1 Lecture 3a: an array of uniformity theorems

recall:

1.1 Theorem (Caporaso-Harris-Mazur)

1. Lang⇒unif. Mordell 2. dependence on K

Assume that Lang’s “weak” conjecture holds true. Let K be a number field. Then there is a number B(K, g), such that the number of K-rational points on any curve of genus g > 1 over K is bounded by B(K, g).

1.2 Theorem (Pacelli)

Assume that Lang’s “weak” conjecture holds true. Then there is a number B(d, g), such that, for any number field K of degree d, the number of K-rational points on any curve of genus g > 1 over K is bounded by B(d, g).

1. geometric case

2. how about “strong”

1.3 Theorem (Caporaso-Harris-Mazur)

higher dim setup

Assume that Lang’s “strong” conjecture holds true. Then there is a number N(g), such that, for any number field extension K, the number of curves of genus g > 1 defined over K which have more than N(g) rational points over L is finite.

1.4 Theorem (Abramovich - Voloch)

There is a number B(d, X → B), such that, for any number field K of degree d, and for any b ∈ B(K), the number X_{b}(K)^{nex} is bounded by B(d, X → B).

1. Lang⇒Unif. Lang 2. integral? stint

1.5 Theorem (Pacelli)

Assume that the Lang-Vojta conjecture holds true. Fix a finite set of primes S in ℤ, and for any number field K denote by S_{K} the set of primes of O_{K} over S. Then there is a number B(d, S), such that, for any number field K of degree d, the number of stably S_{K}-integral points on any elliptic curve K is bounded by

B(d, S). S-unit, surfaces

1.6

“Vojta” implies “uniform Vojta”???

2 Lecture 3b: Structure of the proofs

First idea: all curves of a fixed genus fit in a family.

diag Hilbert

Could start if the family is of general type

{not \atop prod}

Enough if the family dominates a variety of general type

still pencil

Second idea: work with fibered products. ^{diag}

Need:

1. arithmetic/geometric argument for showing that using fibered powers is enough.

2. geometric construction of a varity of general type.

Given a family of varieties X → B and a set of points P ⊂ X(K), we say that P is ncorrelated in X → B if P_{B}^{n} is not Zariski dense in X_{B}^{n}.

1-correlated open set noetherian

2.1 Lemma (CHM)

Suppose X → B is a family of curves and P ⊂ X is n-correlated in X → B. Then there is an open Zariski - dense set U ⊂ B such that the number of points of P_{b} on all fibers of X → B over b ∈ U is uniformly bounded.

1.give proof

2. higher dim. problem

2.2 Theorem

1. who

2. moduli

3. Lemma+Th⇒unif

Let X → B be a family of varieties of general type over an irredicible base variety B. Then for large enough n, there exixts a positive dimensional variety of general type W_{n} and a dominant rational map X_{B}^{n}/G −−→ W.

2.3 Theorem (CHM)

Let g > 1 be an integer, and fix a subvariety X_{0} ⊂ M_{g}. Let X_{n} ⊂ M_{g,n} be the inverse image. Then for large enough n, the variety X_{n} is of general type.

A family of varieties X → B is said to be of maximal variation, if through a general point b ∈ B there is no curve b ∈ C ⊂ B over which the family is birationally isotrivial. ^{explain}

2.4 Theorem

Lem+Theorem⇒ Strong unif

Let X → B be a family of varieties of general type over an irredicible base variety B. Assume X → B is of maximal variation. Let G ⊂ Aut(X → B). Then for large enough n, the variety X_{B}^{n}/G is a variety of general type.

2.5 Exercise

Let X, Y be varieties over K with finite automorphisms, with an isomorphism XK̄ → YK̄. Then for any n there is an isomorphism X^{n}/Aut_{K}(X) → Y^{n}/Aut_{K}(Y), where the automorphism groups act diagonally on the products. ^{descent}

2.6 Exercise

Consider the projection

p_{n,k} : X^{n}/Aut_{K}(X) → X^{k}/Aut_{K}(X)

onto the first k factors. Let P = (P_{1}, …, P_{k}) ∈ X^{k}(K) be a K-rational point, and let Q ∈ X^{k}/Aut_{K}(X)(K) be its image. Assume P is not fixed by any nontrivial element of Aut_{K}(X). Then there is an induced isomorphism

X^{n−k} → p_{n,k}^{−1}{P }.

Let τ : X → B be a projective surjective morphism of reduced schemes of finite type over a field K, with B irreducible. Denote the dimension of the generic fiber of τ : X → B by d. Fix a subset Q ⊂ X(K), and denote by G_{k} the Zariski closure of Q_{B}^{k} in the fibered power X_{B}^{k}. We say that Q is strongly k-correlated with respect to τ : X → B, if every irreducible component of G_{k} which dominates B has relative dimension < kd over B.

strong corr⇐ cor for a related family

Let X = X_{1} ∪ … ∪ X_{c} be a decomposition into irreducible components. Let X′_{i} → X_{i} be the normalization, and let X′_{i} → B′_{i} → B be the Stein factorization. There is a dense open set U_{i} ⊂ X_{i} over which X′_{i} → X is an isomorphism. Therefore the set Q′_{i} = U_{i} ∩ Q sits naturally in X′_{i}(K).

