Éric Gaudron

SOME EXPLICIT COMPUTATIONS IN ARAKELOV GEOMETRY OF ABELIAN VARIETIES

Keywords: Abelian variety, theta function, Gromov lemma, jet of section, canonical polygon, successive slopes of tangent space, Faltings height, injectivity diameter, Matrix Lemma, Zarhin's trick

Given a polarized complex abelian variety (A, L), a Gromov lemma makes a comparison between the sup and L 2 norms of a global section of L. We give here an explicit bound which depends on the dimension, degree and injectivity diameter of (A, L). It rests on a more general estimate for the jet of a global section of L. As an application we deduce some estimates of the maximal slope of the tangent and cotangent spaces of a polarized abelian variety defined over a number field. These results are effective versions of previous works by Masser and Wüstholz on one hand and Bost on the other. They also improve some similar statements established by Graftieaux in 2000.

Introduction

Let (A, L) be a polarized abelian variety over a number field K. Its tangent space at the origin t A is a finite dimensional K-vector space which can be endowed with a structure of Hermitian vector bundle over the ring of integers O K : the integral structure is given by the tangent space at the origin of the Néron model of A over O K and the metrics at infinity are provided by the Riemann forms of the complex line bundles L induced by L on each A = A ⇥ Spec C for : K ,! C. This feature allows to consider the successive Arakelov slopes of t A in the sense of the 1995 Bourbaki seminar of Bost [START_REF] Bost | Périodes et isogénies des variétés abéliennes sur les corps de nombres (d'après D. Masser et G. Wüstholz)[END_REF].

The aim of our article is to provide some sharp and explicit bounds for the slopes of t A , in terms of the dimension, Faltings height and degree of A relative to L. The estimates obtained in Sections 4.2 (upper bound for the maximal slope) and 4.3 (lower bound for the minimal slope) improve on previous results by Graftieaux [Gr] and some of them are even best possible with respect to L. For example, such results enable to have a basis of t A made up of vectors whose height is well controlled (using connections between minima and slopes, see [START_REF] Gaudron | Minima and slopes of rigid adelic spaces[END_REF]§ 4]).

The main motivation for this text lies in the progress made during the last twenty years about the effective aspects of some problems of Diophantine geometry for abelian varieties. Indeed it can be observed that the emergence of the Faltings height of A in the proofs of the major theorems of [START_REF] Bosser | Logarithmes des points rationnels des variétés abéliennes[END_REF][START_REF] Gaudron | Formes linéaires de logarithmes effectives sur les variétés abéliennes[END_REF][START_REF] Gaudron | Théorème des périodes et degrés minimaux d'isogénies[END_REF] only rests on three basic tools:

A matrix lemma which bounds from below the norms of the non-zero periods of A for every : K ,! C, a formula for the Arakelov degree of the space of global sections of L equipped with a suitable Hermitian vector bundle structure over a finite extension of K, an explicit lower bound for the minimal slope of t A . If the origin of the first problem is located in an article of Masser [Mas], it has since been studied by several authors and an almost optimal solution was found by Autissier in 2013 [START_REF] Autissier | Un lemme matriciel effectif[END_REF]. Regarding the formula for the Arakelov degree of H 0 (A, L), it has been established by Bost in 1996 [Bo1] (see the end of § 2.3.2). These first two tools allow to build the third one via evaluation at order 1 of global sections of L as Bost explained in [START_REF] Bost | Périodes et isogénies des variétés abéliennes sur les corps de nombres (d'après D. Masser et G. Wüstholz)[END_REF]§ 5.3.4]. Following this, Graftieaux gave an explicit estimate for the minimal slope of t A , still used until now. Furthermore, the latest accurate period theorems obtained by the author and Rémond [START_REF] Gaudron | Théorème des périodes et degrés minimaux d'isogénies[END_REF] on one hand and the author and Bosser [BG] on the other hand brought to light the need for sharper statements, in particular from a numerical point of view. That is why it seemed useful to us to examine again the work of Graftieaux, taking into account the technological advances yielded by Autissier's matrix lemma. Here the use of the first jet evaluation map is done in a more direct way, without any reference to a so-called Shimura map, by applying a basic slope inequality. To do this, we have been led to consider the square of the polarization on A and to evaluate at a well-chosen torsion point of A so that the jet be onto. On the way, we revisit the so-called Gromov lemma (according to the terminology of Gillet and Soulé [START_REF] Gillet | An arithmetic Riemann-Roch theorem[END_REF]§ 5.2.3]) which makes a comparison between the supremum norm of a section of L (for any : K ,! C) and its Hermitian norm. This type of results turns out to be of great interest in other settings, such as the theory of linear forms in abelian logarithms [START_REF] Gaudron | Formes linéaires de logarithmes effectives sur les variétés abéliennes[END_REF].

Acknowledgements. I thank Pascal Autissier and Gaël Rémond for their numerous remarks which allowed me to significantly improve the results of this article.

Preliminary results

2.1.

Let A be a complex abelian variety of dimension g and L be an ample and symmetric line bundle over A. Let t A be the tangent space at the origin of A and exp A : t A ! A be the exponential map of A. The period lattice ⌦ A = ker exp A of A is a free Z-module with rank 2g. An Appell-Humbert data for L is composed of a complex Hermitian form H : t A ⇥ t A ! C (linear in the second variable), which is positive definite since L is ample and such that Im H(⌦ A , ⌦ A ) ⇢ Z, and a semicharacter :

⌦ A ! ({z 2 C ; |z| = 1}, ⇥) that is, for all !, ⇠ 2 ⌦ A , we have (! + ⇠) = (!) (⇠) exp (⇡i Im H(!, ⇠)).
The Hermitian form H, called the Riemann form of L, induces a Hermitian norm on the tangent space t A : for all z 2 t A , kzk L = H(z, z) 1/2 . Define the factor of automorphy a L :

⌦ A ⇥ t A ! C of L by 8! 2 ⌦ A , 8z 2 t A , a L (!, z) = (!) exp ⇣ ⇡H(!, z) + ⇡ 2 H(!, !)
⌘ and consider a trivialization : exp ⇤ A L ! O t A compatible with this factor. Thus we get an isomorphism s 7 ! # = (exp ⇤ A s) which associates to a global section s 2 H 0 (A, L) a holomorphic map

# : t A ! C satisfying #(! + z) = a L (!, z)#(z) for all z 2 t A and ! 2 ⌦ A . The map z 2 t A 7 ! |#(z)| exp ⇣ ⇡ 2 kzk 2 L ⌘
is invariant by translation by elements of ⌦ A . Hence, given z 2 t A and x = exp A (z) 2 A, we may endow the fiber L x = x ⇤ L with the (cubist) metric:

8s 2 H 0 (A, L), ks(x)k x ⇤ L = |#(z)| exp ⇣ ⇡ 2 kzk 2 L ⌘ .
That induces (at least) two norms on H 0 (A, L): the supremum norm k • k 1 defined by

8s 2 H 0 (A, L), ksk 1 = sup (ks(x)k x ⇤ L ; x 2 A), and the Hermitian norm k • k 2 defined by 8s 2 H 0 (A, L), ksk 2 = ✓Z A ks(x)k 2 x ⇤ L dµ(x) ◆ 1/2
where µ is the normalized Haar measure on the compact group of complex points of A.

2.2. Jets of sections.

2.2.1. Algebraic definition. Let S be a scheme and A ! S be a scheme with base S. Let L be an invertible sheaf on A. We assume that there exists a closed immersion " : S ! A. Let us denote by I the sheaf of ideals on A defined by " and by ⌦ A/S = I/I 2 the O A -module of relative differentials.

Let `be a nonnegative integer. When A ! S is smooth along " (I is regular), the inverse image " ⇤ I `/I `+1 is isomorphic to the symmetric power S ` " ⇤ ⌦ A/S . A section s 2 H 0 (A, L) vanishes along " up to order `when s 2 H 0 (A, I `⌦O A L). In that case, the jet of s of order `along ", denoted by jet `s("), is the image of s by the composed map

H 0 A, I `⌦ L ! H 0 A, I `/I `+1 ⌦ L Projection ? ? y formula H 0 S, S ` " ⇤ ⌦ A/S ⌦ " ⇤ L H 0 S, " ⇤ I `/I `+1 ⌦ " ⇤ L 2.2.2.
Analytic description of the jets. Let us come back to the case of a complex abelian variety: S = Spec C, A = A and L = L. Let x 2 A(C) and z 2 t A be a logarithm of x. This point can be viewed as a closed immersion x : Spec C ! A defined by the sheaf of ideals I. For `2 N and s 2 H 0 (A, I `⌦O A L), the element jet `s(x) is the image of s in the quotient

H 0 Spec C, x ⇤ I `/I `+1 ⌦ O A L ' S `(t v A ) ⌦ C x ⇤
L. This construction is compatible with the map s 7 ! # of § 2.1. If we consider a basis e = (e 1 , . . . , e g ) of t A , the jet of # of order `at z takes the form

jet `#(z) = X i 1 i 1 ! • • • i g ! ✓ @ @z 1 ◆ i1 • • • ✓ @ @z g ◆ ig #(z 1 e 1 + • • • + z g e g ) • (e v 1 ) i1 • • • (e v g ) ig where i = (i 1 , . . . , i g ) 2 N g is of length |i| = i 1 + • • • + i g equal to `. Let k • k be a norm on S `(t v A ).

