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The Relative Manin-Mumford Conjecture

This paper proves the Relative Manin-Mumford Conjecture for families of abelian varieties in characteristic 0 by adapting the Pila-Zannier method with new height inequalities and degeneracy loci results, while also providing a novel proof of the Uniform Manin-Mumford Conjecture for curves that avoids equidistribution techniques.

Original authors: Ziyang Gao, Philipp Habegger

Published 2026-07-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ziyang Gao, Philipp Habegger

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 standing in a vast, invisible library where every book represents a different kind of mathematical shape. Some books describe simple loops, while others describe complex, multi-dimensional donuts that twist and turn in ways our eyes can't see. This is the world of number theory and geometry, a field where mathematicians try to find hidden patterns in the way numbers and shapes interact. One of the most famous puzzles in this library is figuring out where "special" points live. Think of these special points as the "golden tickets" hidden inside a chocolate factory. In the world of these shapes, the golden tickets are called torsion points. They are spots on a shape that, if you keep adding them to themselves over and over again, eventually bring you back to the very beginning (the zero point).

For a long time, mathematicians wondered: If you take a specific path or a smaller shape drawn inside one of these giant mathematical donuts, will it be covered in these golden tickets? Or are the tickets scattered so thinly that you'd never find enough of them to make a pattern? The answer depends on how "twisty" the path is compared to the donut itself. If the path is too simple, it might only touch a few tickets. But if the path is complex enough, it might be covered in them. This question is known as the Manin–Mumford Conjecture. It's like asking if a specific road in a city is paved with gold coins or just a few scattered ones. Solving this helps us understand the deep, hidden rules that govern how numbers and shapes fit together, which is crucial for everything from cryptography to understanding the fundamental structure of the universe.

Now, imagine you have a whole family of these donuts, not just one. Maybe the donuts change shape slightly as you move along a road, getting bigger or twisting differently. This is what mathematicians call a "family of abelian varieties" (specifically, an abelian scheme). The big question becomes: If you draw a path through this entire family of changing donuts, will that path be covered in golden tickets? This is the Relative Manin–Mumford Conjecture, and it's a much harder puzzle because the "donuts" aren't static; they are moving targets.

In this paper, Ziyang Gao and Philipp Habegger finally solve this puzzle for families of abelian varieties (abelian schemes) in characteristic 0 (which is the standard mathematical world we usually work in). They prove a crucial direction: if your path is "Zariski dense" (a fancy way of saying it's not stuck in a tiny, boring corner and actually explores the whole space) AND it is covered in golden tickets, then the path must be at least as complex as the donuts themselves. In other words, you cannot have a simple path that is covered in enough tickets to be interesting; complexity is a necessary condition for being covered.

The authors didn't just guess this; they proved it with absolute certainty using a clever mix of tools. They used a method called the Pila–Zannier method, which is like a high-tech metal detector. This detector helps find rational points (the golden tickets) by counting them in a very specific, organized way. They also used something called the Betti map, which acts like a coordinate system or a GPS for these shapes. Imagine the Betti map as a way to translate the complex, twisting shape of a donut into a simple, flat grid of numbers. By looking at how the path moves on this flat grid, they could see if it was "degenerate" (stuck in a pattern) or "non-degenerate" (moving freely).

A key part of their proof involved checking two scenarios. First, they looked at whether the path was "degenerate," meaning it was stuck in a specific, boring pattern that would make it easy to count the tickets. If it was degenerate, the proof was straightforward. But if the path was not degenerate, they had to use a powerful tool called a height inequality. Think of "height" as a measure of how complicated a number is. The authors showed that if the path is not degenerate, the "height" of the numbers involved is bounded, which limits how many golden tickets can exist. They combined this with a theorem about how these shapes behave (the Ax–Schanuel theorem) to show that if the path is complex enough, it must be covered in tickets.

One of the most exciting results of this paper is that it also proves a "Uniform Manin–Mumford Conjecture" for curves. This means they found a rule that says: for any curve of a certain complexity (genus $g$ ), there is a specific number $c$ (which depends only on $g$ ) such that the curve will never have more than $c$ golden tickets, no matter how you draw it. This is a huge deal because it means the number of these special points is limited and predictable, rather than growing forever.

The authors also tackled a tricky question: Is the opposite true? If a path is complex enough, does it have to be covered in golden tickets? They proved that the answer is "not always." They found a specific condition involving the "Betti rank" (a measure of how much the path twists on the flat grid) that must be met for the path to be fully covered. If this condition isn't met, even a complex path might miss the golden tickets. This shows that the relationship between complexity and golden tickets is subtle and precise: while complexity is required to be covered, it is not always sufficient on its own.

In short, Gao and Habegger have built a complete map of where these golden tickets live in families of abelian varieties. They showed that you can't have a path that is both simple and covered in tickets, and they gave a precise rule for when a complex path will be covered. Their work doesn't just solve a puzzle; it provides a new, powerful toolkit for mathematicians to explore the hidden geometry of numbers, proving that even in the most abstract corners of math, there are strict, beautiful rules that govern where the magic happens.

