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$p$ -adic Waldspurger Formula for Non-split Primes and Converse of Gross--Zagier and Kolyvagin Theorem

This paper generalizes the $p$ -adic Waldspurger formula to non-split primes and establishes a new anticyclotomic local $\pm$ -Iwasawa theory to prove the converse of the Gross--Zagier--Kolyvagin theorem for self-dual CM characters, thereby confirming Sylvester's conjecture on sums of two rational cubes and Goldfeld's conjecture for CM elliptic curves.

Original authors: Yangyu Fan, Xin Wan

Published 2026-03-31

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Yangyu Fan, Xin Wan

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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. The pieces of this puzzle are numbers, shapes, and patterns that have fascinated mathematicians for centuries. The specific puzzle this paper tackles is about elliptic curves—which look like smooth, looping shapes on a graph—and a mysterious property called their "rank."

Think of the rank as the number of independent "generators" or "keys" needed to unlock all the rational solutions (points with nice number coordinates) on the curve.

Rank 0: The curve has very few solutions (just the "trivial" ones).
Rank 1: The curve has one main generator that can produce infinitely many solutions.
Rank 2 or higher: It gets much more complicated.

For decades, mathematicians have had a powerful rule of thumb (the Gross–Zagier and Kolyvagin theorem) that says: If you can find a specific geometric point on the curve (a "Heegner point") that isn't just a trivial repetition, then the curve must have Rank 1.

The Big Question (The Converse):
This paper asks the reverse question: If we know the curve has Rank 1, does that guarantee the existence of this special geometric point? Proving this "converse" is like saying, "If the door is unlocked, there must be a key inside." This is crucial because it connects the algebraic world (counting solutions) with the analytic world (studying complex functions called L-functions).

The Problem: The "Non-Split" Wall

In the past, mathematicians could only prove this connection when a specific prime number $p$ behaved nicely (it "split" in a certain way). But what if $p$ behaves stubbornly? What if it's "inert" (refuses to split) or "ramified" (behaves chaotically)? This is the "Non-Split" scenario.

For a long time, the tools used to prove the connection broke down in these stubborn cases. It was like trying to use a standard key in a lock that had been welded shut.

The Solution: Building a New Toolkit

Authors Yangyu Fan and Xin Wan have built a brand new toolkit to crack these stubborn locks. They did two main things:

1. The "Magic Lens" (The p-adic Waldspurger Formula)

Imagine you have a blurry photograph of a distant mountain (the L-function). You want to see the details of a specific flower growing on it (the Heegner point).

Old Method: The lens only worked if the mountain was facing a certain way (split primes).
New Method: Fan and Wan built a super-lens (a new type of $p$ -adic L-function) that works no matter how the mountain is facing. They used a technique called "iterating Gauss-Manin connections," which is like taking a blurry photo, sharpening it, moving the camera slightly, sharpening it again, and repeating this until the flower is crystal clear.
The Twist: They realized that by "twisting" the image with a specific mathematical matrix (a "test vector"), they could force the blurry flower to land in a spot where the lens works perfectly, even for the most stubborn primes (including the tricky prime number 2).

2. The "Two-Track System" (The $\pm$ -Iwasawa Theory)

Once they had the clear image, they needed to organize the data.

The Analogy: Imagine a river flowing through a landscape. Sometimes the river splits into two channels (split primes), making it easy to study. But sometimes it's a single, turbulent, winding river (non-split primes).
The Innovation: Fan and Wan developed a new way to map this single, turbulent river. They divided the river into two distinct "tracks" (called $+$ and $-$ subspaces). Even though the water looks chaotic, they proved that you can always separate it into these two clean, predictable streams. This allows them to track the "flow" of solutions (Selmer groups) with precision, regardless of how the prime number behaves.

The Grand Result: Unlocking Ancient Mysteries

By combining their new "Magic Lens" with their "Two-Track System," they proved the Converse Theorem for a huge class of curves (those with Complex Multiplication).

