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A refined non-vanishing of the $p$ -adic logarithm of a rational point on an abelian variety

Inspired by the BDP formula, this paper utilizes the $p$ -adic analytic subgroup theorem to establish refined non-vanishing results for the $p$ -adic logarithms of non-torsion rational points on abelian varieties of $\mathrm{GL}_2$ -type associated with Hilbert modular newforms and Heegner points.

Original authors: Ashay Burungale, Christopher Skinner, Xin Wan

Published 2026-03-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ashay Burungale, Christopher Skinner, Xin Wan

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 Detective Story

Imagine you are a detective trying to solve a mystery about a very special kind of mathematical object called an Abelian Variety. Think of these objects as multi-dimensional, doughnut-shaped landscapes (like a donut, but with many holes and dimensions).

On these landscapes, there are specific points that have "rational" coordinates—think of them as points that can be described using simple fractions. The paper asks a crucial question about these points: If a point is not "stuck" in a loop (non-torsion), does it leave a unique, non-zero "fingerprint" when we measure it with a special tool called the p-adic logarithm?

The authors, Burunga, Skinner, and Wan, prove that yes, the fingerprint is always unique and non-zero, provided the landscape has certain geometric properties. This confirms a hunch that mathematicians had for a long time, which is vital for solving deep mysteries in number theory.

The Characters and Tools

To understand the story, let's meet the cast:

The Abelian Variety ( $A$ ): Imagine a complex, multi-dimensional playground. In the simplest case, it's just a circle (an elliptic curve). In this paper, it's a much more complicated shape, but it still has a "group" structure, meaning you can add points together like numbers.
The Rational Point ( $x$ ): This is a specific location on the playground that the mathematicians are interested in. The paper assumes this point is "non-torsion," which means if you keep adding it to itself (like $x + x + x...$ ), you never return to the starting point (zero). It keeps wandering forever.
The $p$ -adic Logarithm ( $\log$ ): This is the detective's magnifying glass. In regular math, a logarithm turns multiplication into addition. In this "p-adic" world (a different way of measuring distance based on prime numbers), the logarithm turns the complex movement of points on the playground into a simple vector (an arrow) in a flat space.
- The Mystery: If the point $x$ is wandering forever, does its "arrow" (the logarithm) point to a non-zero direction? Or does it somehow vanish into the void (become zero)?
The BDP Formula: This is a famous equation discovered by other mathematicians (Bertolini, Darmon, Prasanna). It's like a treasure map that links the "fingerprint" (the logarithm) to a value called an $L$ -function. The map works beautifully for simple circles (elliptic curves), but the authors wanted to see if it works for the complex, multi-dimensional playgrounds too.

The Core Question

For simple circles, we already know the answer: If a point wanders forever, its fingerprint is never zero. It's like saying if you walk in a straight line forever, you will never end up exactly where you started.

But for the complex, multi-dimensional playgrounds (associated with Hilbert modular newforms), things get tricky. The playground has many different "directions" or "lenses" (mathematically called embeddings $\sigma$ ). The question is:

If we look at the point through every single one of these lenses, will the fingerprint be non-zero in all of them?

The authors prove that it is. No matter which lens you use, the point leaves a mark.

The Magic Weapon: The "p-adic Analytic Subgroup Theorem"

How did they prove this? They used a powerful tool from a branch of math called Transcendence Theory.

Think of the p-adic analytic subgroup theorem as a "Law of Conservation of Structure." It says:

"If a wandering point's fingerprint (logarithm) accidentally lands in a specific, smaller, flat subspace (like a shadow on a wall), then the point itself must actually be trapped inside a smaller, simpler playground hidden inside the big one."

The Logic of the Proof (The "Gotcha" Moment):

Assume the opposite: Suppose the fingerprint does vanish (become zero) for one of the lenses.
Apply the Law: If the fingerprint vanishes, the theorem says the point must be hiding inside a smaller, simpler playground (a sub-variety).
The Contradiction: The authors show that the specific playgrounds they are studying are "rigid." They are built in such a way that they cannot contain these smaller, hidden playgrounds without breaking their own rules (specifically, rules about their "trace fields" and how they are constructed from number fields).
Conclusion: Since the point cannot be hiding in a smaller playground (because none exist that fit the rules), the assumption that the fingerprint vanished must be false. Therefore, the fingerprint must be non-zero.

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

This isn't just a game of logic; it has real-world implications for the Birch and Swinnerton-Dyer (BSD) Conjecture, which is one of the seven "Millennium Prize Problems" in mathematics.

