Claus Lehr

Michel Matignon

Maximal wild

come

Introduction

Let R be a complete discrete valuation ring of mixed characteristic (0, p) with field of fractions K containing the p-th roots of unity. This work is about semi-stable models of p-cyclic covers of the projective line C -→ P 1 K . It is a continuation of our paper . Denote by K the minimal extension of K over which C has a stable model. Using the techniques developed in that paper we study the wild finite monodromy extension, i.e. the extension corresponding to the Sylow p-subgroup of Gal(K /K).

In section 2, we review some basic definitions and the results from that are essential to this paper. Among these are the stably marked models and the monodromy polynomial. Associated with C/K are the finite monodromy group and related Galois groups which will be recalled briefly.

The Galois group of the finite monodromy extension acts on the special fiber C R ⊗ R k of the stable model:

Gal(K /K) → Aut(C R ⊗ R k)
The bounds derived from Theorem 1.1 for the right hand side also limit the possible size of Gal(K /K). Hence one can ask if v p (|Gal(K /K)|) can be maximal? (Here v p is the p-adic valuation with v p (p) = 1). This is one central question of the paper and we study it in section 3 for covers that are defined over Q tame p . The reason for working over this ground field is that we are mostly interested in the wild part of the monodromy. We encounter various situations where the above can be affirmatively answered. (cf. Prop.3.2b,Rem.3.4,Prop.3.6,Exa.3.8,Prop.3.10 and Exa.3.12). Section 3.1 provides families of curves with potentially good reduction and a criterion for when there is maximal finite monodromy. In this case the finite monodromy group turns out to be an extraspecial p-group.

Section 3.2 deals more specifically with the case of genus 2 curves and residue characteristic 2. Here we have shown that any type of degeneration with maximal wild monodromy is possible.

Section 3.3 relates our work to results by Silverberg and Zarhin on the monodromy of abelian surfaces (cf. , ). In particular we give an example of nontrivial monodromy in unequal characteristic; something they were asking for.

The Magma code we have used to construct the examples is available from the authors upon request. To give an idea we have included it here only in one example (cf. Exa. 3.8).

Background

This is a brief review of the results proven in . The reader is encouraged to take a look at that paper for more details and motivation. Let R be a complete mixed characteristic (0, p) discrete valuation ring with field of fractions K and algebraically closed residue field k. Denote by v the valuation defined by R on K and by π a uniformizer. We will always assume that R contains a primitive p-th root of unity ζ and define λ = ζ -1. Then v(λ p-1 ) = v(p). Let C -→ P 1 K be a p-cyclic cover of smooth projective geometrically irreducible K-curves with genus(C) ≥ 2. We assume that the branch locus B of the cover consists of K-points, write m = |B| -1, and assume that B has equidistant geometry in the sense of the following definition.

Definition 2.1. We say that B has equidistant geometry if there exists a smooth R-model Proj(R[u, w]) for P 1 K such that the points of B have distinct specializations on the special fiber of the model.

We can assume that, with respect to the coordinate X 0 := u/w, the cover is given birationally by the equation Z p 0 = f (X 0 ) with f (X 0 ) ∈ R[X 0 ] monic, n = deg(f (X 0 )), (n, p) = 1 and n ≤ m(p-1). Then any two distinct zeros of f (X 0 ) specialize to distinct elements in k. We denote by K(C)/K(X 0 ) the function field extension corresponding to the cover C -→ P 1 K . With the above notation, by Deligne and Mumford (cf. [De-Mu] Cor.2.7 and [Des] Prop.5.7, Lemme 5.16), there exists a minimal extension K /K such that C K has a stable model C • R over the integral closure R of R in K . Further K /K is Galois and acts faithfully on the special fiber:

Gal(K /K) → Aut k (C • R ⊗ R k) ( 1 )
Definition 2.2. Let K /K be the minimal extension over which C has a stable model. We call Gal(K /K) the finite monodromy group associated to C. The extension K /K is called the finite monodromy extension, Gal(K /K) has a unique p-Sylow subgroup which we call the wild monodromy group Gal(K /K) w . The field extension K /K Gal(K /K)w is called the wild monodromy extension.

Let K /K be as above. Then by Prop.2.4 there exists a semi-stable model model C R , defined over the integral closure R of R in K minimal with the property that the points in the ramification locus of C K -→ P 1 K specialize to distinct smooth points on the special fiber. Further the finite monodromy group operates faithfully on its special fiber:

Gal(K /K) → Aut k (C R ⊗ R k) ( 2 )
Definition 2.3. The semi-stable model C R , given above, is called the stably marked model. We refer to the component to which the ramification points specialize as original component. Any model obtained from the stably marked model by a base change to a ring R ⊃ R we refer to as a stably marked model.

In [START_REF] Raynaud | p-groupes et réduction semi-stable des courbes[END_REF] Thm.2 Raynaud shows that the dual graph of the special fiber of a stably marked model of C is an oriented tree whose origin corresponds to the original component.

We will frequently use the following result which was shown in Prop.2.7.

Proposition 2.4. Let C -→ P 1 K be as above. Assume that for each irreducible component E of genus > 0 in the stable reduction there exists a flat quasi-projective

R-model C for C, such that C ⊗ R k is birational to E. Then C has a stable model over R.
As stated before, we shall always assume that the branch locus of C -→ P 1 K has equidistant geometry and consists of rational points. This has consequences for the image of the injection (2). Namely, any element of its image will have trivial action on the original component of the stably marked model. The group of such automorphisms we denote by Aut

k (C R ⊗ R k) # . Hence we have Gal(K /K) → Aut k (C R ⊗ R k) # (3)
Definition 2.5. If the injection (3) is surjective we say that C has maximal monodromy.

If v p (|Gal(K /K)|) = v p (|Aut k (C R ⊗ R k) # |)
we say that C has maximal wild monodromy.

The rest of this section introduces the monodromy polynomial and its main properties. (cf. section 3). Let m be the number of distinct zeros of f (X 0 ) in an algebraic closure of frac(R) and r the greatest integer such that rp < n. For

X 0 = X + Y Taylor expansion yields f (X 0 ) = f (X + Y ) = s 0 (Y ) + s 1 (Y )X + s 2 (Y )X 2 + • • • + s n (Y )X n ∈ R[X, Y ] (4)
which we view as a polynomial in two variables.

Definition 2.6. If f (X 0 ) as above can be written

f (X + Y ) = s 0 (Y ) H(X, Y ) p - n i=r+1 A i (Y )X i (5) with H(X, Y ) = 1 + a 1 (Y )X + a 2 (Y )X 2 + • • • + a r (Y )X r ∈ K(Y )[X], A i (Y ) ∈ K(Y )
and r as introduced earlier, we call this a special decomposition of f (X 0 ).

The existence of such a decomposition is shown in

Lemma 3.3. We set S 1 (Y ) = s 1 (Y )/gcd(s 0 (Y ), s 1 (Y )) and S 0 = s 0 (Y )/gcd(s 0 (Y ), s 1 (Y )). Then S 1 (Y ), S 0 (Y ) ∈ R[Y ], f (Y )/f (Y ) = S 1 (Y )/S 0 (Y ), deg(S 1 (Y )) = m-1, deg(S 0 (Y )) =
m and (S 0 (Y ), S 1 (Y )) = 1. Also, for t ∈ N, we will use the following notation

1 p t = t-1 i=0 1 p -i t! .
For later use we note that the p-adic value of these binomial coefficients is given by

-v p ( 1 p t ) = t + [t/p] + [t/p 2 ] + . . . ( 6
)
where v p (p) = 1.

Lemma 2.7. There exist

c i ∈ K, T (Y ) ∈ R[Y ] and N i (Y ) ∈ R[Y ] monic such that A i (Y ) = c i N i (Y ) s 0 (Y ) i for r + 1 ≤ i ≤ n and A p α (Y ) = - 1 p p α-1 p S 1 (Y ) p α + pT (Y ) S 0 (Y ) p α with v p (c p α ) = v p ( 1 p p α-1 p ) ≤ 0
where p α is the power determined by p α < n < p α+1 .

Definition 2.8. In the situation of Lemma 2.7 we define

L(Y ) = S 1 (Y ) p α + pT (Y ).
It is a polynomial of degree p α (m -1) and we call it the monodromy polynomial of f (Y ).

From the definition one immediately obtains L(Y

) ≡ S 1 (Y ) p α mod p. Notice that f (Y )/f (Y ) = S 1 (Y )/S 0 (Y ).
Definition 2.9. The valuation on K(X 0 ) corresponding to the DVR R[X 0 ] (π) is called the Gauss valuation v X 0 with respect to X 0 .

We then have From C R we obtain the stably marked model C R for C by a series of blowups, after passing to a finite extension R . The uniqueness of C R implies that the p-cyclic group action on C extends to C R and the quotient by this action is a semi-stable model for P 1 K , where K = frac(R ) (cf. [START_REF] Raynaud | p-groupes et réduction semi-stable des courbes[END_REF] appendice). This semi-stable model is obtained from Proj (R [u, w]) by a series of blowups. By construction, these are all centered on the affine patch Spec(R [X 0 ]) and hence correspond to ideals of the form (X 0y, ρ) ⊂ R [X 0 ]. We call y the center and ρ the radius, and often speak of the Gauss valuation v X with X = (X 0y)/ρ instead. All blowups we will consider are of this type and on Proj(R alg [u, w]). Blowing up (X 0y, ρ) ⊂ R [X 0 ] gives an exceptional divisor which in turn, by normalization in K (C), yields irreducible components of the stably marked model. The DVRs at their generic points are the ones that dominate the DVR of v X .

v X 0 m i=0 a i X i 0 = min{v(a i )|0 ≤ i ≤ m}.
A local computation shows that the centers specialize to points below the singularities (cusps) of C R ⊗ R k and are outside the specialization of the branch locus. Their X 0coordinates are zeros of f (X 0 ), hence also zeros of S1 (X 0 ) = f (X 0 )/gcd( f (X 0 ), f (X 0 )). In particular, if X 0 = y is such a center, then y ∈ R alg and f (ȳ) ∈ k-{0}. A schematic picture of this situation is given in figure 1.

b) It is immediately derived from the equation that the cover of special fibers C R ⊗ R k -→ P 1 k is purely inseparable. Therefore, to yield a component of genus > 0 in the stable reduction of C, a blowup (on Proj(R[u, w])) must have a radius ρ with v(ρ) > 0.

Figure 1

: C R ⊗ R k -→ P 1
k with singularities and branch locus Now we can state the main theorem of , describing a stably marked model. Suppose that the cover is given birationally by Z p 0 = f (X 0 ) with f (X 0 ) monic of degree n prime to p and such that, with respect to X 0 , the branch points have distinct specializations. Let r be the greatest integer such that rp < n and let α be determined by p α < n < p α+1 . We write |B| -1 = m and can assume that f (X 0 ) has been chosen such that m(p -1) ≥ n. Choosing f (X 0 ) this way limits the size of L(Y ). We point out that, due to the freedom in choosing f (X 0 ), the polynomial L(Y ) is not uniquely determined by the cover C -→ P 1 K .

Theorem 2.11. The components of genus > 0 of the stably marked model of C correspond bijectively to the Gauss valuations v X j with ρ j X j = X 0y j where y j is a zero of the monodromy polynomial L(Y ) and ρ j ∈ R alg is such that

v(ρ j ) = max{ 1 i v λ p A i (y j ) for r + 1 ≤ i ≤ n}.
The dual graph of the special fiber of the stably marked model of C is an oriented tree whose ends are in bijection with the components of genus > 0.

Corollary 2.12. Let K /K be the minimal extension such that C K has a stable model over the integral closure R of R in K and denote by F the splitting field of L(Y ) over K. Let E be the field obtained from F by adjoining ρ i as well (s 0 (y i )) 1/p for each i. Then K ⊆ E. In particular C has a stable model over the integral closure of R in E.

3 Maximal wild monodromy over Q tame p In this section, we illustrate the effectiveness of our methods in order to construct covers with maximal wild monodromy over a local field K ⊂ Q tame p , i.e. a finite tamely ramified extension of Q p . We work under the hypotheses of section 2. In particular C -→ P 1 K is a p-cyclic cover of the projective line with branch locus B. Recall that there exists a minimal extension K /K such that C has a stably marked model

C R over the integral closure R of R in K. The group Gal(K /K) injects into Aut k (C R ⊗ R k) # and, if v p (Gal(K /K)) = v p (Aut k (C R ⊗ R k) #
), we say that the cover has maximal wild monodromy (cf. Definition 2.5).

Potentially good reduction with m = 1 + p s

In this section we analyze the finite monodromy for a family of curves with potentially good reduction. We will see that the wild finite monodromy group can, but needs not to be maximal with respect to the injection (3). We denote by m the maximal ideal of R alg and assume that C -→ P 1 K is given birationally by the equation

Z p 0 = f (X 0 ) = 1 + cX q 0 + X q+1 0 for c ∈ R, q = p n and n ≥ 1.
Clearly the branch locus has equidistant geometry and m = 1+q. In the following, when considering α ∈ R alg , we will write α 1/p for a p-th root of α. The formulas we get will be valid independently of the choice of the p-th roots.

Definition 3.1. For p > 2 we define the two groups E(p 3 ) and

M(p 3 ) by E(p 3 ) = x, y|x p = y p = [x, y] p = [x, [x, y]] = [y, [x, y]] = 1 and M(p 3 ) = x, y|x p 2 = y p = 1, y -1 xy = x 1+p
. These are the two non-abelian p-groups of order p 3 . The group E(p 3 ) has exponent p and M(p 3 ) has exponent p 2 . We write Q 8 and D 8 for the quaternion and dihedral groups of order 8. If G is an extraspecial p-group then |G| = p 2n+1 for some n > 0 and the following four types occur:

I.a) If p > 2 and exponent(G) = p then G is isomorphic to the central product of n copies of E(p 3 ). I.b) If p > 2 and exponent(G) = p 2 then G is isomorphic to the central product of M(p 3 ) with n -1 copies of E(p 3 ). If p = 2 then either II.a) E = D 8 * D 8 * • • • * D 8 is the central product of n factors D 8 . Or II.b) E = Q 8 * D 8 * • • • * D 8 is the central product with n -1 factors D 8 .
For the purposes of and this paper we call this groups extraspecial group of type I.a, I.b, II.a and II.b.

