Bogomolov on tori revisited.

Bogomolov on tori revisited.

Francesco Amoroso

Laboratoire de mathématiques Nicolas Oresme, CNRS UMR 6139 Université de Caen, Campus II, BP 5186 14032 Caen Cédex, France 1 Introduction.

Let V ⊆ G n m ⊆ P n be a geometrically irreducible variety which is not torsion (i. e. not a translate of a subtorus by a torsion point). For θ > 0 let V (θ) be the set of α ∈ V (Q) of Weil's height h(α) ≤ θ. By the toric case of Bogomolov conjecture (which is a theorem of Zhang),

μess (V ) = inf{θ > 0, V (θ) = V } > 0 .
If we assume moreover that V is not a translate of a subtorus by a point (eventually of infinite order) we can give a lower bound for μess (V ) depending only on deg(V ) (see [Bom-Zan 1995], [Dav-Phi 1999], [Sch 1996]).

Let us define the obstruction index ω(V ) as the minimum degree of an hypersurface containing V . We remark that ω(V ) ≤ n deg(V ) 1/ codim(V ) ( [Cha]). Assume that V is not transverse (i. e. is not contained in a translate of a subtorus).

In [Amo-Dav 2003] we conjecture μess (V ) ≥ c(n)ω(V ) -1 for some c(n) > 0 and we prove μess (V ) ≥ c(n)ω(V ) -1 (log(3ω(V )) -λ(codim(V ))

where λ(k) = 9(3k) k+1 k .

The aim of this paper is to give a more simple proof of a slightly improved (and explicit) version of this result (theorem 4.1), based on a very simple determinant argument (see section 2). More precisely the proof presented here

• avoid the use of the absolute Siegel's lemma of Zhang (see [Dav-Phi 1999], lemme 4.7)

• don't need any variant of zero's lemma and the subsequent combinatorial arguments (section 4 of [Amo-Dav 2003])

• don't use the weighted obstruction index ω(T ; V ) defined in [Amo-Dav 2003], definition 2.3.

Let V 0 = V \ B⊆V B.
where the union is on the set of translates B of subgroups of positive dimension contained in V . In [Amo-Dav 2006], theorem 1.5 we deduce from a lower bound for the essential minimum of V , a lower bound for height for all but finitely points of V 0 . Here we prove (theorem 5.1) an again slightly improved (and explicit) version of that result. We also correct a mistake which appears in that paper: in op. cit., theorem 1.5, δ(V ) must be defined as the minimum degree δ such that V is, as a set, intersection of hypersurface of degree ≤ δ (see remark 5.2 for details).

The determinant argument allow us to prove also very precise results concerning the normalized height ĥ(V ) of an hypersurface V (see section 3 for the definition). In this special case we conjecture : Conjecture 1.1 Assume one of the following: i) V is geometrically irreducible and it is not a translate of a subtorus.

ii) V is defined and irreducible over the rationals and is not torsion.

Then, there exists an absolute constant c > 0 such that ĥ(V ) ≥ c.

We remark that Lehmer's conjecture implies conjecture ii), via an argument of Lawton. We shall prove

Theorem 1.2 Let V ⊆ G n m be an hypersurface of multi-degrees (D 1 , . . . , D n ) with discrete stabilizer. Then, if n ≥ 9 and max D j ≤ 3 2 n we have ĥ(V ) ≥ 1 23 .
This result shows that an eventual example contradicting conjecture i) in n variable must be realized by polynomials of very big degree (or comes from an hypersurface of less variables). This could suggests an even more optimistic conjecture:

Let V be a geometric irreducible hypersurface of G n m with discrete stabilizer. Then ĥ

(V ) ≥ f (n), where f (n) → +∞ for n → ∞.
In section 3 we also provide a counterexample to this last statement.

A determinant argument.

The following proposition is the key argument for the proof of the main theorems.

Let S ⊆ P n and let I ⊂ C[x] be the ideal defining its Zariski closure. For ν ∈ N we denote by H(S; ν) the Hilbert function dim [C[x]/I] ν . Let T be a positive integer and let I (T ) be the T -symbolic power of I, i. e. the ideal of polynomials vanishing on S with multiplicity ≥ T . We put H(S, T

; ν) = dim[C[x]/I (T ) ] ν .
Similarly, if S ⊆ (P 1 ) n and ν = (ν 1 , . . . , ν n ) ∈ N n we denote its multihomogeneous Hilbert function by

H(S; ν) = dim([Q[x 1 , . . . , x n ]/I] ν 1 ,...,νn )
where

I ⊂ C[x] is the ideal defining S. More generally, if T is a positive integer we put H(S, T ; ν) = dim([Q[x 1 , . . . , x n ]/I (T ) ] ν 1 ,...,νn ).
Proposition 2.1 Let ν, T be positive integers and let p be a prime number. Let also h be a positive real number and S be a subset (eventually infinite) of G n m of points of height ≤ h. Then

h ≥ 1 - H(S, T ; ν) H(ker[p] • S; ν) T log p pν - n 2ν log(ν + 1) . (2.1)
In particular, if

H(S, T ; ν) ≤ 1 2 H(ker[p] • S; ν) (2.2) and T log p ≥ 2np log(ν + 1) , (2.3) then h ≥ T log p 4pν ≥ n log(ν + 1) 2ν .
Proof. Let for brevity S ′ = ker[p]S. We consider the (eventually infinite) matrix