The trivialization of exp ⇤

A L which induces the isomorphism s 7 ! # allows to connect the norm of jet `s(x) (with respect to S `(t v A ) ⌦ x ⇤ L) to that of jet `#(z) by the formula:

(1)

k jet `s(x)k S `(t v A )⌦x ⇤ L = k jet `#(z)k • k1k z = k jet `#(z)k exp ⇣ ⇡ 2 kzk 2 L ⌘ .
Afterward the tangent space t A will be endowed with a Hermitian inner product. If e is an orthonormal basis of t A , the chosen norm on S `(t v A ) will be defined by:

8 (x i ) |i|=`2 C ( `+g 1 g 1 ) , k X |i|=`x i (e v 1 ) i1 • • • (e v g ) ig k 2 S `(t v A ) = X |i|=`| x i | 2 • i 1 ! • • • i g ! `!
.

2.3. Elements of slope theory. We make some very brief reminders about Hermitian vector bundles over the spectrum of the ring of integers O K of a number field K and their slopes. In a second time we shall mention some classical statements about several Hermitian vector bundles attached to a polarized abelian variety. We refer to [START_REF] Bost | Périodes et isogénies des variétés abéliennes sur les corps de nombres (d'après D. Masser et G. Wüstholz)[END_REF]Appendix A] or to [START_REF] Gaudron | Minima and slopes of rigid adelic spaces[END_REF] for more details ⇤ .

A Hermitian vector bundle

E over O K is the data of a locally free O K -module E of finite type with Hermitian norms k • k E,v : E ⌦ v C ! R + at archimedean places v of K. The O K -module E brings about norms k • k E,v : E ⌦ O K K v ! R + at ultrametric places v of K, defined by: 8 ⇠ 2 E ⌦ OK K v , k⇠k E,v = inf (|x| v ; x 2 K v , ⇠ 2 x.E). Here K v is the completion of K at v equipped with the absolute value | • | v normalized in such a way that |p| v 2 {1, p, p 1
} for every prime p. The standard example is

E = O n K with |(x 1 , . . . , x n )| 2,v = ( P i |x i | 2 v ) 1/2 (x i 2 K v ) at the archimedean places. In that case one has |(x 1 , . . . , x n )| 2,v = max i |x i | v
when v is ultrametric. Every Hermitian vector bundle E can be obtained from this canonical one via an adelic matrix a = (a v ) v 2 GL n (A K ) and after the choice of a K-basis e = (e 1 , . . . , e n ) of E ⌦ O K K:

8 x = (x 1 , . . . , x n ) 2 K n v kx 1 e 1 + • • • + x n e n k E,v = |a v x| 2,v .
The datum of E amounts to the data of a K-vector space (still denoted by) E of finite dimension (namely E ⌦ O K K) with norms on the topological space E ⌦ K K v satisfying the above equality. A Hermitian subbundle of E is a subspace of E endowed with the restricted norms of E. The dual E v of a Hermitian vector bundle E is comprised of the dual space Hom K (E, K) with the dual norms of k • k E,v 's. If F ⇢ E is a subbundle, the quotient space E/F inherits of (quotient) norms from E, giving it a Hermitian vector bundle structure over O K . The (normalized Arakelov) degree of a Hermitian vector bundle E is

deg E = 1 [K : Q] X v place of K [K v : Q v ] log | det a v | v and its slope is µ(E) = (deg E)/ dim E.
It does not depend on the choice of (a, e) and the degree is invariant by finite extension of K. When F ⇢ E is a subbundle, we have deg E/F = deg E deg F . The canonical polygon P E of E is the piecewise linear function delimiting from above the convex hull of the set (dim E 0

, deg E 0

) where E 0 runs over the non-zero subbundles of E. It satisfies the noteworthy duality relation: for all 0  x  n = dim E, we have

P E v (x) = P E (n x) deg E.
⇤ Following the latter reference, we omit the traditional bar over Hermitian vector bundles and the hat over their degree and slopes.

We also have P E⌦L (x) = P E (x) + x deg L for any Hermitian line bundle L. The maximal slope of E is the real number µ max (E) = P E (1), also equal to max (µ(E 0

) ; E 0 ⇢ E). These notions are invariant by scalar extension. A Hermitian vector bundle E such that µ max (E) = µ(E) is called semistable. A K-linear map ' : E ! F between two Hermitian vector bundles E and

F over O K induces local maps ' v : E ⌦ K K v ! F ⌦ K K v at the places v of K, having operator norms k'k v = sup ✓ k' v (x)k F,v kxk E,v ; x 2 (E ⌦ K K v ) \ {0} ◆ .
By bringing together these norms, we form the height of ':

h(') = 1 [K : Q] X v [K v : Q v ] log k'k v .
This number allows to compare the canonical polygons of E and F . Proposition 2.1. Let E and F be some Hermitian vector bundles over O K and ' : E ! F be a K-linear map. If ' is injective then, for all x 2 [0, dim E], we have

P E (x)  P F (x) + xh('). If ' is surjective then, for all y 2 [0, dim F ], we have P F (y)  P E (y + dim ker ') + (dim F y)h(') deg E + deg F . Proof. Since the canonical polygon is linear on [i 1, i],
it is enough to prove these properties for x or y equal to an integer i. According to [START_REF] Gaudron | Minima and slopes of rigid adelic spaces[END_REF]Theorem 43], the j-th slope µ j (E) = P E (j) P E (j 1) of E satisfies the inequality µ j (E)  µ j (F ) + h(') when ' is injective and j 2 {1, . . . , dim E}. We add them for j ranging from 1 to i to get the first statement. The second one is a consequence: we apply the first result to the one-to-one dual map ' v : F v ! E v , using the duality formula for the canonical polygon and noting that ' v has the same height as '. ⇤ 2.3.2. Let A be an abelian variety over a number field K of dimension g. For every finite extension K 0 /K and every embedding : K 0 ,! C, we denote by A the complex abelian variety deduced from A K 0 = A ⇥ K K 0 by scalar extension through . Over a certain finite extension

K 0 /K, A K 0 is semistable. The sheaf of relative differentials ! A 0 /O K 0 of the Néron model A 0 ! O K 0 of A K 0 over K 0 , endowed with the norms ksk 2 = 2 g i g 2 R
A s ^s is a Hermitian vector bundle over O K 0 whose degree is the (stable) Faltings height h F (A) of A. Corollary 8.4 of [START_REF] Gaudron | Théorème des périodes et degrés minimaux d'isogénies[END_REF] states that h F (A) g log(⇡ p 2), named Bost's inequality afterwards. Furthermore, given an ample and symmetric line bundle L on A (the pair (A, L) is said to be a polarized abelian variety), we can consider the Hermitian tangent bundle (t A , k • k L ) over O K , also denoted t A when there is no ambiguity on L, composed of the tangent space at the origin of the Néron model of A over Spec O K and with the norms k • k L induced by the Riemann forms of the complex line bundles L ! A for all : K ,! C. Its degree is linked to the Faltings height of A by the formula

deg(t A , k • k L ) = ✓ h F (A) + 1 2 log h 0 (A, L)
◆ where h 0 (A, L) denotes the dimension of H 0 (A, L). In the same vein, the space of global sections H 0 (A, L) of L can be equipped with a Hermitian vector bundle structure over a suitable finite extension K 0 /K. The integral model is composed of H 0 (A 0 , L 0 ) where L 0 is a cubist Hermitian line bundle over the Néron model A 0 of A K 0 over K 0 . The metrics at the complex embeddings of K 0 are the norms k • k 2 on H 0 (A 0 , L 0 ) ⌦ C defined at the end of § 2.1 (see [START_REF] Gaudron | Formes linéaires de logarithmes effectives sur les variétés abéliennes[END_REF]§ 4.2.2]). Building on deep results by Moret-Bailly, Bost proved that the Hermitian vector bundle H [START_REF] Bost | Intrinsic heights of stable varieties and abelian varieties[END_REF]§ 4.3] and [Bo2, p. 129]).

0 (A 0 , L 0 ) (over O K 0 ) is semistable with slope equal to (1/2)h F (A) + (1/4) log(h 0 (A, L)/(2⇡ 2 ) g ) (see

Gromov lemma for the jets

Let us start with a Gromov lemma in its initial acceptance (according to [START_REF] Gillet | An arithmetic Riemann-Roch theorem[END_REF]§ 5.2.3]) before generalizing it to higher order jets.

3.1. Let (A, L) be a polarized complex abelian variety of dimension g. For s 2 H 0 (A, L), we always have ksk 2  ksk 1 . A Gromov lemma makes a comparison in the other direction. The following statement is an unpublished result of Autissier. It involves the injectivity diameter

⇢(A, L) = min (k!k L ; ! 2 ⌦ A \ {0}) of (A, L),
where k • k L is the Hermitian norm associated to the Riemann form of L.

Proposition 3.1. Assume (A, L) is principally polarized. Then, for all s 2 H 0 (A, L), we have

ksk 1  ksk 2 ⇥ max ✓ 1, 1 ⇢(A, L) ◆ g/2 (1.7g) g/2 .
The proof is based on some estimates for explicit theta functions.

