Generalized generalized Vojta inequality (d'apr\es Ange)

GENERALIZED GENERALIZED VOJTA INEQUALITY ` ANGE) (D’APRES

arXiv:1811.07784v1 [math.NT] 19 Nov 2018

GABRIEL A. DILL Abstract. Following and generalizing unpublished work of Ange, we prove a generalized version of Rémond’s generalized Vojta inequality. This generalization can be applied to arbitrary products of irreducible positive-dimensional projective varieties, defined over the field of algebraic numbers, instead of powers of one fixed such variety. The proof runs closely along the lines of Rémond’s proof.

Let m ≥ 2 be an integer and let X1 , . . . , Xm be a family of irreducible ¯ We wish to extend positive-dimensional projective varieties, defined over Q. Rémond’s results of [8] to the case of an algebraic point x = (x1 , . . . , xm ) in the product X1 × . . . × Xm . The following article is a further generalization of a generalization of these results by Thomas Ange in [1]. It draws heavily on a written account of this generalization by Ange. In work in progress, we apply our generalized Vojta inequality to a relative π version of the Mordell-Lang problem in an abelian scheme A → S, where S is ¯ In the problem, one an irreducible variety and everything is defined over Q. ¯ ¯ fixes an abelian variety A0 , defined over Q, a finite rank subgroup Γ ⊂ A0 (Q) and an irreducible closed subvariety V ⊂ A and studies the points p ∈ V of the form φ(γ) for an isogeny φ : A0 → Aπ(p) , Aπ(p) denoting the fiber of the abelian scheme over π(p), and γ ∈ Γ. In this application, it is crucial that we allow the Xi to lie in different fibers of the abelian scheme. If the abelian scheme A is constant, an analogue of the intended height bound has been obtained by von Buhren in [10]. In his case, the generalized Vojta inequality from [8], where X1 = X2 = · · · = Xm = X, was sufficient, however for our intended application it is necessary to allow the Xi to be different. Let us recall the hypotheses which come into play. We use the same notation as in [8] and we refer to that article for the history of Vojta’s inequality. For every m-tuple a = (a1 , . . . , am ) of positive integers, we write Na =

m O

i p∗i L⊗a i ,

i=1

where Li is a fixed very ample line bundle on Xi and pi : X1 ×. . .×Xm → Xi is the natural projection. We fix a non-empty open subset U ⊂ X1 ×. . .×Xm and relate a to an irreducible projective variety X , provided with an open immersion U ⊂ X and a proper morphism π : X → X1 × . . . × Xm such that 2010 Mathematics Subject Classification. 11G35, 11G50, 14G25, 14G40. Key words and phrases. Heights. 1

2

GABRIEL A. DILL

π|U = idU , as well as to a line bundle M on X which satisfies some further conditions, specified below. We assume that there exists a very ample line bundle P on X , an injection P ,→ Na⊗t1 which induces an isomorphism on U and a system of homogeneous coordinates Ξ for P which are (by means of the previously mentioned isomorphism) monomials of multidegree t1 a in the homogeneous coordinates W (i) ⊂ Γ(Xi , Li ), fixed in advance (we identify p∗i W (i) with W (i) ). We also assume that there exists an injection (P ⊗ M⊗−1 ) ,→ Na⊗t2 which induces an isomorphism on U and that P ⊗ M⊗−1 is generated by a family Z of M global sections on X which are polynomials of multidegree t2 a in the W (i) such that the height of the family of coefficients P of all these polynomials, seen as a point in projective space, is at most i ai δi . The height of any polynomial is defined by considering the family of its coefficients as a point in an appropriate projective space. On projective space, the height is defined as in Definition 1.5.4 of [2] by use of the maximum norm at the archimedean places. The integral parameters t1 , t2 , M and the real parameters δ1 , . . . , δm (all at least 1) are fixed independently of the triple (a, X , M). This triple permits ¯ to define the following two notions of height for an algebraic point x ∈ U (Q): hM (x) = h(Ξ(x)) − h(Z(x)), hNa (x) = a1 h W (1) (x) + . . . + am h W (m) (x) . Our goal is to prove an inequality among these two numbers under certain assumptions about the intersection numbers of M. Let therefore θ ≥ 1 and ω ≥ −1 be two integer parameters and put (with ω 0 = 3 + ω) Y m u0 Λ = 2θ(2t1 u0 ) max Ni + 1 deg(Xi ), 1≤i≤m