Technical Summary: The Relative Manin–Mumford Conjecture

Problem Statement
This paper addresses the Relative Manin–Mumford Conjecture for families of abelian varieties in characteristic 0. Let $S$ be a regular, irreducible, quasi-projective variety over an algebraically closed field $L$ of characteristic 0, and let $\pi: A \to S$ be an abelian scheme of relative dimension $g \ge 1$ . Let $A_{tor}$ denote the union of torsion points in the fibers of $A$ . The conjecture posits that if $X$ is an irreducible subvariety of $A$ such that the union of its translates by torsion points, $Z_X = \bigcup_{N \in \mathbb{Z}} [N]X$ , is Zariski dense in $A$ , and if the set of torsion points $X(L) \cap A_{tor}$ is Zariski dense in $X$ , then the dimension of $X$ must satisfy $\dim X \ge g$ .

The paper also investigates the converse: under the same density assumptions on $Z_X$ , if $\dim X \ge g$ , is $X(L) \cap A_{tor}$ necessarily Zariski dense in $X$ ? The authors provide a criterion for this density involving the "Betti rank" of $X$ .

Methodology
The authors employ the Pila–Zannier method, a strategy for solving special point problems in Diophantine geometry, adapted to the setting of abelian schemes. The proof strategy is divided into four main steps, heavily relying on the Betti map introduced by Masser and Zannier:

Large Galois Orbits: The authors establish a lower bound on the size of the Galois orbits of torsion points. This relies on a height inequality (Theorem B.1 in [DGH21]) proved by the authors and Dimitrov, which bounds the height of the abelian variety itself, combined with quantitative isogeny estimates (specifically R´emond's refinement of Masser–Wüstholz).
Point Counting in O-minimal Structures: The torsion points are encoded as rational points in a set definable in the o-minimal structure $\mathbb{R}_{an,exp}$ . The authors utilize a semi-rational variant of the Pila–Wilkie counting theorem (due to Habegger and Pila) to count these points.
Functional Transcendence: The authors apply the mixed Ax–Schanuel theorem for the universal abelian variety (proved by the first-named author in [Gao20b]). This step involves analyzing the "degeneracy locus" of the subvariety $X$ .
Degeneracy Loci Analysis: A crucial innovation is the detailed study of the degeneracy loci $X^{deg}(t)$ $X^{d e g} (t)$ , defined as the union of subvarieties where the dimension of the bi-algebraic closure drops relative to the base. The authors distinguish two cases:
- If the 0-th degeneracy locus is not Zariski dense, the height inequality applies, allowing the Pila–Zannier method to proceed.
- If the 0-th degeneracy locus is dense, the authors show that the 1-st degeneracy locus is also dense, which directly implies the conjecture via known properties of the Betti rank.

Key Contributions and Results

Proof of the Relative Manin–Mumford Conjecture (Theorem 1.1): The paper proves that for any irreducible subvariety $X \subseteq A$ where $Z_X$ is Zariski dense in $A$ , if $X(L) \cap A_{tor}$ is Zariski dense in $X$ , then $\dim X \ge g$ . This result holds for any algebraically closed field $L$ of characteristic 0.
Uniform Manin–Mumford for Curves (Corollary 1.2): As a corollary, the authors provide a new proof of the Uniform Manin–Mumford Conjecture for curves embedded in their Jacobians. They establish that for a genus $g \ge 2$ curve $C$ , the number of torsion points in the image of the Abel–Jacobi embedding is bounded by a constant $c(g)$ depending only on the genus. Notably, this proof does not rely on equidistribution results, distinguishing it from previous proofs by K¨uhne.
Criterion for Torsion Density (Theorem 1.3): The authors prove a criterion for the Zariski density of torsion points in $X$ . Specifically, if $Z_X$ is Zariski dense in $A$ , then $X(L) \cap A_{tor}$ is Zariski dense in $X$ if and only if the generic Betti rank of $X$ equals $2g$ . This result clarifies that the converse of the main conjecture is false in general, as there exist cases where $\dim X \ge g$ but the Betti rank is less than $2g$ .
Reduction from $\mathbb{C}$ to $\mathbb{Q}$ : The paper details a specialization argument (inspired by Masser and refined by Barroero and Dill) to extend the results from the case where the base field is $\mathbb{Q}$ to arbitrary algebraically closed fields of characteristic 0. The proof for $L=\mathbb{Q}$ is established first (Sections 1–7), and then extended via induction on the transcendence degree of the field over $\mathbb{Q}$ .

Significance
The paper claims to resolve the Relative Manin–Mumford Conjecture in full generality for abelian schemes over characteristic 0. Its significance lies in:

Unification: It unifies previous partial results for curves, surfaces, and specific families of elliptic curves into a single theorem for arbitrary abelian schemes.
Methodological Refinement: It introduces a refined application of the Pila–Zannier method that avoids the use of equidistribution theorems, relying instead on a careful analysis of degeneracy loci and the height inequality.
New Tools: It leverages the mixed Ax–Schanuel theorem and the study of degeneracy loci to handle the "degenerate" cases that previous approaches might have struggled with.
Clarification of Converse: By providing the Betti rank criterion, the paper clarifies the precise geometric conditions under which torsion points are dense, showing that dimension alone is insufficient without the non-degeneracy condition (Betti rank $2g$ ).

The work builds upon and synthesizes techniques from Masser, Zannier, Pila, Wilkie, and the authors' own previous work on height inequalities and Betti maps, offering a comprehensive solution to a central problem in Diophantine geometry.

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