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A proof of $p$ -adic Gross--Zagier theorem via BDP formula

This paper presents a novel proof of the $p$ -adic Gross--Zagier theorem for base changes of cuspidal eigen-newforms to imaginary quadratic fields, covering both ordinary and non-ordinary cases (including $k>2$ with $\mathrm{ord}_p(a_p(f)) > 0$ ) by utilizing a "wall-crossing" strategy based on the BDP formula and Beilinson--Flach elements rather than traditional kernel comparisons.

Original authors: Kâzım Büyükboduk, Peter Neamti

Published 2026-07-24

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Kâzım Büyükboduk, Peter Neamti

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 giant, cosmic puzzle where the pieces are numbers, but not just any numbers—these are the mysterious "prime numbers" that act like the atoms of arithmetic. For centuries, mathematicians have been trying to connect two very different worlds: the world of geometry (shapes, curves, and points floating in space) and the world of analysis (smooth curves, derivatives, and the flow of functions). It's a bit like trying to prove that the sound of a violin string vibrating is exactly the same as the pattern of ripples spreading across a pond.

In this specific corner of math, called number theory, there is a famous bridge called the Gross–Zagier formula. It connects the "height" of a special point on a geometric shape (called a Heegner point) to the slope of a complex function (an L-function) at a specific spot. Think of the "height" as how far a hiker has climbed up a mountain, and the "slope" as how steep the mountain is at the peak. If the mountain is flat (the slope is zero), the hiker is at the bottom; if the mountain is steep, the hiker is high up. This connection is crucial because it helps mathematicians understand deep secrets about how numbers behave, particularly regarding elliptic curves (which are fancy, donut-shaped equations) and their L-functions (which are like secret codes that tell us how many solutions the equations have).

For a long time, proving this connection required comparing two massive, complicated machines: one built from geometry and one built from analysis. But what if there was a shortcut? What if you could use a "wall-crossing" strategy, like walking through a secret door in a castle to get from one room to another without climbing the whole tower? That is exactly what this paper does.

The Paper: A New Shortcut Through the Castle Walls

This paper, written by Kâzım Büyükboduk and Peter Neamti, offers a fresh, unified proof of the p-adic Gross–Zagier theorem. In the world of math, "p-adic" is a special way of measuring numbers that focuses on a specific prime number (let's call it p) rather than the usual distance we use in everyday life. The authors are trying to prove that the "slope" of a special p-adic function is directly linked to the "height" of a geometric point, even in tricky situations where the numbers behave badly (what mathematicians call "non-ordinary" scenarios).

The Old Way vs. The New Way

Previously, mathematicians proved this theorem by building two separate, massive engines. One engine calculated the geometric height, and the other calculated the analytic slope. They then had to compare the outputs of these two engines to see if they matched. It was like trying to verify that two different clocks show the same time by checking every single gear in both of them. This was hard, especially when the "clocks" were broken or behaving strangely (like when the weight of the form is greater than 2, or when the prime number p divides the level of the form in a messy way).

The authors of this paper say, "Let's try a different approach." Instead of comparing the two engines directly, they use a wall-crossing strategy. Imagine you are in a castle with many rooms. The old way was to walk from the Geometric Room to the Analytic Room by walking around the entire castle. The new way is to find a secret door (a "wall") that connects them directly.

The Secret Door: The BDP Formula and Beilinson–Flach Elements

The key to this secret door is a famous formula called the BDP formula (named after Bertolini, Darmon, and Prasanna). Think of the BDP formula as a magical map that tells you exactly how to translate a message from one language to another.

To use this map, the authors introduce two main characters:

Heegner Cycles: These are the geometric hikers climbing the mountain.
Beilinson–Flach Elements: These are the analytic messengers carrying the slope data.

