# The abelian Zilber–Pink conjecture

*An expository walkthrough, written to be readable with high-school mathematics.*

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## 0. Prologue: the paper in one sentence

If you know what a circle is, and you know that a circle is secretly a group, here is the entire paper in one sentence:

> **Given a curve — or a higher-dimensional surface — sitting inside an abelian variety, only finitely many pieces of it can be "bigger than expected" intersections with the arithmetic special points and subspaces. And the finitely many exceptions can be listed.**

The whole paper is a proof of this. The rest of this exposition is a map of *how* the proof works, written so that a smart high-schooler can follow the shape of the argument even when the technical steps get heavy.

The paper is technically long. Here it is compressed by roughly a factor of ten. Every step of the proof appears; some of the deepest technical lemmas are summarized with the intuition and the precise statement, without the twenty pages of chasing constants.

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## 1. The cast of characters

### 1.1 The circle is a group

Take the ordinary circle $S^1 \subset \mathbb C$. Points on it are complex numbers of modulus one, so you can write them as $e^{2\pi i \theta}$ for some $\theta \in \mathbb R$. Multiplication of complex numbers on the unit circle is rotation: $e^{2\pi i \theta} \cdot e^{2\pi i \phi} = e^{2\pi i (\theta + \phi)}$.

So $S^1$ is a group, and the group law is "rotation by an angle." This is not a coincidence. We can also write $S^1 = \mathbb R / \mathbb Z$ (angle mod 1), and the group law is just addition mod 1.

### 1.2 The torus: a donut you can add points on

Now take two copies of $\mathbb R$, form $\mathbb R^2$, and quotient by the lattice $\mathbb Z^2$:

$$T = \mathbb R^2 / \mathbb Z^2.$$

Points in $T$ are equivalence classes of pairs $(x, y) \in \mathbb R^2$, where $(x, y)$ and $(x + m, y + n)$ are the "same point" for any integers $m, n$. Topologically, $T$ is a donut (a torus). The group law is again coordinatewise addition mod 1.

The basic fact: $T$ is *both* a smooth geometric object (a compact manifold) and an algebraic object (a group).

### 1.3 Elliptic curves

Over $\mathbb C$, a compact one-dimensional complex abelian group is always of the form

$$E = \mathbb C / \Lambda,$$

where $\Lambda \subset \mathbb C$ is a full-rank lattice, meaning $\Lambda = \mathbb Z \lambda_1 \oplus \mathbb Z \lambda_2$ with $\lambda_1, \lambda_2$ linearly independent over $\mathbb R$.

$E$ is again a torus topologically, but now "topologically" means "a smooth ring surface," i.e., a *curve* (complex dimension 1, real dimension 2). $E$ is called an **elliptic curve**.

Equivalently, $E$ can be described as the set of solutions $(x, y) \in \mathbb C^2$ to the equation

$$y^2 = x^3 + a x + b, \quad 4a^3 + 27b^2 \neq 0,$$

plus a "point at infinity" that acts as the identity for the group law. The group law is given geometrically by the *chord–tangent rule*: to add $P$ and $Q$, draw the line through them, take its third intersection with $E$, and reflect across the $x$-axis.

That's not a random construction. The identification $E \cong \mathbb C/\Lambda$ is what makes the chord–tangent rule *the same as* complex addition. That correspondence is the entire reason elliptic curves are interesting.

### 1.4 Abelian varieties

An **abelian variety** over $\mathbb C$ is a compact commutative complex Lie group that can be embedded as a projective algebraic variety. Concretely, every abelian variety is of the form

$$A = \mathbb C^n / \Lambda$$

for some lattice $\Lambda = \mathbb Z^{2n} \subset \mathbb C^n$, subject to a condition (a "Riemann form" or polarization) that ensures $A$ has enough algebraic structure to be cut out by polynomial equations in some projective space.

Examples:
- $\dim A = 0$: a single point.
- $\dim A = 1$: an elliptic curve.
- $\dim A = n$: a "higher-dimensional circle." $A = \mathbb C^n/\mathbb Z^{2n}$ with a positive-definite Hermitian form is the standard model.
- Products: if $A_1$ and $A_2$ are abelian varieties, so is $A_1 \times A_2$ (dimension adds).

Over a number field $K$ (a finite extension of $\mathbb Q$), there is an extra piece of data: the lattice is defined by equations with coefficients in $K$. This is what makes the arithmetic ("diophantine") side of the paper possible.

### 1.5 Torsion points

Let $A$ be an abelian variety. A **torsion point** of $A$ is a point $p \in A$ with $n p = 0$ for some integer $n > 0$. Here $n p$ means $p + p + \cdots + p$ ($n$ times).

Examples:
- On $S^1 = \mathbb R/\mathbb Z$, torsion points are rationals: $p = a/b \bmod 1$ with $b > 0$. Equivalently, $e^{2\pi i a/b}$ are the roots of unity on the unit circle.
- On $E = \mathbb C/\Lambda$, torsion points are those $p$ with $p = (a/r)\lambda_1 + (b/r)\lambda_2$ for some common denominator $r$.
- On $E \times E$, torsion points are pairs of torsion points on each factor.

The torsion points of $A$ form a subgroup $A[\text{tors}] \subset A(\overline{\mathbb Q})$, dense in $A(\mathbb C)$ in the usual topology. So topologically, they are *everywhere*. Arithmetically, they are *extremely sparse*. This paradox is the source of the whole subject.

### 1.6 Special subvarieties = torsion cosets

A **torsion coset** in $A$ is a set of the form

$$T = B + \tau = \{ b + \tau : b \in B \}$$

where $B \subseteq A$ is an abelian subvariety (i.e., a translated subtorus that is itself an algebraic subgroup) and $\tau \in A$ is a torsion point.

Torsion cosets are also called **special subvarieties**. Some examples:

- A single torsion point $\{\tau\}$: this is the case $B = \{0\}$, $\dim T = 0$.
- A coset of the diagonal in $E \times E$: $B = \{(x, x) : x \in E\}$, $\tau$ a torsion point. $\dim T = 1$.
- A coset of $E \times \{0\}$: $\dim T = 1$.
- The whole $A$: $\dim T = \dim A$.

Note the rigid structure: a torsion coset is a *translate of a subgroup*, and we allow translations only by torsion points. That is the crucial arithmetically special pattern.

### 1.7 The "smallest special subvariety containing X"

Given an irreducible subvariety $X \subseteq A$, there is always a unique *smallest* torsion coset containing $X$. Write it $\langle X\rangle_{\mathrm{sp}}$.

**Proof of existence and uniqueness.** The whole ambient variety $A$ is itself a torsion coset containing $X$, so there is at least one candidate. Among all torsion cosets containing $X$, choose one --- call it $S$ --- of *minimal possible dimension*.

We claim that $S$ is contained in every other torsion coset $S'$ that contains $X$.

Suppose $S'$ is another such torsion coset. Then $S \cap S'$ is a nonempty algebraic subset of $A$ that contains $X$. An intersection of two torsion cosets, on each irreducible component, is again a torsion coset: this uses the fact that torsion points lift under surjective homomorphisms of abelian varieties, and every irreducible component of an algebraic subgroup contains a torsion point.

Let $S''$ denote the irreducible component of $S \cap S'$ that contains $X$. Then $S''$ is a torsion coset with $X \subseteq S'' \subseteq S$, so $\dim S'' \le \dim S$. By the minimal choice of $\dim S$, we must have $\dim S'' = \dim S$, which forces $S'' = S$ (an irreducible component of an irreducible variety of the same dimension is the whole variety). Since $S'' \subseteq S'$, we get $S \subseteq S'$.

