1 Lecture 2: Vojta’s conjectures

1.1 Vojta’s setup

deffers from Soulé

Notation: K = a number field 𝕄_{K} = the places of K

(abs val or emb? norm or norm^{2}?)

k(v) = residue field at v ∈ 𝕄_{K} (𝕄_{K})_{na} = nonarchimedean ^{maxspec} X = variety over K X = model of X over K.

arithmetic variety nice properties 𝕄_{K} itself is

D = a Cartier divisor on X, endowed with Green functions g_{D,v} on X for every archimedean place v ∈ 𝕄_{K}:

g_{(f)} = − log ||f||^{2} factor 2

dd^{c}g + δ_{D} = [ω]

dd^{c} = −(2πi)^{−1}∂ ¯∂

Assume D a cartier divisor on 𝕄_{K}, assume v ∈ (𝕄_{K})_{na}, the multiplicity of D at v is denoted n_{v}(D). We denote by deg_{v}(D) = n_{v} log #k(v).

If, on the other hand v is archimedean, denote deg_{v}(D) = g_{D,v} (this is a real number!)

For S ⊂ 𝕄_{K} a subset (not necessarily finite yet), denote

deg_{S}(D) = ∑_{v∈S} deg_{v}(D)

and deg D = deg_{𝕄_{K}} D.

Product formula implies: deg D is invariant under linear equivalences (where the Green function of (f) at an archimedean v is − log ||f||_{v}^{2?}).

inv of metr. lb

Let L be a metrized line bundle on X, and let P be an algebraic point on X, defined over E. there is an associated morphism σ : 𝕄_{E} → X. Define

h_{L,K(P)} = \frac{1}{[E : K]} deg σ^{∗}L.

If P ∉ Supp D, then h_{O(D),K(P)} = \frac{1}{[E:K]} deg σ^{∗}D.

Now S will be a finite set of places of K, containing all infinite places. For E ⊃ K a finite extension, we denote S_{E} the set of places of E above S.

We break the height into S-part and non-S-part.

m_{K,S}(D, P) = \frac{1}{[E:K]} deg_{S_{E}} σ^{∗}D

= “proximity function”.

N_{K,S}(D, P) = \frac{1}{[E:K]} deg_{𝕄_{K} S_{E}} σ^{∗}D

= “counting function”.

A closely related quantity is the “truncated counting function”

N_{K,S}^{(1)}(D, P)

= \frac{1}{[E : K]} ∑_{w∉S_{E}} min(1, n_{D,w}(P)) log #k(w)

FACT: All these quantities depend on the choice of the model X of X only up to adding a function which is bounded over P ∈ X(K̄).

Relative discriminant: if E ⊃ K be a finite extension. Denote d_{K}(E) = \frac{1}{[E:K]} log |D_{E}| − log |D_{K}|. It can be identified as

d_{K}(E) = \frac{1}{[E : K]} deg ω_{O_{E}/O_{K}},

the latter having zero Green functions ^{𝕄_{E} → 𝕄_{K} decl. unram. at ∞.} We can analogously define

d_{K,S}(E) = \frac{1}{[E:K]} deg_{𝕄_{E} S_{E}} ω_{Spec O_{E}/ Spec O_{K}}.

If P ∈ X and K(P) = E we denote d_{K}(P) = d_{K}(E).

1.2 Vojta’s conjecture

K = a number field; S = a finite set of places, containing archimedean X = smooth complete variety over K; D ⊂ X a normal crossings divisor; A = a big line bundle on X; r = positive integer; ϵ = positive real

Then there exists a proper Zariski closed subset Z (depending on all the data) such that

h_{ω_{X}}(P)+m(D, P) ≤ d_{K}(P)+ϵh_{A}(P)+O(1)^{nevanlinna}

for all P ⊂ X(K̄) Z s.t. [K(P) : K] ≤ r.

Equivalently: N(D, P) + d_{K}(P) ≥

h_{ω_{X}(D)}(P) − ϵh_{A}(P) − O(1).

1.3 Proposition: Vojta implies Lang-Vojta 1.4 Vojta’s abc conjecture

N^{(1)}(D, P) + d_{K}(P) ≥ ^{abc}

h_{ω_{X}(D)}(P) − ϵh_{A}(P) − O(1).

1.5 Proposition

Vojta’s abc implies abc. ^{of M-Oé}

1.6 Theorem

Vojta’s conjecture implies Vojta’s abc conjecture.

1.7 Corollary

Vojta’s conjecture implies abc