MOTIVIC SERRE INVARIANTS OF CURVES by ...

MOTIVIC SERRE INVARIANTS OF CURVES by Johannes Nicaise∗ & Julien Sebag

Abstract. — In this article, we generalize, to a relative point of view, the notion of motivic Serre invariant and we compute relative motivic Serre invariant for curves defined over the field of fractions of a complete discrete valuation ring R of equicharacteristic zero. In particular, we develop a relative theory of motivic integration for formal schemes topologically of finite type. One aim of this study is to understand the behaviour of motivic Serre invariants under ramified extension of the ring R. Thanks to our constructions, we obtain an expression for the generating power series, whose coefficients are the motivic Serre invariant associated to a curve, computed on a tower of ramified extensions of R. We give an interpretation of this series in terms of the motivic zeta function of Denef and Loeser.

1. Introduction In [18], F. Loeser and J. Sebag have constructed a motivic analog of the Serre invariant of a smooth compact p-adic analytic variety (see [24]), for smooth rigid analytic spaces Xη . This object, defined in some quotient of the Grothendieck ring of varieties K0 (V ark ), depends only on the rigid analytic space Xη , but can be expressed in terms of the special fiber of any weak Néron model of Xη . In this article, we generalize the theory of motivic integration for formal schemes, initiated in [23], as well as the notion of motivic Serre invariant, to a relative point of view. Consider a separated formal scheme X∞ , flat, generically smooth, and topologically of finite type over a complete discrete valuation ring R. We associate to any such X∞ , an element of a quotient of the relative Grothendieck ring of varieties over the special fiber Xs of X∞ . Specializing to a quotient of the Grothendieck ring of varieties over the residue field k of R, we recover the (absolute) motivic Serre invariant of the generic fiber Xη of X∞ , defined previously in [18].

∗

Research Assistant of the Fund for Scientific Research – Flanders (Belgium)(F.W.O.).

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JOHANNES NICAISE & JULIEN SEBAG

The key point of this article is the computation of (relative) motivic Serre invariants of curves in equicharacteristic zero, i.e. generically smooth algebraic curves over a complete discrete valuation ring R of equicharacteristic zero, where we suppose that the residue field of R is algebraically closed. We study the behaviour of motivic Serre invariants under ramified extension of R. An explicit answer is obtained by constructing weak Néron models from an appropriate resolution of singularities for the curve. As a consequence, we prove that the generating power series, whose coefficients are the (relative) Serre invariants associated to our curve, calculated on a tower of ramified extension of R, coincides with Denef and Loeser’s motivic zeta function, if our R = k[[t]]-curve is obtained by base change from a regular dominant k[t]-curve Y . As explained in [20], this fact makes it possible to consider the zeta function as a Weil zeta function, and to apply the philosophy of the solution of the Weil Conjectures to the study of the Monodromy Conjecture. The main idea is that the geometric properties of the singularities of the morphism Y → Spec k[t] interact with the arithmetic properties of the generic fiber of t-adic completion of Y . Many generalizations should be studied. Does the theory extend to arbitrary generically smooth R-varieties, instead of only curves? What happens if we allow any complete discrete valuation ring as base ring R? And does the theory also hold for formal R-schemes? These and other questions will be studied in forthcoming papers, in particular in [21]. To conclude this introduction, we give an overview of the paper. The first Section contains the essential notations and definitions that will be used throughout the paper, as well as some preliminary results. Section 3 recalls the definition of the Greenberg schemes associated to a formal scheme. In Section 4, we develop a theory of motivic integration on a formal scheme, relative to the special fiber of the scheme, essentially by re-examining the constructions in [23] with extra care. The principal part of the paper starts with Section 5. We recall the notion of “weak Néron model”, and we construct weak Néron models for a generically smooth algebraic curve over R, and for its base changes to the finite ramifications of R, from appropriate resolutions of singularities for the curve. In Section 6, we use the material in Section 4 to define the relative motivic Serre invariant of a generically smooth formal R-scheme, generalizing the constructions in [18]. As in the absolute case, this Serre invariant can be expressed in terms of a weak Néron model for the scheme. Finally, in Section 7, we use the results in Section 5 to give an expression for the relative motivic Serre invariant of a generically smooth algebraic curve over R, and of its base changes to the finite ramifications of R, in terms of a resolution of singularities for the curve. Encrypting the Serre invariants in a generating power series, we obtain the Serre Poincaré series associated to the curve, and we give an explicit expression of this series in terms of a resolution of singularities. In Theorem 7, we compare this series to Denef and Loeser’s motivic zeta function. The second author thanks Qing Liu for very helpful discussions.