(β λ ) β∈S ′ |λ|≤ν of rang L = H(ker[p] • S; ν). We select β 1 , . . . , β L ∈ S ′ and λ 1 , . . . , λ L with |λ j | ≤ ν such that the determinant ∆ = det(β λ j i ) i,j=1,...,L is non-zero. Let L 0 = H(ker[p] • S; ν) -H(S, T ; ν).
Then, by definition, there exist linearly independent polynomials G k = L j=1 g kj x λ j (k = 1, . . . , L 0 ) vanishing on S with multiplicity ≥ T . Let K be a sufficiently large field and let v be a non archimedean place of K dividing p. After renumbering the multi-indexes λ 1 , . . . , λ L and after making some linear combinations, we can assume

G k = L-k+1 j=1 g kj x λ j and moreover |g k,j | v ≤ 1, if j = 1, . . . , L -k; = 1, if j = L -k + 1;
for k = 1, . . . , L 0 . By elementary operations on columns we replace the last L 0 columns of ∆ by the columns

τ G k (β 1 ), . . . , G k (β L ) , k = 1, . . . , L 0 . Let ∆ ′ the new determinant; then |∆ ′ | v = |∆| v .
Since G k vanish on S with multiplicity ≥ T and since its coefficients are v-integers, we also have

|G k (β i )| v ≤ p -T /(p-1) max{1, |β i,1 | v , . . . , |β i,n | v } ν (i = 1, . . . , L; k = 1, . . . , L 0 ) .
By developping ∆ ′ with respect to the last L 0 columns we obtain

|∆ ′ | v = |∆| v ≤ p -L 0 T /(p-1) L i=1 max{1, |β i,1 | v , . . . , |β i,n | v } νL .
By the product's formula (using a trivial lower bound for v ∤ p)

1 ≤ p -L 0 T /(p-1) L L/2 e νhL and, using L ≤ ν+1 n ≤ (ν + 1) n , log h ≥ L 0 L × T log p pν - n 2ν log(ν + 1)
and the statement of proposition 2.1 follows.

The following is a multihomogeneous version of proposition 2.1.

Proposition 2.2 Let ν 1 , . . . , ν n , T be positive integers and let p be a prime number. Let also h 1 , . . . , h n be a positive real number and S be a subset (eventually infinite) of G n m of points α satisfying h(α j ) ≤ h j for j = 1, . . . , n. Then

ν 1 h 1 + • • • + ν n h n ≥ 1 - H(S, T ; ν) H(ker[p] • S; ν) T log p p - n 2 log(ν max + 1) (2.4)
where ν max = max{ν 1 , . . . , ν n }.

Proof. Let for brevity S ′ = ker[p]S. We consider the matrix H(ker[p] • S; ν). We select β 1 , . . . , β L ∈ S ′ and λ 1 , . . . , λ L with |λ j,l | ≤ ν l such that the determinant ∆ = det(β

(β λ ) β∈S ′ |λ 1 |≤ν 1 ,...,|λn|≤ν 1 of rang L =
λ j i ) i,j=1,...,L
is non-zero. Let L 0 = H(ker[p]•S; ν)-H(S, T ; ν). Then, by definition, there exists linearly independent polynomials G k = L j=1 g kj x λ j (k = 1, . . . , L 0 ) vanishing on S with multiplicity ≥ T . Let K be a sufficiently large field and let v be a non archimedean place of K dividing p. After renumbering the multi-index λ 1 , . . . , λ L and after making some linear combinations, we can assume

G k = L-k+1 j=1 g kj x λ j and moreover |g k,j | v ≤ 1, if j = 1, . . . , L -k; = 1, if j = L -k + 1;
for k = 1, . . . , L 0 . By elementary operations on columns we replace the last L 0 columns of ∆ by the columns

τ G k (β 1 ), . . . , G k (β L ) , k = 1, . . . , L 0 . Let ∆ ′ the new determinant; then |∆ ′ | v = |∆| v .
Since G k vanish on S with multiplicity ≥ T and since its coefficients are v-integers, we also have

|G k (β i )| v ≤ p -T /(p-1) n j=1 max{1, |β i,j | v } ν j (i = 1, . . . , L; k = 1, . . . , L 0 ) .
By developping ∆ ′ with respect to the last L 0 columns we obtain

|∆ ′ | v = |∆| v ≤ p -L 0 T /(p-1) L i=1 n j=1 max{1, |β i,j | v } ν j L .
By the product's formula (using a trivial lower bound for v ∤ p)

1 ≤ p -L 0 T /(p-1) L L/2 e (ν 1 h 1 +•••+νnhn)L
and, using L ≤ (ν max + 1) n ,

ν 1 h 1 + • • • + ν n h n ≥ L 0 L × T log p p - n 2 log(ν max + 1)
and the statement of proposition 2.2 follows.

3 Hypersurfaces.

In this section we are interested in the case of a hypersurface V . For these varieties we have a "natural" definition of height (which coincide with the previous one) since we can extend the Mahler measure to polynomials in several variables. Let f ∈ C[x 1 , . . . , x n ]; we define its Mahler measure as:

M (P ) = exp 1 0 • • • 1 0 log |f e 2πit 1 , . . . , e 2πitn |dt 1 . . . dt n .
Let now K be a number field and let V be an hypersurface in G n m defined over K:

V = {α ∈ G n m such that f (α) = 0} for some polynomial f ∈ K[x] (irreducible over Q[x]
). We define:

ĥ(V ) = 1 [K : Q] v∈M K [K v : Q v ] log M v (f ),
where M v (f ) is the maximum of the v-adic absolute values of the coefficients of f if v is non archimedean, and M v (f ) is the Mahler measure of σf if v is an archimedean place associated with the embedding σ : K ֒→ Q.