2.7 Proposition

Assume that for each X_{i} of relative dimension d over B, the set Q′_{i} is k′_{i}-correlated with respect to X′_{i} → B′_{i}. Then for large k, the set Q is strongly k-correlated for X → B.

Step 1. Let B′ ⊂ B be a nonempty open set, X′ ⊂ τ^{−1}B′ a dense open set, and Q′ = Q ∩ X′. Assume Q′ is strongly kcorrelated. Then Q is strongly k-correlated as well.

Step 2. Let X_{1}, …, X_{c′} be the irreducible components of X of relative dimension d over B. Let X′ = X_{1} ∪ … ∪ X_{c′} and let Q′ = Q ∩ X′. If Q′ is strongly k-correlated then so is Q.

Step 3. Let Q_{i} = X_{i} ∩ Q. Assume Q_{i} is strongly l-correlated. Then for k = l · c, Q is strongly k-correlated.

Step 4. Denote by c_{i} the degree of B′_{i} → B. Assume Q′_{i} in the proposition is k′_{i}- correlated with respect to X′_{i} → B′_{i} . Then for k = c_{i}k′_{i} we have that Q_{i} is strongly kcorrelated with respect to X_{i} → B.

from strong correlation to uniformity

Given X → B a family of varieties of general type, defined over K, consider P ⊂ X(K), the set of “nex” rational points. Let E_{n} ⊂ P_{B}^{n} be the set of n-tuples of distinct points. Consider the tower of maps ⋯ → E_{n+1} → E_{n} → E_{n−1} → ⋯ → E_{1}.

Let E_{n}^{(m)} be the image of E_{m} in E_{n} and set

m > n

prolongable

F_{n}^{(m)} = \overline{E_{n}^{(m)}}. We have a descending chain of closed subsets

F_{n}^{(m)} ⊃ F_{n}^{(m+1)} ⊃ ⋯

which must stabilize to a closed set F_{n}, i.e., there exists m_{n} such that

F_{n}^{(m_{n})} = F_{n}^{(m_{n}+1)} = ⋯ = F_{n}.

Accordingly, we have a tower of maps

{want F_{n} = ∅ \atop F_{0} = ..}

⋯ F_{n+1} →^{τ_{n+1}} F_{n} → F_{n−1} →^{τ_{n}} ⋯ → F_{1}.

Note: τ_{n+1} : F_{n+1} → F_{n} surjective for all n. Will show: any fiber of F_{n+1} → F_{n} cannot have dimension 0, 1, …, dim X_{η} and hence all the F_{n} are empty.

2.8 Lemma

Suppose no fiber of τ_{n+1} : F_{n+1} → F_{n} has dimension < d for all n ≥ 1, where 0 ≤ d ≤ dim X_{η}. Assume that the “weak” Lang conjecture holds true. Then no fiber of τ_{n+1} : F_{n+1} → F_{n} has dimension ≤ d for all n ≥ 1.

Step 1. The lemma holds for d = 0.

Step 2. It suffices to consider an irreducible component B_{n} ⊂ F_{n} such that X_{n} = τ_{n−1}^{−1}B_{n} has relative dimension d.

Step 3. It suffices to show that X_{n}(K) is strongly correlated.

Step 4. Proposition ⇒ lemma. ^{“nex”⇒ GT} That shows uniformity in X → B, K. Suppose X → B defined over ℚ. Let [K : ℚ] = d Let P = (P_{1}, P_{2}) ∈ X_{B}^{2}(K). Let σ_{1}, …, σ_{d} : K ↪ ℚ̄ distinct. Then {P^{σ_{1}}, …, P^{σ_{d}}} ∈ Sym^{d}(X_{B}^{2}) (ℚ).

1. If P_{1} not defined over a subfield of K, then (P^{σ_{1}}, …, P^{σ_{d}}) not fixed under any nontrivial element of S_{d}.

2. If so, consider the inverse image Y of {P_{1}^{σ_{1}}, …, P_{1}^{σ_{d}}} ∈ Sym^{d}(X) in Sym^{d}(X_{B}^{2}). Then we have an isomorphism ∏ (X^{σ_{i}} ×_{K^{σi}} ℚ̄) → Y ×_{ℚ} ℚ̄ ^{Weil restriction} 3. If furthermore P_{2} is “nex”, then (P_{2}^{σ_{1}}, …, P_{2}^{σ_{d}}) is “nex”.

4. Apply theorem to the family Sym^{d}(X_{B}^{2}) → Sym^{d}(X)