MOTIVIC SERRE INVARIANTS OF CURVES

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2. Notations, definitions, conventions • Let R be a complete discrete valuation ring, with quotient field K, and perfect residue field k. We fix a uniformizing element π, and we put, for each integer n ≥ 0, Rn := R/(π n+1 ). For any integer e ≥ 1, we put R(e) := R[T ]/(T e − π), and we denote by K(e) its quotient field. Let A be a k-algebra. We put L(A) = A if R is a ring of equal characteristic, and L(A) = W (A), the ring of Witt vectors, when R is a ˆ L(k) L(A). ring of unequal characteristic. We denote by RA the ring RA := R⊗ • An stf t formal R-scheme is a separated formal R-scheme, topologically of finite type over R, in the sense of [9]/10 or [2]. Throughout this paper, we’ll always suppose a formal R-scheme to be stf t. We recall that a formal R-scheme is a locally ringed space (X∞ , O∞ ) in topological R-algebras. It is determined by the system of Rn -schemes Xn = (X∞ , OX∞ ⊗R Rn ), n ≥ 0. The k-scheme X0 is denoted by Xs , and is called the special fiber of X∞ . As a topological space, X∞ is homeomorphic to Xs , and OX∞ = ←− lim OXn . We have n Xn = Xn+1 ×Rn+1 Rn , and X∞ = −→ lim Xn in the category of locally ringed spaces. n To an stf t formal R-scheme X∞ , we can associate a generic fiber Xη , which is a quasi-compact, separated, rigid analytic K-space, in the sense of [2]. If Xη is smooth over K, we call X∞ generically smooth. To a separated R-scheme of finite type X, we can associate its π-adic completion X∞ = −→ lim (X ×R Rn ). This is an stf t formal n R-scheme. Its underlying space is homeomorphic to the special fiber Xs = X ×R k, but its structure sheaf OX∞ contains information on the infinitesimal neighbourhood of Xs inside X. A formal R-scheme X∞ that is isomorphic to the π-adic completion of a scheme X, is called algebrizable, with algebraic model X. In this article, we will consider schemes, formal R-schemes, and rigid analytic spaces. We adopt the following notational conventions: a roman capital (e.g. X) always denotes a scheme, a subindex ∞ (as in X∞ ) always denotes a formal scheme, Xs is always a k-scheme that is realized as the special fiber of an R-scheme X or a formal R-scheme X∞ , Xη is a rigid analytic K-space that is realized as the generic fiber of a formal R-scheme X∞ , and XK is the generic fiber X ×R K of an R-scheme X. If X is a R-scheme, we denote by X∞ its π-adic completion. • For any integer e ∈ N\{0}, we denote X(e) := X ×R R(e) and X∞ (e) := b R R(e). Observe that (X(e))∞ = X∞ (e). X∞ × • If S is a scheme, we denote by Sred its underlying reduced scheme. • Let X be a regular separated flat R-scheme of finite type. We say Xs is a strict normal crossing divisor, if we can find, for each closed point x of Xs , a regular Qmsystem i of local parameters (x0 , . . . , xm ) in the local ring OX,x , such that π = u i=0 xM i , with u a unit, and Mi ∈ N. Let Ei , i ∈ I, be the irreducible components of the underlying reduced scheme (Xs )red of Xs . If Xs is a strict normal crossing divisor, then Ei is smooth over k, for each i ∈ I. We’ll denote by di the length of the local ring of Xs at the generic point of Ei , and we call di the multiplicity of Ei in Xs . For any subset J of I, we define EJ := ∩i∈J Ei and EJo := EJ \ (∪i∈J / Ei ).