ψ(u) =

u0 Y

i=1

(ω 0 j + 1),

j=u+1

c1 = c2 = Λψ(0) , c3 = Λ2ψ(0) (M t2 )u0 max {ai (h(Xi ) + δi )} , 1≤i≤m

where u0 = dim(X1 ) + . . . + dim(Xm ), Ni + 1 = #W (i) and the degrees and heights are computed with respect to the embedding given by W (i) . We use here the (normalized) height of a closed subvariety of projective space as defined in [3] (via Arakelov theory) or [4] (via Chow forms). The two definitions yield the same height by Théorème 3 of [9]. The following theorem therefore generalizes Théorème 1.2 of [8]. ¯ be an algebraic point and (a, X , M) a triple Theorem 0.1. Let x ∈ U (Q) as defined above. Suppose that, for every subproduct of the form Y = Y1 × . . . × Ym , where Yi ,→ Xi is a closed irreducible subvariety that contains xi , we have the following estimate m Y dim(Yi ) · dim(Y ) −1 (M · Y) ≥ θ (deg(Yi ))−ω ai , i=1

` ANGE) GENERALIZED GENERALIZED VOJTA INEQUALITY (D’APRES

3

where Y denotes the closure of π −1 (Y ∩ U ) in X . Then we have hNa (x) ≤ c1 hM (x) if furthermore c2 ai+1 ≤ ai for every i < m and c3 ≤ ai h W (i) (xi ) for every i ≤ m. Naturally, we follow the proof in [8] very closely with some minor changes: Firstly, the term 12uψ(u) that appears in the last equation of [8] should be replaced by 4ω 0 uψ(u); that’s why we don’t use Lemme 5.4 of [8] and define Λ slightly differently. Secondly, Corollaire 5.1 of [8] doesn’t apply if (i) xj = 0, which means that Corollaire 3.2 of [8] has to be made more precise. Thirdly, in the last inequality in the proof of Proposition 4.2 of [8], a term bounding the contribution of the archimedean places when the Pi are raised to the d-th power is missing. Fourthly, the factor 8 in the upper bound 8(N + 1)Di log(N + 1)Di for log 2f2 (ui , Di ) given in the proof of Proposition 5.3 of [8] has to be increased. We first consider a subproduct Y = Y1 × · · · × Ym of minimal total dimension u = u1 + · · · + um , satisfying the following conditions: ¯ for all 1 ≤ i ≤ m; (i) xi ∈ Yi (Q) (ii) di ≤ deg(Xi )Λψ(u)−1 for all 1 ≤ i ≤ m; Q Q ψ(u)−1 ; di ≤ ( m (iii) m i=1 deg(Xi ))Λ i=1 Pm Pm −1 2ψ(u) (M t )u0 −u (iv) 2 i=1 (ai (h(Xi ) + δi )), i=1 ai (hi + δi ) ≤ 2 Λ where ui = dim(Yi ), di = deg(Yi ) and hi = h(Yi ) (in the projective embedding defined by the W (i) ). Such a subproduct certainly exists, since X1 × . . . × Xm satisfies these conditions. Furthermore, we have u > 0 since otherwise Y = {x} and therefore mc3 ≤ hNa (x) ≤

m X

ai hi ≤

i=1

mc3 . 2

We use the definition of an adapted projective embedding on p. 466 of [8]. By Proposition 2.2 of [8], we may define an embedding adapted to the closed subvariety Yi of Xi by putting (i) Vj