The BSD Conjecture tries to predict how many rational points exist on these shapes based on the behavior of their $L$ -functions (complex formulas).
The BDP formula (mentioned at the start) connects these $L$ -functions to the $p$ -adic logarithms.
If the logarithm were zero, the connection would break, and the BSD formula would fail or become useless for these complex shapes.
By proving the logarithm is never zero, the authors confirm that the connection holds. This allows mathematicians to use these formulas to prove that certain points exist and to calculate the "size" of the group of rational points.

A Creative Analogy: The Echo Chamber

Imagine you are shouting into a massive, multi-walled canyon (the Abelian Variety).

The Point ( $x$ ): You are the shout.
The Logarithm: The echo you hear.
The Lenses ( $\sigma$ ): Different microphones placed in different corners of the canyon.

For a simple canyon (an elliptic curve), we know that if you shout, every microphone picks up an echo.
For this complex, multi-dimensional canyon, there was a fear that maybe, if you shouted, one of the microphones might pick up silence (zero) because of a weird acoustic trick in the geometry.

The authors proved that no matter how complex the canyon is, if you shout (have a non-torsion point), every single microphone will hear an echo. There is no silence.

Summary

This paper is a triumph of rigidity. It shows that certain mathematical structures are so tightly woven that they cannot "hide" points in a way that makes them invisible to specific mathematical tools. By using a deep theorem about the geometry of these shapes, the authors confirmed that the "fingerprint" of a wandering point is always visible, securing a vital link in the chain of logic that helps us understand the fundamental nature of numbers.

1. Problem Statement

The paper addresses a fundamental question in the arithmetic of abelian varieties regarding the non-vanishing of $p$ -adic logarithms of rational points.

Context: The inquiry is inspired by the Bertolini–Darmon–Prasanna (BDP) formula, which relates special values of $p$ -adic $L$ -functions to the square of the image of a Heegner point under a $p$ -adic Abel–Jacobi map.
The Core Question: Let $A$ $A$ be an abelian variety over a number field $L$ $L$ (typically $\mathbb{Q}$ $Q$ ) associated with a modular form or Hilbert modular form. Let $x \in A(L)$ $x \in A (L)$ be a non-torsion rational point. Is the $p$ $p$ -adic logarithm $\log_\omega(x)$ $lo g_{ω} (x)$ non-zero for all non-zero differentials $\omega$ $ω$ in specific eigenspaces?
- Specifically, if $A$ has "GL $_2$ -type" structure (associated with a number field $F$ of endomorphisms), does $\log_{\omega_\sigma}(x) \neq 0$ hold for all embeddings $\sigma$ of $F$ into $\mathbb{Q}$ and all corresponding differentials $\omega_\sigma$ ?
Motivation:
- For elliptic curves ( $F=\mathbb{Q}$ ), the answer is trivially "yes" because the kernel of the $p$ -adic logarithm consists only of torsion points.
- For higher-dimensional abelian varieties, the kernel of the logarithm map $\log_A: A(\mathbb{Q}_p) \to V_p$ is still the torsion subgroup, but the projection $\log_\omega$ onto a specific 1-dimensional subspace might vanish even if the full vector $\log_A(x)$ is non-zero.
- This non-vanishing is crucial for the Birch and Swinnerton-Dyer (BSD) conjecture (specifically the $p$ -part) and for Iwasawa theory, as it ensures that $p$ -adic heights and regulators are non-degenerate.

2. Methodology

The authors employ tools from $p$ -adic transcendence theory, specifically the $p$ -adic analytic subgroup theorem.

The $p$ -adic Analytic Subgroup Theorem (Waldschmidt/Masser): This theorem states that if a rational point $x$ on an abelian variety $A$ has a $p$ -adic logarithm $\log_A(x)$ lying in a specific rational subspace $W \subset t_A \otimes \mathbb{Q}_p$ , then $x$ must lie (up to torsion) on a proper algebraic subgroup $B \subset A$ whose tangent space is contained in $W$ .
Strategy of Proof:
1. Assume for contradiction that $\log_\omega(x) = 0$ for some non-zero differential $\omega$ .
2. This implies $\log_A(x)$ lies in the kernel of the linear functional associated with $\omega$ .
3. Apply the $p$ -adic analytic subgroup theorem to deduce the existence of a proper, non-zero abelian subvariety $B \subset A$ containing a multiple of $x$ .
4. Analyze the constraints imposed by the endomorphism algebra (the field $F$ or $E$ ) on such a subvariety $B$ .
5. Show that the existence of such a $B$ contradicts the structural hypotheses of the abelian variety (e.g., the non-existence of certain extensions of the endomorphism algebra or the nature of the trace field).

