# The abelian Zilber–Pink conjecture



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## 1. The cast of characters

### 1.1 The circle is a group

Take the ordinary circle S^1 subset ℂ. Points on it are complex numbers of modulus one, so you can write them as e^{2pi i theta} for some theta in ℝ. Multiplication of complex numbers on the unit circle is rotation: e^{2pi i theta} · e^{2pi i phi} = e^{2pi 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 = ℝ / 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 ℝ, form ℝ^2, and quotient by the lattice ℤ^2:

T = ℝ^2 / ℤ^2.

Points in T are equivalence classes of pairs (x, y) in 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 ℂ, a compact one-dimensional complex abelian group is always of the form

E = ℂ / Λ,

where Λ subset ℂ is a full-rank lattice, meaning Λ = ℤ lambda_1  ⊕  ℤ lambda_2 with lambda_1, lambda_2 linearly independent over 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 ℂ^2 to the equation

y^2 = x^3 + a x + b,  4a^3 + 27b^2 != 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 ~= ℂ/Λ 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 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 = ℂ^n / Λ

for some lattice Λ = ℤ^{2n} subset 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 = ℂ^n/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 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 + ... + p (n times).

Examples:
- On S^1 = ℝ/ℤ, torsion points are rationals: p = a/b bmod 1 with b > 0. Equivalently, e^{2pi i a/b} are the roots of unity on the unit circle.
- On E = ℂ/Λ, 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[tors] subset A(Qbar), dense in A(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 ⊆ 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 ⊆ A, there is always a unique *smallest* torsion coset containing X. Write it < X>_{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 ⊆ S'' ⊆ S, so dim S'' ≤ 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'' ⊆ S', we get S ⊆ 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. QED

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

Let A be an abelian variety of dimension n, let X ⊆ A be irreducible of dimension d_X, and put S = < X>_{sp}, of dimension d_S.

Let T ⊆ 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.  (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.  (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 ⊆ 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 Qbar, and X ⊆ 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(Qbar) cap A[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,  equivalently  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 ⊆ A_{Kbar} an irreducible subvariety. Let S = < X>_{sp}. Then the number of **maximal atypical subvarieties** of X in S is finite.

Equivalently: there are finitely many proper atypical subvarieties Y_1, ..., Y_m ⊊ 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 ⊆ 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 {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 : ... : a_N] in ℙ^N(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 hhat_L — the **canonical height** or **Néron–Tate height** — with the following crucial property:

hhat_L(np) = n^2 hhat_L(p)   (for any integer n).

It's a *quadratic form* on A(Qbar). Its associated bilinear form

< p, q > = 1/2(hhat_L(p + q) - hhat_L(p) - hhat_L(q))

is positive definite on the group A(Qbar) / A[tors].

**Consequences:**

- Torsion points have height exactly 0. (Because if np = 0, then n^2 hhat(p) = hhat(np) = hhat(0) = 0.)
- For points with hhat_L(p) > 0, hhat_L(np) grows like n^2.
- The "height space" Hcal(A) = A(Qbar)  ⊗ _{ℤ} 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 hhat = 0; a generic rational point of E has hhat growing as its coordinates' numerators and denominators grow.

For X ⊆ A, one of the key players is the *minimum height* of a point of X(Qbar), or the minimum of hhat on a dense subset. The paper uses essential minima e_1(X, Dbar) — 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 = ℝ/ℤ. Take a straight line through the origin with slope alpha: the set { (alpha t, t) : t in ℝ } in ℝ^2.

- If alpha is **rational**, its image mod 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 ⊆ G. Then:

dim < Z >_{geo} - dim Z ≤ dim U - dim J,

where < Z>_{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< Z>_{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 = ℂ and Z is a single point, then < Z>_{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 ⊆ A_ell of complex abelian varieties. Then for each ell there is a fixed finite set Hcal_ell of abelian subvarieties of A_ell with the following property: whenever Z_ell ⊆ Y_ell, y_ell in Z_ell, C_ell ⊆ A_ell abelian subvarieties, and g : ⊕_ell 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 rank(g|_{⊕ Lie(C_ell)}) + ε,

then there exist H_ell in Hcal_ell with the same inequality for Z_ell ⊆ 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, ..., X_s be projective varieties over Qbar, with very ample bundles L_ell and positive integer weights a_ell. Let Mcal be a nef bundle on the product, and Pcal a suitable power of the tensor product of the L_ell. Suppose Pcal  ⊗  Mcal^{-1} is generated by J global sections admitting an injection into another power of L_ell's with bounded coefficient height ≤ sum_ell a_ell delta_ell. Then for every point x = (x_ell) in X_1(Qbar) times ... times X_s(Qbar) that lies in a certain open set and satisfies the *intersection lower bound*