=

Ni X

(i)

(i)

Mjk Wk

k=0 (i) with M (i) ∈ GLNi +1 (Q), where the coefficients of the matrix M are inte gers and bounded by max 1, d2i in absolute value, at least if Yi 6= PNi . If

Yi = PNi , then the notion of an adapted projective embedding is not defined (i) (i) in [8], but we may set Vj = Wj (j = 0, . . . , Ni ) and check that all the assertions about adapted embeddings made in this article also hold true in this case. We now prove the equivalent of Proposition 3.1 in [8], introducing Λh =

m X i=1

ai (hi + δi + di (ui + 1) log 2di (Ni + 1)),

4

GABRIEL A. DILL

which we will prove to verify m X 2ψ(u) u0 −u Λh < Λ (M t2 ) (ai (h(Xi ) + δi )) ≤ Λ2ψ(u)−2ψ(0) (M t2 )−u mc3 . i=1

(1) In order to show the first inequality (given condition (iv) from above), it suffices to show that m m X X ai di (ui + 1) log 2di (Ni + 1) < 2−1 Λ2ψ(u) (M t2 )u0 −u (ai (h(Xi ) + δi )) i=1

i=1

Pm

−1 2ψ(u) (M t )u0 −u . But since or even 2 i=1 di (ui + 1) log 2di (Ni + 1) < 2 Λ log 2di (Ni + 1) < 2di (Ni + 1), it follows from (ii) that the left-hand side is at most

(u + m)Λ2ψ(u)−2 max {2 deg(Xi )2 (Ni + 1)} < 2−1 Λ2ψ(u) 1≤i≤m

and now the claim is obvious. Proposition 0.2. There doesn’t exist any pair (l, U ) such that 1 ≤ l ≤ m (l) and U V is a homogeneous polynomial in the first adapted coordinates (l)

(l)

V0 , . . . , Vul satisfying (a) U V (l) (xl ) = 0; (b) U is not the zero polynomial; 0 (c) deg(U ) ≤ Λω uψ(u) ; t2 Λh . (d) al h(U ) ≤ Λ2ψ(u−1)−2ψ(u) M 4dl Proof. We assume the contrary and define Yl0 as an irreducible component containing xl of the closed subvariety of Yl defined by the equation U V (l) = 0 and we verify that the subproduct Y 0 obtained by replacing Yl by Yl0 in Y contradicts the minimality of the latter. We have Y 0 = Y10 ×· · ·×Ym0 with Yi0 = Yi for all i 6= l. By (a), condition (i) holds for Y 0 . By (b) and the definition of an adapted embedding, Y 0 is a proper subvariety of Y . The polynomial U (V (l) ) corresponds by means of M (l) to a polynomial U 0 (W (l) ), where deg(U 0 ) = deg(U ) and dl deg(U ) + ul 0 h(U ) ≤ h(U ) + deg(U ) log(Nl + 1) max 1, + log . 2 deg(U ) As deg(U Y ) deg(U ) + ul ul = 1+ ≤ (1 + ul )deg(U ) , deg(U ) i i=1

it follows that h(U 0 ) ≤ h(U ) + deg(U ) log dl (Nl + 1)(ul + 1). The (arithmetic as well as geometric) theorems of Bézout yield deg(Yl0 ) ≤ deg(U 0 )dl and

p h(Yl0 ) ≤ deg(U 0 )hl + dl h(U 0 ) + Nl .