The paper shows that these two characters are actually the same person wearing different costumes. By using a tool called the Rubin-style formula, the authors show that the "height" of the geometric hiker is exactly equal to the "slope" of the analytic messenger, once you account for some specific algebraic factors (like the size of the mountain or the weight of the hiker's backpack).

The Big Discovery

The main finding of the paper is a precise equation (Theorem A) that links the derivative of the p-adic L-function to the p-adic height of the Heegner cycle. The formula looks like this:

$\text{Slope} \propto \text{Height}$

But the authors don't just say "they are related." They give you the exact recipe for how they are related, including specific numbers that depend on whether the prime p divides the level of the form or not.

What the paper proves:

It works for "bad" cases too: Previous proofs struggled when the weight of the form was greater than 2 or when the prime p behaved in a "non-ordinary" way (meaning the numbers didn't play nice). This paper proves the formula works even in these difficult scenarios, including cases where the weight $k > 2$ and the p-adic valuation of a specific number is positive.
It unifies the theory: It treats the "easy" cases (where the numbers behave well) and the "hard" cases (where they don't) with the exact same method. You don't need a different tool for different problems anymore.
It uses "Big" elements: The authors use "Big" Heegner classes and "Big" Beilinson–Flach elements. Think of these as super-versions of the standard elements that can handle families of numbers all at once, rather than just one number at a time. This allows them to prove the result for a whole family of forms simultaneously.

What the paper rules out

The paper is very careful about what it doesn't cover. It explicitly states that their method does not work for a very specific, tricky case: when the prime p divides the level of the form ( $p | N_f$ ) AND a specific number called $a_p(f)$ equals $p^{k/2}$ . The authors admit this case is still a mystery and is being worked on by other mathematicians. They also note that if certain conditions (labeled NV) are not met, the formula might just say "0 = 0," which is technically true but not very helpful. However, they show that if those conditions are met, the formula gives a meaningful, non-zero answer.

How sure are they?

The authors are extremely confident. They don't just suggest this might be true; they provide a rigorous, step-by-step proof. They use established tools like Selmer complexes (which are like safety nets that catch the important parts of the numbers) and Euler systems (which are like a chain of dominoes that, once you push the first one, proves the whole chain falls). The logic is built on solid ground, relying on the "Big Image" condition (a technical requirement that ensures the numbers are complex enough to be interesting) and the "Heegner Hypothesis" (a rule that ensures the geometric points exist).

In short, Büyükboduk and Neamti have built a new, elegant bridge between the geometric and analytic worlds. They didn't just cross the bridge; they showed us a secret tunnel that works even when the bridge is under construction or the weather is stormy. This proof confirms that the deep connection between the height of a point and the slope of a function is a fundamental truth of the universe of numbers, holding true even in the most chaotic and non-ordinary scenarios.

Technical Summary: A Proof of the p-adic Gross–Zagier Theorem via the BDP Formula

Problem Statement
The paper addresses the proof of the $p$ -adic Gross–Zagier formula for the $p$ -adic $L$ -function associated with the base change of a normalized cuspidal eigen-newform $f$ of weight $k \geq 2$ to an imaginary quadratic field $K$ . The authors aim to establish a precise relationship between the derivative of this $p$ -adic $L$ -function at the central critical point and the $p$ -adic height of a Heegner cycle.

While the formula was previously established for classical $p$ -ordinary cases (where $p \nmid N_f$ and the form is ordinary) by Perrin-Riou, Nekovář, and Kobayashi, this work extends the result to non-ordinary scenarios. Specifically, it covers cases where $k > 2$ and the valuation of the Hecke eigenvalue satisfies $\text{ord}_p(a_p(f)) > 0$ (non-ordinary), as well as cases where $p \mid N_f$ (semi-stable reduction). The proof operates under the "strong Heegner hypothesis" and specific "big image" conditions.

