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$L$ -functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings

This paper derives new integral representations for the central derivative values of $L$ -functions of elliptic curves over $\mathbb{Q}$ twisted by ring class characters of a real quadratic field, expressing them via automorphic Green's functions on products of modular curves and relating these results to Birch-Swinnerton-Dyer constants and periods.

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

The Big Picture: A Mathematical Treasure Hunt

Imagine you are trying to find a hidden treasure (a specific number related to an elliptic curve) buried deep underground. For decades, mathematicians have had a map (the Birch and Swinnerton-Dyer conjecture) that tells them exactly where to dig and how big the treasure chest should be.

However, the map has a tricky rule:

If you are looking for treasure in a "negative" world (imaginary numbers), the map works beautifully. You can find "Heegner points" (like GPS coordinates) that lead you straight to the treasure.
If you are looking for treasure in a "positive" world (real numbers, specifically real quadratic fields), the GPS signal is dead. There are no known "Heegner points" to guide you. The map says the treasure exists, but it doesn't tell you how to find it or what it looks like.

This paper is about building a new kind of GPS for that "positive" world.

The Main Characters

The Elliptic Curve ( $E$ ): Think of this as a complex, wavy shape drawn on a piece of paper. It has a secret "rank" (how many independent loops it has). The paper tries to figure out this rank.
The L-Function: This is a magical calculator. If you feed it a number, it spits out a result. If the result is zero, it tells you something important about the shape of the curve.
The "Forced Vanishing": In this specific mathematical setting, the calculator is forced to spit out zero. This is like a door that is locked shut. Since the door is locked (the value is zero), the mathematicians can't look at the door itself; they have to look at how fast the door is trying to open (the "derivative" or the rate of change).
The Real Geodesic Cycles: In the "negative" world, the treasure is found at specific points (like islands). In this "positive" world, the treasure isn't on an island; it's hidden along a long, straight line stretching across the ocean. These lines are called "geodesic cycles."

The New Discovery: The "Regularized Theta Lift"

The author, Jeanine Van Order, has invented a new tool to measure these lines.

The Old Way (Gross-Zagier Formula): In the imaginary world, you calculate the treasure's value by measuring the distance between two specific islands.
The New Way (This Paper): In the real world, you can't measure islands. Instead, you have to measure the vibrations along the long, straight lines (geodesics) that run through the mathematical landscape.

The paper derives a formula that says:

"The speed at which the locked door is trying to open (the central derivative) is exactly equal to the sum of the 'vibrations' (Green's functions) measured along these specific straight lines."

The Analogy: The Symphony of Strings

Imagine the mathematical world is a giant concert hall.

The Elliptic Curve is a specific musical instrument.
The L-function is the silence of the room.
The "Forced Vanishing" means the room is perfectly silent (zero volume).
The "Central Derivative" is the rate at which the silence is breaking.

In the imaginary world, you find the sound by plucking a specific string (a point).
In this paper, the author realizes that in the real world, the sound isn't coming from a single string. Instead, it's coming from the entire length of a long, vibrating wire (the geodesic cycle).

The paper provides a formula to calculate the "volume" of this sound by integrating (summing up) the vibrations along that wire. It's like saying, "To know how loud the music is, don't listen to one note; listen to the hum of the entire wire."

Why Does This Matter? (The "So What?")

The paper connects this new "wire vibration" formula to the Birch and Swinnerton-Dyer (BSD) conjecture.

The BSD conjecture predicts that the "rank" of the curve (how complex it is) is related to a specific number involving:

The Regulator: A measure of the "size" of the points on the curve.
The Tate-Shafarevich Group: A mysterious group of "phantom" points that exist in theory but might not be visible.
Periods: Specific numbers derived from integrals.

The Paper's Claim:
The author shows that the "vibration sum" (the new formula) is mathematically identical to the product of these mysterious BSD numbers.

In simple terms:

Before: We knew the treasure existed, but we had no way to measure it in the "real" world.
Now: We have a formula that says, "If you measure the vibrations along these specific lines, you get the exact number that describes the size of the treasure chest and the phantom points."