(2)


5

For the arithmetic Bézout theorem, we use Théorème 3.4 and Corollaire 3.6 of [6], where the modified height hm used there can be bounded thanks to Lemme 5.2 of [7]. Together with (c), the first line implies that deg(Yl0 ) ≤ dl Λψ(u−1)−ψ(u) , since by definition ψ(u − 1) = (ω 0 u + 1)ψ(u). This shows that Y 0 satisfies conditions (ii) and (iii). From the second line together with (2), (c) and (d), we deduce that m X

0

ai (h(Yi0 ) + δi ) ≤ dl al h(U ) + dl al Λω uψ(u) log dl (Nl + 1)(ul + 1)

i=1

+dl al

p

Nl + Λ

ω 0 uψ(u)

m X

0

ai (hi + δi ) ≤ dl al h(U ) + 3Λω uψ(u) Λh

i=1

M t2 0 Λh + 3Λω uψ(u) Λh . ≤Λ 4 0 0 Finally, we have 3Λω uψ(u) ≤ Λ2ω uψ(u) M4t2 = Λ2ψ(u−1)−2ψ(u) M4t2 . It then follows from (1) that Y 0 satisfies condition (iv) as well and we get the desired contradiction. 2ψ(u−1)−2ψ(u)

We proceed to deduce from this an equivalent of Corollaire 3.2 in [8] (with a modification of the last assertion). Let us mention that by Lemme 2.3 of [8], there exist polynomial relations (i) (i) (i) (i) (i) (i) ⊗di (i) Pj = Q V , . . . , V , W = 0 in Γ Y , L V0 , . . . , Vu(i) , V i ui 0 j i j j i (i)

(i)

for all 1 ≤ i ≤ m and all 0 ≤ j ≤ Ni . The polynomials Pj (T ) and Qj (T ) are homogeneous of degrees di , monic in their last variable Tui +1 and equal to a power of an irreducible polynomial (we denote the corresponding exponent (i) for Qj by bi,j ). Furthermore, we know from the same lemma that the height (i)

(i)

of the family Bi of all the coefficients of the Pj and the Qj for fixed i (seen as a point in projective space) can be estimated from above as h(Bi ) ≤ hi + di (ui + 1) log di (Ni + 1).

(3)

Corollary 0.3. For every index 1 ≤ i ≤ m, we have that (1) the morphism ρi : Yi → Pui , defined by the first adapted coordinates (i) (i) ¯ V0 , . . . , Vui , is finite, surjective and étale at x ∈ Yi (Q); (i) (2) V0 (xi ) 6= 0; (i) (3) for every index 0 ≤ j ≤ Ni , such that Wj 6= 0 in Γ(Yi , Li ), we have (i) ! (i) (i) bi,j Q(i) W ∂ V V u j j (i) i 1, 1(i) , . . . , (i) , (i) (xi ) 6= 0. Wj bi,j ∂Tui +1 V0 V0 V0 Proof. That the morphism ρi is finite and surjective follows from the definition of adapted embeddings (see [8], Section 2.1). If oneof the three assertions weren’t true, we could construct a pair i, U V (i) that would contradict Proposition 0.2, with deg(U ) ≤ 2d2i and h(U ) ≤ 6Ni d3i +2di h(Bi ). (i) We refer to Corollaire 3.2 of [8] for the proof – in the case that Wj 6= 0, (i) (i) (i) (i) Wj (xi ) = 0 it suffices to take U V (i) = Qj (V0 , . . . , Vui , 0). Note that

6

GABRIEL A. DILL (i)

Pui +1 is not only a power of an irreducible polynomial, but in fact irreducible, since its degree is equal to the degree of Yi , which is also equal to the degree (i) of any irreducible factor of Pui +1 . Hence, its discriminant doesn’t vanish identically. That the morphism is étale in x is proved in the same way as in the proof of Lemme 4.3 in [5]. The line bundle π ∗ Na on X restricts naturally to the line bundle Na |U on U ⊂ X . We denote it also by Na and identify π ∗ W (i) with W (i) for all i. Following Section 4 of [8], we put =

m 1 1 Y −1−ω di 2uθ (t1 m)u i=1

and define a family of sections Zd0 ⊂ Γ(X , M⊗−d ⊗P ⊗d ⊗Na⊗d ) of cardinality M 0 = M (N1 + 1) . . . (Nm + 1) for every d ∈ −1 N ⊂ N by (1)da1

Zd0 = {ζ ⊗d ⊗ Wj1

(m)dam

⊗ . . . ⊗ W jm

(i)

; ζ ∈ Z, Wji ∈ W (i) }.