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Definition 1. — Let J be a non-empty subset of I. We say that an integer e ≥ 1 is J-linear if there exists, for each j ∈ J, an integer αj ∈ N\{0}, such that X αj dj e= j∈J

We say that an integer e ≥ 1 is Xs -linear if there exists a subset J ⊂ I, with |J| > 1 and EJo 6= ∅, such that e is J-linear. e → X the normalization of X. • If X is an R-scheme of finite type, we denote by X • If S is a scheme, an S-variety is a separated, reduced S-scheme of finite type. A K-curve is a one dimensional K-variety. • For a formal R-scheme X∞ , we denote by Sm(X∞ ) the open formal subscheme of X∞ defined as the smooth locus of the structural morphism X∞ → Spf(R). The complement of Sm(X∞ ) in X∞ (with the induced reduced structure), is denoted by Sg(X∞ ). We’ll denote by Sm(Xs ) the open subscheme of smooth points of Xs over k, and by Sg(Xs ) its complement. For any stf t formal R-scheme X∞ , we have Sm(X∞ )s = Sm(Xs ). For any separated flat R-scheme of finite type X, we have Sm(X)∞ = Sm(X∞ ). 3. Greenberg Schemes We recall some material from [4]/9.6 and [14] (see also [23]/§3 or [18]/§2.3). For every n ≥ 0, we consider the functor h∗n , from the category of k-schemes to the category of locally ringed spaces, defined (locally) by   Spec(Wn (A)) if R a ring of unequal characteristic h∗n (Spec(A)) =  Spec(A[[t]]/tn+1 )) if R a ring of equal characteristic where Wn (A) is the ring of Witt vectors of length n with coefficients in A. By a fundamental result of M. J. Greenberg (see [14]), which, in the equal characteristic case, amounts to Weil restriction of scalars, for any Rn -scheme Xn , locally of finite type, the functor T 7→ HomRn (h∗n (T ), Xn ) from the category of k-schemes to the category of sets, is represented by a k-scheme Grn (Xn ), which is locally of finite type. Hence, for every perfect k-algebra A, ¡ ¢ Grn (Xn )(A) ' Xn Rn ⊗L(k) L(A) and, in particular, setting A = k, we have Grn (Xn )(k) ' Xn (Rn ) We obtain a functor Grn from the category of Rn -schemes locally of finite type, to the category of k-schemes locally of finite type. Now let us consider an stf t formal R-scheme X∞ . The canonical adjunction morphism h∗n+1 (Gr (Xn+1 )) → Xn+1 gives rise, by tensoring by Rn , to a canonical morphism of Rn -schemes h∗n+1 (Gr (Xn+1 )) → Xn . From this morphism, one derives,


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n+1 (or simply πnn+1 ) again by adjunction, a canonical morphism of k-schemes πn,X ∞ n+1 πn,X : Grn+1 (Xn+1 ) → Grn (Xn ) ∞

In this way, we attach to the formal scheme X∞ a projective system (Grn (Xn ))n∈N n+1 of k-schemes of finite type. The morphisms πn,X being affine, the projective limit ∞ Gr(X∞ ) := ←− lim Grn (Xn ) n

exists in the category of k-schemes. It is not of finite type, in general. We’ll denote the canonical projection morphisms by πn,X∞ : Gr(X∞ ) → Grn (Xn ) (or simply πn ). We obtain a functor Gr from the category of stf t formal R-schemes, to the category of separated and quasi-compact k-schemes. We’ll write Grn (X∞ ) instead of Grn (Xn ). If X is a separated R-scheme of finite type, and X∞ is its π-adic completion, we write Grn (X), resp. Gr(X), instead of Grn (X∞ ), resp. Gr(X∞ ). Proposition 1. — 1. The functors Grn (the functor Gr) respect open and closed immersions and fiber products, and they send affine Rn -schemes of finite type (affine stf t formal R-schemes) to affine k-schemes of finite type (to affine kschemes). (i) 2. Let X∞ be an stf t formal R-scheme, and let (X∞ )i∈I be a finite covering by (j) (i) formal open subschemes. There are canonical isomorphisms Gr(X∞ ∩ X∞ ) ' (j) (i) Gr(X∞ ) ∩ Gr(X∞ ), and the scheme Gr(X∞ ) is canonically isomorphic to the (i) scheme obtained by gluing the schemes Gr(X∞ ). Proof. — See, for example, [23]/§3. Remark 1. — When R = k[[t]], and Z is a k-variety, the schemes Grn (Z ×k R) and Gr(Z ×k R) are canonically isomorphic to the schemes Ln (Z) and L(Z), defined in [7].