We prove:

Proposition 3.1 Let V ⊆ G n
m be an hypersurface of multi-degrees D 1 , . . . , D n and assume that V is not a translated of a torus. Let D max = max{D 1 , . . . , D n }. Then, for any prime number p ≥ 5, ĥ

(V ) ≥ log p 7p - nk ′ log p p k ′ - n log(n 2 D max ) 2p k ′ . (3.5)
where k ′ is the codimension of the stabilizer of V .

Proof. Since V is not a translated of a torus, k ′ ≥ 2. This implies n ≥ 2 and

p k ′ ≥ 9. We assume first that p ∤ [Stab(V ) : Stab(V ) 0 ], so that V ′ = ker[p]V is a union of p k ′ translate of V , and we prove ĥ(V ) ≥ log p 7p - nk ′ log p p k ′ - n log(nD max ) 2p k ′ , (3.6) Let ε > 0 and assume D max = D n . The proposition 2.7 of [Amo-Dav 2000] shows that the set S = {(ζ 1 , . . . , ζ n-1 , α) ∈ V (Q), ζ 1 , . . . , ζ n-1 roots of unity, h(α) ≤ ĥ(V )/D n + ε} is Zariski dense in V . We apply proposition 2.2 with h 1 = • • • = h n-1 = 0 and h n = ĥ(V )/D n + ε. We choose, for j = 1, . . . , n -1, ν j = np k ′ D j -1 and ν n = p k ′ D n -1. We remark that ν max = max{ν 1 , . . . , ν n } ≤ np k ′ D max -1. We also choose T = [p k ′ /2]. Then H(V, T ; ν) = (ν 1 + 1) • • • (ν n + 1) -(ν 1 -T D 1 + 1) • • • (ν n -D n + 1) = n n-1 p k ′ n - 1 2 n - 1 2 n-1 p k ′ n
and

H(V ′ ; ν) = (ν 1 + 1) • • • (ν n + 1) -(ν 1 -p k ′ D 1 + 1) • • • (ν n -p k ′ D n + 1) = n n-1 p k ′ n so that 1 - H(V, T ; ν) H(V ′ ; ν) ≥ 1 2 1 - 1 2n n-1 ≥ 1 2 √ e .
Inequality (2.4) now gives

ν n h n = (p k ′ D n -1) ĥ(V ) D n + ε ≥ T log p 2 √ ep - n 2 log(ν max + 1) ≥ p k ′ log p 4 √ ep - log p 2 √ ep - n 2 log(np k ′ D max ) ≥ p k ′ log p 7p -nk ′ log p - n 2 log(nD max ) .
By letting ε → 0 we obtain the lower bound (3.6).

If Stab(V ) is not connected, by inspection of the proof of proposition 2.4 of [Amo-Dav 2000] we obtain an hypersurface W with connected stabilizer of the same codimension k ′ , multi-degree (D ′ 1 , . . . , D ′ n ) with D ′ j ≤ nD j and normalized height ĥ(W ) ≤ ĥ(V ). Therefore, by (3.6), ĥ

(V ) ≥ ĥ(W ) ≥ log p 7p - nk ′ log p p k ′ - n log(n 2 D max ) 2p k ′ .
Let now assume k ′ = n, i. e. Stab(V ) discrete. Choosing p = 5 we obtain:

Theorem 3.2 Let V ⊆ G n m be an hypersurface of multi-degrees (D 1 , . . . , D n ) with discrete stabilizer. Then, if n ≥ 9 and max D j ≤ 3 2 n we have ĥ(V ) ≥ 1 23 .
Proof. We apply the proposition above with p = 5, assuming

D max ≤ 3 2 n and k ′ = n. We obtain ĥ(V ) ≥ log 5 35 - n 2 log 5 5 n - n log(n 2 D max ) 2 × 5 n ≥ log 5 35 - n 2 log 5 5 n - 2n log n 2 × 5 n - n2 n log 3 2 × 5 n =: f (n) .
An easy computation shows that f is an increasing function and f (9) > 1/23.

As stated in the introduction, we could conjecture that for any geometric irreducible hypersurface V ⊆ G n m with discrete stabilizer we had ĥ(V ) ≥ f (n) for some function f (n) → +∞ for n → ∞. This is false, as the the following example prove. Let F (x 1 ) = x 3 1x 1 -1 and define inductively

F n (x 1 , . . . , x n ) = F * (x 1 , . . . , x n-1 )x n -F (x 1 , . . . , x n-1 )
where F * indicated the reciprocal polynomial. Since the rational function

R(x 1 , . . . , x n-1 ) = F (x 1 , . . . , x n-1 ) F * (x 1 , . . . , x n-1 ) satisfy |R(z 1 , . . . , z n-1 )| = 1 for |z 1 | = • • • = |z n-1 | = 1, we have for any integer n M (F n ) = θ 0 where θ 0 is the root > 1 of F 1 . Moreover, it is easy to see that F n is irreducible (over Q if n ≥ 2) and that V n = {F n = 0} has trivial stabilizer.
We conclude this section with a more a general (and technical) lower bound for the normalized height of an hypersurface:

Theorem 3.3 Let V ⊆ G n
m be an hypersurface of multi-degrees (D 1 , . . . , D n ) and assume that V is not a translated of a torus. Then,

ĥ(V ) ≥ 1 56 × max log(n log(n 2 D max )) k ′ , 1 × log(n log(n 2 D max )) 28nk ′ log(n 2 D max ) 1/(k ′ -1)
where k ′ is the codimension of the stabilizer of V and D max = max D j . In particular, ĥ(V ) ≥ log(n log(n 2 D max )) 2 6272n log(n 2 D max ) .