Proof. Let ⌧ 2 M g (C) be a Siegel-reduced element of the Siegel upper half plane such that ⌦ A identifies with Z g + ⌧ Z g . Put Y = Im ⌧ . Then the norm on t A induced by L can be written as kzk L = t zY 1 z 1/2 and the one dimensional space H 0 (A, L) is generated by the theta function

8 z 2 C g , # ⌧ (z) = exp ⇣ ⇡ 2 t zY 1 z ⌘ X n2Z g exp (i⇡ t n⌧ n + 2i⇡ t nz). It is known that k# ⌧ k 2 = det(2Y ) 1/4 (see, for instance, [GR1, Lemma 8.2 (1)]). Moreover [Au2, Lemma 8.2] yields k# ⌧ k 1  # iY (0) = # iY 1 (0) (det Y ) 1/2 , from which we deduce k# ⌧ k 1 k# ⌧ k 2  (2 g det Y ) 1/4 # iY (0) = 2 g/4 (# iY (0)# iY 1 (0)) 1/2 . Besides [Au2, Proposition 8.4] applied to Y and Y 1 , with (Y ) = min n2Z g \{0} t nY n p 3/2 and (Y 1 ) ⇢(A, L) 2 , gives # iY (0)# iY 1 (0)  ✓ g + 2 2 ◆ 2 ✓ max ✓ g + 2 ⇡ p 3 , 1 ◆ max ✓ g + 2 2⇡⇢(A, L) 2 , 1 ◆◆ g/2 .
We combine with the previous estimate to get, for all s 2 H 0 (A, L) = C.# ⌧ ,

ksk 1  ksk 2 ⇥ max ✓ 1, 1 ⇢(A, L) ◆ g/2 ⇥ ✓ g + 2 2 ◆ ✓ 2 max ✓ g + 2 ⇡ p 3 , 1 ◆ max ✓ g + 2 2⇡
, 1

◆◆ g/4
.

The latter constant is smaller than (1.5g) g/2 when g 2. Otherwise, if g = 1, we proceed as above except that we keep the exact value of # iY (0):

k# ⌧ k 1 k# ⌧ k 2  (2Y ) 1/4 X n2Z exp ⇡n 2 Y =: ✓(Y ). Since Y = ⇢(A, L) 2 (see [GR1, Remark 3.3]), it is enough to prove that 8 Y p 3 2 , (2 min (1, Y )) 1/4 X n2Z exp ⇡n 2 Y  (1.7) 1/2 .
If

(Y ) = ✓(1/Y ) implies ✓ 0
(1) = 0 and so ✓ 0 (Y ) < 0 for all Y 2 ( p 3/2, 1), yielding the decreasing property of ✓. ⇤ Proposition 3.1 is best possible with respect to ⇢(A, L). Indeed [Bo2, Inequality (C.9)] claims that k# ⌧ k 1 1, that is, ksk 1 ksk 2 det(2 Im ⌧ ) 1/4 for all s 2 H 0 (A, L). When dim A = 1, the imaginary part of ⌧ is precisely ⇢(A, L) 2 as stated above. In this case, a global section s of the g-th external tensor power (B, M) of (A, L) satisfies ksk 1 ksk 2 p 2/⇢(A, L) g/2 . The relation ⇢(B, M) = ⇢(A, L) then shows that the exponent g/2 of the injectivity diameter in Proposition 3.1 cannot be improved in general. The same example also indicates that the numerical constant 1.7 is close to the best one, necessarily bounded from below by p 2 > 1.4.

Generalization to jets.

The keystone of this article is the following statement.

Theorem 3.2. Let (A, L) be a polarized complex abelian variety of dimension g. Let x 2 A, `2 N and s 2 H 0 (A, L) which vanishes at x up to order `. Then we have

k jet `s(x)k S `(t v A )⌦x ⇤ L  ksk 2 ⇥ h 0 (A, L) 1/2 max ✓ 1, 1 ⇢(A, L) ◆ g/2 (5(g + `)) g/2 ✓ ⇡ ! ◆ 1/2
(the norm on S `(t v A ) is the one induced by k • k L as explained at the end of § 2.2.2). Note that the dependence in h 0 (A, L) is accurate (at least for `= 0) since we always have

h 0 (A, L) 1/2  sup ✓ ksk 1 ksk 2 ; s 2 H 0 (A, L) \ {0} ◆ .
Indeed, for any orthonormal basis s 1 , . . . , s

⌫ of H 0 (A, L), consider b(x) = P ⌫ i=1 ks i (x)k 2
x ⇤ L for x 2 A, which does not depend on the choice of the basis. Let x 2 A such that b(x) = sup y2A b(y) and choose s 1 , . . . , s ⌫ such that s i (x) = 0 for 2  i  ⌫. Then we have

h 0 (A, L) = ⌫ = Z A b(y) dµ(y)  b(x)  ks 1 k 2 1 = ✓ ks 1 k 1 ks 1 k 2 ◆ 2
.

The proof of Theorem 3.2 is different from that of Proposition 3.1. It is inspired by the analytic estimates of [START_REF] Viada | Slopes and abelian subvariety theorem[END_REF]§ 3.1], partly derived from some lecture notes by Bost and it uses some ideas of Autissier and Rémond.

3.2.1.

Let us begin with some auxiliary results. Given `2 N, an element ' of the symmetric product S `(t v A ) is a sum of products of `linear forms on t A . It defines a polynomial map z 2 t A 7 ! '(z) by evaluating at z the linear forms which compose '. Given a real number r > 0, let us define the sesquilinear form on S `(t v A ) (linear on the right):

8 ', 2 S `(t v A ) h', i = Z B(0,r) '(t) (t) exp ⇡ktk 2 L dt
where B(0, r) = {u 2 t A ; kuk L < r} and dt is the normalized Haar measure of (t A , k • k L ).

Lemma 3.3. The product h•, •i is a Hermitian inner product on S `(t v A ). The proof is straightforward. The group G of isometries of the Hermitian space (t A , k • k L ) acts on S `(t v A ) by composition on the right: if g 2 G and ' 2 t v A then ' g 2 t v A . By [START_REF] Igusa | Theta functions. Die Grundlehren der mathematischen Wissenschaften[END_REF]§ I.4], this action is irreducible. Moreover the norms k

• k (associated to h•, •i) and k • k S `(t v
A ) are both invariant under the action of G. From these facts comes the following result.

Lemma 3.4. There exists a constant C > 0, which depends on g, `, r, such that k

• k = Ck • k S `(t v A ) . Proof. Let a = inf ⇣ k'k ; ' 2 S `(t v A ) and k'k S `(t v A ) = 1 ⌘ .
Since the infimum is a minimum by continuity on a compact space, the set S = {' 2 S `(t v A ) ; k'k = a} is not empty. This set and, a fortiori, the subspace Vect S of S `(t v A ) generated by S are G-invariant. The irreducibility property gives Vect S = S `(t v A ). Moreover if ' and belong to S then h', i = ah', i `where h•, •i `is the Hermitian inner product associated to the norm k

• k S `(t v A ) . Indeed, for all ⇠ 2 C, we have k' + ⇠ k ak' + ⇠ k S `(t v A )
. Squaring and expanding, we get Re ⇠h', i a Re ⇠h', i `and we conclude taking ⇠ 2 {±1, ±i}. This property entails that the set

V = u 2 S `(t v A ) ; 8' 2 S, h', ui = ah', ui ` is a linear subspace of S `(t v
A ), containing S and so Vect S. We deduce V = S `(t v A ) which leads to h', ui = ah', ui `for all ', u 2 Vect S = S `(t v A ). We put C = p a to conclude. ⇤ Lemma 3.5. The constant C in the previous lemma is characterized by

C 2 = `! ⇡ `(g 1 + `)! Z ⇡r 2 0 x `+g 1 exp ( x) dx
(here dx is the Lebesgue measure on R).

Proof. Let (e 1 , . . . , e g ) be an orthonormal basis of (t A , k • k L ). The dual basis (e v 1 , . . . , e v g ) yields an orthogonal basis (e v 1 )

i1 • • • e v g ig , i = (i 1 , . . . , i g ) 2 N g with |i| = `, of S `(t v A ). We have k (e v 1 ) i1 • • • e v g ig k S `(t v A ) = ✓ i 1 ! • • • i g ! `! ◆ 1/2
and, writing t = t 1 e 1 + • • • + t g e g with (t 1 , . . . , t g ) 2 C g , we also have

k (e v 1 ) i1 • • • e v g ig k 2 = Z |t1| 2 +•••+|tg| 2 r 2 |t 1 | 2i1 • • • |t g | 2ig exp ⇡ktk 2 L dt.
Thus we get the formula

C 2 = `! i 1 ! • • • i g ! Z ktk L r |t 1 | 2i1 • • • |t g | 2ig exp ⇡ktk 2 L dt
for all (i 1 , . . . , i g ) 2 N g of length `. Summing over all such i's and using the multinomial theorem, we obtain

✓ g 1 + `◆C 2 = Z B(0,r) ktk 2L exp ⇡ktk 2 L dt.
To get the desired formula, we compute the latter integral with Fubini's theorem ( R

B(0,r) = R r 0 ⇣ R ktk L =x ⌘
dx) using that the volume of the sphere of radius x:

lim "!0 + vol (B(0, x + ")) vol (B(0, x)) " = vol(B(0, 1)) ⇥ lim "!0 + (x + ") 2g x 2g " equals ⇡ g g! ⇥ 2gx 2g 1 (vol stands for the Lebesgue measure on (t A , k • k L )). We get C 2 = 2⇡ g `! (g 1 + `)! Z r 0 x 2`+2g 1
exp ⇡x 2 dx and we conclude with the change of variables x 7 ! y = ⇡x 2 . ⇤ 3.2.2. Let F be a fundamental domain of t A with respect to the period lattice ⌦ A . Given a positive real number r and z 2 t A , let us define N (u) = Card ⌦ A \ B(z u, r) for any u 2 t A (where B(z u, r) = z u + B(0, r) is the ball of radius r centered at z u). This integer only takes a finite number of values.