4. Relative Motivic Integration In [23], the second author has developed a theory of motivic integration for stf t formal R-schemes, taking values in a completion of the localized Grothendieck ring of k-varieties Mk (we recall its definition in Section 4.1). We generalize this construction to a relative point of view, adopting the strategy followed in [23]. Throughout this section, we endow the schemes Grn (Xn ), and Gr(X∞ ) with their reduced structure. We obtain new functors Grn and Gr, which verify, for any perfect field F ⊃ k, Grn (X∞ )(F ) ' Xn (Rn ×L(k) L(F )) and Gr(X∞ )(F ) ' X∞ (RF )

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ˆ L(k) L(F ), where L(A) = A when R Recall that we denote by RF the ring RF := R⊗ is a ring of equal characteristic and L(A) = W (A), the ring of Witt vectors, when R is a ring of unequal characteristic. 4.1. The relative Grothendieck ring of varieties. — Let F be a field, and let Z be a variety over F . Consider the free Abelian group, generated by the isomorphism classes [X] of Z-varieties X. We take the quotient of this group w.r.t. the following relations: whenever X is a Z-variety, and Y is a closed subvariety of X, we have [X] = [X \ Y ] + [Y ]. This quotient is called the Grothendieck group of varieties over Z, and is denoted by K0 (V arZ ). We denote the class [A1Z ] of the affine line over Z by LZ . A constructible subset C of a Z-variety X can be written as a disjoint union of locally closed subsets, and defines unambiguously an element [C] of K0 (V arZ ). When Z is a separated scheme of finite type over k, we will write K0 (V arZ ) instead of K0 (V arZred ). For any separated scheme X of finite type over Z, we will write [X] instead of [Xred ]. We can define a product on K0 (V arZ ) as follows: for any pair of Z-varieties X, Y , we put [X].[Y ] = [X ×Z Y ]. This definition extends bilinearly to a product on K0 (V arZ ), and makes it into a ring, the Grothendieck ring of varieties over Z. The localized Grothendieck ring MZ is obtained by inverting LZ in K0 (V arZ ). We denote by IZ the ideal of K0 (V arZ ) generated by [A1Z ] − [Z]. Observe that, for any subvariety V of Z, the element [A1V ] − [V ] = ([A1Z ] − [Z]).[V ] belongs to IZ . For any m ∈ N, consider the subgroup F m MZ of MZ , generated by [S]L−i with cZ the completion of the ring MZ with respect to dim S − i ≤ −m. We’ll denote by M • cZ , which is metrisable, the filtration F MZ . This filtration induces a topology on M by the norm cZ → R≥0 kk :M defined by kak :=

 cZ and a 6∈ F n+1 M cZ  2−n , if a ∈ F n M 

0, if a = 0 A morphism of k-varieties f : W → Z induces base-change ring morphisms K0 (V arZ ) → K0 (V arW ) and MZ → MW , as well as forgetful morphisms of Abelian groups K0 (V arW ) → K0 (V arZ ) and MW → MZ . The definition of the latter morphism deserves some care: an element [X]L−m W of MW , for some W -variety X and some positive integer m, is mapped to [X]L−m in MZ . One checks that this yields Z a well-defined morphism of Abelian groups MW → MZ . This morphism induces a cW → M cZ on the completions. forgetful morphism M If Z = Spec k, we write K0 (V ark ), L, and Mk , rather than K0 (V arSpec k ), MSpec k , and LSpec k . The Grothendieck group K0 (V arZ ) is a universal additive invariant for Z-varieties: if A is an Abelian group, and χ is an invariant of Z-varieties taking values in A, such that, for any Z-variety X and any closed subvariety Y ⊂ X, χ(X) = χ(X \Y )+χ(Y ), then χ factors uniquely through a group morphism χ : K0 (V arZ ) → A, defined by