(Mcal^{dim Z} · Z) ge theta^{-1} prod_ell (deg Z_ell)^{-omega} a_ell^{dim Z_ell}

for every subproduct Z through x, we have:

∑_ℓ a_ℓ h(x_ℓ) ≤ c_1 · big( h(Xi(x)) - h(Zcal(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 Dbar on a projective variety X of dimension n with D geometrically big,

e_1(X, Dbar)/(n + 1) ≤ (Dbar^{n+1} · X)/((n+1)(D^n · X)) ≤ e_1(X, Dbar),

where e_1 is the *essential minimum* of the point height. The paper uses **Siu's arithmetic inequality**: for nef adelic bundles Dbar, Nbar,

volhat(Dbar - Nbar) ge (Dbar^{n+1} · X) - (n+1)(Dbar^n Nbar · 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 = ℙ^{d_1} times ... times ℙ^{d_m}. Let G be a nonzero multihomogeneous polynomial with multidegrees D = (D_1, ..., 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** Ical_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 < ε ≤ 1, and suppose

D_ℓ/D_{ℓ+1} ge (mN/ε)^N

where N = sum d_ell. Then any irreducible component J that is *common to Ical_sigma and Ical_{sigma + ε}* is a product of subvarieties of the individual ℙ^{d_ell}.

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

q · prod_ell deg J_ell ≤ C(N, ε),

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 ⊆ A be irreducible and contained in no proper torsion coset. Then X cap A^{[r]} is not Zariski dense in X, where

A^{[r]} = ∪_{T  special,  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 ⊆ A irreducible, delta(Z) = dim < Z>_{sp} - dim Z. Z ⊆ X is **optimal in X** if every proper enlargement U ⊋ Z inside X has delta(U) > delta(Z).

**Key claim:** Maximal atypical ⇒ optimal.

**Sketch.** If Y is maximal atypical, witnessed by special T, then delta(Y) ≤ dim T - dim Y < dim S - dim X = delta(X). So Y can be strictly contained in a proper U ⊆ X only if delta(U) > delta(Y) — otherwise the component of X cap < U>_{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 ⊆ 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** ⇐ **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 diag(H_{iell}^{-1/2}) 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 Acal. 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 Ebar_i^{d+1} · 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.

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## 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(Qbar) 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 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 hhat on A = ℂ^n/Λ is a positive-definite quadratic form. Choose coordinates x_i in Hcal(A) ~= 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 = pr_{≤ell}(X) ⊆ A_ell. Then the marked points satisfy

dim_{y_{i,ell}}(Y_ell cap (y_{i,ell} + W_ell)) ≤ 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 ⊆ Lie(W) be the image of an idempotent (a projecting operator). Put Phi = orthogonal projector with kernel V. Then the *forbidden condition* is:

codim_{Lie(W)} V > sum_ell m_ell  and  |Phi(q_i)| to 0,  (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.

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## 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 dimAcal. 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  (ell in I)

and

sum_ell a_ell ge rank(Phi|_{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)) < rank(Phi|_{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(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:

codim_{Lie(W')} V' = codim_{Lie(W)} V + sum_ell e_ell - rank(Phi|_{Lie(C)}).

Combining with the "excluded" hypothesis sum_ell a_ell ge rank(Phi|_{Lie(C)}), we get

codim_{Lie(W')} V' = codim_{Lie(W)} V + sum_ell e_ell - rank(Phi|_{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 dimAcal.

The "quotient and shrink" argument is forced by minimality of dimAcal.

### 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 Ucal of Phi in End(Acal)_{ℝ} and an index i_0 such that P_Z(g) > 0 for all i ge i_0, all g in Ucal, all Z.
2. For every sufficiently large i and every such Z, P_{Z^r}(DeltaPhi) > 0, where DeltaPhi sends (x_1, ..., x_r) to (Phi(x_2) - Phi(x_1), ..., 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 DeltaPhi, 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 ∞.

### 11.1 Setup

Assume we have a minimal counterexample. Take a rational box Bcal 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 Bcal, not on Z.