For a complete account on extraspecial p-groups we refer to [Su]

chapter 4. Proposition 3.2. a) If v(c) ≥ v(λ p/(q+1) ) then C has good reduction over K and the wild monodromy is trivial. b) If v(c) < v(λ p/(q+1) ) let s i (Y ) = f (i) (Y )/i! and define L(Y ) = s 1 (Y ) q -a n s 0 (Y ) q-1 s q (Y ) where a n = (-1) q (-p) p+p 2 +•••+p n . Let y be a root of L(Y ). Then C has good reduction over M := K(y, f (y) 1/p ). Further, if L(Y ) is irreducible over K, then M/K is Galois
with the extraspecial group of order pq 2 and exponent p if p > 2, and type II.b. if p = 2. In this case M/K is the finite monodromy extension denoted K /K in section 2 and the finite monodromy is maximal in the sense of Definition 2.5.

If

v(c) = (a/b)v(p) < v(λ p/(q+1) ) with a, b both prime to p then L(Y ) is irreducible. (This is the case e.g. for c = p p/(q+1) ). c) If L(Y ) is irreducible over K so is L(Y )
, and the finite monodromy extension M/K is obtained from the splitting field of L(Y ) by adjoining f (y) 1/p for y any zero of L(Y ).

Remark 3.3. The polynomial L(Y ) appearing in the proposition is closely related to L(Y ). In fact it turns out that for this family all terms s i (Y ) for 2 ≤ i ≤ q -1 yield p-adically small contributions in L(Y ) and hence are negligible (cf. Lemma 2.7 and Def.2.8). Consequently in this case it is sufficient to work with L(Y ). Notice q+1) ). In this case consider X = λ p/(q+1) T and Z 0 = λZ 1 + 1. With T and Z 1 the defining equation of the cover becomes

though that deg(L(Y )) = deg(L(Y )). Proof. a) v(c) ≥ v(λ p/(
Z p 0 = (λZ 1 + 1) p = λ p Z p 1 + pλ p-1 Z p-1 1 + • • • + pλZ 1 + 1 = cλ pq/(q+1) T q + λ p T q+1 .
The assumption made about c implies that this equation is divisible by λ p and after reducing it to k we get Z p 1 -Z 1 = aT q + T q+1 for some a ∈ k. Using Artin-Schreier Theory, we see that this equation has the same geometric genus as the generic fiber. By Proposition 2.4 we conclude that C has good reduction over K. Therefore, in this case the finite monodromy is trivial. b) The claim is that the cover has potentially good reduction. We therefore look for y ∈ R alg such that T defined by X 0 = y + λ p/m T , with m = q + 1, induces the good reduction. The proof is divided into several steps:

I. If y is a root of L(Y ) we claim that v(y) = v(a n c)/q 2 . One has s 0 (Y ) = 1 + cY q + Y q+1 , s 1 (Y ) = qcY q-1 + (q + 1)Y q and s q (Y ) = c + (q + 1)Y . Then L(Y ) = [(q + 1) q -a n (q + 1)]Y q 2 + • • • -a n (q + 1)Y -a n c
and we look at its Newton Polygon. The coefficient of Y q 2 is a unit, the coefficients of the linear and the constant term have valuation v(a n ) and v(ca n ), respectively. Moreover for 0 < i ≤ q the coefficient of Y q 2 -i has valuation ≥ min{v((cq) i ), v(a n )} and for q < i < q 2 it has valuation ≥ v(a n ). By assumption v(c) < v(λ p/(q+1) ) = v(a n )/(q 2 -1), hence v(a n c)/q 2 ≤ v(a n )/(q 2 -1) and it follows that the Newton Polygon is the line joining the points (0, 0) and (q 2 , v(a n c)/q 2 ). We conclude that v(y) = v(a n c)/q 2 . II. We show the irreducibility of

L(Y ) if v(c) = (a/b)v(p)
with a, b prime to p. Under these hypothesis we have v(y) = ((p + ... + p n ) + (a/b))/q 2 v(p). Therefore the ramification index in L/K is q 2 and so L(Y ) is irreducible in this case.

III. For y a root of L(Y ) and X, T defined by X 0 = X + y and X = λ p/m T we claim that

f (X + y) = s 0 (y) + s 1 (y)X + s q (y)X q + X q+1 + r with r ∈ λ p m[T ].
Using the general formula

(A + B) p n = (A p n-1 + B p n-1 ) p mod(p 2 ) we compute f (y + λ p/m T ) modulo (λ p m[T ]): f (λ p/m T + y) = 1 + c(y + λ p/m T ) q + (y + λ p/m T ) q+1 = 1 + c(y p m-1 + λ p n /m T p n-1 ) p = (y + λ p/m T )(Y p n-1 + λ p n /m Y p n-1 ) p = f (y) + y p n X + py p n-1 (p-1) (c + y)X p n-1 + X p n (y + c + X) = s 0 (y) + s 1 (y)X + s q (y)X q + X q+1 mod (λ p m[T ]).
IV. Again let y be a root of L(Y ). We claim that for 0 ≤ i ≤ n there exist

A i (y, X) ∈ m[T ] and B i (y) ∈ R alg such that f (X + y) = (s 0 (y) 1/p + XA i (y, X)) p + s 1 (y)X + B i (y)X q/p i + X q+1 + r i (7) with r i ∈ λ p m[T ] and v(B i (y)) ≥ (1 + 1 p + ... + 1 p i-1 )v(p) for i ≥ 1 ( 8 )
The proof works by induction. Clearly A 0 (y, X) = 0 and B 0 (y) = s q (y) work. For i = 1 one can take XA 1 (y, X) = B 0 (y) 1/p X q/p and B 1 (y) = -ps 0 (y) (p-1)/p B 0 (y) 1/p . Then (7) follows as pX 2q/p ∈ λ p m[T ] (one can assume q > 2) and ( 8) is trivially satisfied.

Assume that for some i with 1 ≤ i < n conditions ( 7) and ( 8) are both satisfied. We define A i+1 and B i+1 by XA i+1 (y, X)

= XA i (y, X) + B i (y) 1/p X q/p i+1 and B i+1 (y) = -ps 0 (y) (p-1)/p B i (y) 1/p . Observe that v(pB i (y) 1/p ) ≥ (1 + 1 p + ... + 1 p i )v(p). Hence it follows that v(pB i (y) 1/p λ pq (q+1)p i+1 +1 ) ≥ ((1 + 1 p + ... + 1 p i ) + (( pq (q + 1)p i+1 + 1) 1 p -1 ))v(p) > v(λ p )
and so pB i (y) 1/p X q/p i+1 +1 ∈ λ p m[T ]. This proves ( 7) and ( 8) for i + 1. Note that (7) does not depend on the choice of the p-th root

B i (y) 1/p , since for ζ a p-th root of 1 one has v(p(ζ -1)) = v(λ p ). V.
Still denoting by y a root of L(Y ) we claim that C has good reduction over M := K(y, f (y) 1/p ). We apply IV. formula (7) for i = n. Then we get

f (X + y) = (s 0 (y) 1/p + XA n (y, X)) p + (s 1 (y) + B n (y))X + X q+1 + r n (9) with r n ∈ λ p m[T ].
Note that in (9) one can replace B n (y) by ζ n B n (y) where

ζ q n = 1 as v((ζ n -1)B n (y)) ≥ v(λ p ).
Recall that B 0 (y) = s q (y) and B i+1 (y) = -ps 0 (y) (p-1)/p B i (y) 1/p . It follows that

B n (y) = (-p) 1+ 1 p +...+ 1 p n-1 s 0 (y) 1-1 q s q (y) 1 q .
By definition of L(Y ) we have s 1 (y) q = (-B n (y)) q and it follows that (9) can be written in the following way:

f (X + y) = (s 0 (y) 1/p + XA n (y, X)) p + X q+1 + rn with rn ∈ λ p m[T ] ( 10
)
It follows that the Gauss valuation for T = (X 0 -y)/λ p/(q+1) induces at the special fiber the cover Z p 1 -Z 1 -T 1+q whose genus is g(C) and the corresponding model is defined over K(y, f (y) 1/p ). VI. Let y i , y j be two distinct roots of L(Y ). We claim that v(y i -

y j ) = v(λ p/(q+1) ). One has L (Y ) = qs 1 (Y ) q-1 s 1 (Y )-a n (s 0 (Y ) q-1 s q (Y )+(q-1)s 0 (Y ) q-2 s 1 (Y )s q (Y )).
For y a root of L(Y ) we remark that s q (y) = q + 1 and v(s 1 (y)) > 0 so we get v(L (y)) = v(a n ). Further the uniqueness of the model yielding good reduction implies that for all the roots y of L(Y ), (X 0y)/λ p/(q+1) defines the same Gauss valuation. It follows that if y i , y j are two such roots then v(y iy j ) ≥ v(λ p/(q+1) ). Finally as v(L (y i )) = v(a n ) = (q 2 -1)v(λ p/(q+1) ) it follows that we have equality v(y iy j ) = v(λ p/(q+1) ) for any choice of two distinct roots y i , y j of L(Y ).

VII. Assuming that L(Y ) ∈ K[Y ] is irreducible, we will show that the extension M/K is the finite monodromy. Further, the extension is Galois with the extraspecial group of order pq 2 and exponent p if p > 2 and type II. b. if p = 2.

Denote by y i , for 0 ≤ i < q 2 , the q 2 roots of L. As L(Y ) is assumed to be irreducible over K, we can consider elements σ i ∈ Gal(K alg /K) for which σ i (y 0 ) = y i for 0 < i < q 2 . Then σ i = σ j for i = j. Consider the equation Z p 0 = f (X 0 ) for C -→ P 1 K . By Proposition 4.4 we know that the change of coordinates

X 0 = λ p/(q+1) X + y 0 and Z 0 /(s 0 (y 0 ) 1/p ) = λZ 1 + H(Xλ p/(q+1) , y 0 ) (11)
induces the component of genus > 0 in the stably marked model. We now look at the action of σ i on this component.

σ i (X) = X 0 -y i λ p/(q+1) hence X -σ i (X) = y i -y 0 λ p/(q+1) .
We conclude that the σ i act non-trivially on the special fiber and so K(y i , 0 ≤ i < q 2 ) ⊂ K where K is the minimal extension of K such that C has a stably marked model. By V. we know that K ⊆ M := K(y 0 , s 0 (y 0 ) 1/p ) and so we also have K(y i , 0 ≤ i < q 2 ) ⊆ M. In V. we have seen that the curve C R ⊗ R k is isomorphic to the curve C f : w pw = f (x) where f (x) = x q+1 . We denote by G ∞ the inertia group of C f at infinity, and by G ∞,1 its p-Sylow subgroup. The image of (3) fixes ∞ and is a p-group. Therefore we have

Gal(K /K) → G ∞,1 ⊂ Aut k (C R ⊗ R k) # .
We will use a group theoretical argument in order to prove that the injection is actually surjective onto G ∞,1 and hence [K : K] = pq 2 . In we have shown that the group G ∞,1 is an extension of Z/pZ = τ (normal subgroup) by (Z/pZ) 2n where τ (x) = x, and τ (w) = w + 1. Further τ generates the center Z(G ∞,1 ). The quotient (Z/pZ) 2n is the group of translations x → x + y which extend to elements of G ∞,1 . The above injection therefore induces a homomorphism

Gal(K /K) → G ∞,1 / τ
whose image contains the translations by the q 2 elements (y iy 0 )/λ p/(q+1) . It follows that this map is surjective, and Gal(K /K) is a subgroup of G ∞,1 of index p or 1. Then it is a maximal subgroup (strict or not) of G ∞,1 and so it contains the Frattini subgroup which in this case is the center. This implies the surjectivity of (3). As [K : K] = pq 2 and K(y i , 0 ≤ i < q 2 ) ⊂ K ⊂ M = K(y 0 , s 0 (y 0 ) 1/p ) it follows that K = M. c) Let y be a root of L(Y ) and E = K(y, f (y) 1/p ). By b) we already know that we have potentially good reduction with M = K . Hence by Theorem 2.11 E contains K and [E : K] ≤ pq 2 . We conclude that L(Y ) is irreducible and E = M = K . Observe that then E also contains the splitting field of L(Y ) and the construction is independent of the choice of the zero y of L(Y ).

Remark 3.4. The main interest of Proposition 3.2 is that it works uniformly for all p. The proof is facilitated by the fact that all roots of f (X 0 ) have multiplicity 1. This implies n = m which is the smallest possible value for n for a given genus.

Nevertheless, we can show similar results as well in the case where there are zeros of f (X 0 ) of multiplicity > 1: for example take p = 3,

K ⊂ Q tame p and f (X 0 ) = (1 + X 0 + X 3 0 )(1 + X 0 ) 2 (resp. f (X 0 ) = (1 + X 0 + X 9 0 )(1 + X 0 ) 2 ). Then f (X 0 ) f (X 0 ) = 2X 3 0 (1+X 0 +X 3 0 )(1+X 0 ) mod 3 (resp. f (X 0 ) f (X 0 ) = 2X 9 0 (1+X 0 +X 9 0 )(1+X 0 )
mod 3) and, in the notation of Remark 2.10, it follows that C ⊗ R k has only one cusp and hence is a good candidate for having potentially good reduction. Note that m = 4, n = 5 and so p α = 3 (resp. m = 10, n = 11 and so p α = 9). In fact, following the proof of Proposition 3.2, one can show that both examples have potentially good reduction, and using Magma we check that the monodromy polynomials are irreducible over Q tame 3 . Then the padic valuation of the discriminant of L(Y ) shows that the wild monodromy group is maximal, isomorphic to the extraspecial group of order 3 3 and exponent 3 (resp. of order 3 5 and exponent 3). Note that the two curves Z 3 = 1 + X 3 0 + X 4 0 (appearing in Prop.3.2) and Z 3 = (1 + X 0 + X 3 0 )(1 + X 0 ) 2 both have genus 3, maximal wild monodromy over Q tame 3 , and the same type of reduction. Yet, as covers of P 1 K , these two curves have different branch cycle descriptions originating from the multiplicities in their defining equations. This suggests that we can refine the problem of realizing maximal wild monodromy groups over Q tame p and also prescribe the branch cycle description.