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χ([X]) = χ(X). If A is a ring, and χ is multiplicative, i.e. χ((X ×Z Y )red ) = χ(X).χ(Y ) for any pair of Z-varieties X, Y , then χ : K0 (V arZ ) → A is a morphism of rings. For instance, for any k-variety X, we can consider its topological Euler characteristic χtop (X). Fix a prime `, invertible in k. Then χtop (X) is defined as χtop (X) :=

X

(−1)i dim Hci (X ×k k s , Q` )

i≥0

where Hci ( . , Q` ) is `-adic étale cohomology with proper support, and k s is a separable closure of k. This is an additive invariant, hence defines a morphism of groups χtop : K0 (V arZ ) → Z for any base variety Z. It is multiplicative for Z = Spec k, so we get a morphism of rings χtop : K0 (V ark ) → Z.

4.2. The construction of the relative motivic integral. — Let X∞ be an stf t formal R-scheme, with generic fiber Xη purely of dimension d, and special fiber Xs . Definition 2. — Let S a scheme, X, Y and F three S-schemes, A ⊂ X and B ⊂ Y two constructible subsets of X and Y , respectively. A piecewise S-morphism π : B → A, consists of the following data: a finite partition {Vi }i∈I of B into locally closed subsets Vi of X, a finite partition {Wi }i∈I of A into locally closed subsets Wi of Y , and a set of morphisms of S-schemes {πi : Wi → Vi }i∈I . We identify this piecewise S-morphism with any piecewise S-morphism obtained by refining {Vi }i∈I and corestricting the morphisms πi . We say that a piecewise S-morphism π : B → A is piecewise trivial S-fibration with fiber F , if we can represent it by a set of morphisms of S-schemes {πi : Wi → Vi }i∈I , such that there exists, for each i ∈ I, an isomorphism Wi ∼ = Vi ×S F , such that πi corresponds to the projection Vi ×S F → Vi . Lemma 1. — Let S and F be two k-schemes, let X and Y be two S-schemes, and let A ⊂ X and B ⊂ Y be two constructible subsets of X and Y , respectively. If π : Y → X is a piecewise morphism of S-schemes, which is a piecewise trivial kfibration with fiber F , then π : B → A is a piecewise trivial S-fibration with fiber F ×k S. Proof. — By definition, we can represent π by a set of morphisms of S-schemes {πi : Wi → Vi }i∈I , such that there exists, for each i ∈ I, an isomorphism of k-schemes

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Wi ∼ = Vi ×k F , such that the following diagram of k-morphisms commutes : ' / Vi × k F Wi8 88 88 88 π 88 i 88 88 8¿ ² Vi

Of course, Vi ×k F carries the structure of an S-scheme, induced by the S-structure of Vi , and this diagram is a diagram of S-morphisms. We can conclude by the canonical isomorphism of S-schemes Vi ×k F ' Vi ×S (S ×k F )

Example 1. — If X∞ is a formal R-scheme, smooth over R, with d-equidimensional generic fiber Xη , then the transition morphisms πnn+1 : Grn+1 (X∞ ) → Grn (X∞ ) are locally trivial (so, in particular, piecewise trivial) fibrations with fiber Adk (see [23]/Lemme 3.4.2). We recall some useful definitions from [23] or [18], which are similar to definitions in [7] : Definition 3. — A subset B of Gr(X∞ ) is a n-cylinder of Gr(X∞ ), with n ∈ N, if there exists a constructible subset of Grn (X∞ ) such that B = πn−1 (C). A subset B of Gr(X∞ ) is a cylinder of Gr(X∞ ), if it is a n-cylinder for some n ∈ N. The set CX∞ of cylinders is a Boolean algebra. Definition 4. — A subset B of Gr(X∞ ) is a stable n-cylinder of Gr(X∞ ), with n ∈ N, if it is a n-cylinder of Gr(X∞ ), and if, for each integer m ≥ n, the truncation morphism πm+1 (B) → πm (B) is a piecewise trivial Xs -fibration with fiber AdXs . By Lemma 1, this definition is equivalent to [23], Déf. 4.3.19. Remark 2. — Let n ∈ N, and let C be an constructible subset of Grn (Xn ). Then a fundamental result of Greenberg (see [13] or [22] for a generalization in the rigid setting) ensures that the subset ¡ ¢ πn πn−1 (C) of Grn (X∞ ) is constructible. Lemma 1 allows us to generalize [23]/Lemme 4.5.4 (or [18]/Proposition 3.2.1). Lemma 2. — Let X∞ be a flat, stf t formal scheme purely of dimension d. There exists an integer c ≥ 1 such that, for each pair e, n ∈ N with n ≥ ce, the projection πn+1 (Gr (X∞ )) → πn (Gr (X∞ ))