What does this mean in plain English?
If you look at an elliptic curve and its associated L-function (the analytic side) and see that it vanishes exactly once (Rank 1), you can now be 100% sure that there is a geometric "key" (Heegner point) that generates all the solutions. The algebraic and analytic worlds are perfectly synchronized.

Real-World Impact: Solving 1879 and 1979

The paper doesn't just stay in the abstract; it solves famous historical riddles:

Sylvester's Conjecture (1879): Can every prime number of the form $9k+4, 9k+7,$ or $9k+8$ be written as the sum of two rational cubes ( $x^3 + y^3 = p$ )?
- The Answer: YES. The authors proved this by showing that for these specific primes, the associated curve has Rank 1, meaning there are infinitely many ways to write the prime as a sum of two cubes.
Goldfeld's Conjecture: If you take an elliptic curve and twist it in every possible way, do 50% of them have Rank 0 and 50% have Rank 1?
- The Answer: YES (for curves with Complex Multiplication). This confirms a deep statistical prediction about how these curves behave on average.

Summary

Fan and Wan took a problem that was stuck behind a wall of "stubborn" prime numbers. They built a new, flexible lens to see through the wall and a new map to navigate the chaotic terrain. Their success not only proves a major theoretical theorem but also settles centuries-old questions about how numbers can be added together to form cubes. It's a triumph of connecting the invisible patterns of numbers to the visible shapes of geometry.

1. Problem Statement

The paper addresses two interconnected central problems in modern number theory:

Generalization of p-adic L-functions: Extending the construction of anticyclotomic p-adic L-functions and p-adic Waldspurger formulas to the case where the prime $p$ is non-split (inert or ramified) in an imaginary quadratic field $K$ . Previous work (e.g., Andreatta-Iovita) was restricted to split primes or required strong assumptions on ramification and weights.
The p-converse Theorem: Proving the converse of the Gross–Zagier and Kolyvagin theorem for self-dual Hecke characters (including CM elliptic curves). Specifically, if the $p$ -Selmer rank of a character $\psi$ is 1, then the analytic rank of the associated L-function $L(\psi, s)$ at $s=1$ is also 1. This is a crucial step toward the full Birch and Swinnerton-Dyer (BSD) conjecture.

2. Methodology and New Ideas

The authors develop a novel framework combining $p$ -adic automorphic forms, Iwasawa theory, and arithmetic geometry.

A. Generalized p-adic Waldspurger Formula

To handle non-split primes, the authors overcome geometric convergence issues that plagued previous constructions.

Nearly Overconvergent Modular Forms: They utilize the theory of Vector Bundles with Marked Sections and Marked Splittings (VBMSS) to define spaces of nearly overconvergent quaternionic modular forms. This allows for the $p$ -adic iteration of Gauss-Manin connections in a broader context than the ordinary locus.
Half-Canonical Locus and Test Vectors: A key innovation is the introduction of a "half-canonical locus" on the Shimura curve. By choosing specific local test vectors $\phi_p$ and applying a translation matrix $g_{\ell, n}$ , they ensure that CM points (which usually have large Hasse invariants in the non-split case) land in a region where the Gauss-Manin connection converges.
Syntomic Formalism: To relate the $p$ -adic L-function to Heegner cycles, they employ the syntomic formalism with coefficients (Andreatta-Bertolini-Seveso-Venerucci). This avoids the need for semi-stable models of Kuga-Sato varieties, which are difficult to construct in full generality. Instead, they work directly with local systems on Shimura curves.
Result: They establish a $p$ -adic Waldspurger formula relating the $p$ -adic logarithm of Heegner points to special values of the anticyclotomic $p$ -adic L-function for all ramification types of $p$ (split, inert, ramified) and all weights, provided the automorphic representation is a principal series at $p$ .

B. New $\pm$ -Iwasawa Theory

The authors develop a new type of anticyclotomic $\pm$ -Iwasawa theory valid for all ramification types of $p$ .

