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Generalised height pairings and the Albanese kernel

This paper investigates the domain and computability of the generalized $p$ -adic height pairing used in the depth-two Chabauty–Coleman–Kim method by linking it to Beilinson–Bloch conjectures and demonstrating that if the Albanese kernel of $X \times X$ is torsion, an algorithm exists to compute this pairing on rational points of the Jacobian.

Original authors: Netan Dogra

Published 2026-04-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Netan Dogra

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: Hunting for Hidden Numbers

Imagine you are a detective trying to find all the "rational points" on a specific, twisted shape (a mathematical curve). In plain English, a rational point is just a coordinate $(x, y)$ where both numbers are simple fractions (like $1/2$ or $3/4$ ).

For centuries, mathematicians have had a toolkit called the Chabauty–Coleman–Kim method to solve this mystery. Think of this method as a high-tech metal detector. It doesn't just scan the ground; it scans the "shape" of the universe around the curve to see where the hidden treasure (the rational points) could possibly be.

However, this metal detector has a glitch. It works great for simple shapes, but for more complex curves, it gets confused. It produces a "noise" signal that looks like a treasure, but might just be a rock. To fix this, the method needs a new, more sensitive calibration tool.

This paper provides that new tool. It explains how to build a "universal translator" that can decode the noise and tell us exactly where the rational points are hiding.

The Problem: The "Height" is Too Complicated

In this mathematical world, we assign a "height" to every point. You can think of height as a measure of how "complicated" a fraction is.

$1/2$ has a low height.
$1,000,000/999,999$ has a high height.

The Chabauty–Coleman–Kim method relies on a special kind of "height pairing." Imagine you have two points, $A$ and $B$ . The method asks: "If I combine $A$ and $B$ , what is their combined height?"

The Old Way (Simple): For simple curves, this is like adding two numbers. Easy.
The New Way (Complex): For the complex curves this paper tackles, the "height" isn't just a number; it's a complex structure involving layers of hidden information (mathematicians call these "mixed extensions").

The problem is: We don't know how to calculate this complex height. It's like having a recipe for a cake that requires an ingredient you can't find in any store. Without this ingredient, the recipe (the algorithm) stops working.

The Solution: The "Albanese Kernel" Shortcut

The author, Netan Dogra, proposes a clever workaround. He suggests that if we assume a famous, unproven mathematical guess (the Beilinson–Bloch conjecture) is true, we can bypass the missing ingredient entirely.

Here is the analogy:

Imagine you are trying to build a bridge across a deep canyon (the gap in our knowledge).

The Direct Route: You try to build a bridge using a specific, hard-to-find type of steel (the complex height calculation). You can't find the steel, so you are stuck.
The Detour (This Paper): Dogra says, "Wait! If we assume the ground on the other side is solid (the conjecture), we can build a bridge using a different material: Algebraic Cycles."

What is an Algebraic Cycle?
Think of the curve as a landscape. An algebraic cycle is just a collection of paths or loops drawn on that landscape.

The Albanese Kernel is a specific "trash can" for loops that look like they should be zero but aren't quite.
Dogra's discovery is that if this "trash can" is empty (or contains only trivial items), then any complex loop we need can be broken down into a sum of simple, easy-to-calculate loops.

The Magic Trick:
The paper proves that if the "trash can" is empty, we can take our impossible-to-calculate complex height and rewrite it as a sum of heights of simple loops drawn on smaller curves inside the big shape.

Before: "I need to calculate the height of this giant, invisible monster."
After: "I can calculate the height of this monster by adding up the heights of 50 tiny, visible ants."

The "Motivic Avatar": A Digital Twin

The paper also introduces a concept called a "Motivic Avatar."

Imagine you have a complex machine (the mathematical object we are studying). It's too big to fit in your lab.

The Avatar: This is a small, simplified 3D-printed model of the machine that behaves exactly like the real one for the specific task you need.
The Paper's Contribution: Dogra builds a "digital twin" of the problem. Instead of trying to solve the problem in the messy, real world of numbers, he solves it in the clean, structured world of "motives" (a theoretical framework for shapes).