Genus 2-curves and wild monodromy

Now we consider the case of p-cyclic covers where p = 2 and m = 5, i.e. genus 2 curves over a 2-adic field in Q tame 2 . In this case, there are three possible types for the degeneration of the stably marked model (cf. figure 2). We assume that the cover

C -→ P 1 K is given birationally by Z p 0 = f (X 0 ) and can choose f (X 0 ) of the form f (X 0 ) = 1 + b 2 X 2 0 + b 3 X 3 0 + b 4 X 4 0 + X 5 0 ∈ R[X 0 ].
In the following we shall also write C k for the special fiber of the stably marked model. In this case we obtain for the monodromy polynomial L(Y ) = (s 1 (Y ) 2 -4s 0 (Y )s 2 (Y )) 2 -64s 0 (Y ) 3 s 4 (Y ), a polynomial of degree 16. The condition on the branch locus to have equidistant geometry translates to disc( f (X 0 )) = 0. Consider the normal R-model C for C given by Z p 0 = f (X 0 ). Its cusps are above the points with f (X 0 ) = b3 X 2 0 + X 4 0 = 0. (See Remark 2.10 and figure 1). Therefore we have two cusps iff b3 = 0. This case we call type 1, and the stably marked reduction consists of the original component intersected by two elliptic curves E 1 and E 2 which are isomorphic to the unique supersingular curve E in characteristic 2 given by w 2 + w = x 3 . The p-Sylow of the group of automorphisms of E leaving ∞ fixed is

genus 1 curves genus 1 curves genus 2 curve P 1 k P 1 k Type 1 Gal(K /K) w → Q 8 × Q 8 Gal(K /K) w → (Q 8 × Q 8 ) Z/2Z Gal(K /K) w → Q 8 * D 8 original component Type 2 original component P 1 k original component P 1 k Type 3
Q 8 and Q 8 /Z(Q 8 ) = (Z/2Z) 2 is the group of induced translations x → x + y for y a constant. We have Syl p (Aut k (C k ) # ) = Q 8 × Q 8 .
Type 2 is the case where C has one cusp and the stably marked reduction C k has two components of genus zero and two of genus one, isomorphic to E. In this case Syl p (Aut(C k ) # ) = (Q 8 × Q 8 ) Z/2Z (here Z/2Z acts on Q 8 × Q 8 in exchanging the factors) because the crossing points with the rational component and the original component are left fixed.

Type 3 is the case of potentially good reduction with group Q 8 * D 8 and has been treated in more generality in section 3.1. The following is a group theoretical lemma we shall need at various places. Lemma 3.5. i) Let H ⊆ Q 8 be a subgroup of index smaller or equal to 2 and assume that the natural projection

H -→ Q 8 /Z(Q 8 ) is onto. Then H = Q 8 . ii) Let H ⊆ Q 8 × Q 8 be
a subgroup of index smaller or equal to 2 and assume that the projection

H -→ Q 8 /Z(Q 8 ) × Q 8 /Z(Q 8 ) is onto. Then H = Q 8 × Q 8 . iii) Same statement for (Q 8 × Q 8 ) Z/2Z -→ ((Q 8 × Q 8 ) Z/2Z)/(Z(Q 8 ) × Z(Q 8 )).
(Here the action of Z/2Z is the one exchanging the factors Q 8 ).

Proof. i) If the index is equal to 1, there is nothing to show. We hence assume that H has index 2. Then H is a maximal subgroup and therefore contains the Frattini subgroup Φ(Q 8 ) which is Z(Q 8 ). This proves the claim. ii) We point out that the Frattini subgroup Φ

(Q 8 × Q 8 ) = Z(Q 8 ) × Z(Q 8 ) and the same proof as in i) works. iii) By ii) H contains Q 8 × Q 8 .
On the other hand it has to be strictly bigger.

Degeneration of type 1

We show that the wild monodromy can be maximal in the case of degeneration of type 1.

Proposition 3.6. Let C -→ P 1 K be given by the Kummer equation

Z 2 0 = f (X 0 ) = 1 + b 2 X 2 0 + b 3 X 3 0 + b 4 X 4 0 + X 5 0 where v(b i ) ≥ 0, v(b 3 ) = 0 and K = Q 2 (ρ, b 2 , b 3 , b 4
) with ρ = 2 2/3 . We write u for a uniformizer of K and assume

1 + b 3 b 2 + b 2 3 b 4 ≡ 0 mod u. Then C has stably marked reduction of type 1. Let T (Y ) = s 1 (Y ) 2 -4s 0 (Y )s 2 (Y ). Then T (Y ) factorizes over K in T (Y ) = T 1 (Y )T 2 (Y ) with T1 (Y ) = Y 4 and T2 (Y ) = Y 4 + b2
3 . For 1 ≤ i ≤ 4 denote by y i the zeros of T 1 (Y ) and for 5 ≤ i ≤ 8 the zeros of T 2 (Y ). If K(y 1 ) and K(y 5 ) are linearly disjoint over K then the finite monodromy extension is maximal, equal to the splitting field of

T (Y ) 8 i=1 (Y 2 -y i ). Its Galois group is Q 8 × Q 8 .
Remark 3.7. The proof will show that the centers for the blowups to obtain the stably marked model are given by the zeros of T (Y ). It turns out that in this case T (Y ), which is of degree 8, is easier to handle than the polynomial L(Y ), which has degree 16. Observe, though, that T (Y ) 2 and L(Y ) are congruent modulo 2 6 . We point out that the fact that here we can use the polynomial T (Y ) of degree strictly smaller than deg(L(Y )) is not in contradiction to what was said in Remark 5.4. The reason for this is that in the present situation the type of degeneration has been fixed. This implies that the possible maximum size of the right hand side group in ( 3) is smaller than in the general situation, i.e. the situation where no supplementary assumptions on the special fiber are made. Hence following the same reasoning as in Remark 5.4 one expects to be able to find polynomials that have smaller degree than L(Y ) but can function in the same way in Theorem 2.11.

Proof. Choose y ∈ K with T (y) = 0. Then for X 0 = X 1 + y we get

f (X 1 + y) = s 0 (y) + s 1 (y)X 1 + • • • + X 5 1 = (s 0 (y) 1/2 + s 2 (y) 1/2 X 1 ) 2 + s 3 (y)X 3 1 + s 4 (y)X 4 1 + X 5 1
where s 3 (y) = b 3 + 4b 4 y + 10y 2 , s 4 (y) = b 4 + 5y and we have chosen the roots s 0 (y) 1/2 and s 2 (y) 1/2 such that 2s 0 (y) 1/2 s 2 (y) 1/2 = s 1 (y). We write X = X 1 /ρ and obtain

f (X 1 + y) = (s 0 (y) 1/2 + s 2 (y) 1/2 ρX) 2 + s 3 (y)ρ 3 X 3 + s 4 (y)ρ 4 X 4 + ρ 5 X 5 . Now define Z by Z 0 = Z + s 0 (y) 1/2 + s 2 (y) 1/2 ρX. From Z 2 0 = f (X 0 )
we get, after a small calculation,

Z 2 + 2(s 0 (y) 1/2 + s 2 (y) 1/2 ρX)Z = s 3 (y)ρ 3 X 3 + s 4 (y)ρ 4 X 4 + ρ 5 X 5 . With W = Z/2 we get W 2 + (s 0 (y) 1/2 + s 2 (y) 1/2 ρX)W = s 3 (y)X 3 + s 4 (y)ρX 4 + ρ 2 X 5 . ( 12
)
This last equation defines a model over K(y, s 0 (y) 1/2 ) with special fiber given by W 2 + s0 (y) 1/2 W = b3 X 3 . This equation has genus 1 and therefore also defines a component in the stable reduction. Now T1 (Y ) = Y 4 and using the uniqueness of the stable reduction we conclude that the four roots of T 1 (Y ) induce the same ideal (ρ, X 0y) ⊂ R alg [X 0 ]. The corresponding statement for the roots of T2 (Y ) = ( b3 + Y 2 ) 2 holds. Hence we conclude

v(y i -y j ) ≥ v(ρ) for i = j ∈ {1, 2, 3, 4} or i = j ∈ {5, 6, 7, 8} (13)
This implies disc(T i (Y )) ≥ 12v(ρ) = 8v(2) for i = 1, 2 and we have equality iff ( 13) are all equalities. On the other hand, using Maple, 2

-16 disc(T (Y )) = b 8 3 (1 + b 2 b 3 + b 2 3 b 4 ) 4 mod 2. Now observe that disc(T 1 (Y )) + disc(T 2 (Y )) = disc(T (Y )) and hence disc(T i (Y )) = 8v
(2) and so ( 13) are all equalities. (This geometry is illustrated in figure 3). Therefore 0,

y 2 -y 1 ρ , y 3 -y 1 ρ , y 4 -y 1 ρ
are all distinct modulo (u). Hensel's Lemma therefore shows that K(y 1 )/K is Galois, i.e. T 1 (Y ) is totally split over K(y 1 ). The same holds for K(y 5 ) and T 2 (Y ). By Proposition 2.4 a stably marked model is defined over L := K(y 1 , y 5 , s 0 (y 1 ) 1/2 , s 0 (y 2 ) 1/2 , . . . , s 0 (y 8 ) 1/2 ) and we will show that L is equal to the finite monodromy extension. Let E 1 and E 2 be the two elliptic curves that are the components of genus > 0 in the stably marked model. Let i 1 , i 2 be their hyperelliptic involutions and ∞ 1 , ∞ 2 be the crossing points of E 1 and E 2 with the original component. We have the following diagram where j is given by the action on the special fiber of the stably marked model defined over L.

Gal(L/K) j -→ Syl 2 (Aut(E 1 ) ∞ ) × Syl 2 (Aut(E 2 ) ∞ ) ∼ = Q 8 × Q 8 ↓ pr Syl 2 (Aut(E 1 ) ∞ )/ i 1 × Syl 2 (Aut(E 2 ) ∞ )/ i 2 ∼ = (Z/2Z) 2 × (Z/2Z) 2
Here the index ∞ means that we consider only automorphisms that leave the infinite points ∞ 1 and ∞ 2 fixed. We claim that pr • j is onto. By assumption K(y 1 ) and K(y 5 ) are linearly disjoint. Hence there exist σ i,j ∈ Gal(L/K) for i ∈ {1, 2, 3, 4} and j ∈ {5, 6, 7, 8} such that σ i,j (y 1 ) = y i and σ i,j (y 5 ) = y j .

Now pr • j operates on E 1 / i 1 × E 2 / i 2 in the following way. pr • j(σ i,j ) X 0 -y 1 ρ - = X 0 -y 1 ρ - + y 1 -y i ρ - pr • j(σ i,j ) X 0 -y 5 ρ - = X 0 -y 5 ρ - + y 5 -y j ρ - .
Notice that X 0 -y 1 ρ and X 0 -y 5 ρ are local coordinates on E 1 / i 1 and E 2 / i 2 and that the action is non trivial. This shows the claim and in particular we have 2 4 |Im(j). Now consider the obvious maps

Syl 2 (Aut(E 1 ) ∞ ) × Syl 2 (Aut(E 2 ) ∞ ) pr 1 -→ Syl 2 (Aut(E 1 ) ∞ ) -→ Syl 2 (Aut(E 1 ) ∞ )/ i 1
Their composition is surjective by what was said about the σ i,j above. Now applying Lemma 3.5 i) to the second arrow shows that pr 1 is onto. We conclude that 2 5 |Im(j). Further Lemma 3.5 ii) applied to pr implies that j is onto. This shows that the curve C has maximal monodromy.

We proceed to show that j is an isomorphism. Let σ ∈ ker(j). By what was said above about the σ i,j we know that σ(y i ) = y i for all i ∈ {1, . . . 8}. If σ(s 0 (y i ) 1/2 ) = -s 0 (y i ) 1/2 for some i, then from the equation of the model ( 12), we get

σ(W ) = W + s 0 (y i ) 1/2 + s 2 (y i ) 1/2 ρ X 0 -y i ρ .
Note that the action on s 2 (y i ) 1/2 is known via the above equality 2s 0 (y) 1/2 s 2 (y) 1/2 = s 1 (y) which now holds for y = y i . In reduction this yields

σ( W ) = W + s 0 (y i ) 1/2 = W .
Hence such a σ would induce a non-trivial action on the special fiber of the stably marked model. We conclude that σ acts trivially on all s 0 (y i ) 1/2 , hence σ = id and so j is injective.

We next give an example where the conditions of Proposition 3.6 hold.

Example 3.8. Let f (X 0 ) = 1 + 2 3/5 X 2 0 + X 3 0 + 2 2/5 X 4 0 + X 5 0 . One checks, using Magma, that T (Y ) decomposes into two irreducible factors T 1 (Y ), T 2 (Y ) of degree 4 over K = Q 2 (2 1/15 ). Then one checks that T 2 (Y ) is irreducible over the splitting field of T 1 (Y ). Hence we satisfy the linear disjointness condition of the proposition. Below we have included the Magma and Maple programs used in this example: The two degree 4 factors obtained from the above program are Eisenstein polynomials for the extensions defined by the polynomials T 1 (Y ) and T 2 (Y ). We calculate the resultant res(Y ) of the Eisenstein polynomial (see the polynomial L1(Y ) above) for the extension defined by T 1 (Y ) and pi 15 -2 using Maple. This resultant gives an equation over Q 2 for the compositum of the extensions defined by T 1 (Y ) and pi 15 -2. This polynomial has too big coefficients to be worked with in Magma and so we use a theorem of Krasner (cf. [Pa-Ro] Theorem 5.2) in order to get an Eisenstein polynomial with smaller coefficients and defining the same extension. We first calculate the 2-adic valuation of the discriminant of res(Y ) using Maple. In the notation of [Pa-Ro] we get v 2 (res(Y )) = 62 = n + j -1 where n is the degree of the polynomial res(Y ). This gives j = 3. Let c be the smallest integer such that c > (n + 2j)/n = 66/60, i.e.

