Diophantine Approximation of non-algebraic points on varieties II Explicit estimates for ar

a r X i v :0711.1667v 1 [m a t h .A G ] 11 N o v 2007Diophantine Approximation of non-algebraic points on varieties II:Explicit estimates for Arithmetic Hilbert Functions Heinrich Massold February 2,2008Contents 1Introduction 12Algebraic Hilbert functions 63Arithmetic varieties 104Arithmetic Hilbert functions 164.1Arithmetic Interpolation .........................204.2The lower bound .............................271Introduction Let k be a number field with ring of integers O k ,X a projective flat scheme of

relative dimension t over Spec O k ,which in this paper is refered to by the term of a t -dimensional arithmetic variety,and ¯L a positive metrized line bundle on X .If for D ∈one defines H X (D )as the dimension of the vector space of global sections of L ⊗D on X =X ×Spec O k Spec k ,and ˆH X (D )as the arithmetic degree of the arithmetic bundle of global sections of ¯L ⊗D over X ,there are the well know algebraic and arithmetic Hilbert-Samuel formulas

H X (D )=deg L X D t

(t +1)!+O (D t log D ).

One can define a third kind of Hilbert function:Letσ:k→be some embedding, andθ∈X(σ)a generic point,i.e.a point whose algebraic closure over the algebraic closure¯k of k is all of X,assume further that X(σ)is endowed with a K¨a hler metric such that the K¨a hler form conincides with the chern form¯c1(¯L),and define

Γ(D,H):={f∈Γ(X,L⊗D)|log|f|L2(X)≤H},

and

¯H X,θ(D,H):=−

min

f∈Γ(D,H)

log|fθ|.(1)

We always assume that H sufficiently big compared with D.For k=,the Theorem of Minkowski together with the algebraic and arithmetic Hilbert-Samuel formulas implies that for sufficiently big D,

¯H X,θ(D,H)≥h(X)

D t+1

t!

+O(D t log(DH)).(2)

Proof Let D∈,and Sθbe the stalk of L⊗D atθ.If f∈Γ(D,H)is a vector of norm one,orthogonal to the kernel of the evaluation map

ϕD:Γ(D,H)→Sθ,f→fθ,

and

c:=

|f|L2(X)

(t+1)!−H deg X

D t

(1+ǫ) −h(X)D t+1t! .

Sinceǫwas chosen arbitrary,the claim follows.

The just proved lower bound(2)is equivalent to arithmetic ampleness as stated in [SABK],ch.8,Theorem2:

Proof The cited Theorem states that

#Γ(D,H−log2)≥h(X)D t+1

t!

.

Since for each f∈Γ(D,H),

log|fθ|≤log|f|L2(X)+D log c≤H+D log c,

there are vectors f,g∈Γ(D,H)such that

log||f|θ−|g|θ|≤−h(X)D t+1t!+H+D log c→

−h(X)D t+1t!.

Thus|f−g|L2(X)≤H,i.e.f−g∈Γ(D,H),and

log|f−g|θ≤log||f|θ−|g|θ|≤−h(X)D t+1t!, which is formula2.

On the other hand if

#Γ(D,H−log2)<h(X)D t+1

t!

,

then for a generic pointθ∈X(σ),there is an f∈Γ(D,H−log2)such that for every g∈Γ(D,H),

log||f|θ−|g|θ|>−h(X)D t+1t!.

Because the set{f+g|g∈Γ(D,H)}contains the setΓ(D,H−log2),this implies that there is no h∈Γ(D,H−log2)with

log|h|θ<−h(X)D t+1

t!

,

and thus

¯H X,θ(D,H)<h(X)

D t+1

t!

+O(D t log(DH)).