The proof of Proposition 4.1 of [8] then goes through without any major modifications and yields a natural number d0 . We obtain the following equivalent of Proposition 4.2 in [8]. Proposition 0.4. For d ∈ −1 N ∩ d0 N sufficiently large, we write Qd M⊗d ⊗ Na⊗−d and fix a basis of Γ(Y, P ⊗d ) that consists of monomials the sections Ξ of degree d. Then there exists a section 0 6= s ∈ Γ(Y, Qd ) such that the height of defined as the height of the coefficients of the sections s ⊗ Zd0 with respect the fixed basis, satisfies h(s) ≤

= in s, to

2M 0 d (t1 + 2t2 + )Λh + o(d). u

Proof. The dimension estimate m

du Y −ω ui dim Γ(Y, Qd ) ≥ di ai + O(du−1 ) 4θu! i=1

given in Proposition 4.1 of [8] is still valid, since the intersection numbers are formally the same. We choose d sufficiently large so that we can choose a basis of Γ(Y, P ⊗d ) that consists of monomials of degree d in the elements of Ξ. In the Faltings complex on Y defined by the family Zd0 of cardinality M 0 ⊕M 0 ⊕(M 0 )2 0 → Qd → P ⊗d → Na⊗d(t1 +t2 +) 0

the image of Γ(Y, Qd ) in F = Γ(Y, P ⊗d )M coincides with the kernel of a family of linear forms (with coefficients in a number field that is independent of d) of total height at most d(t1 +2t2 +)Λh +o(d): in order to show this, we follow the proof of Proposition 4.2 of [8] by applying Lemme 2.5 in [8] with ni = Ni and using the above estimate for h(Bi ). Note that when estimating d ) as in the proof of Proposition 4.2, one obtains by well-known h(P1d , . . . , PM


7

height estimates an upper bound of d

m X

ai δi + dt2

m X

i=1

ai log(Ni + 1)

i=1

(the second summand, coming from the archimedean places, is missing in [8]). Furthermore, the injection P ⊗d ,→ Na⊗dt1 yields that m Y di dim F ≤ M (dt1 ai )ui + o(du ) ui ! 0

i=1

and so log dim F = o(d). Hence, the Dirichlet exponent of the system can be estimated as dim F 2M 0 ≤ + o(1) dim Γ(Y, Qd ) u and the proposition follows from the Siegel lemma (Lemme 2.6 in [8]).

We now replace Y by a sufficiently small open subset of Y that contains x. According to Corollary 0.3, we can in particular assume that each section (i) V0 vanishes nowhere on this subset and suppose that the sheaf of differ(i) (i) entials ΩY/Q¯ is generated by the differentials of the Vj /V0 (i = 1, . . . , m, 1 ≤ j ≤ ui ). We can furthermore suppose that P, M and Na all can be trivialized over this subset. We fix an isomorphism Qd ' OY and consider the index σ (as defined in Section 5.2 of [8]) of the section sd ∈ Γ(Y, Qd ) that was constructed in the preceding proposition with respect to the weight dt1 a in x. Lemma 0.5. With notations as above, we have σ ≤ (4t1 max di (Ni + 1))−1 i

for d ∈ −1 N ∩ d0 N sufficiently large. Proof. We assume that the inequality is false and derive a contradiction. We can estimate m m Y Y −1 −1 σ di ≥ (4t1 max di (Ni + 1)) d−1 i i=1

i

i=1

≥ (8uθtu+1 mu max(Ni + 1))−1 1 i

m Y

0

d−ω . i

i=1

It then follows from (iii) that σ

m Y i=1

d−1 i

≥

(8uθtu+1 mu max(Ni 1 i

+ 1))

−ω 0 ψ(u)