Degeneration of type 2

This case is more subtle and so we only give an example of a cover that has maximal finite monodromy. Proposition 3.10. Let K = Q 2 (a) with a 9 = 2. Consider the cover C -→ P 1 K given by Z 2 0 = f (X 0 ) = 1 + a 3 X 2 0 + a 6 X 3 0 + X 5 0 . This cover has stably marked reduction of type 2 and maximal finite monodromy. Let T (Y ) = s 1 (Y ) 2 -4s 0 (Y )s 2 (Y ) and denote by y i , 0 ≤ i ≤ 8 its zeros in some algebraic closure of K. Then the finite monodromy is the splitting field M of

T (Y ) 8 i=1 (Y 2 -f 0 (y i )).
Its Galois group is isomorphic to

(Q 8 × Q 8 ) Z/2Z.
Proof. We first show that the model is of type 2. Consider

f (X + Y ) = s 0 (Y ) + s 1 (Y )X + • • • + X 5 .
Then s 3 (Y ) = a 6 + 10y 2 , s 4 (y) = 5y and T (Y ) = 15Y 8 + 22a 6 Y 6 + 24a 3 Y 5 + 6a 3 Y 4 + 48Y 3 + 12a 6 Y + 4a 3 . In this case the valuation of disc(T (Y )) has not enough information to determine the relative geometry of the zeros of T (Y ). We therefore proceed as follows. A look at the Newton polygon of T (Y ) shows that the 8 roots all have valuation equal to (1/8)v(4a 3 ) = (7/24)v( 2 Still for y a zero of T (Y ) and X 0 = X 1 + y we have

f (X 1 + y) = (s 0 (y) 1/2 + s 2 (y) 1/2 X 1 ) 2 + s 3 (y)X 3 1 + s 4 (y) 4 X 4 1 + X 5
1 where v(s 3 (y)ρ 3 ) = v(4), v(s 4 (y)ρ 4 ) > v(4) and v(ρ 5 ) > v(4). As in the case of degeneration of type 1 we set Z 0 = 2Z + s 0 (y) 1/2 + s 2 (y) 1/2 X 1 and X = X 1 /ρ. With these coordinates the model has the equation:

Z 2 + 2(s 0 (y) 1/2 + s 2 (y) 1/2 ρX)Z = s 3 (y)ρ 3 X 3 + s 4 (y) 4 ρ 4 X 4 + ρ 5 X 5 .
With W = Z/2 we get

W 2 + (s 0 (y) 1/2 + s 2 (y) 1/2 ρX)W = ρ 3 4 s 3 (y)X 3 + ρ 4 4 s 4 (y) 4 X 4 + ρ 5 4 X 5 . ( 14
)
In reduction we obtain a curve of genus 1 and hence the curve C has stably marked reduction of type 2.

We proceed to compute its finite monodromy using Magma. We first check that T (Y ) is irreducible over K, then we show that if y 1 is a zero of T (Y ) then T (Y ) has the following decomposition into irreducible factors over K(y 1 ).

T (Y ) = 4 i=1 (Y -y i )T 2 (Y ) (15)
From ( 14) and Proposition 2.4 we know that a stably marked model is defined over M and we proceed to show that M is the finite monodromy. As in the case of the degeneration of type 1 we have

Gal(M/K) j -→ Syl 2 (Aut(C k ) # ) ∼ = (Syl 2 (Aut(E 1 ) ∞ ) × Syl 2 (Aut(E 2 ) ∞ )) Z/2Z.
Here E 1 and E 2 are the two elliptic curves, i 1 and i 2 their hyperelliptic involutions, and Aut(C k ) # are the automorphisms that leave the initial component fixed. The index ∞ means that the crossing point of E i , with the rational component that is not the original component in the stably marked reduction, is left fixed. Now Syl 2 (Aut(E i ) ∞ ) ∼ = Q 8 and we have a projection map pr obtained by modding out by the subgroup generated by the involutions i 1 and i 2 .

Gal(M/K) j -→ (Q 8 × Q 8 ) Z/2Z ↓ pr ((Z/2Z) 2 × (Z/2Z) 2 ) Z/2Z
From (15) it follows that there exist σ i,j ∈ Gal(M/K) for i ∈ {1, 2, 3, 4} and j ∈ {5, 6, 7, 8} such that σ i,j (y 1 ) = y i and σ i,j (y 5 ) = y j . Further there is τ ∈ Gal(M/K) with τ (y 1 ) = y 5 . They act on the special fiber via:

j(σ i,j ) X 0 -y 1 ρ - = X 0 -y 1 ρ - + y 1 -y i ρ - j(σ i,j ) X 0 -y 5 ρ - = X 0 -y 5 ρ - + y 5 -y j ρ - j(τ ) X 0 -y 1 ρ - = X 0 -y 1 ρ - + y 1 -y 5 ρ -
This shows that pr • j is onto and hence, by Lemma 3.5 iii), j is onto. To obtain the injectivity of j one follows the same route as in the case of degeneration of type 1.

As before, we have a corollary in Inverse Galois Theory.

Corollary 3.11. Let L/Q tame 2 be the splitting field of

T (Y ) 8 i=1 (Y 2 -f (y i ))
where T (Y ) = 15Y 8 + 22a 6 Y 6 + 24a 3 Y 5 + 6a 3 Y 4 + 48Y 3 + 12a 6 Y + 4a 3 , a 9 = 2, the y i are the zeros of T (Y ) and f (X 0 ) = 1 + a 3 X 2 0 + a 6 X 3 0 + X 5 0 . Then this is a Galois extension with group (Q 8 × Q 8 ) Z/2Z.

Degeneration of type 3

This is the case of potentially good reduction and, as said above, we have treated it in some generality already in section 3.1. Here we add an example that has not been included earlier.

Example 3.12. Let C -→ P 1 K be given by the Kummer equation

Z 2 0 = f (X 0 ) = 1 + X 4 0 + X 5 0 with K = Q tame 2 (i.e. c = 1 in Proposition 3.2).
We claim that this cover has maximal wild monodromy with group Q 8 * D 8 .

Let

a n = (-1) q (-p) p+p 2 +•••+p n = 64. Considering Proposition 3.2 it is sufficient to prove that the modified monodromy polynomial L(Y ) = s 1 (Y ) q -a n s 0 (Y ) q-1 s q (Y ) = s 1 (Y ) 4 -64s 0 (Y ) 3 s 4 (Y ) is irreducible over K. Note that the monodromy polynomial is L(Y ) = (s 1 (Y ) 2 -4s 0 (Y )s 2 (Y )) 2 -64s 0 (Y ) 3 s 4 (Y ).
We can check the irreducibility of L(Y ) using Magma. Let y be a root of L(Y ). Then C has good reduction over M := K(y, f (y) 1/2 ) and M/K is Galois with group Q 8 * D 8 .

Related work and perspective

1. In [Si-Za1] and [Si-Za2] Silverberg and Zarhin study the finite monodromy groups for abelian surfaces. In particular, they classify the finite groups which can occur as finite monodromy groups for abelian surfaces. Their examples are mostly in equal characteristic, and they ask at the end of loc. cit. section 1 for examples in mixed characteristic. Our section 3 provides such examples, if one considers the Jacobians associated to the curves C. Note that realizing wild monodromy groups over Q tame p is a strong condition because Q tame p has trivial absolute wild ramification and this has no counterpart in the equal characteristic case. In particular, the fact that in equal characteristic a group is realized as soon as a bigger one is ([Si-Za2] Remark 1.9.) has no analog over Q tame p .

Motivated by [Si-Za1

] and [Si-Za2] , one can look at generalizations of type 2 degeneration (cf. section 3.2.2). Namely, we have the following:

Let C/k be a stable curve of genus g ≥ 2, over an algebraically closed field of characteristic p > 0. Then Pic 0 (C) is an extension of an abelian variety by a torus. The group Aut k (C) is finite and injects into Aut k (Pic 0 (C)) (cf. [De-Mu] Thm.1.13). It can be described by considering its action on the torsion points of Pic 0 (C). Assume that the dual graph of C is a tree or, equivalently, that Pic 0 (C) is an abelian variety (this is the case for the special fibers of the stable models of the p-cyclic covers which appear in theorem 2.11). Then we can give a bound for a p-Sylow subgroup Syl p (C) of Aut k (C). Define N p (g) := a + [a/p] + [a/p 2 ] + . . . where a :=

[ 2g p-1 ]. We claim that v p (|Syl p (C)|) ≤ N p (g) (16
)

and for g ∈ p N (p -1)/2 this bound is optimal. Note that these bounds are bad when C/k is smooth (compare with Thm.1.1.II.(f)). Yet they can be sharp in the case where C is not assumed to be irreducible, as we now show. Let > 2 be a prime distinct from p and [ ] the multiplication by in Pic 0 (C). Then, by [Se], Aut k (C) → ker[ ] GL 2g (F ). The p-adic valuation v p (|GL 2g (F )|) = v p ( 1≤i≤2g ( i -1)) and, varying , the minimal value for p > 2 is N p (g) (exercise using [We] §2). Hence we obtain (16) for p > 2.

For p = 2, the minimal value is 5gt(2g) = g + N 2 (g) > N 2 (g) (17

)
where t(2g) is the number of digits in the 2-adic expansion of 2g (see [Ca-Fo]).

Let us first assume that p > 2 and k = F alg p . We consider the marked curve (C 1 , ∞ 1 ), birational to the curve y py = x 2 , and ∞ 1 is the point x = ∞. The curve has genus (p -1)/2. Let Syl p (C 1 ) ⊂ Aut Fp (C 1 ) be a (the) p-Sylow subgroup. Then Syl p (C 1 ) = Z/pZ and it fixes ∞ 1 . Now, for j ≥ 1, we build inductively a semi-stable curve C j (with a marked point ∞ j ). The curve C j+1 is obtained from the projective line P 1

Fp by choosing one rational point over F p , say ∞ j+1 and patching transversally to each of the p other F p -rational points a copy of C j at the point ∞ j . The genus is g(C j ) = p j-1 (p -1)/2. Let Syl p (C j ) ⊂ Aut Fp (C j ) be the p-Sylow subgroup fixing ∞ j . Then Syl p (C j+1 ) = Syl p (C j ) Z/pZ, the wreath product (i.e. the semidirect product of p copies of Syl p (C j ) and Z/pZ where this last group acts cyclically on the components). We have v p (|Syl p (C j+1 )|) = pv p (|Syl p (C j )|) + 1 and so v p (|Syl p (C j )|) = 1 + p + p 2 + ... + p j-1 . This gives equality in (16) for p > 2. Moreover, we remark that in this case Syl p (C j ) is isomorphic to a p-Sylow subgroup of the symmetric group S p j of degree p j .

If p = 2, (17) gives v 2 (|Syl 2 (C j )|) ≥ N 2 (1) + 1 = 4 (resp. N 2 (2) + 2 = 9) and it is an exercise to see that the sharp bound for genus 1 (resp. 2) is 3 (resp. 7) as is stated in (16). In order to prove (16) for p = 2, we need to invoke more structure. We remark that one can endow the Tate-module T (Pic 0 (C)) = lim ← -Pic 0 (C)( n ) = Z 2g with the Weil pairing e : T (Pic 0 (C)) × T (Pic 0 (C)) → Z (1) (see [Mi] §16).

Here the polarization is the product of the canonical polarizations of the jacobians of the irreducible components of genus > 0 of C. It is skew symmetric and the automorphisms in Aut k (C) respect this polarization. For ( , |G|) = 1 and > 2 in the above setting we can then replace the linear group GL 2g (F ) by the symplectic group Sp 2g (F ).

This time we need to minimize v p (|Sp 2g (F )|) = v p (( 2 -1)( 4 -1)...( 2i -1)...( 2g -1)) by varying . It is an exercise to see that this minimum is N p (g) for all p (when p > 2 see [We] §3 and for p = 2 this follows from [Ca-Fo]).

In order to show that for p = 2 this is sharp, in the above construction, we replace the curve y py = x 2 (whose genus is 0 when p = 2) by the curve C 2 : y 2y = x 3 whose genus is 1. (Note that there is a shift of the indices.) We proceed in the same way for the patching. The curve C j will have genus g = 2 j-2 = 2 j-1 (2 -1)/2. The Sylow subgroup Syl 2 (C j ) is isomorphic to the wreath product Syl 2 (C j ) Z/2Z. It follows inductively that v 2 (Syl 2 (C j )| = 2 j -1 = N 2 (g). Note that this time Syl 2 (C j ) is not isomorphic to a 2-Sylow subgroup of the symmetric group S 2 j of degree 2 j although it has the same cardinal.

We finish this paper with a question. The curves C j (for all p) look like the special fibers of p-cyclic covers of the projective line (see Thm. 2.11); moreover by [Mau] Cor. 5.6. of Th. 5.4 we know that these curves lift as a p-cyclic cover of the projective line over some finite extension of Q tame p . Hence one can ask if they have a lift to a p-cyclic cover of the projective line over Q tame p with maximal monodromy Syl p (Aut k (C j )) ? (The case p = 2 and genus 2 corresponds to type 2 above).

Thanks. We would like to thank Qing Liu for illuminating discussions, Michel Raynaud for support concerning section 3.3, and Sebastian Pauli for his unfailing help in using the Magma package on factorization over p-adic fields. Part of this paper was written while the authors were guests of the Max-Planck-Institut in Bonn.

Remark 2 .

 2 10. a) In the above situation, consider the flat projective R-model C R for C obtained by normalizing Proj(R[u, w]) in the function field K(C).

Figure 2 :

 2 Figure 2: The three types of degeneration

Figure 3 :

 3 Figure 3: The geometry of the zeros of T (Y ) in Proposition 3.6

Figure 4 :

 4 Figure 4: The geometry of the zeros of T (Y ) in Proposition 3.10

* Work supported in part by the European Community's Sixth Framework Programme under Contract MRTN-CT2003-504917

where

, the y i are the zeros of T (Y ) and f (X 0 ) = 1 + 2 3/5 X 2 0 + X 3 0 + 2 2/5 X 4 0 + X 5 0 . Then this is a Galois extension with group Q 8 × Q 8 .

Claus LEHR Max-Planck-Institut für Mathematik Vivatsgasse 7 53111 Bonn, Germany e-mail : lehr@mpim-bonn.mpg.de Michel MATIGNON Laboratoire de Théorie des Nombres et d'Algorithmique Arithmétique, UMR 5465 CNRS Université de Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France e-mail : matignon@math.u-bordeaux1.fr


Dominique Barbolosi
email: dominique.barbolosi@neuf.fr

Assia Benabdallah

Florence Hubert
email: fhubert@latp.univ-mrs.fr

Federico Verga
email: verga@latp.univ-mrs.fr

Mathematical and numerical analysis for a model of growing metastatic tumors

Keywords: Von Foerster equation, semigroup approach, asymptotic behaviour, characteristic scheme, metastatic tumors

come

Mathematical and numerical analysis for a model of growing metastatic tumors

Dominique Barbolosi * -Assia Benabdallah ** -Florence Hubert ** -Federico Verga *,**

Introduction

Primary tumors after reaching a critical volume may be at the origin of several metastatic tumors disseminated in the human body. At early stages, metastatic tumors are of small size and not detectable by medicinal apparatus, like in the case of the breast cancer, but evidence for the existence of occult micrometastases at diagnosis is overwhelming [START_REF] Harris | Malignant tumors of the breast[END_REF]. Metastases grow rapidly, escape any therapeutic treatment and often lead to the patient's death. This is the fate of an uncontrolled disease process.

Because metastases, the systemic therapy is the only therapeutic treatment which may have an impact on the problem of disseminated tumors [START_REF]Systemic treatment of early breast cancer by hormonal, systemic or immune therapy: 133 randomized trials involving 31,000 recurrences and 24,000 deaths among 75,000 women[END_REF]. The adequate systemic therapy is chemotherapy based on the combination of cytotoxic, cytostatic or other biologic agents. Any chemotherapeutic treatment must be initiated before intensive disseminating process, in order to prevent the emergence of micrometastases. It is the so-called adjuvant chemotherapy given despite undetectable tumor masses. This is the by the chemotherapy disturbance of tumor growth process.