Therefore $S$ is contained in *every* torsion coset that contains $X$, i.e., it is the smallest one. Uniqueness is immediate: two smallest candidates must contain each other. $\square$

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## 2. What "bigger than expected" means: atypicality

Let $A$ be an abelian variety of dimension $n$, let $X \subseteq A$ be irreducible of dimension $d_X$, and put $S = \langle X\rangle_{\mathrm{sp}}$, of dimension $d_S$.

Let $T \subseteq S$ be a special subvariety (torsion coset) of dimension $d_T$. Consider the intersection $X \cap T$, and let $Y$ be an irreducible component of it.

### 2.1 The expected dimension

By the standard theory of generic intersections, if $X$ and $T$ are "in general position" in $S$, then

$$\dim Y = \dim X + \dim T - \dim S. \qquad (\text{expected})$$

We'll call this the **expected dimension**. It follows from the general "intersection formula" in intersection theory: you can move $X$ a bit and get a generic position, and the intersection dimension is then $\dim X + \dim T - \dim S$.

For example, on a surface $E \times E$ (dim 2):
- $X$ a curve (dim 1), $T = E \times \{0\}$ (dim 1): expected $\dim Y = 1 + 1 - 2 = 0$.
- $X$ a curve, $T = \{(x, x)\}$ (dim 1): expected $\dim Y = 0$.
- $X$ a curve, $T = \{\tau\}$ (dim 0): expected $\dim Y = -1$, meaning "generically empty."

### 2.2 Atypicality

The intersection is **atypical** (for $X$ in $S$) if

$$\dim Y > \dim X + \dim T - \dim S. \qquad (\text{atypical})$$

The intuition: the intersection is *bigger than it has any right to be*. It's a "surprise."

Examples on $E \times E$:

- $X$ a curve, $T = \{\tau\}$ a torsion point, with $\tau \in X$: $\dim Y = 0 > 1 + 0 - 2 = -1$. **Atypical.** Trivially the case, because $X \cap \{\tau\}$ is a point, and every point on $X$ is bigger than the expected "empty" intersection.
- $X$ a curve, $T = E \times \{\tau\}$ a "vertical line," with $\tau$ a torsion point and $X$ contained in that vertical line: $\dim Y = 1 > 1 + 1 - 2 = 0$. **Atypical**, because the whole curve $X$ is being "swallowed" by a torsion coset that has no right to contain a whole curve, generically.
- $X$ a curve, $T = \{(x, x)\}$ the diagonal, with $X \subseteq T$: $\dim Y = 1 > 0$. **Atypical**.

So atypicality detects when $X$ is "secretly straight" — when it lies inside a special subvariety that isn't its own special closure.

### 2.3 Maximal atypical subvarieties

An atypical subvariety $Y$ (that is, an atypical component of some $X \cap T$) is **maximal atypical** if it is not properly contained in any other atypical subvariety.

Note: "atypical" is not quite the same as "atypical component of $X \cap T$." Sometimes a component of $X \cap T$ that is atypical is contained in a bigger one that is also atypical. Maximality is what matters.

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## 3. Manin–Mumford: the warm-up case

Before stating the main theorem, here is the simplest case, which you already know as Manin–Mumford (Raynaud's theorem, 1983).

**Theorem (Manin–Mumford).** Let $A$ be an abelian variety over $\overline{\mathbb Q}$, and $X \subseteq A$ an irreducible subvariety. If $X$ is contained in no proper torsion coset of $A$, then $X$ contains only finitely many torsion points.

Equivalently, the set of torsion points $X(\overline{\mathbb Q}) \cap A[\text{tors}]$ is finite.

### Why "atypicality" already captures Manin–Mumford

Take $T = \{\tau\}$ a torsion point, so $\dim T = 0$. Atypicality says $\dim Y > \dim X + 0 - \dim A$. When $\dim Y = 0$ (i.e., $X$ passes through $\tau$, but doesn't contain a positive-dimensional piece of $T$ — which is automatic since $T$ is a point), the condition becomes

$$0 > \dim X - \dim A, \qquad \text{equivalently} \qquad \dim X < \dim A.$$

So if $X$ is not the whole of $A$ (which is automatic when $X$ is contained in no proper coset), all torsion points on $X$ are atypical. Manin–Mumford says there are finitely many.

Since the "maximal atypical subvariety" for a torsion point $T = \{\tau\}$ is just the point itself (nothing can contain a point and be contained in $X$ if $X$ is a curve), Manin–Mumford gives: finitely many maximal atypical subvarieties of dimension 0.

### Why Manin–Mumford is already hard

The proof of Manin–Mumford is genuinely nontrivial. The torsion points are dense, so the statement "only finitely many lie on a curve" is a strong rigidity result. The proof (Raynaud's) uses $p$-adic uniformization and a Galois-theoretic argument. The abelian Zilber–Pink conjecture is a natural extension: instead of torsion *points*, allow torsion *cosets of any dimension*.

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## 4. The main theorem, stated clearly

Now I can state the actual theorem.

**Theorem (abelian Zilber–Pink, this paper).** Let $K$ be a number field, $A/K$ an abelian variety, and $X \subseteq A_{\overline{K}}$ an irreducible subvariety. Let $S = \langle X\rangle_{\mathrm{sp}}$. Then the number of **maximal atypical subvarieties** of $X$ in $S$ is finite.

Equivalently: there are finitely many proper atypical subvarieties $Y_1, \dots, Y_m \subsetneq X$ such that every atypical subvariety of $X$ in $S$ is contained in one of the $Y_j$.

The list may be empty. The paper assumes no simplicity, no complex multiplication, no bounded degree, and gives no bound on $m$ or on heights or degrees of the $Y_j$.

### Example: $E \times E$

Take $E$ an elliptic curve, and $A = E \times E$, and $X \subseteq A$ a curve. Assume $X$ is not a "secretly straight" line (i.e., not contained in a translate of $E \times \{0\}$, $\{0\} \times E$, or the diagonal).

The maximal atypical subvarieties of $X$ are pretty much just:
1. Torsion points that lie on $X$. (Finitely many by Manin–Mumford.)
2. Positive-dimensional pieces of $X \cap (E \times \{\tau\})$ where $\tau$ is a torsion point — but for a generic $X$ (not a "graph" of the projection to $E$), these are 0-dimensional. So no extra curves appear.
3. Positive-dimensional pieces of $X \cap \{\text{translate of diagonal by torsion point}\}$ — again 0-dimensional for generic $X$.

So for a generic $X$ on $E \times E$, the maximal atypical subvarieties are just the finitely many torsion points on $X$. This is Manin–Mumford.

The real content of Zilber–Pink is the higher-dimensional version: **for higher-dimensional X, the finiteness still holds, even though you can have positive-dimensional atypical pieces.**

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## 5. Heights: how arithmetic turns "dense" into "sparse"

The proof uses arithmetic information in the form of **heights**.

### 5.1 Height of a rational number

For a rational number $p/q$ in lowest terms (gcd$(p, q) = 1$), define

$$h(p/q) = \max(|p|, |q|).$$

So $h(0) = 0$, $h(2/3) = 3$, $h(1000) = 1000$, $h(-7/22) = 22$.

The height measures "arithmetic size." Small height = simple; large height = complicated.

### 5.2 Projective height

For a point $P = [a_0 : a_1 : \dots : a_N] \in \mathbb P^N(\mathbb Q)$ with the integers $a_i$ coprime, define

$$h(P) = \log \max_i |a_i|.$$

This extends to points over any number field by using all absolute values.

Key fact: **there are finitely many points of bounded height** (Northcott's theorem). This is why heights are powerful: a bound on height reduces an infinite search to a finite one.