Lemma 3.6. For every integrable function f : t A ! R, periodic with respect to ⌦ A , we have Z

B(z,r) f (u) du = Z F f (u)N (u) du.
Proof. Let 1 denote the characteristic function of B(z, r). By definition we have

N (u) = X !2⌦ A 1(u + !).
Since F is a fundamental domain, the translated F + ! for ! 2 ⌦ A cover t A and their interiors are disjoint sets. We deduce

Z B(z,r) f (u) du = X !2⌦ A Z F+! f (u)1(u) du = Z F X !2⌦ A f (u + !)1(u + !) du.
It remains to observe that the latter sum can be written as

f (u)N (u) since f (u + !) = f (u) for all ! 2 ⌦ A . ⇤
Proposition 3.7. For any z 2 t A , `2 N and s 2 H 0 (A, L) which vanishes at x = exp A (z) up to order `, we have

k jet `s(x)k S `(t v A )⌦x ⇤ L  ksk 2 ⇥ ✓ h 0 (A, L) max u2F N (u) C 2 ◆ 1/2
where C is the constant of Lemma 3.4.

Proof. Let µ denote the normalized Haar measure on A. The inverse measure exp ⇤ A µ is a Haar measure on t A , which, by uniqueness up to a positive scalar, is proportional to the Lebesgue measure

dz associated to k • k L . Thus we have dz = h 0 (A, L) exp ⇤ A µ since Z F dz = vol(F) = covol(⌦ A , k • k L ) = h 0 (A, L).
Let # : t A ! C be the theta function associated to s (see § 2.1) and define

f (u) = |#(u)| 2 e ⇡kuk 2 L for all u 2 t A . The definition of the norm k • k 2 on H 0 (A, L) gives ksk 2 2 = Z A ks(x)k 2 x ⇤ L dµ(x) = Z F f (u) (exp ⇤ A µ) (u) = 1 h 0 (A, L) Z F f (u) du. With n = max u2F N (u), we have n R F f (u) du R F f (u)N (u) du and Lemma 3.6 yields ksk 2 2 1 nh 0 (A, L) Z B(z,r) f (u) du = 1 nh 0 (A, L) Z B(0,r) |#(z + u)| 2
e ⇡kz+uk 2 L du.

In the latter integral, we write kz + uk 2 L = kzk 2 L + 2 Re H(z, u) + kuk 2 L (H is the Riemann form of L) and we expand the holomorphic function ⇥(u) = #(z + u)e ⇡H(z,u) =

P i=(i1,...,ig)2N g a i u i1 1 • • • u ig g
in terms of the coordinates u 1 , . . . , u g of u in an orthonormal basis (e 1 , . . . , e g ) of t A . We get Z B(0,r)

|#(z + u)| 2 e ⇡kz+uk 2 L du = e ⇡kzk 2 L Z B(0,r) |⇥(u)| 2 e ⇡kuk 2 L du
and, since # vanishes at z up to order `, we have jet `#(z) = jet `⇥(0). Using Bessel's inequality, we deduce that ksk 2 2 ⇥ nh 0 (A, L) exp ⇡kzk 2 L is bounded from below by

X i2N g |a i | 2 Z B(0,r) |u 1 | 2i1 • • • |u g | 2ig exp ⇡kuk 2 L du X |i|=`| a i | 2 k(e v 1 ) i1 • • • (e v g ) ig k 2 = C 2 X |i|=`| a i | 2 i 1 ! • • • i g ! `! = C 2 k jet `⇥(0)k 2 S `(t v A ) = C 2 k jet `#(z)k 2 S `(t v A ) .
We conclude with formula (1) that yields the desired upper bound for the norm of jet `s(x). ⇤ 3.2.3. Proof of Theorem 3.2. Let F be a fundamental domain of t A and z 2 t A be a logarithm of the point x considered in Theorem 3.2. Let us choose r > 0 such that ⇡r 2 is the median of the (`+ g, 1) distribution, that is, such that Z ⇡r 2 0

x `+g 1 exp ( x) dx = (`+ g 1)! 2

.

In particular the constant C 2 of Lemma 3.5 equals `!/(2⇡ `). Besides [Ch, Theorem 1] provides the bound ⇡r 2  `+ g 1 + log 2. We apply Proposition 3.7 to these data F, z, r. To get Theorem 3.2, we have to prove that 2 max u2F N (u)  (5(g + `) max (1, 1/⇢(A, L)))

g . This follows from a result by Malikiosis [START_REF] Malikiosis | Lattice-point enumerators of ellipsoids[END_REF]Theorem 1.2], which gives an upper bound of N (u) in terms of the minima

⇢(A, L) = 1 (⌦ A )  • • •  2g (⌦ A ) of the Euclidean lattice (⌦ A , k • k L ): 8 u 2 F, N(u) = Card ⌦ A \ B(z u, r)  2g Y i=1 2r i (⌦ A ) + 1
⌫ (b•c refers to the integer part). We bound from below i (⌦ A ) by ⇢(A, L) for 1  i  g and by 1 for , p. 2072]). We get

g + 1  i  2g (thanks to the property 1  i (⌦ A ) 2g+1 i (⌦ A ) [GR2
max u2F N (u)  ✓ 2r ⇢(A, L) + 1 ◆ g b2r + 1c g  max ✓ 1, 1 ⇢(A, L) ◆ g ((2r + 1)b2r + 1c) g .
With the previous upper bound for r, the conclusion comes from

max x2N 1 2 x ✓ 2 p ⇡ p x 1 + log 2 + 1 ◆ 2 p ⇡ p x 1 + log 2 + 1 ⌫  5
(the worst case for the size of the constant is g = `= 1 corresponding to x = 2). ⇤

When `= 0, we could reduce the constant 5 to 3.9 (leaving 2 1/x instead of the first 2 in the above maximum) where Proposition 3.1 yields 1.7 (but only when h 0 (A, L) = 1) .

3.2.4. Remark. We could choose a (Hermitian) norm k

• k t A on t A different from k • k L provided there exists a positive real number ⇠ such that k • k L = ⇠k • k t A .
In that case, the norm on S `(t v A ) is that of induced by k • k t A and the norm of the jet jet `s(x) is simply multiplied by ⇠ `. This often happens in applications where k • k t A is the norm associated with a polarization L 0 on A while L = L ⌦n 0 (for some positive integer n). Then we have ⇠ = p n and the minima of

⌦ A satisfy i (⌦ A , k • k L ) = p n i (⌦ A , k • k L0 ).
We keep this p n in the Malikiosis bound of N (u) which is used in the proof of Theorem 3.2. Let us introduce further an additional parameter ↵ > 0 to bound

2r ⇢(A, L) + 1  ✓ 2r p n + ↵ ◆ max ✓ 1 ↵ , 1 ⇢(A, L 0 ) ◆ .
Thus, for every u 2 F, we get

N (u)  max ✓ 1 ↵ , 1 ⇢(A, L 0 ) ◆ g ✓✓ 2r p n + ↵ ◆ 2r p n + 1 ⌫◆ g .
In this way, a numerically sharper version of Theorem 3.2 is: For all ↵ > 0, x 2 A, `2 N, n 2 N 1 and s 2 H 0 (A, L) which vanishes at x up to order `, the norm of jet `s(x) relative to

S `((t A , k • k L0 ) v ) ⌦ x ⇤ L is smaller than ksk 2 ⇥ h 0 (A, L) 1/2 max ✓ 1 ↵ , 1 ⇢(A, L 0 ) ◆ g/2 ⇥ ✓ 2(⇡n) ! ◆ 1/2 ⇥ ✓✓ 2r p n + ↵ ◆ 2r p n + 1 ⌫◆ g/2
with r 2 = (`+ g 1 + log 2)/⇡.

4. Application to Arakelov geometry of abelian varieties Henceforth, K is an algebraic closure of a number field K and (A, L) is a polarized abelian variety over K, with dimension g 1.

Let

H = H 0 (A, L ⌦2 ) ⌦ K K.
For every a 2 A(K), the evaluation map (0)

a : H ! a ⇤ L ⌦2 K , s 7 ! s(a), is a non-zero K-linear form since L ⌦2 is generated by global sections [START_REF] Mumford | Abelian varieties[END_REF]p. 60].

Let

(1) a : ker

(0) a ! t v A K ⌦ K a ⇤ L ⌦2 K , s 7 ! jet 1 s(a)
, be the jet of order 1 at a. By duality,

(1) a is surjective if and only if the complete linear system H separates tangent vectors at a: 8 t 2 t A K \{0}, 9 s 2 ker (0) a such that jet 1 s(a)(t) 6 = 0. In other words,

a is surjective when the differential at a of the finite morphism : A K ! P(H) built from H is injective. Such a a exists by generic smoothness of : there exists a non-empty open subset U ⇢ A such that : U ! (U ) is étale. By density of torsion points, we can even assume that a is a torsion point of A. Then let us consider a Moret-Bailly model

(A, L 2 , " a : Spec O K 0 ! A) of ⇣ A K , L ⌦2 K , {a} ⌘ over a finite extension K 0 /K, in the sense of [Bo1, § 4.3.1]. The maps (0) a and
(1) a associated to the polarized abelian variety

(A K 0 , L ⌦2 K 0 ) and a 2 A(K 0
) extend to homomorphisms of O K 0 -modules 0 : H 0 (A, L 2 ) ! " ⇤ a L 2 and 1 : ker 0 ! t v A ⌦ " ⇤ a L 2 . The norms on L 2 equip the modules ker 0 ⇢ H 0 (A, L 2 ) and " ⇤ a L 2 of a structure of Hermitian vector bundle over O K 0 (see § 2.3.2). Although they could endow t A with a Hermitian vector bundle structure over O K 0 , using the Riemann forms of (L 2 ) 's ( : K 0 ,! C), it seems more convenient to choose the structure given by (t A , k • k L ) extended to K 0 . The important point is that the norms given by (L 2 ) and L satisfy the proportionality condition of Remark 3.2.4 (with n = 2). In this context the maps 0 and 1 have some heights which can be evaluated in a simple way thanks to Theorem 3.2. For the approximated form given in the proposition, we set apart the case g = 2 for which we numerically compute the values of c 0 (2) and c 1 (2) (they determine the constants 5 and 10.2).