Disease monitoring joint to chemotherapy constitutes not only a high complex but also constrained process because of the undesirable toxic adverse events that limit the applicability of chemotherapy. Optimal management of cancer requires today thorough and upgraded tools integrating current information about cancer cell kinetic pharmacokinetics and efficacy-toxicity pharmacodynamics.

Since several years numerous authors attempted to optimize anticancer chemotherapy by using mathematical modeling [START_REF] Barbolosi | Optimizing drug regimens in cancer chemotherapy by an efficacy-toxicity model[END_REF]. Dose-dense schedules may have an advantage over conventional schedules of drug administration [START_REF] Norton | Theorical concepts and the emerging role of taxanes in adjuvant therapy[END_REF], and we have already developed mathematical models computing the features of an administration protocol so that the size of primary tumor is minimized while the toxicity effects are controlled. These features concern the drug amounts (intensification) and their distribution over a treatment cycle (densification). This approach was applied to the metastatic breast cancer in phase I trial [START_REF]A mechanistic model predicting hematopoiesis and tumor growth to optimize docetaxel + epirubicin (ET) administration in metastatic breast cancer (MBC): Phase I trial[END_REF].

Recently, disease models were also developed describing the dissemination dynamics of metastatic tumors [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF]. In this line, Iwata et al. developed a metastatic disease model as a partial differential equation with initial and boundary conditions. Starting from a primary tumor with one cell, this equation calculates the colony size distribution v(x, t) of metastatic tumors with cell numbers x at time t. The authors obtain an analytical solution of this equation using Laplace transform; however the obtained solution is not numerically tractable.

In contrast to all these developments, no models are available for the disease and treatment joint processes. Especially for micrometrastases, these models could describe the number and sizes of tumors below the detection threshold, predict the behavior of metastases in patients and thus allow the design of adjuvant chemotherapy.

Accordingly, we propose to revisit the Iwata's model in order to incorporate the effect of chemotherapy and pharmacokinetics and pharmacodynamics of antitumor agents. The purpose of this paper will be a thorough mathematical analysis of the so obtained new partial differential equation. This analysis will be followed by the presentation of a numerical scheme of resolution. Aiming to prevent the emergence of the micrometastases, accordingly the Iwata's model has to be modified to incorporate the effect of chemotherapy in order to to de-sign an administration protocol which will minimize the number of metastases, by using the method of optimization developed in [START_REF]A mechanistic model predicting hematopoiesis and tumor growth to optimize docetaxel + epirubicin (ET) administration in metastatic breast cancer (MBC): Phase I trial[END_REF].

Reader's guide

We precise in section 2 the model under study: a Von Foerster equation with Gompertzian growth. It is a linear transport equation with a non local boundary condition combined with a singular source. The mathematical analysis of such an equation is developped in section 3. We, in particular, derive the asymptotic behaviour of the solution. We exhibit a Malthus parameter λ 0 that gives the growth rate of metastatic sites. This explicit formula that gives this parameter shows a deep dependance on clinical characteristic data of the patient. We propose in section 4 an algorithm to approximate the solutions of such an equation. We give a theoritical validation of the numerical model for long time intervals. We finally give in section 4 some numerical results. Our results are first compared to the ones of [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF]. Numerical simulations also confirm the theoritical asymptotic growth of the number of metastases. We conclude this section by a sensibility analysis with respect to the clinical characteristic data of the patient.

Presentation of the model

We present in this section the mathematical model describing the dynamics of the metastatic colony size distribution introduced in [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF]. In this model we suppose that the primary tumor is generated by a single cell at t = 0 and grows with the rate g(x) per unit time where x is the tumor size represented by the number of cells in the tumor. The growing tumor emits metastatic cells with the rate β(x). Each metastatic cell develops into a new tumor, which also grows at rate g(x) and emits new nuclei of metastasis just as the primary tumor does. Let v(x, t) represent the colony size distribution with cell number x at time t, actually v(x, t)dx means the number of metastatic tumors whose sizes range from x and x + dx at time t. In the case considered here we suppose that the nuclei of colonization are located far enough from each other so that their ranges do not overlap for a long time period. The dynamics of the colony size distribution is given by the following Mac-Kendrick Von Foerster equation:

           ∂ ∂t v(x, t) + ∂ ∂x [g(x)v(x, t)] = 0, x ∈ [1, b), t ≥ 0 g(1)v(1, t) = b 1 β(x)v(x, t)dx + β(x p (t)), v(0, x) = 0. (1)
We have to deal with a transport equation with a non local boundary condition and an initial data equals to zero which supposes that there is no metastatic tumor at time t = 0. The boundary condition means that the number of metastatic cells newly created per unit time at time t is the total rate of occurences of metastases due to metastatic tumors (corresponding to the integral term) and the primary tumor.

In the Iwata's model the data x p , g and β are precised as follows.

1) The number of cells x p (t) in the primary tumor at time t is the solution of the Cauchy problem:

d dt x p (t) = g(x p (t)) x p (0) = 1.
(2)

2) A Gompertzian growth rate g is adopted:

g(x) = ax ln b x , (3)
where a denotes a growth rate constant and b > 1 the maximum tumor size.

We can solve explicitely the equation (2) which gives:

x p (t) = b 1-e -at . (4)
3) The colonization rate β(x) is choosen as:

β(x) = mx α , ( 5
)
where m is the colonization coefficient and α is the fractal dimension of blood vessels infiltrating the tumor. The parameter α expresses how the blood vessels geometrically distribute in or on a tumor. If the vascularity is superficial the fractal dimension α is assigned to be 2/3 because we suppose that the tumor has the shape of a sphere hence the surface area is proportional to 2/3. Else if the vascularization is homogeneously distributed in the whole tumor, α is supposed to be equal to 1.

Mathematical analysis of the Von Foerster equation with Gompertzian growth

This section is devoted to the mathematical analysis of System 1. We are going to use a semigroup approach. This will give the existence and uniqueness of solutions as well as the long time behaviour of these solutions. Apart from the construction of the semigroup, the main ideas can be found in [START_REF] Perthame | Transport Equations on Biology[END_REF], [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]. We would like also to refer to [START_REF] Devys | A model describing the growth and the size distribution of multiple metastatic tumors[END_REF] for another approach.

Let us consider the equation ( 1) in a more general case where the initial condition is non zero:

           ∂ ∂t v(x, t) + ∂ ∂x [g(x)v(x, t)] = 0, x ∈ (1, b), t ≥ 0 g(1)v(1, t) = b 1 β(x)v(x, t)dx + f (t), t > 0 v(x, 0) = v 0 (x). (6)

Existence of solutions

In the sequel, we suppose that g is given by (3), β by (5) and f (t) = β(x p (t)) where x p the number of cells in primary tumor is defined in (4). In particular, we have

b 1 β(x) g(x) dx = +∞. (7)
Let us introduce the operator A and its domain:

A = -∂ x (g(x).), D(A) = v ∈ L 1 (1, b), gv ∈ W 1,1 (1, b), lim x→b -(gv)(x) = 0, (gv)(1) = b 1 β(x)v(x) dx .
In order to prove the existence of solutions for the considered equation ( 6), we first need to state with some spectral properties of the operator A.

Spectral properties of A

We derive in this section some properties of the point spectrum σ p (A), the set of the eigenvalues of (A, D(A)). Theorem 1. There exists a unique real eigenvalue λ 0 ∈]0, +∞[ of the operator (A, D(A)). More precisely :

m < λ 0 < mb α . ( 8
)
Moreover there exists ε > 0 such that

σ p (A) ⊂ {0 < Re λ < λ 0 -ε} ∪ {λ 0 }.
Proof.

Step 1:The eigenvectors are given by V (x) = g(1)V (1) g(x) e We recall that λ ∈ σ p (A) if and only if there exists, V = 0 such that

V ∈ D(A) AV = λV, in (1, b)
The solutions of the differential equation

AV = ∂ x (g(x)V ) = -λV, in (1, b)
are given by:

V (x) = g(1)V (1) g(x) e - x 1 λ g(y) dy . (9)
The boundary condition

g(1)V (1) = b 1 β(x) V (x)dx then reads g(1)V (1) = b 1 g(1)V (1) β(x) g(x) e - x 1 λ g(y) dy dx (10)
First of all, notice that the function gV defined in [START_REF]Systemic treatment of early breast cancer by hormonal, systemic or immune therapy: 133 randomized trials involving 31,000 recurrences and 24,000 deaths among 75,000 women[END_REF] is in W , and therefore

V (x) = V (1) x ln b x ln b λ a -1
.

Thus V ∈ L 1 (1, b) if and only if Reλ > 0.
Therefore, to be in the spectrum of (A, D(A)), Reλ has to be greater than 0. We can remark that if V (1) = 0 then V = 0, so V (1) = 0 to ensure that V = 0 and dim(Ker(A -λI)) = 1.

We deduce that λ ∈ C is an eigenvalue of (A, D(A)) if and only if Reλ > 0 and from [START_REF] Harris | Malignant tumors of the breast[END_REF]:

F (λ) := b 1 β(x) g(x) e - x 1 λ g(y) dy dx = 1. (11)
Let us introduce

G(x) = x 1 dy g(y) . ( 12
)
Thanks to the properties of g the function G is strictly increasing, C 1 and one to one from (1, b) to (0, +∞). As G (x) = 0, ∀x ∈ (1, b), G is a C 1 diffeomorphism. We can rewrite F (λ) as follows:

F (λ) = +∞ 0 β(G -1 (y))e -λy dy.
Then, setting θ = β • G -1 , we remark that:

F (λ) = +∞ 0 θ(y) e -λy dy.
F is the Laplace transform of the function θ = β • G -1 ∈ L ∞ (0, ∞), we deduce that F is well defined and holomorphic (Laplace transform property) in the complex half plane Ω = {λ ∈ C, Re λ > 0}. Thus the solutions of the equation F (λ) = 1 are all isolated. We have:

V (G -1 (x)) = g(1)V (1)G (G -1 (x))e -λx for all x ∈ (0,

+∞).

Step 2: There exists a unique real eigenvalue λ 0 of (A, D(A)). Moreover, its algebraic multiplicity is 1.

We will look at the solutions of (11) on the real line. Remark that F is continuous and strictly decreasing on R + with F (0) > 1 thanks to [START_REF]A mechanistic model predicting hematopoiesis and tumor growth to optimize docetaxel + epirubicin (ET) administration in metastatic breast cancer (MBC): Phase I trial[END_REF] and:

lim λ→+∞ F (λ) = 0.
Therefore by the intermediate value theorem, (11) admits a unique solution λ 0 ∈ R + . As λ 0 > 0, we prove in the last step that ker(A -λ 0 I) = RV for V defined by [START_REF]Systemic treatment of early breast cancer by hormonal, systemic or immune therapy: 133 randomized trials involving 31,000 recurrences and 24,000 deaths among 75,000 women[END_REF]. In the sequel, we fix V (1) > 0.

Let us now show ker(A -λ 0 I) 2 = ker(A -λ 0 I) = RV . A function ϕ ∈ ker(A -λ 0 I) 2 if and only if ϕ ∈ D(A) and there exists c ∈ R such that:

Aϕ = λ 0 ϕ -cV, in (1, b) g(1)ϕ(1) = b 1 β(x) ϕ(x)dx .
As previously this system is equivalent to:

∂ x (g(x)ϕ) = -λ 0 ϕ + cV, in (1, b) (13)
g(1)ϕ(1) = b 1 β(x) ϕ(x)dx (14)
Solving the differential equation ( 13) we obtain for ϕ:

ϕ(x) = g(1)ϕ(1) g(x) e - x 1 λ 0 g(y) dy + c g(x) x 1 e - x τ λ 0 g(y) dy V (τ )dτ.
One can deduce (since

F (λ 0 ) = 1): b 1 β(x) ϕ(x)dx = g(1)ϕ(1) + c b 1 β(x) g(x)
x

1 e - x τ λ 0 g(y) dy V (τ )dτ dx
In order to fulfill the condition ( 14) the condition:

c b 1 β(x) g(x)
x

1 e - x τ λ 0 g(y) dy V (τ )dτ dx = 0 is required. As β, g, V are positive functions on [1, b)
we deduce that c = 0 and:

ϕ(x) = g(1)ϕ(1) g(x) e - x 1 λ 0 g(y) dy = δV (x), with δ = ϕ(1)
V (1) , the conclusion follows.Therefore the algebraic multiplicity of λ 0 is 1.

It remains to prove [START_REF] Engen | One parameter Semigroups for Linear Evolution Equations[END_REF]. From [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF], one has :

F (λ 0 ) = +∞ 0 β(G -1 (x))e -λ0x dx = 1. As m < β(x) < mb α for x ∈ (1, b), one gets that m λ 0 < 1 < +∞ 0 mb α e -λ0x dx = mb α λ 0 , so m < λ 0 < mb α .
As we will see in the sequel, λ 0 gives the asymptotic behaviour.

Step 3:

There is a unique eigenvalue whose real part is equal to λ 0 .

Let λ 0 the real solution of

F (λ 0 ) = 1. If λ = λ 0 +iα, α ∈ R, is an eigenvalue of (A, D(A)) since θ = β(G -1
) is real valued we have:

F (λ) = 1 if and only if    ∞ 0 θ(x)e -λ0x cos(αx)dx = 1 ∞ 0 θ(x)e -λ0x sin(αx)dx = 0 Since F (λ 0 ) = 1, one can deduce that: ∞ 0 θ(x)e -λ0x (1 -cos(αx))dx = 0.
Considering that θ > 0 almost everywhere, this can be true if and only if 1 -cos(αx) = 0, for all x ∈ R. Therefore α = 0.

Step 4:

There is no eigenvalue whose real part is bigger than λ 0 .

Indeed if λ ∈ {Re λ > λ 0 } then |F (λ)| < F (λ 0 ) = 1 and then λ / ∈ σ p (A).
Step 5:

There is at most a finite number of eigenvalues of real part in a compact of (0, λ 0 ) and there exists

ε > 0 such that σ p (A)\{λ 0 } ⊂ {0 < Reλ ≤ λ 0 -ε} . Let λ = a + ib with 0 < a < λ 0 , F (λ) = 1 if and only if    ∞ 0 θ(x)e -ax cos(bx)dx = 1, ∞ 0 θ(x)e -ax sin(bx)dx = 0.
According to the Riemann-Lebesgue theorem, we have:

∞ 0 θ(x)e -ax cos(bx)dx → b→+∞ 0,
uniformly in a on any compact of {z ∈ C; Re(z) ∈ (0, λ 0 )}.