3. Key Contributions and Main Results

The paper establishes a series of theorems generalizing the non-vanishing result from elliptic curves to higher-dimensional abelian varieties with specific endomorphism structures.

Theorem 1.1 (The Representative Result)

Let $A/\mathbb{Q}$ be an abelian variety with an embedding of a field $F \subset \text{End}^0_{\mathbb{Q}}(A)$ such that:

$\dim(A) = [F : \mathbb{Q}]$ .
$F$ has at least one real embedding.
Then, for any non-torsion point $x \in A(\mathbb{Q})$ , $\log_\omega(x) \neq 0$ for all non-zero differentials $\omega$ in the eigenspaces corresponding to all embeddings of $F$ .

Significance: This confirms the non-vanishing for the abelian varieties associated with weight 2 newforms (where $F$ is the totally real Hecke field).

Theorem 2.3 (General Case with Trace Field Condition)

Let $A/L$ have an endomorphism field $E$ . If $\dim(A) = [E:\mathbb{Q}]$ and a specific condition regarding the trace field of potential $E \times E$ -extensions is met (specifically, the trace field is not contained in the field of definition of the point), then the non-vanishing holds.

Note: The condition on real embeddings in Theorem 1.1 ensures that no such $E \times E$ extension exists, making the trace field condition vacuously true.

Theorem 2.5 (CM Case)

If $A$ admits complex multiplication (CM) by a CM field $E'$ (where $E'$ is a quadratic extension of $E$ ), the non-vanishing holds for all non-torsion points.

Theorem 2.6 (Minimal Subvariety)

In cases where the previous conditions might fail, the paper proves that for any non-torsion point $x$ , there exists a minimal $E$ -stable abelian subvariety $B \subset A$ containing a multiple of $x$ . The non-vanishing holds for all differentials that do not vanish on the tangent space of this minimal $B$ .

Theorem 2.8 & 2.9 (Twists and Heegner Points)

The results are extended to twists of Heegner points by finite-order Hecke characters. This is critical for applications to the BSD conjecture for twists of modular abelian varieties. The authors show that if the $L$ -value does not vanish (or vanishes to order 1), the corresponding twisted Heegner point has a non-vanishing $p$ -adic logarithm in the relevant eigenspaces.

Theorem 1.2 (Dimension of $p$ -adic Closure)

The paper connects these results to a question posed by Bjorn Poonen regarding the dimension of the $p$ -adic closure of a finitely generated subgroup.

The authors show that under the Structural Rank Conjecture, the dimension of the $p$ -adic closure of a set of $F$ -linearly independent points is exactly $\min\{r \cdot \deg(F), \dim(A)\}$ (for totally real $F$ ).
This provides a concrete verification of Poonen's conjecture in the GL $_2$ -type case.

4. Applications and Significance

BSD Conjecture (p-part):
The non-vanishing of the $p$ -adic logarithm is a necessary condition for the validity of the $p$ -part of the BSD conjecture for abelian varieties of GL $_2$ -type. Specifically, it ensures that the $p$ -adic regulator is non-zero, allowing the $p$ -adic $L$ -function to predict the rank and the size of the Tate-Shafarevich group correctly. This paper provides the rigorous proof of non-vanishing that was previously assumed or implicitly used in works by Skinner, Wan, and others.
Iwasawa Theory:
The results support the main conjectures of Iwasawa theory for these varieties. The non-vanishing of the logarithm is required to establish the non-triviality of $p$ -adic heights, which are central to the formulation of $p$ -adic Gross-Zagier formulas.
Generalization of BDP Formula:
While the BDP formula was originally formulated for elliptic curves, this paper validates the extension of the underlying non-vanishing principles to higher-dimensional abelian varieties associated with Hilbert modular forms.
Transcendence Theory in Arithmetic Geometry:
The paper demonstrates the power of the $p$ -adic analytic subgroup theorem in solving concrete arithmetic problems regarding rational points on abelian varieties, bridging the gap between transcendence theory and the arithmetic of modular forms.

5. Conclusion

The paper resolves a subtle but critical gap in the arithmetic of abelian varieties: proving that $p$ -adic logarithms of non-torsion rational points do not vanish in specific eigenspaces, provided the endomorphism algebra has a real embedding (or satisfies specific CM conditions). This result solidifies the theoretical foundation for the $p$ -adic aspects of the BSD conjecture and Iwasawa theory for GL $_2$ -type abelian varieties.