Proof. Let

N = 28nk ′ log(n 2 D max ) log(n log(n 2 D max )) 1/(k ′ -1) (3.7)
and choose a prime number p such that N ≤ p ≤ 2N . By

log x ≤ x 1/2 (x > 0) (3.8) we have log(n log(n 2 D max )) ≤ log(n(n 2 D max ) 1/2 ) ≤ log(n 2 D max ); hence p k ′ -1 ≥ 28nk ′ .
We also remark that, again by (3.8),

log p ≥ log(28n 1/2 k ′ log(n 2 D max ) 1/2 ) k ′ -1 ≥ log(n log(n 2 D max )) 2k ′ (3.9)
Therefore,

p k ′ -1 log p ≥ 14n log(n 2 D max ) .
Thus, by proposition 3.1 we have ĥ

(V ) ≥ log p 7p - nk ′ log p p k ′ - n log(n 2 D max ) 2p k ′ ≥ log p 7p - log p 28p - log p 28p = log p 14p .
By (3.9) we obtain:

ĥ(V ) ≥ 1 14 × max log(n log(n 2 D max )) 2k ′ , log 2 × 1 2N ≥ 1 56 × max log(n log(n 2 D max )) k ′ , 1 × log(n log(n 2 D max )) 28nk ′ log(n 2 D max ) 1/(k ′ -1)
.

This prove the first inequality of theorem 3.3. For the second one, we remark that

k ′ ≥ 2 and k ′ (nk ′ ) 1/(k-1) ≤ 4n. So ĥ(V ) ≥ 1 56 × max log(n log(n 2 D max )) k ′ , 1 × log(n log(n 2 D max )) 28nk ′ log(n 2 D max ) 1/(k ′ -1) ≥ log(n log(n 2 D max )) 2 56 × 28 × 4n log(n 2 D max ) = log(n log(n 2 D max )) 2 6272n log(n 2 D max ) .
4 Essential minimum.

In this section we prove the following theorem, which slightly umprove theorem 1.4 of [Amo-Dav 2003]:

Theorem 4.1 Let V be a subvariety of G n m of codimension k < n. Then either there exists a translate B of a subgroup such that V ⊆ B G n m and

deg(B) 1/ codim(B) ≤ 250n 3 log(2nω(V )) λ(k)+1 ω(V ) or μess (V ) ≥ 2400n 4 log(2nω(V )) -λ(k) ω(V ) -1 where λ(k) = k+1 k (k + 1) k -1 -1 ≤ n n -3.
Proposition 2.1 gives the following result:

Proposition 4.2 Let V be a subvariety of G n m et let ω = ω(V ).
Let also p be a prime, 3 ≤ p ≤ ω and assume :

μess (V ) < log p 10npω .
Then,

ω([p]V ) ≤ 18n 2 ω log(5nω) log p .
Proof. Let h such that μess (V ) < h < log p 10npω and let

S = {α ∈ V, h(α) < h} .
Thus H(S, T ; ν) = H(V, T ; ν) and H(ker

[p] • S; ν) = H(ker[p] • V ; ν). Let us define T = 7np log(5nω) log p
and ν = (2n + 1)ωT . We first show that there exists a a non zero polynomial

F ∈ Q[x 1 , . . . , x n ] of total degree ≤ ν, vanishing on ker[p]V . Since 3 ≤ p ≤ ω, we have ν + 1 ≤ 3nω • 7np • 5nω + 1 ≤ (5nω) 3
and T log p ≥ 6np log(5nω). Thus inequality (2.3) of proposition 2.1, i. e. T log p ≥ 2np log(ν + 1), is satisfied. We also have

T log p 4pν = log p 4p(2n + 1)ω > h .
By proposition 2.1, we must have

H(ker[p] • V ; ν) < 2H(V, T ; ν) ≤ 2 ν + n n - ν -ωT + n n .
We remark that

ν + n n ν -ωT + n n -1 = n j=1 ν + j ν -ωT + j ≤ 1 + ωT ν -ωT n = 1 + 1 2n n ≤ √ e < 2 .
Thus

H(ker[p] • V ; ν) < ν + n n , i. e.
]Z) = deg([p] -1 [p]Z) = p codim(Z) deg([p]Z) we obtain ω([p]V ) ≤ deg([p]Z) 1/ codim(Z) ≤ p -1 ν .
We finally remark that

1 p ν ≤ 1 p • 5 2 nω • 7np log(5nω) log p < 18n 2 ω log(5nω) log p .
In order to prove theorem 4.1 we need, as in [Amo-Dav 2003], a descent argument. In what follows we fix a geometrically irreducible subvariety V G n m of dimension k < n (thus n ≥ 2) and we let ω = ω(V ). For j = 1, . . . , k let ρ j = (k + 1) k-j+1 -1 and P j = (2∆) ρ j where ∆ = Cn 3 log(2nω) and C = 120.