Then, for g 3, we use b⇠c  ⇠ and ` 1  0 in the last two logarithms of c `(g). We obtain a new simplified constant c 0 `(g) for which the function g 7 ! (c 0 `(g) (3g/4) log g)/g is clearly decreasing and we estimate c 0 0 (3) and c 0 1 (3) to conclude. The same method applies when g 16, distinguishing the cases g = 16 and g 17. As for the variant with max (1, h F (A)), it is the one we have in the matrix lemma. We multiply 2.3 + 5.5g by 8 (twice) in the constant c `(g) and we proceed as above. Moreover the same bound holds with max (1, h F (A)) instead of max 1, h F (A) + (1/2) log h 0 (A, L) if we change 15.4 by 26.6. When g 16 we can substitute the numerical constant 15.4 by 5.9.

.

  Then we obtain the desired estimate for the height of `with the constant c `

  ⇤ 4.2. Maximal slope of the tangent space.Proposition 4.2. The maximal slope of (tA , k • k L ) satisfies µ max (t A )  h F (A

The author was supported by the ANR grant Gardio 14-CE25-0015.

Moreover we can replace max 1, h F (A) + (1/2) log h 0 (A, L) by max (1, h F (A)) in these bounds provided we change the couple (5, 10.2) of numerical constants by (8.5, 18). Besides, when g 16, the couple (5, 10.2) can also be substituted by (2.58, 2.84).

The proof of this statement relies on a so-called matrix lemma, involving the injectivity diameters ⇢(A , L ) of the complex abelian varieties (A , L ) (where is an embedding of K into C), deduced from the polarized abelian variety (A, L) defined over K by scalar extension via .

Matrix Lemma. We have

In addition, the latter maximum can be replaced by 8 max (1, h F (A)).

In the main, this result is due to Autissier [START_REF] Autissier | Un lemme matriciel effectif[END_REF]. More precisely this is a consequence of Corollary 1.4, ibid., where " = 1 6/(2.3⇡) is chosen. Then the general case is deduced from [GR1, Lemmas 3.4 and 3.5], in the same way as Proposition 3.6, ibid. The variant with 8 max (1, h F (A)) arises from the first four lines of [START_REF] Gaudron | Théorème des périodes et degrés minimaux d'isogénies[END_REF]p. 358] (Zarhin's trick).

Proof of Proposition 4.1. Given : K 0 ,! C, we apply Theorem 3.2 in the form described in Remark 3.2.4 (to get sharper numerical constants) to the polarized abelian variety (A , (L 2 ) ) at 0 for `2 {0, 1}. We choose L 0 = L , n = 2 and ↵ = p e/(2.3 + 5.5g). The definition of L 2 yields h 0 (A , (L 2 ) ) = 2 g h 0 (A, L) and we get

with r 2 = (`+ g 1 + log 2)/⇡. As for the ultrametric norms of `, they are smaller than 1 since ìs a homomorphism of O K 0 -modules, modules from which the norms have been defined. Now we make use of the elementary [START_REF] Bosser | Logarithmes des points rationnels des variétés abéliennes[END_REF]Lemma 3.19] to bound from above

) and the statement occurs from Bost's inequality h F (A) log(⇡ p 2). Thus, from now on, we may assume g 2. Because the map 1 introduced in the previous paragraph is onto by construction, we can apply Proposition 2.1 with the dual map v 1 , which is injective, and x = 1:

. The maximal slope involved here on the left decomposes into a sum of the maximal slope of t A (with the metrics described in the previous paragraph) and deg " ⇤ a L 2 . This last degree equals 0 since it is the Néron-Tate height of the torsion point a relative to L ⌦2 [START_REF] Bost | Intrinsic heights of stable varieties and abelian varieties[END_REF]Theorem 4.10]. Hence we have

To control the other maximal slope, let us now apply the second part of Proposition 2.1 with the surjective restriction map H 0 (A, L 2 ) v ! (ker 0 ) v , whose height is nonpositive, and y = 1. This gives

Since the duality reverses the sign of the degree, the sum of the two last quantities is the degree of H 0 (A, L 2 )/ ker 0 . This Hermitian line bundle is isomorphic to " ⇤ a L 2 via the map induced by 0 , hence its degree equals deg

The semistability of H 0 (A, L 2 ) and the formula for its slope given at the end of § 2.3.2 lead to

We conclude with Proposition 4.1 and g log ⇡ + (3g/4) log(5 ⇥ 10.2)  (3g/2) log 15.4. The other constants stem from the same inequality where (5, 10.2, 15.4) is replaced by (8.5, 18, 26.6) and, when g 16, by (2.58, 2.84, 5.9). ⇤

Taking the g-th external tensor power of an elliptic curve endowed with its principal polarization, we can prove that (3g/2) log 15.4g which is in the upper bound of µ max (t A ) cannot be replaced by a function smaller than g (using Bost's inequality). Furthermore, we can deduce from Proposition 4.2 some simpler but weaker estimates, getting rid of the logarithm with the following elementary result.

Lemma 4.3. Let x 0 2 R and c > 0. Then, for every x 2 R, we have

Proof. When x 0  x  1, we have log max (1, x) cx = cx  cx 0 . When x 1, we observe that the function x 7 ! log x cx has a global maximum at x = max (1, 1/c). ⇤

As a straightforward consequence of this lemma and Proposition 4.2, we deduce:

Corollary 4.4. For all " > 0, the maximal slope

For instance, we have

and, when g 16, µ max (t A )  1.5h F (A) + log h 0 (A, L) + 2g log 3g.

Proof. Apply Lemma 4.3 with x = h F (A)+0.5 log h 0 (A, L), x 0 = g log(⇡ p 2) and c = 2"/g. That gives an upper bound for log max (1, h F (A) + (1/2) log h 0 (A, L)) and we conclude with Proposition 4.2. The next upper bound derives from the choice " = 0.2 (setting apart the trivial case g = 1) whereas we choose " = 1/2 in the last (with 5.9 instead of 15.4). ⇤

Using the second assertion of Proposition 4.2, we can also prove

With Zarhin's trick, the degree of A can even be removed from these estimates.

Corollary 4.5. We have

Proof. The case g = 1 is treated with Bost's inequality, as at the beginning of the proof of Proposition 4.2. For g 2, there exists a principal polarization L 0 on Z(A) = A 4 ⇥ b A 4 ( b A is the dual abelian variety of A) such that, if ◆ denotes the injective map A ,! Z(A) on the first component, [START_REF] Rémond | Conjectures uniformes sur les variétés abéliennes[END_REF]Lemma 4.6]). In particular we get an isometric injection from

16 it remains now to apply the last estimate in the previous corollary to (Z(A), L 0

) whose Faltings height and dimension are eight times those of A in order to conclude. ⇤

This estimate is optimal with respect to h 0 (A, L) in the sense that there exists an abelian variety A but no unbounded from above function f A : N ! R such that (?) µ max (t A )+f A (h 0 (A, L))  0 for any polarization L on A. Indeed, let us fix an elliptic curve E over K with its canonical polarization L 0 . Define the abelian variety A = E ⇥ E endowed with L n = L ⌦n 0 ⇥ L 0 where n is any positive integer. Then h 0 (A, L n ) = n and

Hence, if (?) were true for (A, L n ), the integer n should be bounded.

Maximal slope of the cotangent space.

In this paragraph we bound from above the maximal slope of the dual of the Hermitian vector bundle (t A , k • k L ). First, let us observe that this maximal slope equals to P t A (g 1) deg t A . The choice of the metrics on t A made in § 4.1 ensures that the canonical polygon of t A is P t A ⌦(" ⇤ a L2) v . Let us apply Proposition 2.1 twice: once with the injective map v 1 and again with the restriction map H 0 (A, L 2 ) v ! (ker 0 ) v (like in the proof of Proposition 4.2). This gives 1). The semistability of H 0 (A, L 2 ) allows to compute its canonical polygon and we get

. The formulas given in § 2.3.2 and Proposition 4.1 yield

where

(21.9 is an upper bound for 10.2 ⇥ ⇡ 2/3 ). As in Proposition 4.1, we can improve this estimate with respect to h 0 (A, L) at the cost of a small loss on the constant:

Let us also state the more manageable following result.

Proposition 4.6. We have

Proof. When g = 1 we have µ max (t v A ) = h F (A) + 1 2 log h 0 (A, L) and the statement comes from Bost's inequality. When g 2, we apply Lemma 4.3 with x = h F (A), x 0 = g log(⇡ p 2) and c = 2/(5g). We substitute the upper bound of log max (1, h F (A)) obtained this way in the previous estimate for µ max (t v A ) and we conclude noting that g 2 log 15.2g is an upper bound for (g 2 /4) max 0.2 log 2⇡ 2 , log(5g/2e) + (3g 2 /4) log 38.7g because the maximum is always attained for the second term (since g 2) and (5/2e) 1/4 ⇥ (38.7) 3/4  15.2. ⇤

If we proceed in the same way using the bound with the numerical constant 21.9 instead of 38.7, we get

In terms of Deligne's normalization h(A) = h F (A) + g 2 log ⇡ of the Faltings height, we may replace the constant 10 by 10/⇡ 0.3  7.1 and then compare with [START_REF] Graftieaux | Formal groups and isogeny theorem[END_REF]Proposition 2.14] giving the estimate µ max (t v A )  (g + 1) h(A) + (1/2) log h 0 (A, L) + 2g 5 log 2, less accurate when g 2. Contrary to the maximal slope of t A , the one of t v A cannot be bounded independently from h 0 (A, L) since

However the same technics lead to the following statement.