−1

m Y

0

0

(deg Xi )−ω Λ−ω (ψ(u)−1)

i=1

and hence σ i=1 d−1 . i ≥ σ0 = mΛ Then, we can construct a multihomogeneous polynomial G(V ) of multi(i) degree dt1 (d1 · · · dm )a in the adapted coordinates Vj , 0 ≤ j ≤ ui , of height Qm

8

GABRIEL A. DILL

bounded by h(G) ≤ (d1 · · · dm ) h(sd ) + dt1

m X

! ai (h(Bi ) + log(2(ui + 1)))

+ o(d)

i=1

and of index at least σ with respect to the weight dt1 a. For this, it suffices to consider the homogenization of the norm of sd ⊗ ζ 0 (seen as a dehomoge(i) (i) ¯ nized polynomial in the Wj /V0 ) in the subfield of Q(Y) generated by the (i)

(i)

Vj /V0 (j = 1, . . . , ui ) with ζ 0 ∈ Zd0 which doesn’t vanish at x (see Lemme 5.5 of [8]). We can then apply Théorème 5.6 (Faltings’ product theorem) of [8] with the value of σ0 above and obtain in this way a contradiction with Proposition 0.2. The hypotheses of the theorem are satisfied, since u ai m 2 ≥ c2 ≥ ≥ (2u2 )u ai+1 σ0 and G has indexQat least σ with respect to the weight dt1 a in x, hence has −1 index at least σ m i=1 di ≥ σ0 with respect to the weight dt1 (d1 · · · dm )a in x. We obtain a pair (l, U ) with U V (l) (xl ) = 0, U non-zero, deg(U ) ≤ u 0 m = Λω uψ(u) and σ0 ! u m X h(G) m + ai (ui log(ui + 1) + log 2) al h(U ) ≤ ul σ0 dt1 d1 · · · dm i=1 u u m deg(U ) + ul m (ul + 1) log (ul + 1) + al log + o(1). +al σ0 σ0 ul After some simplification and by using that ul ≥ 1 (which follows from 0 Corollary 0.3(b)) and σm0 = Λω ψ(u) , we deduce that u m ω 0 uψ(u) h(sd ) al h(U ) ≤ ul Λ + 2Λh + 2al log 2 (ul + 1) + o(1) dt1 σ0 u 0 m ω 0 uψ(u) 2M ≤ ul Λ (1 + 2t2 + 3)Λh + 2al log + o(1). u σ0 0

For the last inequality, we used that 2al log 2(ul + 1) ≤ 2Λh and 2M u ≥ 2. We can now estimate u m 0 2al ul log ≤ 2Λh ω 0 uψ(u) log Λ ≤ Λh Λ(ω u−1)ψ(u) , σ0 1

0

since al ul ≤ Λh , 2ω 0 uψ(u) log Λ ≤ Λ 2 ω uψ(u) and 12 ω 0 u ≤ ω 0 u − 1. Thanks to (iii), we can bound the first term as m

Y 2M 0 ul 2M 0 ≤ ≤ 2M max (Ni + 1)m (2uθ)(t1 m)u deg(Xi )1+ω Λ(1+ω)(ψ(u)−1) i u i=1 m (1+ω)ψ(u)

≤ MΛ Λ

0

≤ M Λ(ω u−1)ψ(u) ,

since m + (2 + ω)ψ(u) ≤ ω 0 uψ(u). This last inequality follows from m ≤ uψ(u).


9

Combining these inequalities with the one above, we obtain that (2ω 0 u−1)ψ(u) 2ψ(u−1)−2ψ(u) M t2 al h(U ) ≤ Λ M Λh (2 + 2t2 + 3) ≤ Λ Λh , 4dl ψ(u)

where we used that 2 + 2t2 + 3 ≤ 5t2 ≤ Λ 4dl t2 by (ii). We could get rid of the o(1), since for example this last inequality is in fact strict. We now have established that the section sd ∈ Γ(Y, Qd ) given by Proposition 0.4 has index (in x and with respect to the weight dt1 a) bounded as σ ≤ (4t1 max di (Ni + 1))−1 . i

We write D for a differential operator associated to that index and finish the proof of Theorem 0.1 by considering the following height −hQd (x) = dh(Z(x)) − dh(Ξ(x)) + d =

m X

ai h(W (i) (x))

i=1 0 h(Zd (x))

− dh(Ξ(x)).

