main Characters

The Elliptic Curve (The Mystery): Imagine an elliptic curve as a musical instrument. When you play it, it produces a specific "song" (a mathematical function called an L-function). Sometimes, this song hits a "zero" note at a very specific moment (the center of the song). The paper is interested in what happens right after that zero note—the "derivative." This value tells us deep secrets about the curve, like how many solutions it has.
The Geometric Landscape (The Map): Imagine a giant map made of two overlapping grids (specifically, two copies of a shape called a "modular curve" multiplied together). On this map, there are special paths and regions called Hirzebruch-Zagier divisors. Think of these as "landmarks" or "scenic routes" on the map.

2. The Old Bridge (The Gross-Zagier Formula)

Previously, a famous theorem by Gross and Zagier showed that the "derivative" of the elliptic curve's song is related to the height of a specific point on the map.

The Analogy: Imagine you have a treasure map. The old rule said: "The value of the treasure (the derivative) is equal to the altitude of a specific mountain peak (a Heegner point) on the map."
The Limitation: This only worked for a very specific type of mountain peak.

3. The New Bridge (This Paper's Discovery)

Jeanine van Order says, "Wait a minute! We can measure the treasure's value using entire regions of the map, not just single peaks."

She introduces a new way to calculate that same mysterious number (the derivative) by looking at Hirzebruch-Zagier divisors.

The Analogy: Instead of measuring the altitude of one single mountain peak, she measures the average elevation of a whole valley or a specific scenic route on the map.
The Magic: She proves that if you calculate the "arithmetic height" (a fancy way of measuring the size or complexity) of these specific scenic routes, you get the exact same number as the old mountain peak method.

4. Two Different Terrains (Imaginary vs. Real)

The paper handles two different types of "worlds" (mathematical fields):

The Imaginary World: When the math involves "imaginary" numbers (like the square root of -1), the scenic routes are like closed loops (think of a circle or a donut). The paper shows how to measure the "height" of these loops to find the answer.
The Real World: When the math involves "real" numbers, the scenic routes are like straight lines stretching out to infinity (geodesics). The paper shows how to measure the "length" and "weight" of these lines to find the same answer.

5. Why This Matters (The "Aha!" Moment)

The most exciting part of the paper is the comparison.
Because the author found two different ways to calculate the exact same number (one using the old mountain peaks, one using the new scenic routes), she can now write an equation that links them directly.

The Result: She proves that the "altitude" of the old mountain peaks is mathematically identical to the "elevation" of the new scenic routes.
The Metaphor: It's like discovering that the height of a specific tree in a forest is exactly equal to the average height of a whole meadow nearby. This reveals a hidden symmetry in the forest that no one noticed before.

6. The Bigger Picture (The Birch-Swinnerton-Dyer Conjecture)

The paper connects this to one of the biggest unsolved problems in math: the Birch-Swinnerton-Dyer (BSD) Conjecture.

The Goal: This conjecture tries to predict how many solutions a curve has based on its "song."
The Paper's Contribution: By providing these new formulas, the paper gives us a new way to check the BSD conjecture. It translates the abstract "song" of the curve into concrete "measurements" of geometric shapes. It also suggests that these mysterious numbers are made of "periods"—a special class of numbers that can be calculated using integrals (areas under curves), making them more tangible.

Summary

In short, Jeanine van Order has taken a famous formula that links a musical note to a mountain peak and expanded it. She showed that you can also link that musical note to the shape of entire valleys and highways on a geometric map. By doing this, she proved that these different geometric shapes are secretly the same thing, giving mathematicians a powerful new tool to solve ancient riddles about numbers and curves.

Technical Summary: Arithmetic Hirzebruch-Zagier Divisors and Central Derivative Values of Rankin-Selberg L-Functions

Problem Statement
The paper addresses the arithmetic interpretation of the central derivative values $\Lambda'(E/k, \chi, 1)$ of the twisted Hasse-Weil $L$ -function of an elliptic curve $E/\mathbb{Q}$ over a quadratic field $k$ (imaginary or real), twisted by a character $\chi$ of the ideal class group of $k$ . While the classical Gross-Zagier theorem (Theorem 1.1) relates these derivatives to the Néron-Tate heights of Heegner divisors on the modular curve $X_0(N)$ when $k$ is imaginary quadratic, this work seeks to generalize and refine these relations. Specifically, it aims to express these central derivatives in terms of arithmetic heights of Hirzebruch-Zagier divisors on the product of modular curves $X_0(N) \times X_0(N)$ , and to provide analogous formulae for the case where $k$ is a real quadratic field.

Methodology
The author employs the framework of arithmetic intersection theory on Shimura varieties, specifically utilizing the theory of regularized theta lifts and automorphic Green's functions developed by Borcherds, Bruinier, and Yang.