The "Unconditional" Result

Usually, these formulas rely on assuming the BSD conjecture is true. However, the paper proves a specific, concrete identity (Theorem 5.1) that does not require assuming the conjecture is true.

It says: "If we look at a specific pair of curves (one original, one twisted), the product of their hidden properties (Tate-Shafarevich groups, regulators, etc.) is exactly equal to the sum of the vibrations along the lines we calculated."

Summary in One Sentence

This paper builds a bridge between the "locked door" of a specific mathematical problem and a new way of measuring "vibrations along straight lines," proving that these vibrations perfectly match the mysterious numbers predicted by the famous Birch and Swinnerton-Dyer conjecture for real quadratic fields.

Technical Summary: L-Functions of Elliptic Curves in Ring Class Extensions of Real Quadratic Fields via Regularized Theta Liftings

Problem Statement
The paper addresses the analytic behavior of the Hasse-Weil $L$ -function $L(E/K, \chi, s)$ associated with an elliptic curve $E/\mathbb{Q}$ of conductor $N$ , base-changed to a real quadratic field $K = \mathbb{Q}(\sqrt{d})$ and twisted by a ring class character $\chi$ of conductor $c$ . Specifically, the work focuses on the central derivative value $\Lambda'(E/K, \chi, 1)$ (equivalently $\Lambda'(1/2, \Pi \otimes \chi)$ ) in the "forced vanishing" setting. This setting occurs when the root number of the functional equation is $-1$, forcing the central value $\Lambda(1/2, \Pi \otimes \chi)$ to vanish.

While the Gross-Zagier formula and its generalizations (e.g., by Waldspurger, Popa, and Bruinier-Yang) successfully relate central derivatives to heights of Heegner points (CM cycles) in the context of imaginary quadratic fields, the real quadratic case presents a significant obstruction: there are no CM points in the real quadratic setting. Consequently, the geometric objects required to interpret the derivative via arithmetic intersection theory are absent. The paper seeks to derive an integral presentation for these central derivatives in the real quadratic case, replacing CM cycles with real geodesic cycles on the product of modular curves.

Methodology
The author employs the theory of regularized theta lifts and automorphic Green's functions, adapting the framework of Bruinier and Yang (who treated the imaginary quadratic case) to the signature $(2,2)$ and $(1,1)$ settings relevant to real quadratic fields.

Geometric Setup: The work utilizes the exceptional isomorphism between the spin group $GSpin(V_A)$ of a rational quadratic space $V_A$ of signature $(2,2)$ and $GL_2 \times_{G_m} GL_2$ . This identifies the associated spin Shimura variety $X_A$ with the product of modular curves $Y_0(N) \times Y_0(N)$ .
Anisotropic Subspaces: The author introduces anisotropic subspaces $(V_{A,1}, Q_{A,1})$ and $(V_{A,2}, Q_{A,2})$ of signature $(1,1)$ , defined via ideal representatives of the ring class group. These subspaces give rise to real geodesic cycles $G_A$ on $Y_0(N) \times Y_0(N)$ , which serve as the real quadratic analogues to CM cycles.
Regularized Theta Lifts: The core calculation involves evaluating regularized theta lifts $\Phi(f_0, z, h)$ along these geodesic cycles. Here, $f_0$ is a harmonic weak Maass form of weight 0. These lifts are identified as automorphic Green's functions for Hirzebruch-Zagier divisors on $X_A$ .
Siegel-Weil and Maass Operators: The derivation relies on the Siegel-Weil formula to relate theta integrals to Eisenstein series. Crucially, the author utilizes Maass weight-raising and lowering operators ( $R_l$ $R_{l}$ and $L_l$ $L_{l}$ ) to relate the derivative of an incoherent Eisenstein series (associated with the signature $(1,1)$ $(1, 1)$ subspace) to the holomorphic part of the derivative Eisenstein series. This allows the decomposition of the theta lift integral into three distinct terms:
- A constant term pairing involving the holomorphic part of the Maass form and the holomorphic part of the derivative Eisenstein series.
- A regularized Rankin-Selberg integral involving the "shadow" of the Maass form.
- The integral of the Green's function along the real geodesic cycle.