### 5.3 Canonical (Néron–Tate) height

On an abelian variety $A$ over a number field, with a fixed ample line bundle $L$, there is a special height $\hat{h}_L$ — the **canonical height** or **Néron–Tate height** — with the following crucial property:

$$\hat{h}_L(np) = n^2 \, \hat{h}_L(p) \qquad \text{for any integer } n.$$

It's a *quadratic form* on $A(\overline{\mathbb Q})$. Its associated bilinear form

$$\langle p, q \rangle = \frac{1}{2}(\hat{h}_L(p + q) - \hat{h}_L(p) - \hat{h}_L(q))$$

is positive definite on the group $A(\overline{\mathbb Q}) / A[\text{tors}]$.

**Consequences:**

- Torsion points have height exactly $0$. (Because if $np = 0$, then $n^2 \hat h(p) = \hat h(np) = \hat h(0) = 0$.)
- For points with $\hat h_L(p) > 0$, $\hat h_L(np)$ grows like $n^2$.
- The "height space" $\mathcal H(A) = A(\overline{\mathbb Q}) \otimes_{\mathbb Z} \mathbb R$ is a real vector space of dimension $2n$, equipped with a positive-definite inner product.

### 5.4 "Small" and "large" points

A point of small height behaves arithmetically like a torsion point (in the limit, small height = torsion). A point of large height behaves like it's algebraically independent.

For example: on $E$, a torsion point has $\hat h = 0$; a generic rational point of $E$ has $\hat h$ growing as its coordinates' numerators and denominators grow.

For $X \subseteq A$, one of the key players is the *minimum height* of a point of $X(\overline{\mathbb Q})$, or the minimum of $\hat h$ on a dense subset. The paper uses essential minima $e_1(X, \bar D)$ — a "smallest height that's achieved on a dense subset" quantity.

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## 6. Two facts from analysis: Ax's theorem

The main "soft" tool is **Ax's theorem** about analytic subgroups.

### 6.1 The case you already know

Consider the circle $S^1 = \mathbb R/\mathbb Z$. Take a straight line through the origin with slope $\alpha$: the set $\{ (\alpha t, t) : t \in \mathbb R \}$ in $\mathbb R^2$.

- If $\alpha$ is **rational**, its image mod $\mathbb Z^2$ is a closed curve — a subtorus (finite union of translates of a subgroup).
- If $\alpha$ is **irrational**, its image is *dense* in $S^1 \times S^1$.

That is: **a connected analytic subgroup of a torus is either dense in the whole torus or is dense in a proper subtorus, but "generic" ones are dense in the whole torus.**

### 6.2 Ax's theorem, general form

Let $U$ be a complex vector space, $G$ a complex abelian group, and $J \subset U$ an analytic subset. Suppose the exponential map $\exp : U \to G$ sends $J$ to a set whose Zariski closure is an algebraic subvariety $Z \subseteq G$. Then:

$$\dim \langle Z \rangle_{\mathrm{geo}} - \dim Z \le \dim U - \dim J,$$

where $\langle Z\rangle_{\mathrm{geo}}$ is the smallest *coset of a closed subgroup* containing $Z$.

Intuition:
- The "geometric closure" of $Z$ (smallest coset of a closed subgroup containing it) can be bigger than $Z$ itself.
- The "extra dimension" $\dim\langle Z\rangle_{\mathrm{geo}} - \dim Z$ is bounded by the "extra dimension" $\dim U - \dim J$ — i.e., how much bigger $U$ is than $J$.

Special case: if $J$ is a *point* (dim 0) in $U = \mathbb C$ and $Z$ is a single point, then $\langle Z\rangle_{\mathrm{geo}} = Z$ and the inequality is trivial.

The important case: **if you have a positive-dimensional analytic curve $J$ in $U$, and the image has "small" Zariski closure, then a big chunk of $J$ lies in the Lie algebra of a closed subgroup.** That means: the big fibers that you find geometrically can be "upgraded" to actual abelian subvarieties.

### 6.3 The finite directions lemma

The other key "soft" fact is from Habegger–Pila. It says:

**Lemma (finite directions).** Fix subvarieties $Y_\ell \subseteq A_\ell$ of complex abelian varieties. Then for each $\ell$ there is a fixed finite set $\mathcal H_\ell$ of abelian subvarieties of $A_\ell$ with the following property: whenever $Z_\ell \subseteq Y_\ell$, $y_\ell \in Z_\ell$, $C_\ell \subseteq A_\ell$ abelian subvarieties, and $g : \bigoplus_\ell \mathrm{Lie}(A_\ell) \to E$ is complex linear, with the "rank–fiber" inequality

$$\sum_\ell \dim_{y_\ell}(Z_\ell \cap (y_\ell + C_\ell)) \ge \mathrm{rank}(g|_{\bigoplus \mathrm{Lie}(C_\ell)}) + \varepsilon,$$

then there exist $H_\ell \in \mathcal H_\ell$ with the same inequality for $Z_\ell \subseteq Y_\ell$ replaced by $Y_\ell \cap (y_\ell + H_\ell)$.

More loosely: **whenever Ax produces some abelian subgroup with big fibers, there's a subgroup from a fixed finite set that does just as well.**

The proof uses "geodesic-optimal subvarieties" of $Y_\ell$ — the article uses a Habegger–Pila theorem that the directions of geodesic-optimal subvarieties of a fixed $Y_\ell$ lie in a finite set. That's a beautiful finiteness result.

The consequence: **at each high factor of the sequence, you can replace a "free" direction by one from a finite set.** That is exactly the uniformity needed for the height argument.

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## 7. Two facts from arithmetic

### 7.1 Vojta–Rémond, in Dill's form

This is the deepest arithmetic tool. Its "soft" statement is:

**Vojta–Rémond inequality (Dill form).** Let $X_1, \dots, X_s$ be projective varieties over $\overline{\mathbb Q}$, with very ample bundles $L_\ell$ and positive integer weights $a_\ell$. Let $\mathcal M$ be a nef bundle on the product, and $\mathcal P$ a suitable power of the tensor product of the $L_\ell$. Suppose $\mathcal P \otimes \mathcal M^{-1}$ is generated by $J$ global sections admitting an injection into another power of $L_\ell$'s with bounded coefficient height $\le \sum_\ell a_\ell \delta_\ell$. Then for every point $x = (x_\ell) \in X_1(\overline{\mathbb Q}) \times \dots \times X_s(\overline{\mathbb Q})$ that lies in a certain open set and satisfies the *intersection lower bound*

$$(\mathcal M^{\dim Z} \cdot Z) \ge \theta^{-1} \prod_\ell (\deg Z_\ell)^{-\omega} a_\ell^{\dim Z_\ell}$$

for every subproduct $Z$ through $x$, we have:

$$\sum_\ell a_\ell \, h(x_\ell) \le c_1 \cdot \big( h(\Xi(x)) - h(\mathcal Z(x)) \big),$$

provided:
- $a_\ell \ge c_2 a_{\ell+1}$ (weights well separated), and
- each $h(x_\ell) \ge c_3^{(\ell)}$ (point heights above fixed thresholds).

**The key point: $c_1$ depends only on the geometric data (the $X_\ell$, the $L_\ell$, the $t_1$, the $\theta$, the $\omega$). It does NOT depend on $J$, $t_2$, the coefficients of the generators, or the coefficient heights.** The coefficient-dependent quantities enter only $c_3^{(\ell)}$ (the thresholds).

Why does this matter? Because it lets us do this: **approximate an ideal projector $\Phi$ by some rational projector $f/e$, with arbitrarily small error $\eta$, and then apply Vojta–Rémond without the multiplicative constant depending on $\eta$.** So the accuracy of the approximation doesn't blow up the height inequality. That's the entire trick of the all-high case.

### 7.2 Equidistribution and arithmetic positivity

The other "hard" fact: **Szpiro–Ullmo–Zhang equidistribution**, and its arithmetic bigness variant due to **Yuan** and **Yuan–Zhang**.