### 11.2 Approximate Phi and construct the bundles

Choose a rational projection f/e in the box with |f/e - Phi| ≤ 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 · 2^k with u in [N, 2N] — so that

1 - eta ≤ (v_{i,ℓ} sqrt(H_{i,ℓ}))/R_i ≤ 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}) ≤ c_1 ( h(Xi_i(y_i)) - h(Zcal_i(y_i)) ) + (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:

(∑_ℓ a_{i,ℓ} h(y_{i,ℓ}))/(e^2 R_i^2) ge (1 - eta)^2 sum_ell kappa_ell.  (L)

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

1/(e^2 R_i^2) ( h(Xi_i(y_i)) - h(Zcal_i(y_i)) ) ≤ Ceta^2 + o(1).

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

1/4 sum_ell kappa_ell ≤ 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 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 codim · h - a, which for large h exceeds h · (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 ∞} (Ebar_i^{d+1} · Z_i)/(M_i^2 B_i) > 0. (**)

Here Ebar_i = F_i^* Lbar, B_i = prod_ell b_{i,ell}^{2 d_ell}, and the b's are chosen so that F_i 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) = (∫_{Z_i^r} p_0^* f · β_j(φ_i)^u)/(∫_{Z_i^r} β_j(φ_i)^u),  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 = dimAcal + 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, ..., z) of Z_0^r, because Phi is injective on T_z Z_0  ⊕  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, ..., 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. (gap)

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

### 12.4 From failure of (astast) 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 Ebar_i (and also for Ebar_{i,1}, Ebar_{i,2}).

Then use **arithmetic bigness** and **Siu's inequality** to show: the arithmetic volume of Ebar_i - c M_i^2 1bar (for a suitable fixed c > 0) is positive. Consequently, on a resolution Ztilde_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 Ebar_i whose adelic sup norms satisfy

‖s_i‖_{v, sup} ≤ exp(-c n_i M_i^2) for archimedean v, and ≤ 1 for finite v.

### 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 Ebar_i, small section, no bounded chain on degrees of fields), there exists eta > 0 such that

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 ≤ 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_{Ebar_i}(y_i) = o(M_i^2) (using the smallness of the marked points).
4. The two metric-variation constants: at infinity, kappa_v ≤ C xi^2 M_i^2; at finite primes, kappa_v ≤ C_v.
5. Conclude: n_i((c - Cxi^2)M_i^2 - C_0 - o(M_i^2)) ≤ 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 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 ≤ e_{max} (bounded degrees), and t_0 is fixed, the multidegrees satisfy

D_{i,ell} / D_{i,ell+1} to ∞

(weight separation).

### 12.7 Faltings' product theorem

Apply the "separated index loci" argument:
1. Choose N+2 levels sigma_j = sigma_0 + j ε all below eta. (Careful: N = sum d_ell.)
2. The point z_i lies in every corresponding index locus Ical_{sigma_j}.
3. Pick components J_j with J_{N+1} ⊆ J_N ⊆ ... ⊆ J_0, all proper.
4. By pigeonhole, two consecutive components coincide: a **common component** J_i of Ical_sigma and Ical_{sigma + ε}.
5. Faltings' product theorem: J_i = prod_ell J_{i,ell}, with q_i prod_ell deg J_{i,ell} ≤ C(N, ε).
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 ⊆ 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' : 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 ⊆ A irreducible, contained in no proper torsion coset,

X cap A^{[dim X + 1]}  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 ⊆ A and every irreducible Z ⊆ 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 ⇒ 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 Qbar, Gamma ⊆ A(Qbar) a subgroup with dim_{ℚ}(Gamma  ⊗ _{ℤ} ℚ) < ∞ (finite rank over ℚ), and X ⊆ A irreducible, in no translate of a proper algebraic subgroup. Then

X(Qbar) cap (A^{[d+1]}(Qbar) + Gamma)

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

**Idea.** Take generators a_1, ..., a_r of Gamma modulo torsion. Build the ambient product B = A times C where C ⊆ A^r is the Zariski closure of the subgroup generated by (a_1, ..., 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(Qbar) cap (A^{[d+1]} + Gamma)) times {a} ⊆ Y cap B^{[d+1]}.

The inclusion uses the relation ngamma = 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 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 = ℂ^n / Λ 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 ⊆ A that is itself an abelian variety.

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

- **< X>_{sp}**: the smallest torsion coset containing X.

- **Atypical**: for T ⊆ S = < X>_{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 hhat with hhat(np) = n^2 hhat(p).

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

- **Essential minimum** e_1(X, Dbar): the infimum of Dbar-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.

---

## 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.