Proposition 4.7. We have

Again this estimate is best possible with respect to h 0 (A, L) by the same argument given for the optimality of Corollary 4.5.

Proof. As in the proof of Corollary 4.5, let us assume g 2 and consider a principal polarization L 0 on Z(A) = A 4 ⇥ b A 4 such that (t A , k • k L ) is seen as a subbundle of (t Z(A) , k • k L 0 ). In particular we have P t A (x)  P t Z(A) (x) for all 0  x  g. Then µ max (t v A )+deg t A = P t A (g 1)  P t Z(A) (g 1). We estimate this last number following the same method as above, paying attention to the fact that (A, L 2 , 0 , 1 ) are now built with (Z(A), L 0

) and therefore the dimension and Faltings height of the abelian variety are multiplied by 8. Then Proposition 4.7 follows with the constant 4g 2 log ⇡ + 2g 2 log 8 + 6g log(2.58 ⇥ 8g) + 6g(g 1) log(2.84 ⇥ 8g).

We conclude with ⇡ 2/3 ⇥ 2 ⇥ 2.84 ⇥ 8 < 98. ⇤

Éric Gaudron

Université Clermont Auvergne CNRS, LMBP F-63000 Clermont-Ferrand France Eric.Gaudron@uca.fr


Ryota Mori
email: r-mori@mpat.go.jp

Daniel Delahaye
email: delahaye@recherche.enac.fr

Simulation-Free Runway Balancing Optimization Under Uncertainty Using Neural Network

Keywords: simulated annealing, arrival manager, convolutional neural network I

This paper proposes a new optimization scheme using neural network for runway balancing to minimize departure and arrival aircraft delay. While other researchers have proposed solutions to the runway balancing problem using a simulationbased technique to calculate aircraft delay, the proposed method replaces the simulation by a neural network model-based estimation using the actual operational data, thus providing the following two advantages. First, accurate estimation of aircraft delay can improve the solution of the runway balancing problem. Second, the simulation process is not required in the optimization. Although it is difficult to develop an accurate simulation model especially under uncertain environment, the neural network model can estimate the average delay without explicitly modeling uncertainty. In this paper, as a first step, the effectiveness of the proposed method is validated through simulations. First, simulations considering uncertainty are used to generate the data, which are then used to train the neural network. The neural network predicts the delay under the current traffic and only this predicted delay is used for the runway balancing optimization with simulated annealing. The simulation result shows that the result by neural network outperforms the one by the simulationbased method under uncertainty. This means that the neural network can accurately estimate the delay under uncertainty environment, and is applicable in the optimization process.

INTRODUCTION

Airport runway is a main bottleneck of air traffic control system, and thus more efficient runway usage is needed to maximize capacity and optimize traffic handling [START_REF]2016-2030 Global Air Navigation Plan[END_REF]. Apart from new runway construction, there are two main approaches to increase runway capacity: take-off/landing separation reduction and runway balancing. The former is a straightforward idea to increase the runway capacity, and various methods are proposed such as optimal aircraft sequencing considering wake turbulence category, time-based separation applied under strong head wind, and development of new separation standard (RECAT) [START_REF] Delahaye | RECAT-EU, European Wake Turbulence Categorisation and Separation Minima on Approach and Departure[END_REF]. The latter (runway balancing) aims at optimal runway allocation of takeoff/landing aircraft to multiple runways. Although this method works only when there are multiple runways at the airport, it is one of the promising solutions for major airports in the world with multiple runways. This research focuses on runway balancing.

There have been many prior studies to optimize the runway allocation of departures and/or arrivals. To optimize the runway allocation, there are several types of the objective functions to be minimized depending on the problems, such as aircraft delay [START_REF] Cheng | Air traffic control using genetic search techniques[END_REF][8] [START_REF] Xie | A hybrid metaheuristic for multiple runways aircraft landing problem based on bat algorithm[END_REF], makespan [START_REF] Ng | Robust aircraft sequencing and scheduling problem with arrival/departure delay using the min-max regret approach[END_REF], and combination of makespan/aircraft delay and other factors such as noise, fuel consumption, and congestion level [START_REF] Hu | An efficient genetic algorithm with uniform crossover for air traffic control[END_REF] [START_REF] Fritzsche | Potential of Dynamic Aircraft to Runway Allocation for Parallel Runways[END_REF][6] [START_REF] Liang | Integrated sequencing and merging aircraft to parallel runways with automated conflict resolution and advanced avionics capabilities[END_REF][10] [START_REF] Ma | Integrated optimization of terminal maneuvering area and airport at the macroscopic level[END_REF] [START_REF] Malaek | A new scheduling strategy for aircraft landings under dynamic position shifting[END_REF]. The runway allocation is a combinatorial optimization, and various optimization methods are applied such as mixed integer linear programming [START_REF] Roling | Flexible Arrival & Departure Runway Allocation Using Mixed-Integer Linear Programming[END_REF], genetic algorithm [START_REF] Cheng | Air traffic control using genetic search techniques[END_REF][4], simulated annealing [START_REF] Salehipour | An efficient hybrid metaheuristic for aircraft landing problem[END_REF][9] [START_REF] Ma | Integrated optimization of terminal maneuvering area and airport at the macroscopic level[END_REF], ant colony optimization [START_REF] Bencheikh | Improved ant colony algorithm to solve the aircraft landing problem[END_REF], swarm intelligence algorithm [START_REF] Ng | Robust aircraft sequencing and scheduling problem with arrival/departure delay using the min-max regret approach[END_REF], bat algorithm [START_REF] Xie | A hybrid metaheuristic for multiple runways aircraft landing problem based on bat algorithm[END_REF], greedy method [START_REF] Fritzsche | Potential of Dynamic Aircraft to Runway Allocation for Parallel Runways[END_REF], and dynamic programming [START_REF] Malaek | A new scheduling strategy for aircraft landings under dynamic position shifting[END_REF]. Also, many researchers optimize the aircraft sequencing and runway allocation simultaneously. Operational constraints, such as conflict on air routes and taxiing, are also considered in some studies [START_REF] Ma | Integrated optimization of terminal maneuvering area and airport at the macroscopic level[END_REF]. Most researchers consider deterministic environment only, and just a few researches are found to take uncertainties into account. However, uncertainty exists in real world, and the uncertainty of take-off time is usually greater than that of landing time because the boarding process includes large uncertainty. Therefore, uncertainty becomes critical when the runway handles departure aircraft.

To tackle uncertainty, several approaches are considered. One is robust optimization. The robust optimization assumes a certain time window of uncertainty, and the best strategy is found under the worst scenario. Runway sequencing has been optimized with robust optimization as shown in [START_REF] Heidt | Robust Runway Scheduling Using a Time-Indexed Model[END_REF]. With this method, the optimization is converted to a deterministic optimization problem, so the deterministic optimization method can be applied. The optimal parameters found for the worst case scenario are not necessarily optimal for various uncertain scenarios, however. Another method is to run numerous simulations by changing uncertainty parameters, and sets of simulation results are evaluated by the expected value (sample average approximation) or its alternatives. This method is referred to as a simulation method here. There are some researches using the simulation method for runway balancing and sequencing [7][15]. This method can consider many uncertain scenarios, but requires large computational time.

However, both methods discussed above have a common problem: they require an accurate runway simulation model, and especially the following important parameters should be found: take-off/landing separation, interaction of departure and landing, and their uncertainty. The optimal runway balancing, for example, is usually obtained by minimizing the objective function that is estimated based on such simulation models. Intricate airport operations make it difficult to develop an accurate simulation model. If the simulation model is inaccurate, the obtained optimal solutions becomes less reliable. To tackle this issue, the authors propose a new optimization approach without using a simulation or parameter estimation of airport operations. The objective function must be calculated for optimization, but the authors propose that this be done by a neural network (NN) trained with the actual operational data directly. Since the NN training process requires both inputs and outputs from actual data, the NN can learn the actual operation environment directly by the actual data. In addition, the actual operation includes various types of uncertainties, which are difficult to be handled in the optimization. However, the NN is trained to minimize the error between the model output and data output, which means that the NN estimates output of the expected value considering uncertainty. This approach should work in theory, but investigation is necessary to evaluate its performance in practice. Therefore, in this paper, the above simulation-free approach is formulated and the result is compared to the simulation-based approach. One of the advantages of the proposed method is the use of the actual operational data directly with no simulation environment required. However, to validate the proposed method, as a first step, the simulated data including uncertainty instead of realworld data is used for NN training, which makes it possible to evaluate the obtained NN performance. This paper starts with the problem formulation in Sec. II, and the NN model is developed in Sec. III. Sec. IV shows the simulation environment and results, and Sec. V concludes this paper.

II. PROBLEM FORMULATION

A. Airport operation and simulation method

This time, Tokyo International Airport (Haneda Airport) is set as a target airport, and the runway balancing problem is considered. Fig. 1 shows the airport layout and runway operations under north wind. Runway A is used for arrival only, and is independent of traffic on other runways. On the other hand, Runway C is used for both departure and arrival i.e. mixed-mode operation. Runway D is used for departure only, but the arrival to Runway C also affects the departure from Runway D. Under this condition, the runway balancing for arrivals between Runways A and C is considered.

In this paper, as a first step, the above runway operation is simply described in the following mathematical forms.

                          Earliest landing time , nom i i i i r r t PLDT w i A    where w r1,r2 1 2 , A r r R   :
Additional flight time required when the nominal runway is r 1 and the landing runway is r 2 .

1 2 1 2 , 0 120 r r if r r w otherwise       Earliest take-off time i i t PTOT i D  
 Arrival is always given priority over departure.