Geometric Setup: The paper constructs rational quadratic spaces $(V_A, Q_A)$ of signature $(2,2)$ associated with each ideal class $A$ in the class group $C(\mathcal{O}_k)$ . The associated spin Shimura variety $X_{K_A}$ is identified with the product of modular curves $Y_0(N) \times Y_0(N)$ (and its compactification $X_0(N) \times X_0(N)$ ).
Automorphic Forms: The author utilizes harmonic weak Maass forms $f$ of weight $1 - n/2$ and their images under the Bruinier-Funke differential operator $\xi_l$ to construct automorphic Green's functions $\Phi(f, z, h)$ . These functions serve as the Green's functions for special divisors (Hirzebruch-Zagier divisors) on the Shimura variety.
Summation Formulas: The core technical machinery involves computing the sums of these Green's functions along two types of cycles:
- CM Cycles ( $Z(V_0)$ ): Zero-cycles associated with imaginary quadratic fields (signature $(0,2)$ subspaces).
- Real Geodesic Cycles ( $G(W)$ ): One-dimensional cycles associated with real quadratic fields (signature $(1,1)$ subspaces).
  These computations yield integral presentations involving Rankin-Selberg $L$ -functions and their derivatives.
L-function Equivalences: The paper establishes equivalences between the standard Rankin-Selberg $L$ -functions $\Lambda(s, \phi \times \theta(\chi))$ (where $\phi$ is the newform for $E$ and $\theta(\chi)$ is the theta series) and the $L$ -functions associated with vector-valued cusp forms on the Shimura variety via the Doi-Naganuma lift and quadratic basechange.

Key Contributions and Results

Arithmetic Height Formulae for Imaginary Quadratic Fields:
The paper derives a new formula for the central derivative $\Lambda'(E/k, \chi, 1)$ (Theorem 1.4, Theorem 7.7). It expresses this value as a sum of Faltings heights of arithmetic Hirzebruch-Zagier divisors $\hat{Z}_A(f_{0,A})$ on $X_0(N) \times X_0(N)$ evaluated along CM cycles $Z(V_{A,0})$ .
$\Lambda'(1/2, \phi \times \theta(\chi)) = -2 \sum_{A \in C(\mathcal{O}_k)} \chi(A) [\hat{Z}_A(f_{0,A}) : Z(V_{A,0})]$
This provides a "higher Gross-Zagier" formula for the product of modular curves.
Real Quadratic Field Case:
For real quadratic fields, the paper provides a distinct integral presentation for the central derivative (Theorem 1.2 (ii), Theorem 6.6 (ii)). This formula involves sums of automorphic Green's functions along real geodesic cycles, combined with constant terms of theta lifts and derivatives of regularized integrals.
Relation to Classical Gross-Zagier:
By comparing the new formulae with the classical Gross-Zagier theorem, the author establishes an identity between the arithmetic heights of Heegner divisors on $X_0(N)$ and the arithmetic heights of Hirzebruch-Zagier divisors on $X_0(N) \times X_0(N)$ (Corollary A.9). This offers a new proof of the Gross-Zagier formula via regularized theta lifts on the modular curve (Appendix A).
Birch-Swinnerton-Dyer (BSD) Constants:
The results are applied to the refined conjecture of Birch and Swinnerton-Dyer. The paper shows that the product of the BSD constants for $E$ and its quadratic twist $E(d_k)$ is proportional to the central derivative $\Lambda'(E/k, 1)$ , which is expressed via the arithmetic heights derived above (Theorem 1.5).
Connection to Half-Integral Weight Forms:
The paper links these central derivatives to the Fourier coefficients of half-integral weight Maass forms (Theorem 1.6), extending previous work by Bruinier, Funke, and Imamoglu.

Significance and Claims
The paper claims to provide a "completely new perspective on the geometric side" of the Gross-Zagier phenomenon by utilizing the quadratic basechange equivalence of $L$ -functions to refine higher Gross-Zagier formulae for spin Shimura varieties of any dimension, specifically applied to $X_0(N) \times X_0(N)$ .

The significance lies in:

Unification: It unifies the treatment of imaginary and real quadratic fields within the framework of automorphic Green's functions on Shimura varieties.
Geometric Interpretation: It provides a geometric interpretation of central derivatives in terms of arithmetic intersections on the product of modular curves, distinct from the classical Heegner point interpretation.
Refinement of BSD: It offers explicit arithmetic height formulae for the leading terms of $L$ -functions in the rank one case, contributing to the understanding of the refined BSD conjecture.
New Proofs: It supplies a distinct proof of the Gross-Zagier formula and establishes relations between different types of arithmetic divisors (Heegner vs. Hirzebruch-Zagier).

The author notes that while the results refine existing higher Gross-Zagier formulae (Bruinier-Yang, Andreatta et al.), the specific application to the product of modular curves and the explicit derivation of the real quadratic case via geodesic cycles represent novel contributions. The paper does not claim to prove the full BSD conjecture but rather provides the arithmetic height formulae that would constitute the geometric side of the conjecture in the rank one setting.