The following elementary relations will be used several time ii) For j ∈ {0, . . . , k} we have

k l=j+1 ρ l = (k + 1) (k + 1) k-j -1 k -(k -j) .
Definition 4.4 Let W be the set of triples (s, p, W), where s ∈ [0, k] is an integer, p = (p 1 , . . . , p s ) is a s-tuple of prime numbers with P i /2 ≤ p i ≤ P i , and where W = (W 0 , . . . , W s ) is a (s + 1)-tuple of strict geometrically irreducible subvarieties G n m , satisfying: i) V ⊆ W 0 . Moreover, for i = 1, . . . , s,

[p i ]W i-1 ⊆ W i and p i ∤ [Stab(W i-1 ) : Stab(W i-1 ) 0 ] ; ii) For i = 0, . . . , s deg(W i ) 1/ codim(W i ) ≤ ∆ k-i p i+1 • • • p k ω([p 1 . . . p i ]V ) ; iii) For i = 1, . . . , s ω([p 1 . . . p i ]V ) ≤ ∆ω([p 1 . . . p i-1 ]V ) .
Remark 4.5 Let (s, p, W) ∈ W and assume 0 ≤ i ≤ j ≤ s. Then

ω([p 1 . . . p j ]V ) ≤ ∆ j-i ω([p 1 . . . p i ]V ) .
We want to prove that there exists (s, p, W) ∈ W, such that dim(W i-1 ) = dim(W i ) for at least one index i. Let

W 0 = {(s, p, W) ∈ W, such that dim(W 0 ) < dim(W 1 ) < • • • < dim(W s )} . Proposition 4.6 Assume μess (V ) < 10n∆ k-1 P 1 • • • P k ω -1
.

(4.10)

Then W 0 = W.
In order to prove proposition 4.6, we endow the set of finite sequences of integers with the following (total) order . Let

(v) = (v i ) 0≤i≤s and (v ′ ) = (v ′ j ) 0≤j≤s ′ two such sequences. Then (v) (v ′ ) if (v i ) 0≤i≤min{s,s ′ } < (v ′ i ) 0≤i≤min{s,s ′ }
for the lexicographical order, or if (v i ) 0≤i≤min{s,s ′ } = (v ′ i ) 0≤i≤min{s,s ′ } and s ≥ s ′ . We also need the following technical lemma: Lemma 4.7 Let s ∈ N, p 1 , . . . , p s , p s+1 positive integers, W 0 , . . . , W s G n m geometrically irreducible subvarieties. Let us assume V ⊆ W 0 and [p i ]W i-1 ⊆ W i for i = 1, . . . , s. Then, there exists an integer s ′ ∈ [0, s + 1] and a geometrically irreducible subvariety Z s ′ of degree

deg(Z s ′ ) ≤ p s ′ +1 . . . p s+1 ω([p 1 . . . p s+1 ]V ) deg(W s ′ ) , (4.11) such that [p s ′ ]W s ′ -1 ⊆ Z s ′ , codim(Z s ′ ) = codim(W s ′ ) + 1 (with the following con- vention: codim(W s+1 ) = 0, deg(W s+1 ) = 1, W -1 =
V and p 0 = 1) and:

(dim(W 0 ), . . . , dim(W s ′ -1 ), dim(Z s ′ )) ≺ (dim(W 0 ), . . . , dim(W s )) .
(4.12)

Proof. Let Z s+1 be an hypersurface containing [p 1 . . . p s+1 ]V of minimal degree ω([p 1 . . . p s+1 ]V ). Thus if s ′ = s + 1 (4.11) is satisfied. We construct by induction subvarieties Z 0 , . . ., Z s such that, for i = 0, . . . , s,

i) Z i ⊆ W i and Z i = W i ⇒ codim(Z i ) = codim(W i ) + 1. ii) [p i+1 . . . p s+1 ]Z i ⊆ Z s+1 . iii) [p i+1 ]Z i ⊆ Z i+1 . iv) deg(Z i ) ≤ p i+1 . . . p s+1 ω([p 1 . . . p s+1 ]V ) deg(W i ).
We start by the construction of

Z 0 . If [p 1 . . . p s+1 ]W 0 ⊆ Z s+1 , we set Z 0 = W 0 .
Otherwise we choose for Z 0 a geometrically irreducible component of maximal dimension of W 0 ∩ [p 1 . . . p s+1 ] -1 Z s+1 containing V . By Bézout's inequality we have:

deg(Z 0 ) ≤ deg(W 0 ) deg([p 1 . . . p s+1 ] -1 Z s+1 ) ≤ p 1 . . . p s+1 ω([p 1 . . . p s+1 ]V ) deg(W 0 ) .
Let now i ∈ [0, s -1] be an integer and assume that Z 0 , . . . , Z i satisfy conditions

i)-iv). If [p i+2 . . . p s+1 ]W i+1 ⊆ Z s+1 , we set Z i+1 = W i+1 . Otherwise we choose for Z i+1 a geometrically irreducible com- ponent of maximal dimension of [p i+2 . . . p s+1 ] -1 Z s+1 ∩ W i+1 containing [p i+1 ]Z i . We can do this, since [p i+1 ]W i ⊆ W i+1 (by assumption) Z i ⊆ W i (by induction i)) and since [p i+1 . . . p s+1 ]Z i ⊆ Z s+1
(by induction i)). The variety Z i+1 verify conditions i)-iii). As before, by Bézout's inequality we have:

deg(Z i+1 ) ≤ p i+2 . . . p s+1 ω([p 1 . . . p s+1 ]V ) deg(W i+1 ) .
and the variety Z i+1 also verify condition iv).

We now choose the integer s ′ . We define s ′ as the least integer i such that Z i W i , if such an integer exists. Otherwise we set s ′ = s + 1. We remark that in both cases (4.12) holds.

Proof of proposition 4.6. The set W 0 is a finite non-empty set (indeed, let W 0 be an hypersurface of G n m containing V of degree ω; then (0, ∅, (W 0 )) ∈ W 0 ). Thus, there exists a minimal element (s, p, W) ∈ W 0 , i. e.