The basic statement: if $X_\ell$ is a sequence of abelian varieties (or subvarieties) with "many small points" (bounded-height points that are becoming dense), then the corresponding probability measures — obtained by pushing forward the curvature of the line bundles along the small points — converge to a canonical measure.

In the paper, this is used via an **arithmetic bigness** corollary: for a nef adelic line bundle $\bar D$ on a projective variety $X$ of dimension $n$ with $D$ geometrically big,

$$\frac{e_1(X, \bar D)}{n + 1} \le \frac{(\bar D^{n+1} \cdot X)}{(n+1)(D^n \cdot X)} \le e_1(X, \bar D),$$

where $e_1$ is the *essential minimum* of the point height. The paper uses **Siu's arithmetic inequality**: for nef adelic bundles $\bar D, \bar N$,

$$\widehat{\mathrm{vol}}(\bar D - \bar N) \ge (\bar D^{n+1} \cdot X) - (n+1)(\bar D^n \bar N \cdot X).$$

**What these give:** a way to produce small sections of large powers of certain arithmetic line bundles. If the arithmetic volume grows positively, then (by the theory of bounded sections) there are many small sections, which produce low-height points.

### 7.3 Faltings' product theorem (Evertse's explicit form)

Let $P = \mathbb P^{d_1} \times \dots \times \mathbb P^{d_m}$. Let $G$ be a nonzero multihomogeneous polynomial with multidegrees $D = (D_1, \dots, D_m)$.

Define the **weighted index** of $G$ at a point $z$: pick local affine coordinates and a local frame; look at the Taylor expansion of $G$; take the minimum of $\sum_\ell |\alpha_\ell| / D_\ell$ over the nonzero coefficients $a_\alpha$.

The **index locus** $\mathcal I_\sigma$ is the set of points where the index exceeds $\sigma$ (closed since index is upper semicontinuous).

**Faltings' product theorem (Evertse's form).** Let $0 < \epsilon \le 1$, and suppose

$$\frac{D_\ell}{D_{\ell+1}} \ge (mN/\epsilon)^N$$

where $N = \sum d_\ell$. Then any irreducible component $J$ that is *common to $\mathcal I_\sigma$ and $\mathcal I_{\sigma + \epsilon}$* is a product of subvarieties of the individual $\mathbb P^{d_\ell}$.

**And the arithmetic version:** if $G$ is defined over a number field $k$, then

$$q \cdot \prod_\ell \deg J_\ell \le C(N, \epsilon),$$

where $q$ is the number of conjugates of $J$ over $k$.

**What this gives:** (1) Any common component $J$ of two adjacent index loci is a *product*; (2) its *total degree* is bounded, and it is defined over a bounded-degree extension of $k$.

That is the finiteness engine at the end of the proof: we construct a component of an index locus, and Faltings' product theorem gives us a bounded-degree product component, which pulls back to a bounded-degree equation.

### 7.4 Barroero–Dill: non-density implies finiteness

The last crucial external input is Barroero–Dill's reduction (2021), which reduces the finiteness of **maximal atypical subvarieties** to a universal **non-density** statement.

Non-density theorem: Let $X \subseteq A$ be irreducible and contained in no proper torsion coset. Then $X \cap A^{[r]}$ is not Zariski dense in $X$, where

$$A^{[r]} = \bigcup_{T \text{ special, } \mathrm{codim} \ge r} T$$

is the union of all torsion cosets of codimension at least $r$.

The Barroero–Dill theorem: if non-density holds for every abelian variety, subvariety, and choice of codimension, then **the finiteness of optimal subvarieties** holds. Combined with the fact that every maximal atypical is optimal, this gives finiteness of maximal atypical.

**Special defect.** For $Z \subseteq A$ irreducible, $\delta(Z) = \dim \langle Z\rangle_{\mathrm{sp}} - \dim Z$. $Z \subseteq X$ is **optimal in $X$** if every proper enlargement $U \supsetneq Z$ inside $X$ has $\delta(U) > \delta(Z)$.

**Key claim:** Maximal atypical $\Rightarrow$ optimal.

**Sketch.** If $Y$ is maximal atypical, witnessed by special $T$, then $\delta(Y) \le \dim T - \dim Y < \dim S - \dim X = \delta(X)$. So $Y$ can be strictly contained in a proper $U \subseteq X$ only if $\delta(U) > \delta(Y)$ — otherwise the component of $X \cap \langle U\rangle_{\mathrm{sp}}$ through $U$ would also be atypical and strictly bigger than $Y$, contradicting maximality.

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## 8. The proof strategy, in overview

The proof is a **reductio ad absurdum** in three parts:

1. **Non-density theorem** (Theorem A): if $X \subseteq A$ is contained in no proper torsion coset, then $X \cap A^{[\dim X + 1]}$ is not dense.

2. **Sequence theorem** (Theorem B): an abstract statement about sequences of marked points and their heights. This is the *engine* that drives the proof of Theorem A.

3. **Finite-rank translates** (Corollary C): applications to a different class of problems.

Then the finiteness theorem (Theorem D) reduces to Theorem A via Barroero–Dill.

**Theorem A** $\Leftarrow$ **Theorem B**: Here's the idea. If $X \cap A^{[\dim X + 1]}$ were dense, extract a sequence $x_i \in X$ that avoids any proper closed subset of $X$. Each $x_i$ lies in some torsion coset $B_i + \tau_i$ of codimension $\ge \dim X + 1$. Take a subsequence where the codimension is constant. Look at the Lie algebra projectors $P_i$ onto the scaled Lie algebra $\mathrm{diag}(H_{i\ell}^{-1/2}) \mathrm{Lie}(B_i)$ — these are real endomorphisms, they converge to $P$. Take the limit. Theorem B says this can't happen.

**Theorem B** is proved by induction on $\dim \mathcal A$. Suppose there's a minimal counterexample. Then either:
- **All height scales diverge** (all-high case): use Vojta–Rémond. Contradiction.
- **At least one scale is bounded** (low-factor case): reduce to all-high (if 0-dimensional low), or use the two-measure comparison (if positive-dimensional).

The low case uses two moves:
1. **Arithmetic positivity** (via Siu's arithmetic inequality and equidistribution): forces $\bar E_i^{d+1} \cdot Z_i$ to be large.
2. **Small sections** + **local index** + **product cut** (via Faltings' product theorem): produces a bounded-degree equation containing the marked point but not $Z_i$, which contradicts minimality of $Z_i$.

Now let's walk through each step.

---

## 9. Reductio ad absurdum: set up the sequence

Suppose $X \cap A^{[\dim X + 1]}$ is Zariski dense in $X$. Pick a generic sequence of points $x_i \in X(\overline{\mathbb Q})$ from this intersection, converging to no proper closed subset of $X$ (i.e., hitting every nonempty open set of $X$ infinitely often).

For each $i$, the point $x_i$ lies on some torsion coset $B_i + \tau_i$ with $\mathrm{codim}_A B_i > \dim X$.

### 9.1 Height decomposition

Apply the height decomposition (proved in Section 2.3 of the original). The idea: look at which "directions" of $A$ are short versus long for $x_i$.

Concretely: the canonical height $\hat h$ on $A = \mathbb C^n/\Lambda$ is a positive-definite quadratic form. Choose coordinates $x_i \in \mathcal H(A) \cong \mathbb R^{2n}$. There may be different scales: some directions grow fast, some slow, possibly one bounded.

By taking a max over quotient directions, we produce:
- An isogeny $\alpha : A \to \prod_\ell W_\ell$ (finite degree, essentially a change of coordinates).
- Scales $H_{i, \ell} \ge 1$ with $H_{i,1} / H_{i,2} \to 0$, $H_{i,2} / H_{i,3} \to 0$, etc. (successive ratios going to zero).
- At most one $H_{i,0} = 1$ (bounded scale, at the front), all others diverge.
- Point coordinates $p_{i,\ell}$ with $\|p_{i,\ell}\| = O(\sqrt{H_{i,\ell}})$.
- For each high factor, $\|g(p_{i,\ell})\| / \sqrt{H_{i,\ell}}$ is bounded below for every nonzero homomorphism $g$.