By definition, there exists such a D with D(sd )(x) 6= 0 and we have D0 (sd )(x) = 0 for every operator D0 of index σ 0 < σ, hence by the product formula h(Zd0 (x)) = h (D(sd ) ⊗ Zd0 )(x) = h (D(sd ⊗ ζ 0 )(x))ζ 0 ∈Zd0 . In order to define the right-hand side, one has to fix an isomorphism P ⊗d ' OY . The right-hand side is however independent of the choice of isomorphism, precisely since D is an operator associated to the index of sd . Let us recall that the sections sd ⊗ ζ 0 ∈ Γ(Y, P ⊗d ) are homogeneous polynomials of degree d in the sections Ξ and that log dim Γ(Y, P ⊗d ) = o(d). Furthermore, the sections Ξ themselves are monomials of multidegree t1 a in the coordinates W (i) . Hence, we can choose the isomorphism such that h(Zd0 (x)) ≤ h(sd ) + h ((D(ξ ν )(x))ξν ) + o(d), where ξ ν runs over the monomials of degree d in the sections Ξ (seen as monomials of multidegree dt1 a in the W (i) ) divided by appropriate products (i) of the V0 (i = 1, . . . , m). We can estimate the height of the D(ξ ν )(x) by using Leibniz’ formula as well as Corollaire 5.1 and Lemme 5.2 of [8] (corrected). For 1 ≤ i ≤ m and l ∈ (N ∪ {0})ui , we define the operator lj ui Y 1 i,l ∂ = ∂V (i) /V (i) : OY,x → OY,x . lj ! 0 j j=1

Lemma 0.6. Let 1 ≤ i ≤ m be an integer and let K be a number field that (i) contains the coordinates Wj (x), the families Bi and the products 

(i)

(i) bi,j

Y  Wj /V0 ci =  bi,j ! j

(i) ∂ bi,j Qj b ∂Tuii,j+1

1,

(i) V1 ,..., (i) V0

(i) Vui , (i) V0

b1

(i) ! Wj  (xi ) (i) V0

i,j

,

10

GABRIEL A. DILL (i)

where j runs over the indices satisfying Wj 6= 0 in Γ(Yi , Li ). Then for every place v of K and every multiindex l ∈ (N ∪ {0})ui , we have i,l (i) (i) (i) (i) |l| Wj /V0 (x) ≤ Wj /V0 (x) (|ci |−2 ∂ v Ci,v ) v

v

with Ci,v = 2−v

di 2(Ni +1) (i) (i) (di (Ni + 1))6di v max |b|v max Wk /V0 (x) b∈Bi

0≤k≤Ni

v

and v = 1 if v is infinite, 0 if v is finite. ¯ Proof. Recall that by Corollary 0.3, the number ci ∈ Q\{0} is well defined (up to the choice of root which can be made arbitrarily). (i) (i) (i) If Wj = 0 in Γ(Yi , Li ), the derivative ∂ i,l Wj /V0 is zero and the inequality holds. Otherwise, we may apply Corollaire 5.1 of [8] and follow the proof of Lemme 5.2 of [8] with N = Ni , using at the end that 1+ 32 di

2f2 (ui , di ) = 2(2ui +4)

di + ui 2(Ni +2) di + 1 2 di (2(Ni +1)di )2(Ni +1)di ui 2

is bounded from above by 3

3

22+ 2 di +2(Ni +1)di (Ni + 2)1+ 2 di (Ni + 1)2(Ni +2)di d5i ((Ni + 1)di )2(Ni +1)di 3