He shows that this digital twin has a specific structure (a "torsor," which is like a twisted box) that we can actually open and measure. By measuring the box, we get the answer to the original problem.

Why Does This Matter?

Solving the Unsolvables: This method allows mathematicians to find rational points on curves that were previously thought to be too difficult to crack.
The Algorithm: The paper doesn't just say "it's possible"; it gives a step-by-step recipe (an algorithm).
- Step 1: Check if the "trash can" (Albanese Kernel) is empty.
- Step 2: If yes, break the problem down into smaller curves.
- Step 3: Calculate the simple heights.
- Step 4: Add them up to get the answer.
The "Recurring Fantasy": The author mentions a dream in mathematics: that these deep conjectures should lead to a computer program that can list all the rational points on any curve. This paper is a major step toward making that dream a reality, even if we have to assume the conjecture is true to run the program.

Summary in One Sentence

This paper shows that if we trust a deep mathematical guess about the "shape" of numbers, we can turn a terrifyingly complex calculation into a simple puzzle of adding up smaller, manageable pieces, finally allowing us to find all the hidden rational points on complex curves.

1. Problem Statement

The paper addresses a critical bottleneck in the Chabauty–Coleman–Kim method (specifically in "depth two" or "quadratic Chabauty") for determining rational points on curves.

Context: The Chabauty–Coleman–Kim method uses non-abelian Galois cohomology to bound the set of rational points $X(\mathbb{Q})$ on a curve $X$ . In the "full-fat" quadratic Chabauty setting, one studies iterated extensions (mixed extensions) of Galois representations with graded pieces $\mathbb{Q}_p$ , $V = H^1_{\text{et}}(X, \mathbb{Q}_p(1))$ , and $W = \text{Coker}(H^1 \otimes H^1 \to H^2)^*$ .
The Obstruction: To compute the equations defining the set of rational points $X(\mathbb{Q})$ , one requires a generalised $p$ -adic height pairing $h$ . This pairing maps pairs of mixed extensions (or points on the Jacobian) to a cohomology group.
The Difficulty: While the existence of this pairing is predicted by the Bloch–Kato conjectures, its domain of definition is unclear, and there is no known algorithm to compute it explicitly for arbitrary points. Unlike classical heights, which can be computed via local data on divisors, the generalised height involves complex motivic structures that are difficult to access computationally.

2. Methodology

The author proposes a strategy to compute the generalised height pairing by reducing the problem to the study of algebraic cycles and the Albanese kernel, relying on the Beilinson–Bloch conjectures.

A. Theoretical Framework

Mixed Extensions: The paper formalizes the category of mixed extensions $M(C; V_0, \dots, V_n)$ and their relation to non-abelian cohomology $H^1(G, U)$ .
Generalised Heights: It defines a global generalised height $h$ as a sum of local heights on mixed extensions. The bilinearity of this height is linked to reciprocity laws in Galois cohomology.
The Albanese Kernel: The core of the methodology relies on the Albanese kernel $T(X^2)$ , defined as the kernel of the map from the group of homologically trivial codimension-2 cycles on $X \times X$ to the product of Jacobians:
$T(X^2) := \text{Ker}\left( CH^2(X^2)_0 \to \text{Jac}(X)^2 \right)$
The Beilinson–Bloch Conjecture: The paper assumes the conjecture that $T(X^2)$ is a torsion group. This implies that any element in the kernel is torsion, meaning some multiple of it is rationally equivalent to zero (i.e., it is a principal divisor).