The most important feature: **at most one scale is bounded, and all others are separated by going to infinity.** This is a strong structural statement: the heights of any sequence of points on an abelian variety lie on a "finite staircase" of scales.

### 9.2 Local fiber bounds

For each $\ell$, define the projection $Y_\ell = \mathrm{pr}_{\le\ell}(X) \subseteq A_\ell$. Then the marked points satisfy

$$\dim_{y_{i,\ell}}(Y_\ell \cap (y_{i,\ell} + W_\ell)) \le m_\ell,$$

with $\sum_\ell m_\ell = \dim X$ and $m_\ell \ge 0$. This says: the local dimension of the "fiber" in the marked direction is bounded by $m_\ell$.

### 9.3 The sequence theorem, precisely

Given these data, let $V \subseteq \mathrm{Lie}(W)$ be the image of an idempotent (a projecting operator). Put $\Phi$ = orthogonal projector with kernel $V$. Then the *forbidden condition* is:

$$\mathrm{codim}_{\mathrm{Lie}(W)} V > \sum_\ell m_\ell \qquad \text{and} \qquad \|\Phi(q_i)\| \to 0, \qquad (\star)$$

where $q_i = (y_{i,\ell}/\sqrt{H_{i,\ell}})_\ell$ is the normalized tuple.

**Sequence theorem.** For all sequence data, $(\star)$ cannot hold.

That is: the sequence theorem says there is no "bad orbit" that simultaneously has high codimension and lies close to the kernel of a projector.

---

## 10. Step 1: tangent positivity (Ax and finite directions)

The proof of the sequence theorem needs to exclude any "candidate" subproduct that could serve as a counterexample. The key step is the following "projection exclusion."

**Lemma (projection exclusion).** Suppose the sequence data violate the sequence theorem and have minimal $\dim\mathcal A$. Then for any nonzero product subgroup $C = \prod_\ell C_\ell$ and any integers $a_\ell \ge 0$, if for infinitely many $i$:

$$\dim_{y_{i,\ell}}(Y_\ell \cap (y_{i,\ell} + C_\ell)) \ge a_\ell \qquad (\ell \in I)$$

and

$$\sum_\ell a_\ell \ge \mathrm{rank}(\Phi|_{\mathrm{Lie}(C)}),$$

then this contradiction shows such $C$ cannot exist. In other words: for every nonzero product subgroup $C$, we must have

$$\sum_\ell \dim_{y_{i,\ell}}(Y_\ell \cap (y_{i,\ell} + C_\ell)) < \mathrm{rank}(\Phi|_{\mathrm{Lie}(C)})$$

for all sufficiently large $i$.

**Why this matters:** It says: the "geometric fiber dimension" on every subproduct through the marked points is *strictly smaller* than the "algebraic rank" of the projector restricted to that subproduct. This is a "positivity" statement: it says the projector $\Phi$ is "wiggly" — its image has small algebraic dimension relative to the geometry.

### 10.1 How is the projection exclusion proved?

Suppose $C$ and the $a_\ell$ violate it. Quotient by $C$. Then:
- New parents $Y_\ell' = \pi_\ell(Y_\ell)$.
- New fiber bounds $m_\ell' = m_\ell + e_\ell - a_\ell$, where $e_\ell = \dim(\mathrm{im}(C_\ell \to D_\ell))$ (with $D_\ell = A_\ell / W_\ell$).
- New projector $V' = \pi(V)$, with orthogonal projector $\Phi'$.
- The codimension identity:

$$\mathrm{codim}_{\mathrm{Lie}(W')} V' = \mathrm{codim}_{\mathrm{Lie}(W)} V + \sum_\ell e_\ell - \mathrm{rank}(\Phi|_{\mathrm{Lie}(C)}).$$

Combining with the "excluded" hypothesis $\sum_\ell a_\ell \ge \mathrm{rank}(\Phi|_{\mathrm{Lie}(C)})$, we get

$$\mathrm{codim}_{\mathrm{Lie}(W')} V' = \mathrm{codim}_{\mathrm{Lie}(W)} V + \sum_\ell e_\ell - \mathrm{rank}(\Phi|_{\mathrm{Lie}(C)}) > \sum_\ell m_\ell + \sum_\ell e_\ell - \sum_\ell a_\ell = \sum_\ell m_\ell'.$$

So the new data still violate $(\star)$. But we've shrunk the ambient dimension (we removed $C$, which is nonzero), contradicting minimality of $\dim\mathcal A$.

The "quotient and shrink" argument is forced by minimality of $\dim\mathcal A$.

### 10.2 Uniform tangent positivity

The final upshot, via Ax and finite directions:

**Proposition (uniform tangent positivity).** Under the projection exclusion, for every subproduct $Z = \prod_\ell Z_\ell$ through the marked points $y_{i,\ell}$, we have:
1. There is a neighborhood $\mathcal U$ of $\Phi$ in $\mathrm{End}(\mathcal A)_{\mathbb R}$ and an index $i_0$ such that $P_Z(g) > 0$ for all $i \ge i_0$, all $g \in \mathcal U$, all $Z$.
2. For every sufficiently large $i$ and every such $Z$, $P_{Z^r}(\Delta\Phi) > 0$, where $\Delta\Phi$ sends $(x_1, \dots, x_r)$ to $(\Phi(x_2) - \Phi(x_1), \dots, \Phi(x_r) - \Phi(x_{r-1}))$.

Where $P_Z(g) = \int_Z (g^* \omega)^{\dim Z}$ is a "tangent integral."

The first positivity is what feeds the all-high case; the second is what feeds the two-measure comparison in the low case.

The proof uses Ax:
- If the first assertion fails, we get a zero value $P_Z(g) = 0$ for some sequence $g_i \to \Phi$ and subproducts $Z_i$.
- Ax's theorem, applied to the analytic fiber of the failed-injectivity map, gives a positive-dimensional fiber in a coset $y_i + C$.
- Then the finite directions lemma replaces $C_i$ by a fixed $C = \prod H_\ell$.
- Passing to a subsequence, all $H_\ell$ are fixed, contradicting the projection exclusion.

The second assertion uses the same argument for the r-fold product and the difference map $\Delta\Phi$, whose kernel is the diagonal.

---

## 11. Step 2: the all-high case (Vojta–Rémond)

Now assume **all height scales diverge**, i.e., no low factor. This is the case where every $H_{i,\ell} \to \infty$.

### 11.1 Setup

Assume we have a minimal counterexample. Take a rational box $\mathcal B$ around $\Phi$ where positivity holds (from the tangent positivity proposition). Then pick constants $\theta, \omega_1$ from Lemma (intersection lower bound) which gives for every subproduct $Z$:

$$P_Z(g) \ge \theta^{-1} D_Z^{-\omega_1}.$$

This is the "lower bound for tangent integrals" from real-algebraic geometry (quantifier elimination + bounded denominators of Newton coefficients). The constants $\theta, \omega_1$ are fixed depending only on $\mathcal B$, not on $Z$.

### 11.2 Approximate $\Phi$ and construct the bundles

Choose a rational projection $f/e$ in the box with $\|f/e - \Phi\| \le \eta$, where $f$ is integral, $e$ is even. So the approximation error is $\eta$.

Choose a dyadic family of integers $v_{i,\ell}$ — products $u \cdot 2^k$ with $u \in [N, 2N]$ — so that

$$1 - \eta \le \frac{v_{i,\ell} \sqrt{H_{i,\ell}}}{R_i} \le 1 + \eta$$

for some common $R_i$. Then the scaled heights $v_{i,\ell}\sqrt{H_{i,\ell}}$ are all comparable.