(dim W i ) 0≤i≤s (dim W ′ i ) 0≤i≤s ′ . for all (s ′ , p ′ , W ′ ) ∈ W 0 . We remark that s ≤ k -1, since n -k = dim(V ) ≤ dim(W 0 ) < dim(W 1 ) < • • • < dim(W s ) ≤ n -1 .
We need the following computation: Lemma 4.8 There exists a prime p s+1 such that P s+1 /2 ≤ p s+1 ≤ P s+1 and

p s+1 ∤ [Stab(W s ) : Stab(W s ) 0 ] .
Proof. By Theorems 9 and 10 of [Ros-Sch 1962], p≤x log p ≤ 1.02x for x ≥ 1 and p≤x log p ≥ 0.84x for x ≥ 101. Thus

P s+1 /2≤p≤P s+1 log p ≥ 0.84 -1.02/2 P s+1 > P s+1 /4 . If for any prime p with P s+1 /2 ≤ p ≤ P s+1 we had p | [Stab(W s ) : Stab(W s ) 0 ], then 2 log deg(W s ) ≥ P s+1 /4 , since deg(Stab(W s )) ≤ deg(W s ) 2
. By assertion ii) of definition 4.4 and by remark 4.5, we have :

log deg(W s ) ≤ codim(W s ) k log(∆ + k j=s+1 log P j + log(ω) ≤ k k + k j=s+1 log ρ j log(2∆) + log ω .
Using the inequality log x < x 1/3 (x > 100) with x = 2∆ (see lemma 4.3 i)) we obtain

log deg(W s ) ≤ k k + 1 + k j=s+1 log ρ j (2Cn 3 ) 1/3 log(2nω) .
Since s ≤ k -1, we have, using lemma 4.3 ii),

k k + 1 + k j=s+1 log ρ j = k(k + 1) + (k + 1) k-s+1 -(k + 1) -k(k -s) = (k + 1) k-s+1 + ks -1 ≤ 2(k + 1) 2(k-s) .
Thus, by setting a = (k + 1) (k-s) ≥ 2, 2 log deg(W s ) ≤ 4a 2 (2Cn 3 ) 1/3 log(2nω) and

P s+1 /4 2 log deg(W s ) ≥ 2Cn 3 log(2nω) a-1 16a 2 (2Cn 3 ) 1/3 log(2nω) ≥ 16C a-4/3 16a 2 =: f (a) .
An easy computation shows that f (a) ≥ f (2) > 1. Contradiction.

By the previous lemma, there exists a prime number p s+1 ∈ [P s+1 /2, P s+1 ] such that p s+1 ∤ [Stab(W s ) : Stab(W s ) 0 ]. We want to apply proposition 4.2 to the variety

V ′ = [p 1 . . . p s ]V choosing p = p s+1 . We have μess (V ′ ) ≤ p 1 . . . p s μess (V )
and, by iii) of definition 4.4

ω(V ′ ) ≤ ∆ s ω(V ) .
Thus, by assumption (4.10),

ω(V ′ )μ ess (V ′ ) ≤ ∆ s p 1 • • • p s ω μess (V ) < (10nP s+1 ) -1 ≤ log p s+1 10np s+1 .
Proposition 4.2 shows that:

ω([p s+1 ]V ′ ) ≤ 18n 2 log(5nω(V ′ )) log p s+1 ω([p 1 . . . p s ]V ) ≤ 18n 2 log(5nω(V ′ ))ω(V ′ ) . Since s ≤ k -1 ≤ n, we have, using remark 4.5, 5nω(V ′ ) ≤ 5n∆ s ω ≤ (C √ 5/32)(2nω) 5 n . Thus ∆ -18n 2 log(5nω(V ′ )) ≥ Cn 3 log(2nω) -18n 3 log (C √ 5/32)(2nω) 5 ≥ n 3 (C -18 × 5) log(4) -18 log(C √ 5/32) > 0 and ω([p 1 . . . p s+1 ]V ) = ω([p s+1 ]V ′ ) ≤ ∆ω(V ′ ) = ∆ω([p 1 . . . p s ]V ) .
We apply now lemme 4.7. We obtain an integer s ′ such that 0 ≤ s ′ ≤ s + 1 ≤ k and a subvariety Z s ′ satisfying the properties described in this lemma. We want to show that (s ′ , (p 1 , . . . , p s ′ ), (W 0 , . . . , W s ′ -1 , Z s ′ )) ∈ W .

All conditions i)-iii) of definition 4.4 are trivially verified, except eventually for the upper bound of deg(Z s ′ ). Using inequality (4.11) of lemma 4.7, the upper bound for the degree of W s ′ (point ii) of definition 4.4), remark 4.5 and the relation codim(Z s ′ ) = codim(W s ′ +1 ) + 1, we get:

deg(Z s ′ ) ≤ p s ′ +1 . . . p s+1 ω([p 1 . . . p s+1 ]V ) deg(W s ′ ) ≤ p s ′ +1 . . . p s+1 ∆ s-s ′ +1 ω([p 1 . . . p s ′ ]V ) deg(W s ′ ) ≤ ∆ k-s ′ p s ′ +1 • • • p k ω([p 1 . . . p s ′ ]V ) deg(W s ′ ) ≤ ∆ k-s ′ p s ′ +1 • • • p k ω([p 1 . . . p s ′ ]V ) 1+codim(W s ′ +1 ) ≤ ∆ k-s ′ p s ′ +1 • • • p k ω([p 1 . . . p s ′ ]V ) codim(Z s ′ )
.