To simplify the problem, the take-off and landing separations are set the same regardless of the wake turbulence category, and are equal to 120 s. The departure-arrival (C runway-C runway) or arrival-departure (C runway-C runway and C runway-D runway) separations are also set to 120 s. N(a,b) indicates the normal distribution with average a and standard deviation b, so the separation is not constant but randomly distributed. Arrival is always given priority over departure, so the departure from both C runway and D runway cannot take off if arrival at C runway is approaching and enough separation cannot be established.

The objective function to be minimized is the total delay of all aircraft. However, the landing runway re-assignment in general requires additional ATC workload, so the number of runway re-assignment is also included in the objective function, defined as the following form where  is the weight factor. Here, the runway re-assignment indicates the case where the nominal and the actual runways are different.

     i i i i i i D i A J t ETOT t ELDT            
If PLDT and PTOT of all aircraft are known in advance, it is easy to optimize the runway balancing, which is not the case in the real world. Both have a considerable uncertainty, but departure aircraft tend to have larger uncertainty than arrivals, because the departure aircraft can start pushback only when all passengers are onboard, which causes the large uncertainty. The estimated time of PLDT and PTOT is denoted by ELDT (Estimated Landing Time) and ETOT (Estimated Take-off Time), respectively. This time, the following uncertainty value is assumed according to past researches [16][17].

 (0, 300 ) 15 min (0,120 ) N s i f T ETOT PTOT N s otherwise           (0, 0.02 ) ELDT PLDT N T     
where T  is the time difference between ELDT/ETOT and the current time. This means that the uncertainty of ELDT is linear to the estimated flight time to the runway with the standard deviation of 2% flight time. The uncertainty of ETOT is assumed to become small at 15 minutes before ETOT, because at this time the largest contributor to the uncertainty, i.e. the boarding process, is completed.

The decision variables are the landing runway of each arrival aircraft only. However, the last minutes change of landing runway is not possible considering both ATC and pilot workload perspectives, so it is assumed that the runway decision must be made 30 minutes before ELDT. Therefore, the runway decision must be made under uncertainty.

B. Optimization method

The optimized parameters are i i A    . However, all i 

do not need to be optimized at the same time, because aircraft due to depart further in the future cause little impact on the runway decision of the aircraft 30 minutes before the landing.

As in other researches, this time, the sliding windows approach is applied [START_REF] Ma | Integrated optimization of terminal maneuvering area and airport at the macroscopic level[END_REF]. The sliding window approach optimizes the aircraft with a certain time window of ELDT only, and the target aircraft are changed as time proceeds. This optimization process is done repeatedly, with the latest result being used for decision making.

Assuming that there are n aircraft to be optimized, the possible combination is 2 n . If n is sufficiently small, the objective function for all combinations is calculated, and the best one is chosen. However, when n increases, it is impossible to calculate all combinations. Therefore, simulated annealing is applied to find the best solution. The simulated annealing is a metaheuristic optimization algorithm, and imitates the annealing process in metallurgy. The simulated annealing is often used to solve combinatorial optimization problems, and is therefore applicable here as well. The general optimization process of the simulated annealing is shown as follows.

III. NN DEVELOPMENT FOR OBJECTIVE FUNCTION CALCULATION

A. Inputs and outputs of the network

In this research, the authors propose that the objective function is calculated by data-trained NN. Therefore, first the

Initialization i:=i 0 , T:=T 0 While T > T final do Generate solution j near solution i if f(i) > f(j) then i := j else i := j with probability of ( ) ( ) exp f i f j T        end if Compute T i becomes the final solution.
NN inputs and outputs should be decided. First the inputs are considered. The possible inputs are the ETOT/ELDT of each aircraft and its runway information. This time, to represent both ETOT/ELDT and runway information, the number of aircraft on each runway at each time slot are set as inputs. Fig. 2 shows the image of the inputs. There are four different queues in this runway operation (runway A for arrival, runway C for arrival, runway C for departure, and runway D for departure). In each queue, the number of aircraft is set to each time slot of ETOT/ELDT. The size of time slot is set to 120 s. The time slot starts with the current time and ends at the end of sliding window. Since the runway re-assignment is considered within the sliding window only, the aircraft out of the sliding window do not affect the decision of the runway re-assignment. In this example, 1 hour sliding window is assumed, and 4*30 inputs are made. This input size (1 hour sliding window) is used for NN development. As for the outputs, the objective function is the sum of the delay and the number of runway re-assignment. Only the delay should be calculated by NN, and the number of runway reassignment is implicitly incorporated in the decision variable. This time, the following four output values are set as NN outputs.

   1 { | } j j j j Ar a o t E L D T              2 { | } j A j j j Ar c o t E L D T              3 { | } i D i i i D r c o t E T O T              4 { | } i i i i D r d o t E T O T          
Each output is affected by different inputs (aircraft). Since it is easier to develop several small networks than a single large network, four separate NNs are made. The sum of all four outputs corresponds to the delay of all aircraft, which exactly matches the objective function given in Eq. ( 1) except the number of runway re-assignment. Each network requires only the necessary inputs that affect the output. Table 1 summarizes the used input in each network.

B. NN structures

Next, NN structures are considered. There are various types of possible networks, such as feedforward fully-connected NN (FFNN) and convolutional NN (CNN). To estimate the delay of aircraft, a single aircraft usually affects the nearby aircraft, and queue length is propagated to the later aircraft. As for FFNN, all inputs are connected, but most connections are actually not needed for estimating the delay. The unnecessary connections often cause over-fitting and results in the failure of training a network. On the other hand, CNN connects the neighborhood inputs only, and smaller network can be created. A popular application of CNN is image processing. Although the inputs are 2 dimensional in image processing, time sequence data in this research is considered as 1 dimensional data, and CNN is applied. Fig. 3 summarizes the NN structures used in this research. The first and second layers use a convolutional layer, and the third and fourth layers use a fully-connected layer. Table 2 summarizes the detailed values of the network. Since Network3 and Network4 have more inputs than Network1 and Network2, the NN size is set larger in Network3 and Network4. However, the general NN structures are set the same for all networks.

C. Training data and NN training

Training data sets are required to train the NN. In reality, the actual operational data is used, but this time data must be generated via simulations. The data should include various scenarios, and various patterns of runway balancing, otherwise the data would be biased, and the appropriate NN cannot be made. Therefore, through a data generation process, NN is trained iteratively and its output is used for runway assignment of each arrival aircraft in the simulation. In order to simulate various situations,  is randomly set with distribution of scale parameter 600 s, and the runway assignment is randomly set with a probability of 0.05.

The simulation parameters are given later in Sec. IV A. The data is obtained every minute for 3 hours, so a single simulation can generate 180 data sets. By running about 1700 times of simulations, 300,000 data sets are created.

Once the training data is obtained, NN is trained. The wellknown training algorithm Adam [START_REF] Kingma | Adam: a Method for Stochastic Optimization[END_REF] is applied here. During a training process, the generalization capability is a big issue. Generalization refers to the ability of the NN to produce reasonable outputs for inputs that are not encountered during training. NN tries to minimize the loss function between model output and trained data output. Since both inputs and outputs data usually include noise, minimizing the error loses the generalization (called over-fitting). There are various ways to avoid over-fitting, but one method is to collect sufficient number of data.

However, the possible number of data obtained is also limited if this process is implemented using real operational data. As for the airport operational data, it is assumed that each day consists of 16 operational hours, with data obtained every 1 minute. For 1 year, 60*16*365=350,400 is the maximum possible number of data obtained. However, a good NN could not be made with 350,400 data sets according to the authors' pre-calculation. Consequently, data augmentation technique is used. The data augmentation is the technique where the data is artificially made based on the obtained data. This technique is often used in the image processing field. In this field, the data augmentation is done by various ways such as rotating the image, reflecting the image, and changing the scaling.

In this research, data augmentation is done by the following way. 1) copy the input/output from the original data. 2) pick up one aircraft from the input data. 3) the aircraft moves to the next time slot. This process is very easy, but it should work because ETOT/ELDT includes uncertainty, so similar output is expected even if the ETOT/ELDT is slightly changed. This time, it is assumed that 300,000 data are available in advance. First, the data is augmented by 9 times in each original data set, and the number of data becomes 3,000,000 (300,000*(9+1)). Second, the data is split into 70 % training data (2,100,000) and 30% test data (900,000). The NN is trained with the training data, and the NN where loss function between model output and test data output is the minimum is used as the obtained NN. Also, batch training is applied, and the batch size is set to 2048. As for the loss function, the mean squared error (MSE) is used.

IV. SIMULATION RESULTS

A. Simulation parameters

Before conducting a simulation, several simulation parameters need to be set. First, the traffic volume is decided as shown in Table 3. There are three scenarios considered. Scenario 1 is low traffic, Scenario 2 is heavy traffic, and Scenario 3 covers various traffic. Without the optimization of runway re-assignment, there is an optimal ratio of runway balance for both departures and arrivals, and the nominal ratio should not be significantly different from the optimal ratio. Therefore, the nominal ratio of arrival/departure is also set in the scenario. As for the departure, 0.5 is the best ratio of departure, so it is randomly set between 0.4 and 0.6 in the simulation. As for the arrival, 0.85 is the best ratio of arrival for A Runway, and distributed between 0.75 and 0.95. Scenario 3 is used for NN training as explained in Sec. III, and Scenarios 1 and 2 are used to investigate the simulation results.