Key Contributions and Results

Main Theorem (Theorem 4.18 / Corollary 4.19): The paper establishes a novel integral presentation for the central derivative value $\Lambda'(1/2, \Pi \otimes \chi)$ . The formula expresses this value as a $\chi$ -twisted sum over the ring class group $\text{Pic}(\mathcal{O}_c)$ of:
1. A constant term pairing: $\text{CT}\langle\langle f^+_{0,A}, 1_{L_{A,1}\oplus 0} \otimes E_{L_{A,2}} \rangle\rangle$ .
2. A regularized Rankin-Selberg integral: $I'(0, f_{0,A} \times \xi_0(\theta_{L_{A,1}}))$ .
3. A sum of automorphic Green's functions evaluated along real geodesic cycles: $\frac{\text{vol}(U_{A,2})}{2} \Phi(f_{0,A}, G_A)$ .
The formula is given by:
$\Lambda'(1/2, \Pi \otimes \chi) = -2h_K \log \varepsilon_K \sum_{A \in \text{Pic}(\mathcal{O}_c)} \chi(A) \left( \text{CT}\langle\langle \dots \rangle\rangle + I'(\dots) + \frac{\text{vol}(U_{A,2})}{2} \Phi(f_{0,A}, G_A) \right)$
where $h_K$ is the class number and $\varepsilon_K$ is the fundamental unit of $K$ .
Geometric Interpretation: The work provides a geometric interpretation of the real geodesic cycles $G_A$ as components of the boundary of the Borel-Serre compactification of a twisted Siegel threefold (a spin Shimura variety of signature $(3,2)$ ). This suggests a boundary interpretation for the "Stark-Heegner" points in $E(K[c])$ .
Application to Birch-Swinnerton-Dyer (BSD): The paper relates these analytic formulas to the refined BSD conjecture.
- Conditional Results: Assuming the BSD conjecture, the derived formulas provide explicit expressions for the regulator $R_E(K[c])$ and the order of the Tate-Shafarevich group $\#\text{X}_E(K[c])$ in terms of the automorphic integrals derived above.
- Unconditional Identity (Theorem 5.1): Under specific hypotheses (semistable reduction, irreducibility of residual Galois representations, and a rank condition where exactly one of $L(E,1)$ or $L(E(d_K),1)$ vanishes), the author derives an unconditional identity. This identity equates a product of arithmetic invariants (regulators, Tamagawa numbers, periods, and orders of Tate-Shafarevich groups for $E$ and its quadratic twist) to the sum of the automorphic integrals described in the main theorem.

Significance and Claims
The paper claims to provide the first known formula relating central derivative values of $L$ -functions in the real quadratic setting to geometric invariants associated with real geodesic cycles on the product of modular curves.

Analogue to Gross-Zagier: The result is presented as a real quadratic analogue of the Gross-Zagier formula. Unlike the imaginary quadratic case, where the derivative is related to the height of a CM point, the real quadratic derivative is related to the evaluation of Green's functions along non-compact geodesic cycles.
Novelty of Method: The author notes that the argument does not follow the arithmetic intersection framework of Kudla's program (which relies on CM cycles) nor the calculations of Bruinier-Yang for CM cycles on spin Shimura varieties. Instead, the derivation arises from evaluating Green's functions along geodesic cycles parametrized by Lorentzian quadratic spaces, utilizing Siegel-Weil identities interpreted via Maass operators.
Periods: The paper suggests that the values $\Lambda'(E/K, \chi, 1)$ are periods, deducing this in special cases via Kontsevich-Zagier arguments, consistent with the expectation that special $L$ -values are periods.

The work does not claim to prove the existence of rational points (Stark-Heegner points) but rather provides a formula that, if such points exist and satisfy the BSD conjecture, would relate their heights to these specific automorphic integrals. It offers a new perspective on the provenance of such points via the boundaries of higher-dimensional Shimura varieties.