As the eigenvalues are isolated, there can exist only a finite number over each compact of {0 < Reλ < λ 0 }. Consequently:

∃ε > 0 ; F (λ) = 1 and λ = λ 0 ⇒ Reλ / ∈]λ 0 -ε, λ 0 [. Thus σ p (A)\{λ 0 } ⊂ {0 < Reλ ≤ λ 0 -ε}.
Proposition 2 (see [START_REF] Banks | Transformation semigroups and l 1 approximation for size structured population models[END_REF]).

1) The domain

D(A) is dense in L 1 (1, b). 2) {λ ∈ C, Reλ ∈]λ 0 -ε, λ 0 [ } ∪ {λ ∈ C; Reλ > λ 0 } ⊂ ρ(A), where ρ(A) is the resolvent set of A. As D(A) is dense in L 1 (1, b), one can define (A * , D(A * )), the adjoint of (A, D(A)): A * w(x) = g(x)∂ x w(x) -β(x)w(1), D(A * ) = {w ∈ L ∞ (1, b), g∂ x w ∈ L ∞ (1, b), } .

Moreover, one has

Proposition 3. There exists a unique triplet

(λ 0 , φ, V ) ∈]0, +∞[×D(A * ) × D(A) such that AV = λ 0 V, A * φ = λ 0 φ φ ≥ 0, φ(1) = 1, and b 1 V (x)φ(x)dx = 1. More precisely, as β(x) = mx α then φ ∈ C 0 ([1, b)), φ ≥ m λ 0 > 0. (15)
Proof. We solve A * φ = λ 0 φ and get

φ(x) = e λ0G(x) b x β(y) g(y)
e -λ0G(y) dy , with G defined in [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]. Recall that θ = β(G -1 ), one deduces

φ(x) = e λ0G(x) +∞ G(x) θ(z)e -λ0z dz > 0. ( 16
) So φ ∈ C 0 ([1, b)). Moreover, as β is increasing in [1, b], one has mx α e -λ0G(x) λ 0 ≤ +∞ G(x) θ(z)e -λ0z dz ≤ mb α e -λ0G(x) λ 0 , so one deduces β(x) λ 0 ≤ φ(x) ≤ mb α λ 0 . ( 17
)
We finally choose

V (1) such that b 1 V (x)φ(x)dx = 1.

The homogeneous problem via a semigroup approach

Consider            ∂ ∂t v(x, t) + ∂ ∂x [g(x)v(x, t)] = 0, x ∈ (1, b), t ≥ 0, g(1)v(1, t) = b 1 β(x)v(x, t)dx t > 0, v(x, 0) = v 0 (x). (18)
We recall at first some properties about semigroups, let (X, . ) a Banach space (see [START_REF] Engen | One parameter Semigroups for Linear Evolution Equations[END_REF] for more details):

Definition 4. An operator (A, D(A)) on a Banach space X is said to be dis- sipative if: (λI -A)x ≥ λ x ,
Actually we have a more practical criterion in order to establish the dissipativity of an operator. For all x ∈ X, we define the dual set of x by:

J(x) = {x ∈ X ; x, x X,X = x 2 = x 2 },
where X is the dual set of X and we recall the result of [START_REF] Engen | One parameter Semigroups for Linear Evolution Equations[END_REF]:

Proposition 5. (A, D(A)) is dissipative if and only if for all x ∈ D(A), there is j(x) ∈ J(x) such that: Re Ax, j(x) X,X ≤ 0. (19)
Let us show now the main theorem Theorem 6. The operator (A, D(A)) generates a semigroup on

L 1 (1, b).
The theorem 6 is a consequence of the following theorem:

Theorem 7 ([8]
). Let (A, D(A)) a dissipative operator on a Banach space X such that there exists λ > 0 such that (λI -A) is onto, then the restriction of A to D(A) generates a contraction semigroup on D(A).

In particular:

D(A) = X ∀λ > 0, R(λI -A) = X ∃ω ∈ R; (A -ωI)is dissipative    ⇒ (A, D(A)) generates a semigroup on X.
The first two points come from Proposition 2. We just have to check the dissipativity condition:

Proposition 8. For all ω ≥ β L ∞ , A -ωI is dissipative. Proof. Let v ∈ D(A) and E = {x ∈ (1, b); v(x) = 0}. The function j(v) = v |v| 1 E c v L 1 (1,b) belongs to J(v). We have j(v) ∈ L ∞ with j(v) L ∞ ≤ v L 1 . In addition: v, j(v) L 1 (1,b),L ∞ (1,b) = b 1 v 2 |v| 1 E c v L 1 (1,b) = v 2 L 1 (1,b) = j(v) 2 L ∞ (1,b) .
Furthermore,

Av -ωv, j(v) L 1 (1,b),L ∞ (1,b) = - b 1 j(v)∂ x (gv)dx -w b 1 vj(v)dx = - b 1 j(v)∂ x (gv)dx -w v 2 L 1 (1,b) .
We use now lemma 25 proved in the appendix and we get:

Av -ωv, j(v) L 1 (1,b),L ∞ (1,b) = g(1) v L 1 (1,b) |v(1)| -ω v 2 L 1 (1,b)
Using the boundary condition (v ∈ D(A)), we get

Av -ωv, j(v) L 1 (1,b),L ∞ (1,b) = v L 1 (1,b) b 1 β(x)v(x)dx -ω v 2 L 1 (1,b) ≤ β L ∞ (1,b) -ω v 2 L 1 (1,b) • It is sufficient to choose β L ∞ (1,b)
≤ ω in order to obtain (19). Hence we have proved that (A, D(A)) generates a semigroup denoted e tA .

Let us recall some definitions of semigroup solutions (see [START_REF] Pazy | Semigroups of Linear 0perators and Applications to Partial Differential Equations[END_REF] for more details). Definition 9. Let X a Banach space and (A, D(A)) a generator of a C 0semigroup on X, denoted e tA . Let

y (t) = Ay(t), t ∈ (0, T ) y(0) = x . ( 20
)
We call weak solution or mild solution of (20), a function y ∈ C([0, +∞[; X) given by

y(t) = e tA x. (21)
strong solution or classical solution of (20) a function y ∈ C([0, +∞]; X) ∩ C 1 ((0, +∞[; X) given by ( 21) such that y(t) ∈ D(A) for all t ∈ (0, +∞ [ and satisfying (20) in [0, +∞[.

Moreover one has

Theorem 10. [START_REF] Pazy | Semigroups of Linear 0perators and Applications to Partial Differential Equations[END_REF] Let e tA a C 0 -semigroup on X of generator (A, D(A)). Then:

-for all x ∈ X (20) has a unique weak solution, -for all x ∈ D(A) (20) has a unique strong solution.

Therefore we have the main result of this subsection : Theorem 11.

-For any v 0 ∈ D(A) and T > 0 there exists a unique strong solution

v ∈ C([0, T ], D(A)) ∩ C 1 (]0, T ], L 1 (1, b)). It satisfies the system (18) in L 1 (1, b) -For any v 0 ∈ L 1 (1, b) and T > 0, there exists a unique weak solution v ∈ C([0, T ], L 1 (1, b)) such that (18) is verified in the distribution sense.
This ends the existence of solutions of the homogeneous problem associated to [START_REF] Devys | A model describing the growth and the size distribution of multiple metastatic tumors[END_REF]. In the next subsection we will construct a solution (weak) of ( 6) for zero initial data. Then we conclude by noticing that the unique solution of ( 6) is the sum of the homogeneous and this previous.

The non homogeneous problem via a fixed point argument

Consider    ∂ t v + ∂ x (gv) = 0 , x ∈ (1, b), t > 0, g(1)v(1, t) = b 1 β(x)v(x, t)dx + f (t), t > 0, v(x, 0) = 0. (22)
Definition 12. We call a strong solution on [0, T ] of system (22

) a function v ∈ C 1 ([0, T ]; L 1 (1, b)) such that gv ∈ C([0, T ]; W 1,1 (1, b)) , lim x→b -(gv)(x, t)
= 0 for all t ∈ [0, T ] and v satisfies (22). This function v can be viewed as a linear function T of the source term f : v = T f .

One has

Theorem 13. For any f ∈ C 1 ([0, +∞[) such that f (0) = 0 there exists a unique strong solution of system (22). Moreover, the operator T defined on {f ∈ C 1 ([0, +∞[); f (0) = 0} satisfies the positivity property:

f ≥ 0 ⇒ T f ≥ 0.
Proof. We are going to construct the strong solution by a fixed point argument. As we have to use a bootstrap argument in time, we have to consider a more general initial data than 0.

Lemma 14. For any

v 0 ∈ L 1 (1, b) such that gv 0 ∈ W 1,1 (1, b), lim x→b - (gv 0 ) = 0 and for all function f ∈ C 1 ([0, +∞[) such that f (0) = g(1)v 0 (1)- b 1 β(y)v 0 (y)dy, there exists a unique strong solution v of    ∂ t v + ∂ x (gv) = 0 g(1)v(1, t) = b 1 β(y)v(y, t)dy + f (t) v(., 0) = v 0 .

Proof.

For 0 ≤ T 0 < T , and u ∈ L 1 (1, b) let us define

X T0 T (u) = {w ∈ C 1 ([T 0 , T ]; L 1 (1, b)); w(., T 0 ) = u}. and X 0 T (v 0 ) = {w ∈ C 1 ([0, T ]; L 1 (1, b)); w(., 0) = v 0 } = X T .
We endow the set X T with the distance associated by :

v X T = sup t∈[0,T ] v L 1 (1,b) + sup t∈[0,T ] ∂ t v L 1 (1,b)
The space X T is a complete metric space. For w ∈ X 0

T (v 0 ), define T v0,f (w) = v by: v(x, t) =      1 g(x) b 1 β(y)w (y, t -G(x)) dy + f (t -G(x)) , G(x) < t 1 g(x) (gv 0 )(G -1 (G(x) -t)), G(x) > t,
where we recall that G(x) =

x 1 dy g(y) . Direct computations show that

v(t) L 1 (1,b) ≤ t β L ∞ (1,b) sup t∈[0,T ] w(t) L 1 (1,b) + T sup t∈[0,T ] |f (t)| + v 0 L 1 (1,b) ,
lim G(x)<t (x,t)→(x0,G(x0)) (gv)(x, t) = lim G(x)<t (x,t)→(x0,G(x0)) b 1 β(y)w (y, t -G(x)) dy + f (t -G(x)) = b 1 β(y)w (y, 0) dy + f (0) .
By the compatibility conditions satisfied by v 0 , f, w(., 0), one gets

lim G(x)<t (x,t)→(x0,G(x0)) (gv)(x, t) = (gv 0 )(1).

Besides as gv

0 ∈ W 1,1 (1, b), one has lim G(x)<t (x,t)→(x0,G(x0)) (gv)(x, t) = lim G(x)<t (x,t)→(x0,G(x0)) (gv 0 )(G -1 (G(x) -t)) = (gv 0 )(G -1 (0)). Recall that G -1 (0) = 1, so one deduces lim G(x)>t (x,t)→(x0,G(x0)) (gv)(x, t) = (gv 0 )(1). Therefore gv ∈ C([1, b) × [0, T ]) and v(., 0) = v 0 . Furthermore, as w ∈ X T and gv 0 ∈ W 1,1 (1, b), we have for x < b ∂ t v(x, t) =          1 g(x) b 1 β(y)∂ t w (y, t -G(x)) dy + f (t -G(x)) , if G(x) < t -1 g(x) (g(G(x) -t))∂ x (gv 0 )(G -1 (G(x) -t))), if G(x) > t and ∂ x (gv)(x, t) =          -1 g(x) b 1 β(y)∂ t w (y, t -G(x)) dy + ∂ t f (t -G(x)) , if G(x) < t 1 g(x) (g(G(x) -t))∂ x (gv 0 )(G -1 (G(x) -t))), if G(x) > t. Therefore ∂ t v, ∂ x (gv) ∈ C([0, T ]; L 1 (1, b)), (24)
and

∂ t v + ∂ x (gv) = 0. ( 25
)
So v = T v0,f (w) ∈ X T . Moreover, we remark

(T v0,f (w 1 ) -T v0,f (w 2 ))(x, t) = 1 g(x) b 1 β(y)(w 1 -w 2 )(y, t -G(x))dy 1 G(x)≤t , and
∂ t (T v0,f (w 1 ) -T v0,f (w 2 ))(x, t) = 1 g(x) b 1 β(y)∂ t (w 1 -w 2 )(y, t -G(x))dy 1 G(x)≤t , what gives T v0,f (w 1 ) -T v0,f (w 2 ) X T ≤ T β ∞ w 1 -w 2 X T .
Consequently, for T < 1 β ∞ , the considered application is a contraction on X T and has therefore a unique fixed point. Define

T 1 = 1 2 β ∞ . Let us denote by v this unique fixed point on [0, T 1 ]. It satisfies T v0,f (v) = v on X T1 . One has that f (T 1 ) = g(1)v(1, T 1 ) - b 1 (β(y)v(y, T 1
)dy, so one can define the operator T v(.,T1),f on X T1 2T1 (v(., T 1 )) and, as T 1 depends only on β ∞ , the same arguments show that it has a unique fixed point on [T 1 , 2T 1 ]. We can iterate the procedure in any interval of length T 1 and therefore in [0, +∞[. This ends the proof of the lemma.

The positivity property is a direct consequence of the fact that w ≥ 0 and, f ≥ 0 implies that T 0,f (w) ≥ 0. This ends the proof of the theorem. Proposition 15. For any T > 0, T is a linear continuous map from the space

{f ∈ C 1 ([0, T ]); f (0) = 0}, endowed with the L 1 (0, T ) norm, to C([0, T ]; L 1 (1, b)). Proof. By theorem 13, T is defined from {f ∈ C 1 ([0, +∞)); f (0) = 0} to C([0, T ], L 1 (1, b)) by T f = v
where v is the solution constructed above. Moreover from (24), one can apply lemma 25, and as v satisfies (23), (25), we get that

0 = ∂ t v + ∂ x (gv), j(v) L 1 ,L ∞ .
thus,

∂ t b 1 |v(x, t)| dx - b 1 β(y)v(y, t)dy + f (t) = 0. So v(t) L 1 (1,b) ≤ β L ∞ (1,b) t 0 v(s) L 1 (1,b) ds + t 0 |f (s)| ds.
Therefore, by Gronwall lemma, we have

v(t) L 1 (1,b) ≤ e t β L ∞ (1,b) T 0 |f (s)| ds, ∀t ≤ T.
Thus one gets

T f C([0,T ];L 1 (1,b)) ≤ e T β L ∞ (1,b) f L 1 (0,T ) . (26)
Now we are able to construct a weak solution of system (22).