In the simulation, first, PTOT/PLDT are randomly distributed for 3 hours, and the simulation is conducted until all departure and arrival aircraft take off or land. The optimization of runway assignment of each arrival aircraft is conducted every 10 minutes, and the runway decision is made 30 minutes before ELDT. Since the time window for the optimization is set for the next 60 minutes, the runway is optimized for arrival aircraft where ELDT is between t + 30 and t + 60. The runway actually assigned is the one obtained in the latest optimization process. Next, Table 4 shows the parameters of simulated annealing. If the number of possible combinations is less than the number of iterations, the objective functions of all possible combinations are calculated, and the best one is chosen, which is actually a deterministic method.

B. Calculation of objective function without considering uncertainty

To proceed with the optimization process, the objective function must be calculated. In this research, the authors propose to calculate it by NN. However, the simulation method is also used here as a benchmark. To calculate the objective function, the simulation method developed in Sec. II A is used, but no uncertainty is considered, i.e. the simulation runs assuming that

i i PLDT ELDT i A    , i i PTOT ETOT i D    , and
120 ij S s 
for given conditions. Since this simulation does not consider uncertainty, the objective function calculated by this simulation differs from that by the simulation including uncertainty. This discrepancy can cause the inappropriate assignment of arrival runway for each aircraft. In reality, the separation 120 ij S s  is also sometimes difficult to be identified, but this time it is assumed that the "average" separation is already known in advance. From now, the case where the objective function is calculated by the simulation without considering uncertainty is denoted by SIM method.

C. Comparison SIM method and NN method

To compare the results obtained by SIM method and NN method, 100 times simulations are conducted in each scenario (1 and 2). Since the same random seed is used between SIM method and NN method, completely the same initial values, the same randomized separations, and the same randomized ETOT/ELDT errors are used in both methods. The optimization calculation is conducted with various  in the objective function between 0 s and 2400 s, which models the cost of a single runway re-assignment.

First, the accuracy of NN performed estimations are discussed. Fig. 4 shows the delay estimation result of both methods by arrival on Runway A (which corresponds to o 1 ), while Fig. 5 shows the result by departure on Runway C (which corresponds to o 3 ). The other two values (o 2 and o 4 ) also show a similar trend. The total delay in this result indicates the sum of the delay for next 60 minutes time windows, and is used to calculate the objective function. Also, this is the result of Scenario 2 where  is 60 s. The root mean square error (RMSE) values are also included in the result. As expected, the RMSE by NN is smaller than that by SIM for all cases, which means that the NN can estimate the total delay more accurately than SIM method. The delay of arrival is estimated better than that of departure, because the uncertainty of arrival is smaller than that of departure. The SIM method estimates the relatively smaller delay than the actual delay. This can be understood by the queueing theory. According to the queueing theory, the arrival process on Runway A can be described as M/G/1 queue [START_REF] Bhat | An Introduction to Queuing Theory[END_REF], and the average waiting time increases if there exists uncertainty of the service time (which corresponds to the landing separation here). SIM method does not consider the uncertainty, so its estimate of the waiting time tends to be lower than the actual one. On the other hand, the NN can estimate the delay better than the SIM method, because NN models the delay considering uncertainty. As for the departure, the delay is estimated less accurately than the departure, because the uncertainty of the departure is assumed to be much larger than that of the arrival. However, the NN can estimate the delay slightly better than SIM method when measured by RMSE. To analyze the estimation accuracy in more detail, Fig. 6 shows the RMSE of each method in each range of actual total delay by arrivals on Runway A, while Fig. 7 shows the result by departures on Runway C. In general, the RMSE tends to increase with the actual total delay. As for arrivals, NN always estimates the delay better than SIM method. However, as for departures, SIM method sometimes estimates the delay better than NN when the actual total delay is large. This can also be understood by Figs. 4 and5. As for arrivals, the number of plots becomes small as the actual total delay increases, but there are enough number of plots even when the actual total delay is large. However, as for departures, the number of plots gets significantly small when the actual total delay is larger than 5000 s. Since NN is trained by the data, the training process does not work well if there is not enough training data. According to Fig. 7, NN can always estimate the delay better than the SIM method where there seems to be sufficient data. However, NN can estimate the delay better than SIM method in most cases, and the difference of the delay accuracy between two methods (SIM and NN) can affect the solution of the runway balancing.

Next, Fig. 8 shows the total delay after the optimization by both methods under scenario 1 (low traffic), while Fig. 9 shows the result under scenario 2 (heavy traffic). Here, the total delay means that the sum of delay for all aircraft for a single simulation. The dotted line indicates the "no optimization" result. Therefore, the results of both methods converge to "no optimization" when the number of runway re-assignment is 0. In each parameter of , the optimization is conducted in each method, and the following 12 values of  are used. [0, 60s, 120s, 180s, 240s, 360s, 480s, 600s, 900s, 1200s, 1800s, 2400s].

When traffic volume is small (Fig. 8), the average flight delay is also small compared to that in the heavy traffic case. Also, the possible reduction of delay is also limited. Since there are 20*2(dep/arr)*3(hours)=120 aircraft in a single simulation, 1 s average flight delay corresponds to 2 minutes total delay. SIM method does not estimate the objective function very accurately, so the delay increases when  is small because unnecessary runway re-assignment can increase the delay due to the term 1 2 , r r w . However, that is not the case when using NN, and the minimum delay is observed when  is 0. While SIM method can reduce the waiting time by 8.7 s per aircraft via runway balancing, NN can reduce the waiting time by 11.8 s per aircraft, or additional 36% savings. On the other hand, when the number of runway re-assignment is small, the possible delay reduction is almost the same for both SIM and NN methods. That could be because "large benefit" runway reassignment strategy is the same regardless of the consideration of uncertainty. If the number of runway re-assignment is limited, the advantage of the proposed method may be limited.

As for heavy traffic (Fig. 9), the delay per aircraft is large, and the possible delay reduction is also large. However, the difference between SIM and NN methods is not quite big. SIM method can reduce the delay by 64.0 s per aircraft, while the NN method can reduce the delay by 68.2 s per aircraft, about 7 % larger. This could be because many departure aircraft exist anyway under heavy traffic, so even if one departure aircraft has large uncertainty, another departure aircraft can compensate it, and the uncertainty effect could be reduced. However, NN method still outperforms the SIM method. These results infer that the proposed method using NN is very promising. This time, simulation environment is developed in a simple manner, which is advantageous for SIM method. However, in reality, much more complicated airport operation is conducted, so the advantage of the proposed data-driven NN method will be significant in the real world. V. CONCLUSIONS This paper proposed a new scheme to optimize a runway balancing problem using NN. NN modeling approach does not require any explicit operational models and simulations, and relies on actual operational data only. This means that simulation parameters such as departure/arrival separations and its interactive effect are not required, because these characteristics are expected to be modeled by NN. This paper showed the effectiveness of the proposed method via simulations. The uncertainty effect was appropriately modeled by NN, and the solution of the runway balancing was improved. This time, relatively simplified airport operation was assumed, so it will be interesting to investigate how much NN can handle more complicated operation. Also, the proposed method will be applied using the actual operational data, and its ability will be investigated. These will be subjects of the future work.

Figure 1 .

 1 Figure 1. Airport layout and runway configuration under north wind at Tokyo International Airport.

,,

  where c D is Runway C used by departures and d is Runway D.  A: Set of arrival aircraft {1,..., } where a is Runway A and c A is Runway C used by arrivals.  PTOT i : Earliest possible take-off time for aircraft i. i D    PLDT i : Earliest possible landing time for aircraft i. runway for departure and arrival.

  : binary variable for arrival runway decision.

Figure 2 .

 2 Figure 2. Representation of NN inputs.

Figure 4 .Figure 5 .

 45 Figure 4. The estimated total delay by NN and SIM methods compared to the actual delay by arrivals on Runway A. (RMSE=809s by NN, RMSE=1122s by SIM)

Figure 6 .

 6 Figure 6. RMSE in each total delay range by arrivals on Runway A.

Figure 7 .

 7 Figure 7. RMSE in each total delay range by departures on Runway C.

Figure 8 .

 8 Figure 8. Average additional flight time (flight delay) per aircraft on Scenario 1.

Figure 9 .

 9 Figure 9. Average additional flight time (flight delay) per aircraft on Scenario 2.

TABLE I .

 I INPUTS AND OUTPUTS USED IN EACH NETWORK.

	Network name	Inputs		Outputs
	Network1	Arr-A (30 inputs)		o 1
	Network2	Arr-C (30 inputs)		o 2
	Network3	Arr-C, Dep-C (30*2 inputs)	o 3
	Network4 o Layers Arr-C, Dep-D (30*2 inputs) Variables Values
			Network1,2 Network3,4
	1 st	Number of out channels	128		256
	(convolution)	Strides	3		3
		Kernel size	3		3
	2 nd	Number of out channels	128		256
	(convolution)	Strides	2		2
		Kernel size	2		2
	3 rd	Number of hidden nodes 128		256
	(fully-connected)
	4 th	Number of hidden nodes 128		256
	(fully-connected)
	1 st -4 th	Activation function	ReLu[18]	ReLu
	Output	Activation function	Linear		Linear

4 Figure 3. NN structures used in this research.

TABLE II. THE PARAMETERS IN NN USED IN THIS RESEARCH.

TABLE III .

 III TRAFFIC VOLUMES IN EACH SCENARIO.

		Traffic [aircraft/hour]
	Scenario	1	2	3
	Total arrival	20	32	15-34
	Total departure	20	32	15-34
	Nominal ratio of arrival for A	0.75-0.95	0.75-0.95	0.75-0.95
	Runway
	Nominal ratio of departure for C	0.4-0.6	0.4-0.6	0.4-0.6
	Runway
	Total number of aircraft	120	192	90-204

TABLE IV .

 IV PARAMETERS OF SIMULATED ANNEALING.

	Parameters	Values
	Number of iterations	1024
	Initial temperature	15000
	Terminal temperature	7.5