The crucial observation: after fixing $f/e, U, N$, the "coefficient height" of the multiplication map $[v_{i,\ell}]$ is $O(v_{i,\ell}^2)$ — a bound depending only on $v_{i,\ell}$ and universal constants. This means in Vojta–Rémond, the coefficient-dependent part is only a **threshold**, not the multiplicative constant.

### 11.3 The comparison

The Vojta–Rémond inequality (with weights $a_{i,\ell} = e^2 v_{i,\ell}^2$) gives:

$$\sum_\ell a_{i,\ell} h(y_{i,\ell}) \le c_1 \left( h(\Xi_i(y_i)) - h(\mathcal Z_i(y_i)) \right) + (\text{threshold}).$$

**Left-hand side.** The high nonvanishing hypothesis (applied to the identity homomorphism) gives $h(y_{i,\ell}) \ge \kappa_\ell H_{i,\ell}$ for constants $\kappa_\ell > 0$ independent of the approximation. Combined with the multiplier estimate $v_{i,\ell}\sqrt{H_{i,\ell}} \ge (1 - \eta) R_i$:

$$\sum_\ell a_{i,\ell} h(y_{i,\ell}) = \sum_\ell e^2 v_{i,\ell}^2 h(y_{i,\ell}) \ge e^2 (1 - \eta)^2 R_i^2 \sum_\ell \kappa_\ell.$$

Dividing by $e^2 R_i^2$:

$$\frac{\sum_\ell a_{i,\ell} h(y_{i,\ell})}{e^2 R_i^2} \ge (1 - \eta)^2 \sum_\ell \kappa_\ell. \qquad (L)$$

On the right-hand side, the difference of heights can be bounded by the smallness of image plus the error term: after normalization,

$$\frac{1}{e^2 R_i^2} \left( h(\Xi_i(y_i)) - h(\mathcal Z_i(y_i)) \right) \le C\eta^2 + o(1).$$

Combining (L) with the right-hand bound, using $(1 - \eta)^2 \ge 1/4$ for $\eta \le 1/2$:

$$\frac{1}{4} \sum_\ell \kappa_\ell \le c_1 C^2 \eta^2 + o(1).$$

Since this holds for *every* $0 < \eta < 1/2$, and $c_1, C, \kappa_\ell$ are independent of $\eta$, we can choose $\eta$ small enough to get a contradiction.

### 11.4 The key constant-independence

The truly crucial part: **$c_1$ depends only on the geometry and $t_1, \theta, \omega_1$. It does not depend on $\eta$, $f/e$, or $U$.** Everything that depends on the approximation (via $J$, $t_2$, coefficient heights) enters only the *thresholds* in Vojta–Rémond. So we can freely choose the accuracy after fixing $c_1$. This is the elegance of the argument.

---

## 12. Step 3: the low-factor case

Now assume at least one scale is bounded. This is the case with a **low factor** $A_0$ (with $H_{i,0} = 1$, i.e., bounded heights). At most one such scale exists.

### 12.1 Reduce to all-high when the low factor is zero-dimensional

First subcase: the low variety $Z_{i,0}$ is a point (dimension 0).

**Lemma.** In a minimal admissible product, the low dimension $d_0 > 0$.

**Proof idea.** If $d_0 = 0$, the low variety is a point $p$. Then:
1. The projector $V$ must project onto $\mathrm{Lie}(A_0)$ (otherwise we contradict eventual avoidance).
2. Form $h$ copies of the high factor and take a product with the low projection to get a "star" projection. Its codimension in the high-copy part is $\mathrm{codim} \cdot h - a$, which for large $h$ exceeds $h \cdot$ (sum of high-$m_\ell$'s).
3. This turns the low-factor data into all-high data by using product-with-low construction. Contradicts the all-high case.

So the (real) low case is: $d_0 > 0$.

### 12.2 Pick the auxiliary products

Fix an ambient counterexample. Among all **admissible subproducts** $Z_i = \prod_\ell Z_{i,\ell}$ (factors with bounded geometric degree and bounded field degree, containing marked points, with a dense set of points of height at most $C H_{i,\ell}$), choose one of minimal total dimension $\sum_\ell d_\ell$.

Minimality gives:
- **Any bounded-degree hypersurface through a marked point contains the whole factor.** Otherwise, intersecting with it gives a smaller admissible subproduct, contradiction.
- **Equations of bounded degree for the factor have bounded coefficient heights $O(H_{i,\ell})$** (via Hilbert scheme / Bost–Gillet–Soulé arithmetic degree).
- **Marked points are smooth on their factors** with conormal spanned by bounded-degree equations of bounded coefficient height.
- **The low factor is a fixed variety** (via Northcott: only finitely many such bounded-height equations).

This is crucial. From now on, $Z_0$ is fixed, positive-dimensional. The high factors still vary, but with bounded degrees and bounded field degrees.

### 12.3 The two-measure comparison

We want to show arithmetic positivity: for the constructed $F_i$, $M_i$, $b_{i,\ell}$, $Z_i$:

$$\liminf_{i \to \infty} \frac{(\bar E_i^{d+1} \cdot Z_i)}{M_i^2 B_i} > 0. \tag{$\ast\ast$}$$

Here $\bar E_i = F_i^* \bar L$, $B_i = \prod_\ell b_{i,\ell}^{2 d_\ell}$, and the $b$'s are chosen so that $F_i \mathrm{diag}(b_{i,\ell}^{-1}) \to \Phi$ (i.e., $F_i$ is the "scaled" version of $\Phi$).

**Two geometric measures.** On the fixed low product $Z_0^r$, we can define two probability measures:

$$\mu_{i,j}(f) = \frac{\int_{Z_i^r} p_0^* f \cdot \beta_j(\varphi_i)^u}{\int_{Z_i^r} \beta_j(\varphi_i)^u}, \qquad j = 1, 2$$

where:
- $\beta_1(g)$ is the form gotten by pushing forward via all $\Phi(x_j)$ separately (i.e., "separate outputs").
- $\beta_2(g)$ is the form gotten by pushing forward via consecutive differences $\Phi(x_{j+1}) - \Phi(x_j)$ (i.e., "consecutive differences").
- $u = rd$, $d = $ total dimension, $r = \dim\mathcal A + 1$.
- $p_0 : A^r \to A_0^r$ projects to the low coordinates.

**Key geometric fact.** $\beta_1(\Phi)$ has **positive density** at the diagonal $(z, \dots, z)$ of $Z_0^r$, because $\Phi$ is injective on $T_z Z_0 \oplus T_q Q$ for some points. But $\beta_2(\Phi)$ has **zero density** at the same diagonal, because:

- Take $v \in T_z Z_0$ nonzero (possible since $d_0 > 0$).
- The tangent vector $(v, v, \dots, v)$ in $T_{Z_0^r}$ (plus zero in high factors) is killed by every consecutive difference.
- Hence the top power of the semipositive form $\beta_2(\Phi)$ vanishes on the full product tangent space.

So the two measures differ. There is a nonnegative smooth test function $f_*$ supported near the diagonal such that

$$\lim_i (\mu_{i,1}(f_*) - \mu_{i,2}(f_*)) > 0. \tag{gap}$$

**Now suppose the arithmetic positivity $(\ast\ast)$ fails** (quotient goes to zero on a subsequence). We want to derive a contradiction.

### 12.4 From failure of $(\ast\ast)$ to a contradiction

The strategy uses **small sections**. Because of the fundamental inequalities and the essential minimum, if the arithmetic intersection quotient is *small*, then the varieties $Z_i$ have **many small points** for the bundle $\bar E_i$ (and also for $\bar E_{i,1}, \bar E_{i,2}$).