Definition 16. We say that v is a weak solution of system ( 22) if there exists a sequence (f n ⊂ C 1 ([0, +∞)) such that f n (0) = 0 that converges to f in L 1 (0, T ) for any T > 0 and

(T f n ) converges to v in C([0, T ]; L 1 (1, b)).
Notice that by (26) the limit is independently of the choice of the sequence (f n ). Therefore by proposition 15, one has Proposition 17. For any f ∈ L 1 loc (0, +∞) there exists a unique weak solution. We still denote T the extension of T to L 1 loc (0, +∞).

Proof. It suffices to notice that if f ∈ L 1 loc (0, +∞) there exists a sequence

f n ∈ C 1 ([0, +∞[) such that f n (0) = 0 that converges toward f in L 1
loc (0, +∞) and to use (26).

Application to Von Foerster equation with Gompertzian growth

From the last subsections one can deduce an existence and uniqueness result for the global problem (6) : [START_REF] Devys | A model describing the growth and the size distribution of multiple metastatic tumors[END_REF] given by

Theorem 18. For any v 0 ∈ L 1 (1, b) and f ∈ C([0, +∞[), there exists an unique weak solution v ∈ C([0, +∞[; L 1 (1, b)) of system
v(t) = e tA v 0 + T f (t). ( 27
)
Moreover if v 0 ∈ D(A) and if f ∈ C 1 ([0, +∞[) such that f (0) = 0
, there exists a unique strong solution v, still given by ( 27), such that

f (t) = g(1)v(1, t) - b 1 β(y)v(y, t)dy, for all t ∈ [0, +∞[.

Asymptotic behaviour of weak solutions

The homogeneous problem

We start considering the homogeneous problem (18). What follows is an adaptation of ideas that can be found in [START_REF] Perthame | Transport Equations on Biology[END_REF], [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]. Let us decompose the space

L 1 (1, b) into : L 1 (1, b) = RV ⊕ (Rφ) ⊥ ,
where

(Rφ) ⊥ = u ∈ L 1 (1, b), u, φ L 1 ,L ∞ = b 1 uφ = 0 .
and φ is the eigenfunction defined in proposition 3.

Any u ∈ L 1 (1, b), can be decomposed into u = u, φ L 1 ,L ∞ V + (u -u, φ L 1 ,L ∞ V ) Actually let L 1
φ (1, b) be the space:

L 1 φ (1, b) = L 1 (( 1
e -λ0t v(t) L 1 φ (1,b) ≤ v 0 L 1 φ (1,b) .
(28)

2) Conservation of the mean value in

L 1 φ (1, b) b 1 e -λ0t v(t)φ(x)dx = b 1 v 0 (x)φ(x)dx , ∀t ≥ 0. (29) 3) Comparison principle v 0,1 ≤ v 0,2 ⇒ v 1 (t) ≤ v 2 (t). Proof. Let v 0 ∈ D(A) and v(t) = e -λ0t v(t), we have    ∂ t v + ∂ x (g(x) v ) + λ 0 v = 0 , x ∈ (1, b), t > 0, g(1) v(1, t) = b 1 β(x) v(x, t)dx, t > 0, v(x, 0) = v 0 (x). But φ satisfies -g(x)∂ x φ(x) + λ 0 φ(x) = β(x) , x ∈ (1, b), φ > 0, φ(1) = 1, and b 1 V (x)φ(x)dx = 1.

Therefore we have

∂ t ( vφ)(x, t) + ∂ x (g vφ)(x, t) = -β v(x, t), x ∈ (1, b), t > 0. (30)
One can remark that ṽ ∈ C([0, +∞[;

D(A)) ∩ C 1 ([0, +∞[; L 1 (1, b)
) and so as for vφ. Using this fact, and Lemma 25 we obtain

d dt b 1 | v| φdx + b 1 ∂ x (g | vφ|)dx = - b 1 β | v(x, t)| dx, t > 0.
Integrating by parts and taking into account the boundary condition on v , we get Integrating in time we get:

d dt b 1 | v| φ dx - b 1 β v(x, t)dx = - b 1 β | v(x, t)| dx, t > 0, (31)
e -λ0t v(t) L 1 φ (1,b) ≤ v 0 L 1 φ (1,b) , ∀t ≥ 0, ∀v 0 ∈ L 1 (1, b).
Hence the inequality (28) follows immediately for any v 0 ∈ D(A). By density of D(A) in L 1 (1, b) we deduce that (28) holds for any initial data in L 1 (1, b). In order to get (29) one integrates (30) with respect to

x d dt b 1 vφ dx + b 1 ∂ x ( vφ g)dx = - b 1 β v(x, t)dx.
Using the boundary condition, we finally obtain:

d dt b 1 vφ dx = 0.

Let us show the positivity property:

We start again from the equation (30). We first prove that if v 0 ≤ 0 then v(t) ≤ 0, ∀t ≥ 0, for v 0 ∈ D(A). Let us consider the function

sgn + (v) = max(v, 0) |v| 1 E c .
As in lemma 25 one gets:

b 1 ∂ x (gv)sgn + (v)dx = -g(1) v L 1 (1,b) v + (1)
We multiply now the equation (30) by sgn + (ṽ) and we obtain:

d dt b 1 ṽ+ φdx ≤ 0, integrating in time we have b 1 ṽ+ (x, t)φ(x)dx ≤ b 1 v + (x, 0)φ(x)dx,
and as v + (x, 0) = 0, we derive b 1 ṽ+ φdx = 0, ∀t ≥ 0.

As we have already shown that φ ≥ mb α λ0 , one gets ṽ+ (x, t) = 0, ∀t ≥ 0, or v(t) ≤ 0.

Thus we have proved the monotony property for all initial data in D(A) and then by density for any initial data in L 1 (1, b).

We state now our principal result concerning the asymptotic behaviour in the case of the model. As in [START_REF] Perthame | Transport Equations on Biology[END_REF], one has Theorem 20. There exists γ > 0 such that β(x) ≥ γφ(x), ∀x ∈ [1, b]. Moreover for any v 0 ∈ (Rφ) ⊥ , one has

e -λ0t e tA v 0 L 1 φ (1,b) ≤ e -γt v 0 L 1 φ (1,b) .
Therefore for any v 0 ∈ L 1 (1, b) we have:

lim t→+∞ e (-λ0+γ )t e tA v 0 -e λ0t v 0 , φ L 1 ,L ∞ V L 1 φ (1,b) = 0, ∀γ < γ.
Proof. From ( 17)) one can verify that γ ≥ λ0 b α . Let v 0 ∈ (Rφ) ⊥ , as e -λ0t e tA v 0 has zero mean value in L

| v 0 | φdx. ( 32
)
As for all

v 0 ∈ L 1 (1, b) v 0 = v 0 , φ L 1 ,L ∞ V + (v 0 -v 0 , φ L 1 ,L ∞ V ),
one gets

e tA v 0 = v 0 , φ L 1 ,L ∞ e λ0t V + e tA (v 0 -v 0 , φ L 1 ,L ∞ V ).
Then we have:

e -λ0t e tA v 0 = v 0 , φ L 1 ,L ∞ V + e -λ0t e tA (v 0 -v 0 , φ L 1 ,L ∞ V ).
Therefore, taking the norm and using (32), we obtain

e -λ0t e tA v 0 -v 0 , φ L 1 ,L ∞ V L 1 φ (1,b) ≤ e -γt v 0 -v 0 , φ L 1 ,L ∞ V L 1 φ (1,b)
which implies that

e -λ0t e tA v 0 -e λ0t v 0 , φ V L 1 φ (1,b) ≤ e -γt v 0 -v 0 , φ L 1 ,L ∞ V L 1 φ (1,b)
and concludes the proof of the theorem.

The non homogeneous problem

We focus now on the asymptotic behaviour of the weak solutions for the non homogeneous problem.

Theorem 21. For all initial data v 0 ∈ L 1 (1, b) and f ∈ C([0, +∞[) the unique weak solution of ( 22) satisfies for all t ≥ 0:

i) b 1 e -λ0t T f (x, t)φ(x)dx = t 0 e -λ0τ f (τ )dτ. ii) T f (t) L 1 φ (1,b) ≤ e λ0t t 0 e -λ0τ |f (τ )| dτ, ∀t ≥ 0. Proof. Consider f n ∈ {C 1 [0, +∞[; f n (0) = 0} such that f n → f in L 1 loc (0, +∞). From (24), one has d dt b 1 e -λ0t T f n (x, t)φ(x)dx - t 0 e -λ0τ f n (τ )dτ = 0.
Integrating in time, passing to the limit and taking into account the continuity of T , we deduce the first item. Besides, still by (24) one can apply lemma 25 to e λ0t T f n :

b 1 φe -λ0t |T f n (x, t)|dx ≤ t 0 e -λ0τ |f n (τ )|dτ.
Once again, passing to the limit we obtain the second item. This ends the proof of the theorem.

We can deduce now the principal result:

Theorem 22 (Asymptotic behaviour of weak solutions). i) For any v 0 ∈ (Rφ) ⊥ , the unique weak solution of ( 22) verifies

e -λ0t v(t) L 1 φ (1,b) ≤ t 0 e -λ0τ |f (τ )| dτ.
ii) There exists γ > 0 such that for any

v 0 ∈ L 1 φ (1, b) e -λ0t v(t) -v 0 , φ L 1 ,L ∞ V L 1 φ (1,b) ≤ e -γt v 0 L 1 φ (1,b) + t 0 e -λ0τ |f (τ )| dτ.
Proof. Recall that v(t) = e tA v 0 + T f (., t). From theorem 20 and theorem 21, one deduces the two items.

Application to Von Foerster equation with Gompertzian growth

In the model of growing metastatic tumors we have v 0 = 0 and f > 0, the unique solution of ( 1) is nonnegative and we have:

v(t) L 1 φ (1,b) = e λ0t t 0 e -λ0τ f (τ )dτ = e λ0t t 0
e -λ0τ β(x p (τ ))dτ.

We define now an interesting quantity for clinical applications. For a clinician, a useful data to know is without a doubt the total number N 1 of metastases and in particular the number of large size metastases that is to say for example the tumors whose size is larger than 10 9 cells, which correspond approximately to a tumor of 30g.

N bmin (t) = b bmin v(x, t)dx. (33)
We can give an estimate of N 1 . By (17) we have m λ0 ≤ φ ≤ mb α λ0 . Therefore, as

1 max φ v L 1 φ ≤ N 1 (t) = b 1 1 φ(x) v(x, t)φ(x) dx ≤ 1 min φ v L 1 φ we get λ 0 e λ0t t 0 e -λ0τ b -αe -aτ dτ ≤ N 1 (t) ≤ λ 0 b α e λ0t t 0 e -λ0τ b -αe -aτ dτ
We derive that for all t ≥ 0,

mb -1 e λ0t -1 ≤ N 1 (t) ≤ b α e λ0t . (34)
We can bring this to light in next section, with the numerical simulations.

An interesting fact to observe is that if we neglect the effect of the growth due to the metastases itself for small times, that is to say:

             ∂ ∂t ρ(x, t) + ∂ ∂x (g(x)ρ(x, t)) = 0, ∀ x ∈ [1, b], ∀ t ≥ 0 g(1)ρ(1, t) = β(x p (t)), ∀ t ≥ 0, ρ(0, x) = 0, ∀x ∈ [1, b].
, one obtains:

N ρ (t) := b 1 v(x, t)dx = t 0 β(x p (τ ))dτ = m t 0 b α 1 b αe -aτ
dτ.

We will see in the numerical simulations that N 1 (t) does not exceed 1 while more than one year and that the influence of the emission of metastases by metastases itself is also long to appear.

Numerical analysis

We propose in this section a numerical approximation of problem (1) for data coming from clinical observations of a metastatic hepatocarcinoma, (see [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF]):

a = 0.00286, b = 7.3 × 10 10 , α = 2 3 , m = 5.3 × 10 -8 .

A characteristics scheme

Due to various scales imposed by the clinical data, we observed that classical upwind finite volume schemes require a very fine discretization in order to be stable for long time. That is why we prefered to use a different approach using the behaviour of the solution of (1) along the characteristic curves. Such methods have been investigated in [START_REF] Angulo | Numerical scheme for size-structured populations equations[END_REF] for more general growth speed g but for space intervals of length 1. In the case of Gompertz growth, the scheme derivation is made easier, by the explicit expression of the solutions along the characteristics (2). Let k be a constant time step discretization, k = T N on this interval [0, T ]:

0 = t 1 < • • • < t n = (n -1)k < • • • < T N +1 = T.
The spatial discretization can be restricted to the interval [1, x p (T )], since we saw in the section 3.1.3 that the solution vanishes for (t, x), x ≥ x p (T ). Let x n = x p (t n ) this discretization:

1 = x 1 < x 2 < • • • < x N < x N +1 = x p (T ).
We then denote by v n i an approximation of the solution v of problem (1) at the point (x i , t n ).

The key point of the construction of this scheme is that the points (x i , t n ) and (x i+1 , t n+1 ) belong to the same characteristic for any n ≥ 1 and i ≥ 1. For all n ≥ 1, we solve explicitely the transport equation on the time interval [t n , t n+1 ] and we use a first order quadrature method to take into account the boundary condition. Computing the characteristics between the point (t n , x i-1 ) and (t n+1 , x i ), we obtain the following scheme:

                     v 0 1 = 1 g(1) β(x p (0)), v 0 i = 0, 2 ≤ i ≤ N + 1 v n+1 i = v n i-1 e ak f i-1 (k), i = 2, • • • , N + 1, n = 1, • • • , N v n+1 1 = 1 g(1) β(x p (t n+1 )) + 1 g(1) N i=2 h i β(x i )v n+1 i , n = 1, • • • , N (35)
where

f i (k) = xi b 1-e -ak and h i = x i -x i-1 .
Note that v n i = 0 for all i = n + 2, . . . , N + 1.