Methodology
The authors employ a strategy fundamentally distinct from the traditional approach of comparing geometric and analytic kernels (which relies on modular forms constructed from $p$ -adic heights versus Rankin–Selberg convolutions). Instead, they utilize a "wall-crossing" strategy centered on the Bertolini–Darmon–Prasanna (BDP) formula and the theory of Beilinson–Flach elements.

The core methodological steps include:

Beilinson–Flach Elements: The $p$ -adic $L$ -function is described in terms of Beilinson–Flach elements via "first reciprocity laws."
Rubin-Style Formula: A formula is derived expressing the derivative of the $p$ -adic $L$ -function in terms of the $p$ -adic height of a Beilinson–Flach element.
Comparison of Local Images: Using the BDP formula for Heegner cycles and "second reciprocity laws" for Beilinson–Flach elements, the authors compare the $p$ -local images of Heegner points and Beilinson–Flach elements.
Global Comparison via Euler Systems: Relying on Euler system machinery (specifically the "big image" condition), the comparison of local images is refined to a global comparison within the appropriate Selmer group.
Wall-Crossing: The proof leverages the relationship between the $p$ -adic variational properties of explicit Gross–Zagier (GGP) formulae and their arithmetic counterparts across different regions (ordinary vs. non-ordinary).

Key Contributions and Results
The main result is Theorem A, which establishes the following equality for the derivative of the $p$ -adic $L$ -function $L_p(f_\alpha/K, s)$ at $s = k/2 + 1$ :

$\frac{A}{B} \cdot \frac{d}{ds} L_p(f_\alpha/K, s) \Big|_{s= k/2 +1} = (-1)^{k/2} \times \begin{cases} \left(1 - \frac{p^{k/2}}{\alpha}\right)^4 \cdot \frac{h_f(z_f, z_f)}{(4|D_K|)^{k/2}} & \text{if } p \nmid N_f \\ 4 \cdot \frac{h_f(z_f, z_f)}{(4|D_K|)^{k/2}} & \text{if } p \parallel N_f \text{ and } a_p(f) \neq p^{k/2} \end{cases}$

Where:

$z_f$ is the Heegner cycle associated with the base change of $f$ to $K$ .
$h_f$ is the $p$ -adic height pairing.
$A$ and $B$ are non-zero algebraic constants.
The ratio $A/B = 1$ if certain non-vanishing conditions (denoted as NV) hold.

Specific Technical Advances:

Uniformity: The proof is uniform, treating known $p$ -ordinary cases and new non-ordinary cases (including $k > 2$ with $\text{ord}_p(\alpha) > 0$ ) simultaneously.
Generalized Heegner Cycles: The results specialize to $p$ -adic Gross–Zagier formulae for generalized Heegner cycles for $f/K \times \chi$ , reproving and generalizing results from Shnider (2016).
Handling Non-Ordinary Cases: The paper successfully navigates the difficulties of non-ordinary $p$ -adic heights, a scenario where previous vanishing arguments for local contributions (relying on Perrin-Riou's trick) were insufficient or required different techniques.

Significance and Claims
The authors claim that this work provides a new proof of the $p$ -adic Gross–Zagier theorem that unifies disparate cases previously treated separately. By shifting the focus from kernel comparison to the interplay between Beilinson–Flach elements and Heegner cycles via the BDP formula, the paper offers a more robust framework for non-ordinary scenarios.

The paper explicitly notes that it does not treat the case where $\text{ord}_p(N_f) = 1$ and $a_p(f) = p^{k/2}$ , identifying this as a subject for future work. Furthermore, the authors state that if the non-vanishing condition (NV) fails, the formula reduces to the trivial identity $0=0$ , consistent with previous literature, but if (NV) holds, the formula provides a non-trivial relation. The proof relies heavily on the "big image" condition to ensure the necessary global comparison in the Selmer group.

The work is presented as a tribute to Henri Darmon, acknowledging that every step of the argument relies on his foundational ideas, particularly the BDP formula and the conceptual framework of $p$ -adic $L$ -functions and heights.