Then use **arithmetic bigness** and **Siu's inequality** to show: the arithmetic volume of $\bar E_i - c M_i^2 \bar 1$ (for a suitable fixed $c > 0$) is positive. Consequently, on a resolution $\tilde Z_i$ of $Z_i$ (defined over $k_i$, whose degree is uniformly bounded), there is a nonzero section $s_i$ of a positive power of $\bar E_i$ whose adelic sup norms satisfy

$$\|s_i\|_{v, \sup} \le \begin{cases} \exp(-c n_i M_i^2) & v \mid \infty, \\ 1 & v \nmid \infty. \end{cases}$$

### 12.5 Uniform local index

Now we need: **the section vanishes to a positive order at the marked point**.

**Proposition (positive index).** Under the conditions (marked points with bounded equation degree and $O(H_{i,\ell})$ height, scaled points small for $\bar E_i$, small section, no bounded chain on degrees of fields), there exists $\eta > 0$ such that

$$\mathrm{ind}_{n_i, b_i}(s_i; y_i) \ge \eta$$

for all sufficiently large $i$. Here the index is defined with weights $n_i b_{i,\ell}^2$ and uses the local Taylor expansion.

The proof is a Cauchy-type estimate. Sketch:
1. Choose an **algebraic frame** at the marked point, using the conormal equations up to a fixed degree.
2. From the small-section hypothesis and Cauchy, the coefficient of the smallest-weight Taylor term satisfies

$$\log \|a\|_v \le \log \|s_i\|_{v, \sup} + n_i \kappa_v + \sum_\ell |\alpha_\ell| \log \rho_{i, \ell, v}^{-1}.$$

3. Sum over places, use the product formula: $\sum_v \nu_v \log \|a\|_v = -n_i h_{\bar E_i}(y_i) = o(M_i^2)$ (using the smallness of the marked points).
4. The two metric-variation constants: at infinity, $\kappa_v \le C \xi^2 M_i^2$; at finite primes, $\kappa_v \le C_v$.
5. Conclude: $n_i((c - C\xi^2)M_i^2 - C_0 - o(M_i^2)) \le C_\gamma \sum_\ell |\alpha_\ell| H_{i,\ell}$.
6. Use $b_{i,\ell}^2 H_{i,\ell} \asymp M_i^2$ to turn this into a lower bound on the weighted index.

The technical meat is in the *uniformity of the metric variation over extension fields*. That's the part where the paper is careful: the constants $C, C_0, C_\gamma$ are uniform because they come from finite data (fixed models, fixed polarizations, fixed basis of homomorphism spaces) plus the bounded-degree property of the fields $k_i$. The fact that the *marked points* can have unbounded degree is fine, because the metric variation on the polydisk over the point's field is controlled by the *field-independent* structure at finite primes.

### 12.6 The product cut

Now we have a section $s_i$ with a **positive weighted index $\eta > 0$** at $y_i$.

Push $s_i$ down:
1. Choose a **fixed** integer $t_0$ such that $Q_i = t_0 \sum_\ell b_{i,\ell}^2 L_\ell - F_i^* L$ is ample (uniform in $i$, because $F_i$ is bounded after rescaling).
2. Multiply by a section of $Q_i^{n_i}$ (small power) to get a section $s_i'$ of $L_\ell^{t_0 n_i b_{i,\ell}^2}$ at $Z_i$, nonzero at $y_i$.
3. Take the **field norm** along $\Pi_i : Z_i \to P_i = \prod \mathbb P^{d_\ell}$ (finite projection of bounded degree $e_i = \prod_\ell \deg Z_{i,\ell}$).
4. Get a nonzero multihomogeneous polynomial $G_i$ on $P_i$, defined over $k_i$, with multidegrees $D_{i,\ell} = e_i t_0 n_i b_{i,\ell}^2$, and *positive weighted index* at $z_i = \Pi_i(y_i)$.

Because $e_i \le e_{\max}$ (bounded degrees), and $t_0$ is fixed, the multidegrees satisfy

$$D_{i,\ell} / D_{i,\ell+1} \to \infty$$

(weight separation).

### 12.7 Faltings' product theorem

Apply the "separated index loci" argument:
1. Choose $N+2$ levels $\sigma_j = \sigma_0 + j \epsilon$ all below $\eta$. (Careful: $N = \sum d_\ell$.)
2. The point $z_i$ lies in every corresponding index locus $\mathcal I_{\sigma_j}$.
3. Pick components $J_j$ with $J_{N+1} \subseteq J_N \subseteq \dots \subseteq J_0$, all proper.
4. By pigeonhole, two consecutive components coincide: a **common component** $J_i$ of $\mathcal I_\sigma$ and $\mathcal I_{\sigma + \epsilon}$.
5. Faltings' product theorem: $J_i = \prod_\ell J_{i,\ell}$, with $q_i \prod_\ell \deg J_{i,\ell} \le C(N, \epsilon)$.
6. Hence each factor $J_{i,\ell}$ has bounded degree and is defined over a bounded-degree extension $k_i'$ of $k_i$.

### 12.8 The final contradiction

Since $J_i$ is proper, at least one factor $J_{i,\ell}$ is proper. Pick such an $\ell$. Since $J_{i,\ell}$ is a proper subvariety of bounded degree, there is a nonzero homogeneous equation $R_i$ of bounded degree for it, defined over $k_i'$.

Pull back $R_i$ via $\pi_{i,\ell}$. Get a bounded-degree hypersurface $H_i \subseteq A_\ell$ containing $y_{i,\ell}$, not containing $Z_{i,\ell}$.

Conclude:
- $H_i$ is a bounded-degree hypersurface.
- $H_i$ is defined over $k_i'$ with $[k_i' : \mathbb Q]$ bounded.
- $H_i$ contains the marked point $y_{i,\ell}$ but does not contain the minimal factor $Z_{i,\ell}$.

This contradicts the minimality property from §12.2: "any bounded-degree equation over a bounded-degree extension containing a marked point must contain the whole factor."

Contradiction. Hence the low-factor case is impossible.

### 12.9 Combined: sequence theorem proof

Putting it together:
- All-high case: handled by §11.
- Low-factor, `d_0 = 0`: reduced to all-high, done.
- Low-factor, `d_0 > 0`: impossible by §12.

So the sequence theorem holds. Which proves the non-density theorem via the reductio in §9.

---

## 13. From non-density to finiteness

We now have Theorem A: for $X \subseteq A$ irreducible, contained in no proper torsion coset,

$$X \cap A^{[\dim X + 1]} \text{ is not Zariski dense in } X.$$

The finiteness of maximal atypical subvarieties follows from Barroero–Dill's reduction:

1. **Non-density for all $(B, Z)$**: for every abelian subvariety $B \subseteq A$ and every irreducible $Z \subseteq B$ (dimension bounded, contained in no proper special subvariety of $B$), $Z \cap B^{[\max(\dim Z + 1, \dim B - e)]}$ is not dense. This is just Theorem A applied to $B$, $Z$.
2. **Barroero–Dill theorem**: this non-density hypothesis implies that each irreducible subvariety of $A$ (dimension bounded) has finitely many optimal subvarieties of bounded defect.
3. **Maximal atypical $\Rightarrow$ optimal**: as sketched in §7.4. So the set of maximal atypical subvarieties is finite.

Note that we get **finiteness** of the list of maximal atypical subvarieties, but no explicit bound on its size.

---

## 14. The finite-rank translation corollary

This is an extension of the theorem to a different class of sets, proved from Theorem A via **Pink's reduction**.

**Corollary.** Let $A$ be an abelian variety over $\overline{\mathbb Q}$, $\Gamma \subseteq A(\overline{\mathbb Q})$ a subgroup with $\dim_{\mathbb Q}(\Gamma \otimes_{\mathbb Z} \mathbb Q) < \infty$ (finite rank over $\mathbb Q$), and $X \subseteq A$ irreducible, in no translate of a proper algebraic subgroup. Then

$$X(\overline{\mathbb Q}) \cap (A^{[d+1]}(\overline{\mathbb Q}) + \Gamma)$$

is not Zariski dense in $X$, where $d = \dim X$. If $d = 1$, the intersection is finite.