≤ (Ni + 1) 2

log 3 (Ni +1)+ 23 di + log (1+ 32 di )+5(Ni +1)di 5 2 di ((Ni

+ 1)di )2(Ni +1)di

5

≤ (Ni + 1) 2 (Ni +1)+2(Ni +1)di +5(Ni +1)di d5i ((Ni + 1)di )2(Ni +1)di ≤ (di (Ni + 1))12di (Ni +1) . Q i,κi , we obtain the following bound (cf. the proof of For D = m i=1 ∂ Proposition 5.3 in [8]) |D(ξ ν )(x)|v ≤ |ξ ν (x)|v

m Y

|κi | 2(|κi |+dt1 ui ai )v (|ci |−2 v Ci,v )

i=1

and hence thanks to the product formula for the ci ν

h ((D(ξ )(x))ξν ) ≤ dh(Ξ(x)) +

m X

2(Ni + 1)|κi |(h(Bi ) + di h(W (i) (x)))

i=1

+

m X

(dt1 ui ai log 2 + 12di (Ni + 1)|κi | log di (Ni + 1)).

i=1

We proceed with bounding |κi | ≤ dt1 ai σ ≤ (4(Ni + 1)di )−1 dai ,


P (i) which implies that m i=1 2di (Ni + 1)|κi |h(W (x)) ≤ with (3), the bound also implies that m X

d 2 hNa (x).

11

Together

2(Ni + 1)|κi |(h(Bi ) + 6di log di (Ni + 1))

i=1 m X

hi + di (ui + 1) log di (Ni + 1) + 3 log di (Ni + 1) ≤ 4dΛh . ≤ d ai 2di i=1 P Finally we know that m i=1 ui ai log 2 ≤ Λh and putting all these estimates together we get

hNa (x) − hM (x) = −

hQd (x) h(sd ) ≤ + hNa (x) + (t1 + 4)Λh + o(1). d d 2

Thanks to Proposition 0.4 and (1), it follows that 2M 0 hNa (x) − hM (x) ≤ (t1 + 2t2 + )Λh + (t1 + 4)Λh + o(1) 2 u 2M 0 ≤ (2t1 + 2t2 + 5)Λh + o(1) u 0 M t1 2ψ(u)−2ψ(0) −u 8(2 + 2t + 5)Λ (M t ) mc , ≤ 2 2 3 2 4 where the strict inequality in (1) allowed us to sweep the o(1) under the rug (for d large enough). We have 8(2 + 2t2 + 5) ≤ 42t2 ≤ Λ2 t2 and it follows from (iii) that !2(1+ω) m Y (M 0 t1 )−2 ≤ M t1 max (Ni + 1)m (2uθ)2 (t1 m)2u deg(Xi ) Λ2(1+ω)(ψ(u)−1) i

≤ MΛ

max{3,m}+2(1+ω)ψ(u)

i=1 2ω 0 uψ(u)−2

≤ MΛ

= M Λ2ψ(u−1)−2ψ(u)−2 ,

where we used that max{3, m} ≤ 4uψ(u) − 2. Hence, we can deduce that mc3 hN (x) − hM (x) ≤ (M t2 )−(u−1) Λ2ψ(u−1)−2ψ(u) ≤ hNa (x), 2 a 4 4 from which it follows that hNa (x) ≤ 4−1 hM (x). The theorem follows, since by (iii) !1+ω m Y 4−1 ≤ 8uθ(t1 m)u deg(Xi ) Λ(1+ω)(ψ(u)−1) ≤ Λ(1+ω)ψ(u)+2 i=1

and Λ(1+ω)ψ(u)+2 ≤ Λ

ω 0 uψ(u)

≤ c1 .

1. Acknowledgements I thank my advisor Philipp Habegger for his continuous encouragement and for many helpful and interesting discussions. I thank Philipp Habegger and Gaël Rémond for helpful comments on a preliminary version of this article.

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GABRIEL A. DILL

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