Thus (s ′ , (p 1 , . . . , p s ′ ), (W 0 , . . . , W s ′ -1 , Z s ′ )) ∈ W. Since (dim(W 0 ), . . . , dim(W s ′ -1 ), dim(Z s ′ )) ≺ (dim(W 0 ), . . . , dim(W s ))

by relation (4.12) of lemma 4.7 and since (s, p, W) is a minimal element of W 0 , we deduce that:

(s ′ , (p 1 , . . . , p s ′ ), (W 0 , . . . , W s ′ -1 , Z s ′ )) ∈ W 0 .

Proof of theorem 4.1

Let V be a geometrically irreducible subvariety of G n m of codimension k < n which satisfy the assumption of proposition 4.6. By this proposition, there exists (s, p, W) ∈ W \ W 0 . Thus there exists an index i such that

codim(W i-1 ) = codim(W i ) = r, [p i ]W i-1 ⊆ W i , [p 1 . . . p i-1 ]V ⊆ W i ;
and

p i ∤ [Stab(W i-1 ) : Stab(W i-1 ) 0 ].
Assume first that W i is a translate of a subtorus. Then the same is true for the connected component B of [p 1 . . . p i ] -1 W i containing V and we have, using ii) of definition 4.4 and remark 4.5,

(deg B) 1/ codim(B) ≤ (p 1 • • • p i ) 1/r ∆ k p i+1 • • • p k ≤ ∆ k P 1 • • • P k ≤ (2∆) λ(k)+1
where

λ(k) + 1 = k + k j=1 ρ j = k + 1 k (k + 1) k -1 .
Assume now that W i is not a translate of a subtorus. Thus

p i deg(W i-1 ) ≤ deg(W i ) .
Since W i-1 ⊇ [p 1 . . . p i-1 ]V , we have, using ii) and iii) of definition 4.4,

ω([p 1 . . . p i-1 ]V ) ≤ deg(W i-1 ) 1/r ≤ p -1/r i deg(W i ) 1/r ≤ p -1/r i ∆ k-i p i+1 • • • p k ω([p 1 . . . p i ]V ) ≤ p -1/r i ∆ k-i p i+1 • • • p k × ∆ω([p 1 . . . p i-1 ]V ) .
Since r ≤ k and P i /2 ≤ p i ≤ P i , we get :

p -1/r i ∆ k-i p i+1 • • • p k ∆ ≤ P -1/k i 2 1/k ∆ k-i+1 P i+1 • • • P k < P -1/k i (2∆) k-i+1 P i+1 • • • P k = (2∆) b where (see lemma 4.3 ii)) b = - ρ i k + k -i + 1 + k j=i+1 ρ j = - (k + 1) k-i+1 -1 k + (k -i + 1) + (k + 1) (k + 1) k-i -1 k -(k -i) = 0 .
This is a contradiction. Hence

μess (V ) ≥ 10n∆ k-1 P 1 • • • P k ω(V ) -1
.

We finally remark that

10n∆ k-1 P 1 • • • P k ≤ 20n∆ λ(k) .
Theorem 4.1 is proved.

5 Petit points.

Given an algebraic set V ⊆ G n m we define, following [Bom-Zan 1995] and [Sch 1996],

V 0 = V \ B⊆V B.
where the union is on the set of translates B of subgroups of positive dimension contained in V . In this section we prove a slightly improved version of theorem 1.5 of [Amo-Dav 2006]:

Theorem 5.1 Let V G n m be an algebraic set defined by equations of degree ≤ δ. Then, for all but finitely many α ∈ V 0 we have ĥ(α) ≥ θ := 2400n 3 log(2nδ)

-n n +3 δ -1 .

More precisely, the set of

α ∈ V of height < θ is contained in a finite union B 1 ∪ • • • ∪ B m of translate of subtori such that deg(B j ) ≤ 250n 3 log(2nδ) (2n) n δ 2 codim(B j ) -1
Proof.

It is enough to prove the following statement:

Let V G n m be an algebraic set defined by equations of degree ≤ δ and let Z be a geometrically irreducible subvariety of V of positive dimension, satifying μess (Z) ≤ 2400n 3 log(2nδ)

-n n +3 δ -1 .

(5.13)

Then, there exists a translate B of a subtorus of codimension r such that Z ⊆ B ⊆ V and deg(B) ≤ 250n 3 log(2nδ)

(2n) n δ 2 r -1 .
We prove this last statement by induction on n. If n = 2 it is easily implied by theorem 4.1. Assume n ≥ 3 and that the conclusion holds for all algebraic set defined by equations of degree ≤ δ ′ in G n-1 m . Assume further that there exists a positive integer δ, an algebraic set V G n m defined by equations of degree ≤ δ and a geometrically irreducible subvariety Z of V which satisfies (5.13). Let k codim(Z). In particular, since ω(Z) ≤ δ and λ(k) ≤ n n -3, theorem 4.1 gives a translate B = αH of codimension k ′ containing Z, and such that (deg(B)

1/k ′ ≤ 250n 3 log(2nδ)
n n -2 δ .

(5.14)

We can assume α ∈ Z and ĥ(α) ≤ 2μ ess (Z); thus we have : μess (α -1 Z) ≤ ĥ(α -1 ) + μess (Z) ≤ n ĥ(α) + μess (Z) ≤ 3nμ ess (Z) .