Convergence analysis

We prove in this section the convergence of the characteristics scheme. We denote e n i = v(x i , t n )-v n i the error of the scheme. Remarking that if one starts from the exact solution at the points (x i , t n ) then the scheme is exact at the points (x i , t n+1 ):

v(x i , t n+1 ) = v(x i-1 , t n )f i-1 (k)e ak we derive          e n+1 i = e n i-1 f i-1 (k)e ak e n+1 1 = 1 g(1) N i=2 h i β i e n+1 i + r n+1 e 0 i = 0 with r n = 1 g(1) b 1 β(x)v(x, t n ) dx - N i=2 h i β(x i )v(x i , t n ) .
The truncation error of the quadrature r n only comes from the quadrature error; one can estimate r n as soon as the function βv is smooth:

Proposition 23 (Truncation error). Let Ω T = {(x, t) ∈ [1, b) × 0, T a ; x ≤ x p (t)} and v ∈ C 1 (Ω T ). There exists C T = C(T, b) such that |r n | ≤ kC T sup [1,xp(tn)] |∂ x (βv)(. . . , t n )|, ∀t n ≤ T a .
Proof. Using the fact that v(x, t n ) = 0 for all x > x p (t n ), we can write that

r n = 1 g(1) n+1 i=2 xi xi-1 (β(x)v(x, t n ) -β(x i )v(x i , t n )) dx = 1 g(1) n+1 i=2 xi xi-1 x xi-1 ∂ x (β v(• , t n ))(s) ds dx ≤ sup [0,xp(tn)] |∂ x (β v(• , t n ))| 2g(1) n+1 i=2 (x i -x i-1 ) 2
Let us define the function

Ψ(x) = x 1 -b x 1-e ak
and remark that x i -

x i-1 = Ψ(x i ). We derive that n+1 i=2 (x i -x i-1 ) 2 = n+1 i=2 (x i -x i-1 )Ψ(x i-1 ).
The function Ψ is a concave on The convergence of the scheme is a consequence of the following error estimates:

(x i -x i-1 ) 2 ≤ 2(x p (t n ) -1)Ψ(x * ) ≤ (b 1-e -atn -
Theorem 24 (Error estimates). 2) There exists

Let Ω T = {(x, t) ∈ [1, b) × 0, T a ; x ≤ x p (t)} and v ∈ C 1 (Ω T ).
C T = C(T, β, v) such that e n 1,φ def = N i=1 h i |e n i φ(x i )| ≤ C T ke λ0tn .
Note that strong solutions of the problem (1) fullfills the assumptions of this theorem and thus the convergence of the characteristic scheme (35) is proved for the strong solutions of problem (1) for long time intervals.

Proof.

1) Error estimate in L 1 norm

We have

e n+1 1 = h 1 |e n+1 1 | + e ak N i=2 h i |e n i-1 |f i-1 (k) = h 1 |e n+1 1 | + e ak N -1 i=1 h i+1 |e n i |f i (k)
with

h 1 |e n+1 1 | ≤ h 1 a ln b N i=2 h i β i |e n+1 i | + h 1 |r n+1 | ≤ h 1 a ln b N -1 i=1 h i+1 β i+1 e ak f i (k)|e n i | + h 1 |r n+1 |.
Hence we obtain

e n+1 1 ≤ h 1 |r n+1 | + N -1 i=1 h i |e n i |e ak A 1 i A 2 i . (36)
with

A 1 i = f i (k) hi+1 hi and A 2 i = 1 + h 1 βi+1
a ln b . Remark that:

• For any k > 0 and any i ≥ 1, A 1 i ≤ 1. Indeed, we can write h i = e ln(b)(1-e -aik ) 1 -e ln(b)e -aik (1-e ak ) h i+1 = e ln(b)(1-e -aik ) e ln(b)e -aik (1-e -ak ) -1 Hence,

f i (k) h i+1 h i = e -ln(b)e -iak
(1-e -ak ) e ln(b)e -aik (1-e -ak ) -1 1 -e ln(b)e -aik (1-e ak ) = 1 -z e -ak 1 -z ≤ 1

as z = e -ln(b)e -aik (e ak -1) ≤ 1 and then 1 -z e -ak ≤ 1 -z.

• There exists k 0 such that for all k ≤ k 0 , A 2 i ≤ 1 + 2kmb α ≤ e 2kmb α . This is indeed a corollary of the mean value theorem: Using a discrete Gronwall lemma we get:

h 1 = x p (k) -1 = b 1-e -ak - 1
e n 1 ≤ 2ak ln(b) n-1 l=0
r l e (n-1-l)k(a+2mb α ) .

so that

e n 1 ≤ 2ak ln(b) max l=1,...,n
r l e (a+2mb α )tn -1 e (a+2mb α )k -1 ,

We conclude using Proposition 23 and the fact that

2mb α ∼1.851488869 ≤ λ 0 . 2) Error estimate in L 1 φ norm Now we have e n+1 1,φ = h 1 φ(x 1 )|e n+1 1 | + e ak N i=2 h i φ(x i )|e n i-1 |f i-1 (k) = h 1 φ(1)|e n+1 1 | + e ak N -1 i=1 h i+1 φ(x i+1 )|e n i |f i (k)
with as φ(1) = 1

h 1 φ(1)|e n+1 1 | ≤ h 1 g(1) N i=2 h i β i |e n+1 i | + h 1 |r n+1 | ≤ h 1 g (1)
N -1

i=1 h i+1 β i+1 e ak f i (k)|e n i | + h 1 |r n+1 |.
Hence we obtain

e n+1 1,φ ≤ h 1 |r n+1 | + N -1 i=1 h i φ(x i )|e n i |e ak A 1 i A 3 i . (37)
with

A 1 i = f i (k) h i+1 h i , A 3 i = 1 φ(x i ) φ(x i+1 ) + h 1 β i+1 g (1)
.

Remark that -For any k > 0 and any i ≥ 1,

A 1 i ≤ 1 (38)
-There exists k 0 > 0 such that for any k < k 0

A 3 i ≤ e kλ0+C2ak 2 . (39)
Estimate (38) is proved in the first part of this proof. To prove (39), we recall that g(x)φ (x) + β(x) = λ 0 φ(x) so we get

A 3 i = 1 + 1 φ(x i ) φ(x i+1 ) -φ(x i ) + h 1 g(1) (λ 0 φ(x i+1 ) -g(x i+1 )φ (x i+1 )) = 1 + 1 Φ(t i ) Φ(t i+1 ) -Φ(t i ) + h 1 g(1) (λ 0 Φ(t i+1 ) -Φ (t i+1 ))
where Φ = φ • x p . As h1 g(1) = k + ak 2 C 1 and as φ(t i ) ≥ mb α λ0 , we obtain that for k small enough

A 3 i ≤ e kλ0+C2ak 2 .
The estimate (37) becomes for k ≤ k 0 :

e -(λ0+C2ak)tn+1 e n+1 1,φ ≤ 2ak ln(b)e -(λ0+C2ak)tn+1 r n+1 + e -(λ0+C2ak)tn e n 1,φ .

Using a discrete Gronwall lemma we get: We conclude the proof using Proposition 23.

Numerical results

Influence of the production of metastases by metastases previously created

The figure 1 and figure 2 gives some comparisons between N bmin and N ρ (see (33) for their definition) for different times. We observe that the influence of the previously created metastases, that is to say the influence of the integral term in the boundary condition, is hardly visible before almost three years, as we could expect.

Evolution of the metastases number following the tumor size

We compare the evolution of the metastases number for large size tumors. The figure 3 shows that before three years, there are very few metastases greater than 10 8 cells. In the figure 4, we compare the number of metastases N 10 8 and N 10 9 . In these figures the time step is k = 1 hour. We compare in table 1 the values of N 1 (T ), that we obtain with the one obtained in [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF]. We observe a good agreement between our results. Time in years N 1 (T ) [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF] Let us now check the lower bound of N 1 given in (34). We choose two differents values of a (see [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF]) and we compute the approximation of λ 0 obtained in [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF] and [START_REF] Devys | A model describing the growth and the size distribution of multiple metastatic tumors[END_REF] by some Laplace transformations. In the figure 5, we observe that the function ln(N 1 (t)) becomes linear after some time. These slope gives a very close approximation of λ 0 , as shown in Table 2 In figure 6, we compare the solution obtained by the characteristics scheme and the theoritical profile given in [START_REF] Devys | A model describing the growth and the size distribution of multiple metastatic tumors[END_REF] and [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF] For α = .4, we observed in figure 7 the "U-shaped" profile claimed in [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF].

Conclusion

Even if this model is a simplification of a more complicated phenomena, its asymptotic gives interesting informations about the evolution of metastatic sites. In particular it shows that for large time, the exponential growth of the density of metastases is given by a parameter λ 0 . The knowledge of this parameter is therefore fundamental. As its defintion shows (see [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF]) , λ 0 depends on a, b, α, m. An interesting problem would be to identify these coefficents by suitable observations on the patient. The first question could be: can we find a one to one correspondance between clinical observations and the parameters a, b, α, m that we use in the model? This leads to a mathematical uniqueness problem that we plan to investigate. The minimization of λ 0 with respect to the parameters a, m, α is another important question. One can define H(λ, a, m, α) := Direct computations show that H(λ 0 , a, m, α) = 1.

The Malthus parameter λ 0 is then an increasing function of all the parameters a, m, α.

It would be interesting to obtain a sharper estimate where dependence on these parameters is expressed. Finding theoretical estimates is an open problem.

Appendix

We finally give in this section the proof of a technical result that we previously used:

  and lim x→b -(gv)(x, t) = 0. (23) Moreover we observe that v is continuous on [1, b) × [0, T ]\{t = G(x)} and from the hypothesis on w we deduce that for all x 0 ∈ [1, b):

  , b); φdx) endowed with the norm u φ := b 1 |u(x)| φ(x)dx. Thanks to the proposition 3, . φ defines a norm equivalent to the L 1 norm. We state now an important proposition giving some properties of the weak solutions: Theorem 19. For any initial data in L 1 (1, b), the weak solution of the homogeneous problem (18) satisfies 1) Decreasing of the norm L 1 φ (1, b)

hence d dt b 1 | v| φ dx = b 1 β vdx - b 1 β

 111 | v| dx ≤ 0, as β > 0.

dt b 1 e -λ0t e tA v 0 φdx = b 1 βe -λ0t e tA v 0 dx - b 1 β e -λ0t e tA v 0 dx = b 1 ( 1 β e -λ0t e tA v 0 dx ≤ b 1 ( 1 β e -λ0t e tA v 0 dx ≤ b 1 ( 1 β e -λ0t e tA v 0 dx ≤ -γ b 1 e 1 e

 11111111111 1 φ (1, b), from (31) one can deduce d β -γφ)e -λ0t e tA v 0 dx -b β -γφ)e -λ0t e tA v 0 dx -b β -γφ) e -λ0t e tA v 0 dx -b -λ0t e tA v 0 φdx , therefore b -λ0t e tA v 0 φdx ≤ e -γt b 1

[ 1 ,

 1 b], it increases on [1, x * ] with x = be -ak e ak -1 and decreases on [x * , b].While x p (t n+1 ) < x * , that is while at n+1 < ln(ln(b)), n+1 i=2 (x i -x i-1 ) 2 ≤ (x p (t n+1 ) -1)Ψ(x p (t)) ≤ (b 1-e -atn -1)b 1-e -atn 1 -b e -atn (1-e ak ) ≤ 2(b 1-e -T -1)b 1-e -T e -T ak ln(b) If at n+1 ≥ ln(ln(b)), the estimate becomes n+1 i=2

  1)bak. This conclude the proof with C T = (b 1-e -T -1)b 1-e -T e -T if T < ln(ln(b)) and C T = b ln(b) (b 1-e -T -1) else.

1 )

 1 There exists C T = C(T, β, v, b) and k 0 such that for all k ≤ k 0 n i | ≤ C T ke λ0tn .

1 ≤

 1 ≤ ak ln(b) sup x∈[0,ak] (e -x e ln(b)(1-e -x ) ) ≤ 2ak ln(b) as soon as k ≤ k 0 where k 0 = The estimate (36) becomes for k ≤ k 0 : e n+1 2ak ln(b)r n+1 + e n 1 e k(a+2mb α ) .

e n 1 ,

 1 φ ≤ e (λ0+C2ak)tn 2a ln(b)n l=1 ke -(λ0+C2ak)t l r l ≤ 2a ln(b)k e (λ0+C2ak)tn -1 e (λ0+C2ak)k -

Figure 1 .

 1 Figure 1. Number of metastases after 3 years with k = 2 hours (left) and k = 1 hours. (right)

Figure 2 .Figure 3 .

 23 Figure 2. Number of metastases after 5 years

  :

Figure 4 .

 4 Figure 4. Comparison of the number of metastases larger than 10 8 cells and larger than 10 9 cells after 3 and 5 years a λ 0 estimated in [11] λ 0 obtained with scheme (35) a = .00286 0.058 0.055 a = .0143 0.0201 0.0204

Figure 5 .

 5 Figure 5. Log of the number of metastases after 5 years with a = .00286 (left) and a = .0143 (right). The asymptotic slope of these curves corresponds to λ 0 * 365.

Figure 6 .

 6 Figure 6. Comparison of the approximate solution and the theoritical solution after 2000 days with λ 0 = 2.02995

Figure 7 .

 7 Figure 7. Solution with α = .4

  -1 (x))e -λ0x dx. Computing G leads to H(λ, a, m, α) := mb α +∞ 0 b -αe -ax e -λx dx.

Lemma 25 . 1 ∂ 1 if s ≥ 1 n- 1 1 ∂ 1 vg∂

 25111111 For any functionv ∈ L 1 (1, b) such that gv ∈ W 1,1 (1, b), we have: b x (gv)j(v)dx = -g(1) v L 1 (1,b) |v(1)| .Proof. Let us consider the following sequence:Γ n (s) =   if s ≤ -1 n ns otherwise , For any function v ∈ D(A), we have g∂ x v ∈ L 1 (1, b) and g∂ x vΓ n (v) ∈ L 1 (1, b). Furthermore {x ∈ (1, b); Γ n (v(x)) = 0} = { |v| ≤ 1 n }. Hence vg∂ x v Γ n (v) ∈ L 1 (1, b) and b x (gv)Γ n (v) = -b x v Γ n (v)dx -g(1)v(1)Γ n (v(1)).By Lebesgue's theorem, one can deduce:∂ x (gv)Γ n (v) -----→ n→+∞ ∂ x (gv) j(v) v L 1 (1,b)and moreover Γ n (v(1)) -----→ n→+∞ |v(1)| . Moreover, as g∂ x v ∈ L 1 (1, b), we conclude that g(∂ x v)vΓ n (v) -----→ n→+∞ 0 in L 1 (1, b).

Table 1 .

 1 Comparative results of N 1 (T )

			N 1 (T ) authors
	T = 3	135	134
	T = 3.4	263	260
	T = 3.6	396	396
	T = 3.8	712	718

Table 2 .

 2 Comparative results of λ 0