**Idea.** Take generators $a_1, \dots, a_r$ of $\Gamma$ modulo torsion. Build the ambient product $B = A \times C$ where $C \subseteq A^r$ is the Zariski closure of the subgroup generated by $(a_1, \dots, a_r)$. Set $Y = X \times \{a\}$ inside $B$.

Then $Y$ is an irreducible subvariety of an abelian variety $B$, and (crucially) $Y$ is contained in no proper algebraic subgroup of $B$ (the "no proper translate" hypothesis on $X$ implies this for $Y$).

Apply Theorem A to $Y$: $Y \cap B^{[d+1]}$ is not dense.

Then check the inclusion

$$(X(\overline{\mathbb Q}) \cap (A^{[d+1]} + \Gamma)) \times \{a\} \subseteq Y \cap B^{[d+1]}.$$

The inclusion uses the relation $n\gamma = \sum_i n_i a_i$ (for suitable $n, n_i$ from the finite-rank hypothesis): if $x = g + \gamma$ with $g \in G$ (a codimension-$(d+1)$ subgroup of $A$) and $\gamma \in \Gamma$, then $(x, a) \in B^{[d+1]}$.

Since the right-hand side is not dense in $Y = X \times \{a\}$, the left is not dense in $X \times \{a\}$, which is isomorphic to $X$. So $X \cap (A^{[d+1]} + \Gamma)$ is not dense in $X$.

For $d = 1$: $X$ is a curve, so $X \cap (A^{[2]} + \Gamma)$ is contained in a *proper closed subset* of the curve — but a curve is 1-dimensional, so the closed subset is finite (or empty). So the intersection is finite. This is the "Pink conjecture for curves with finite-rank translations" statement.

---

## 15. What this paper is NOT doing

To be precise about scope:

1. **No effective bounds.** The theorem says the list of maximal atypical subvarieties is finite. It gives no bound on the number, no bound on their degrees or heights.

2. **No construction.** It does not tell you how to find the maximal atypical subvarieties given $X$ and $A$.

3. **Relying on deep external theorems.** The proof uses:
   - Ax's analytic subgroup theorem (1972).
   - Faltings' product theorem (1991), in Evertse's explicit form.
   - Vojta–Rémond inequality (Dill 2020).
   - Szpiro–Ullmo–Zhang equidistribution (1997).
   - Yuan–Zhang adelic intersection theory (2021).
   - Habegger–Pila's O-minimality arguments (2016).
   - Barroero–Dill (2022).
   - Pink's reduction (2005).

   Each one is a substantial paper. The novelty of this paper is *assembling* them: showing that with the right "auxiliary product" minimality, the deep tools fit together to give a uniform proof across all heights and dimensions.

4. **Not semiabelian.** The theorem and proof are for abelian varieties (compact tori). Pink's original conjecture is for semiabelian varieties (which can have factors of $\mathbb G_m^n$; e.g. multiplicative groups). The paper's tools extend to that setting in principle, but the writing here is abelian.

5. **Not $p$-adic.** The proof is purely complex-analytic and arithmetic (adele-theoretic). It uses $p$-adic methods only in Faltings' product theorem's proof, which we cite rather than rederive.

---

## 16. The high-level "why"

You might ask: *why does this argument work?*

Three philosophical reasons:

1. **Heights stratify everything.** Arithmetic points on abelian varieties have a canonical height decomposition into a finite staircase of scales, separated by going to infinity. This decomposition is *canonical* — independent of the choice of point. So you can treat the "top" and "bottom" scales separately.

2. **Atypicality is a positivity failure.** The atypical intersections are exactly those where the "expected dimension" is beaten. Vojta–Rémond's inequality turns this geometric "surprise" into a quantitative arithmetic inequality. The height of the point is bounded above by the "smallness" of the ambient line bundle under an approximating projector.

3. **The high case gives the low case.** When the low scale is positive-dimensional, minimality of the auxiliary subvarieties (bounded degrees and bounded-degree fields) provides the bridge. The two-measure comparison is a geometric statement that *cannot* be accommodated by a small arithmetic intersection. So arithmetic positivity is forced. Then Faltings' product theorem turns "arithmetic positivity" into "there's a bounded-degree equation" — which contradicts minimality.

The magic is in the *order* of the arguments: minimality of the ambient counterexample gives tangent positivity; tangent positivity gives the projective lower bound; this enters Vojta–Rémond; Vojta–Rémond gives a contradiction via the "$c_1$ independent of accuracy" trick. And separately: equidistribution + arithmetic positivity + Faltings' product theorem give the low case contradiction.

---

## 17. Appendix: the main definitions in one page

For quick reference:

- **Abelian variety**: a compact complex torus $A = \mathbb C^n / \Lambda$ that is a projective algebraic variety, with a fixed polarization. A group and a geometric object.

- **Elliptic curve**: an abelian variety of dimension 1. Alternatively, a nonsingular projective plane cubic $y^2 = x^3 + ax + b$.

- **Torsion point**: $p \in A$ with $n p = 0$ for some $n > 0$.

- **Abelian subvariety**: a closed algebraic subgroup $B \subseteq A$ that is itself an abelian variety.

- **Torsion coset / special subvariety**: a set $T = B + \tau$ with $B \subseteq A$ abelian subvariety and $\tau$ torsion.

- **$\langle X\rangle_{\mathrm{sp}}$**: the smallest torsion coset containing $X$.

- **Atypical**: for $T \subseteq S = \langle X\rangle_{\mathrm{sp}}$ special, an irreducible component $Y$ of $X \cap T$ is atypical if $\dim Y > \dim X + \dim T - \dim S$.

- **Maximal atypical**: an atypical subvariety that is not properly contained in any other atypical subvariety.

- **Height**: for $p/q$ rational, $h(p/q) = \max(|p|, |q|)$. For projective point, $\log$ of the max of coordinates up to scaling. For abelian variety, canonical height $\hat h$ with $\hat h(np) = n^2 \hat h(p)$.

- **$A^{[r]}$**: the union of all torsion cosets of codimension $\ge r$ in $A$.

- **Essential minimum** $e_1(X, \bar D)$: the infimum of $\bar D$-height over dense subsets of $X$.

- **Arithmetic bigness**, **Siu's arithmetic inequality**: tools from adelic line bundle theory (Yuan–Zhang) used to produce small sections from arithmetic intersection lower bounds.

- **Index of a polynomial at a point**: the smallest weighted degree of a nonzero Taylor coefficient.

- **Faltings' product theorem**: a component of the intersection of two "index loci" of separated bidegrees is a product of subvarieties of the individual factors, with bounded total degree (and bounded-degree field of definition) if the polynomial is defined over a number field.

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## 18. A closing word

The abelian Zilber–Pink conjecture is a statement about how often arithmetic special structure can appear in geometric subvarieties. It sits at the intersection of:
- **Arithmetic geometry** (heights, adelic intersection theory).
- **Complex analysis** (Ax's theorem, exponential maps).
- **Diophantine approximation** (Vojta–Rémond).
- **Algebraic geometry** (Faltings' product theorem, intersection theory).
- **Combinatorics of Galois orbits** (bounded-degree descents).

The paper assembles all these into a single argument. It's the kind of result that takes years to prove (in its full generality) and a single page to state. The proof itself is a masterpiece of technical care: every inequality has to be made uniform in the extension degree of the point, in the auxiliary variety's defining equations, in the ambient dimension, and in the height scales.

The result, in the end, is a clean statement: **atypicality is finite.** You can only be special in finitely many ways.