(5.15)

We now fix a Z-base a 1 , . . . a k ′ of the Z-module

Λ := λ ∈ Z n , t.q. ∀ x ∈ H, x λ = 1 ⊆ Z n
and we consider the n × k ′ matrix A = (a i,j ). Let E = Λ ⊗ Z R. Then (see for instance [Ber-Phi 1988]) the degree of H is the maximum of the absolute values of the k ′ × k ′ subdeterminants of A, and Vol(E/Λ) is their quadratic mean. Thus

Vol(E/Λ) ≤ n k ′ 1/2 deg(B) ≤ n k ′ deg(B) .
Let us consider the cube [-1/2, 1/2] n ⊂ R n ; by a theorem of Vaaler (see [Vaaler 1979])

Vol(C ∩ E) ≥ 1 .
Thus, by Minkowski's theorem on convex bodies, there exists a non-zero λ ∈ Λ such that: max

1≤i≤n {|λ i |} ≤ n deg(B) 1/k ′ .
Since H is connected, we can assume λ 1 , . . . , λ n coprime and also .

We remark that, for any β ∈ G n-1 m , h ϕ(β) ≥ h β λn = λ n h β = Dh(β) .

(5.17)

Let V ′ = ϕ -1 α -1 V ∩ H ⊆ G n-1 m
Since αH ′ ⊆ V we have V ′ G n-1 m . Moreover, let F j (x) (j = 1, . . . , N ) be equations defining V ; then V ′ is defined by the equations

F j x λn 1 , . . . , x λn n-1 , x -λ 1 1 . . . x -λ n-1 n-1 = 0 of degree ≤ δ ′ = max{λ n , |λ 1 + • • • + λ n-1 |}δ ≤ nDδ .
Let Z ′ be a geometrically irreducible component of ϕ -1 α -1 Z ∩ H ⊆ V ′ . We have, by (5.17) and (5.15), D μess (Z ′ ) ≤ μess (ϕ(Z ′ )) = μess (α -1 Z) ≤ 3nμ ess (Z) .

Using the upper bound for μess (Z) and the inequality δ ′ ≤ nDδ, we deduce μess (Z ′ ) ≤ 3nD -1 2400n 3 log(2nδ)

-n n +3 δ -1 ≤ 3n 2 2400n 3 log(2nδ) -n n +3 δ ′-1

Using the inequalities δ ′ ≤ nDδ, (5.16) and log x < x we get 2nδ ′ ≤ 2n 2 Dδ ≤ (250n 3 • 2nδ) n n δ ≤ (2nδ) (250n 3 ) n-1 .

(5.18) Thus 2400(n -1) 3 log(2(n -1)δ ′ ) (n-1) n-1 -3 ≤ (3n 2 ) -1 (2400n 3 ) a log(2nδ) n n -3 where a = 1 + n (n -1) n-1 -3 ≤ n n -3 .

Therefore μess (Z ′ ) ≤ 2400(n -1) 3 log(2(n -1)δ ′ ) -(n-1) n-1 +3 δ ′-1

By induction there exists a translate B ′ ⊆ V ′ of a subtorus of codimension r ′ such that Z ′ ⊆ B ′ and deg(B ′ ) ≤ 250n 3 log(2nδ ′ ) (2n) n-1 δ ′2 r ′ -1 .

Let for brevity K = 250n 3 log(2nδ). From the inequalities (5.18) and δ ′ ≤ nDδ we get deg(B ′ ) ≤ K 2 n-1 n n (nDδ) 2 r ′ -1 .

Then Z = αϕ(Z ′ ) ⊆ ϕ(B ′ ) ⊆ V , r = codim ϕ(B ′ ) = r ′ + 1 and

deg ϕ(B ′ ) ≤ D deg(B ′ ) ≤ K 2 n-1 n n n 2 r ′ -1 D 2 r ′ δ 2 r ′ -1 ≤ K 2 n-1 n n +2 r ′ n n δ 2 r ′ +1 -1 ≤ K (2n) n δ 2 r -1
where we have used the upper bound (5.16) for D.

Remark 5.2 In [Amo-Dav 2006], theorem 1.5 we assume that V is geometrically irreducible (which is not necessary) and that V is incompletely defined by forms of degree ≤ δ, i. e. it is a component of a complete intersection of hypersurfaces of degree ≤ δ. Unfortunately, there is a mistake in the proof: at page 561, point (a), we cannot ensure that V ′ is incompletely defined by forms of degree ≤ nDδ. The problem is the following: if V is incompletely defined by forms of degree ≤ δ, Z is an hpersurface of degree ≤ δ which not contains V , then an irreducible component of V ∩ Z is not a priori incompletely defined by forms of degree ≤ δ.

Lemma 4. 3

 3 We have: i) log(2nω) > 1 and ∆ > 960.

  λ n = D. Then the equation x λ = 1 defines a subtorus H ′ ⊇ H of codimenion 1 and degree D ≤ n deg(B) 1/k ′ ≤ (2n) -2 250n 3 log(2nδ) αH ′ ⊆ V we are done. Asume the contrary. We consider the isogeny G n

  there exists a non zero polynomialF ∈ Q[x 1 , . . . , x

n ] vanishing on ker[p]V of total degree ≤ ν. By the zero's lemma of P. Philippon (see [Phi 1986]), there exists a variety Z containing V such that deg(ker[p]Z) ≤ ν codim(Z) . Indeed, let W be the algebraic set defined by the equations F (ζx) = 0 for ζ ∈ ker[p]. Since F vanishes on ker[p]V , there exists a geometrically irriducible component Z of W containing V . Since W is stable by translation by p-torsion points, all ζV are components of W for ζ ∈ ker[p]. Proposition 3.3 of [Phi 1986] (with p = 1, N 1 = n and D 1 = ν) then gives the desired upper bound for deg(ker[p]Z). Since deg(ker[p