B. The Algorithmic Reduction

The paper constructs an algorithm to compute $h(P_1, P_2)$ for points $P_1, P_2 \in \text{Jac}(X)(\mathbb{Q})$ :

Cycle Representation: Represent the pair of points $(P_1, P_2)$ as a cycle in $CH^2(X^2)_0$ .
Torsion Assumption: By the Beilinson–Bloch conjecture, this cycle is torsion. Therefore, there exists an integer $N$ such that $N \cdot (P_1, P_2)$ is the divisor of a rational function on a curve $C \subset X \times X$ .
Decomposition: The algorithm finds a finite set of curves $\iota_i: C_i \hookrightarrow X \times X$ and principal divisors $D_i$ on these curves such that the generalised height of the original pair can be expressed as a linear combination of heights of these principal divisors:
$h(P_1, P_2) = \sum n_j h(D_j)$
Computation: The height of a principal divisor $D$ on a curve $C$ can be computed using standard techniques (e.g., via the fundamental group of $C$ or local height pairings on $C$ ), effectively reducing the "full-fat" problem to "skimmed" or "semi-skimmed" problems on auxiliary curves.

3. Key Contributions

Algorithm for Generalised Heights: The primary contribution is Theorem 1, which provides an explicit algorithm to compute the generalised height pairing on the Jacobian, conditional on the Beilinson–Bloch conjecture. This transforms the height from a theoretical obstruction into a computable quantity.
Motivic Refinements (Motivic Avatars): The paper introduces the concept of "motivic avatars" for non-abelian cohomology varieties. It constructs a set $M(X \times X)$ that sits in an exact sequence relating algebraic cycles ( $CH^2(X^2, 1)$ ), the Jacobian, and the Albanese kernel. This serves as a "motivic" model for the Selmer scheme, analogous to how the Poincaré torsor models the height pairing in the "skimmed" case.
Connection to Algebraic Cycles: It establishes a rigorous link between the non-abelian Chabauty method and the geometry of cycles on $X \times X$ . Specifically, it shows that the obstruction to computing the height is precisely the torsion-ness of the Albanese kernel.
Correction of Previous Work: The author corrects a claim in a previous paper ([BD21]) regarding the injectivity of maps between local and global mixed extensions, clarifying the conditions under which local extensions come from global ones.

4. Results

Theorem 1 (Main Result): Assuming the Beilinson–Bloch conjecture, there exists an algorithm that, given $P_1, P_2 \in \text{Jac}(X)(\mathbb{Q})$ , returns a decomposition of the generalised height $h(P_1, P_2)$ into a sum of heights of principal divisors on curves embedded in $X \times X$ .
Example (Section 4): The author applies this to a specific hyperelliptic curve $y^2 = 4x^5 + \dots$ . By finding a curve $C \subset X \times X$ and a principal divisor on it (derived from rational points on a related genus-8 curve), they demonstrate how to compute the mixed height $h(P_1 P_2)$ which was previously inaccessible using only rational points on $X$ .
Table of Analogies: The paper provides a comprehensive table comparing different levels of Chabauty (Classical, Skimmed, Semi-skimmed, Full-fat, Mixed Tate) with their corresponding motivic avatars (Jacobian, Poincaré Torsor, Bloch-Suslin complex, etc.) and obstructions (Albanese kernel, $K_2$ torsion).

5. Significance

Practicality of Quadratic Chabauty: This work moves "full-fat" quadratic Chabauty from a theoretical framework to a practical computational tool. By reducing the computation of complex $p$ -adic heights to the search for rational points on auxiliary curves (where standard Chabauty or descent methods apply), it opens the door to determining rational points on curves of higher genus where classical methods fail.
Bridging Arithmetic and Geometry: It deepens the connection between the arithmetic of rational points (via Galois cohomology) and the geometry of algebraic cycles (via the Albanese kernel). It suggests that the "recurring fantasies" of the Beilinson–Bloch conjectures (the existence of algorithms for rational points) are realized through the torsion properties of these kernels.
Motivic Perspective: The introduction of "motivic avatars" offers a new language for non-abelian Chabauty, suggesting that the Selmer schemes can be understood through the lens of mixed motives and algebraic cycles, potentially leading to new structural insights into the $p$ -adic period maps.

In summary, Dogra's paper solves the computational impasse of the depth-two Chabauty–Kim method by leveraging the Beilinson–Bloch conjecture to express generalised heights in terms of algebraic cycles, thereby providing a concrete algorithm for determining rational points on curves.