Decomposition of Cohomology: Instead of relying on the classical decomposition based on the conductor (which fails for non-split primes), they define $\pm$ -submodules ( $H^1_+$ and $H^1_-$ ) based on the Archimedean types of the characters.
Two Approaches:
1. Small Disc Approach: Works over a small disc in the weight space (varying Archimedean weights). This allows the inclusion of $p=2$ and provides a direct link to the $p$ -adic Waldspurger formula via explicit reciprocity laws for elliptic units.
2. Localization Approach: Works over a localization of the Iwasawa algebra (varying finite order twists). This uses "bounded" Heegner families constructed via half-logarithms (generalizing Pollack's theory), avoiding the need for Coleman families in certain steps.
Key Insight: They prove an analogue of Rubin's conjecture: $H^1(K_p, \psi) = H^1_+ \oplus H^1_-$ , where the summands are characterized by specializing to the finite part $H^1_f$ at specific arithmetic points determined by root numbers.

C. The "Virtual Heegner Family" and Regulator Maps

Regulator Principle: In the non-split case, explicit regulator maps are not well-understood. The authors bypass this by working with ratios of families (elliptic units or Heegner points) for the target character and auxiliary characters with known vanishing orders (0 or 1).
Virtual Heegner Family: They construct a "virtual" Heegner family $\kappa_{virt}$ in the Iwasawa cohomology. By applying the $p$ -adic Waldspurger formula and analyzing the factorization of $p$ -adic L-functions (noting that one component vanishes identically due to global root numbers), they show that $\kappa_{virt}$ generates the relevant cohomology module.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

Let $\psi$ be a self-dual Hecke character of an imaginary quadratic field $K$ with Archimedean type $(-1, 0)$ . If the $p$ -Selmer rank of $\psi$ is 1, then the vanishing order of $L(\psi, s)$ at $s=1$ is exactly 1.

Significance: This proves the $p$ -converse theorem for self-dual CM characters unconditionally (removing the finiteness assumption on the Shafarevich-Tate group required in earlier works).

Corollaries

Sylvester's Conjecture (Theorem 1.2): Every prime $p \equiv 4, 7, \text{ or } 8 \pmod 9$ $p \equiv 4, 7, or 8 (mod 9)$ can be written as the sum of two rational cubes ( $x^3 + y^3 = p$ $x^{3} + y^{3} = p$ ).
- Proof Sketch: The associated elliptic curve $E_p$ has root number $-1$. The 3-adic Selmer rank is shown to be 1. By the main theorem, the analytic rank is 1, implying the Mordell-Weil rank is 1, hence infinitely many rational solutions exist.
Goldfeld's Conjecture for CM Curves (Theorem 1.4): For an elliptic curve $E/\mathbb{Q}$ $E / Q$ with Complex Multiplication, 50% of its quadratic twists have analytic rank 0 and 50% have analytic rank 1.
- Condition: This relies on recent results by Alexander Smith regarding the distribution of 2-adic Selmer ranks.

4. Significance and Impact

Removal of Restrictions: The paper removes the "split prime" and "unramified" restrictions that previously limited the applicability of $p$ -adic Waldspurger formulas and Iwasawa theory for CM fields. It handles $p=2$ and all ramification types.
Unconditional Results: By combining the new Iwasawa theory with the $p$ -converse theorem, the authors provide unconditional proofs for the rank part of the BSD conjecture in specific CM cases (where previously finiteness of $\text{III}$ was assumed).
Methodological Advancement: The systematic use of VBMSS and the "half-canonical locus" technique offers a robust blueprint for constructing $p$ -adic L-functions in settings where explicit convergence radii are difficult to estimate.
Arithmetic Applications: The results provide a powerful tool for studying the arithmetic of elliptic curves with CM, leading to the resolution of classical conjectures like Sylvester's on sums of cubes and advancing the understanding of Goldfeld's conjecture.

In summary, Fan and Wan have successfully bridged the gap between $p$ -adic automorphic forms and Iwasawa theory for non-split primes, establishing a comprehensive framework that proves the $p$ -converse theorem for CM characters and yields significant arithmetic consequences.