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³ ´ is a piecewise trivial Xs -fibration over πn Gr(e) (X∞ ) with fiber AdXs , where Gr(e) (X∞ ) := Gr (X∞ ) \πe−1 (Gre (Sg (Xe ))) and where Sg(Xe ) denotes the reduced closed subscheme of Xe , consisting of the points where the structural morphism Xe → Spec Re is not smooth. Proof. — This assertion follows from [23]/Lemme 4.5.4 and Lemma 1, applied to πn+1 (Gr (X∞ )) → πn (Gr (X∞ )) → Xs . In particular, Gr(e) (X∞ ) is a stable cylinder, for any e. So, as in [23]/Proposition 4.5.6 (or [18]/Porposition 3.6.2), we can define the motivic measure of a cylinder. Proposition 2. — Let X∞ be a stf t formal R-scheme purely of dimension d. For any cylinder B of Gr(X∞ ), the limit ´ ³ ³ ´ −(e+1)d lim [πe Gr(e) (X∞ ) ∩ B ]LXs e→+∞

cX , and coincides with exists in M s lim

n→+∞

³

−(n+1)d

´

[πn (B)]LXs

Proof. — The proof of the second point is analogous to the proof of [23]/Proposition 5.4.7. For the first point, see the proof of [23]/Proposition 4.5.6, using Lemma 2 instead of [23]/Lemme 4.5.4. Definition 5. — 1. For any stable n-cylinder B of Gr(X∞ ), we define the na¨ıve motivic measure of B as −(n+1)d

µ ˜X∞ (B) = [πn (B)]LXs

∈ MXs

2. For every cylinder B, we define the motivic volume µX∞ (B) of B as one of the two limits in Proposition 2. It is clear that, for any stable cylinder B of Gr(X∞ ), the motivic volume µX∞ (B) is equal to the image of the na¨ıve motivic measure µ ˜X∞ (B) under the completion cX . morphism MXs → M s Definition 6. — A subset A of Gr(X∞ ) is a measurable subset of Gr(X∞ ) if, for every positive real integer ε, there exists an ε-cylindrical approximation of A, i.e. a family of cylinders (A0 (ε); (Ai (ε))i∈I ), with I be a countable set, such that (A4A0 (ε)) ⊂ ∪i∈I Ai (ε) and kµX∞ (Ai (ε))k < ε, for all i ∈ I. We say that A is strongly measurable if, moreover, we can take A0 (ε) ⊂ A. The set of measurable subset of Gr(X∞ ) forms a Boolean algebra, denoted by DX∞ . Example 2. — Every cylinder is measurable. So, as in [23]/Théorème 6.2.2 (see also [18]/ Theorem 3.7.2), Proposition 2 implies the following statement.

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Theorem 1. — Let X∞ be a flat stft formal scheme purely of dimension d. Then, for any measurable subset A, the limit µX∞ (A) := lim µX∞ (A0 (ε)) ε→0

cX and is independent of the choice of the cylindrical approximations of exists in M s A. Proof. — The assertion follows from the same arguments as in [23]/Théorème 6.2.2. Definition 7. — We define an additive map on the boolean ring DX∞ with values cX in M s cX µX∞ : DX∞ → M s that we call the motivic measure on X∞ . As in [23] or [18], we can define motivic integrals of measurable functions. Let A be a measurable subset of Gr(X∞ ). A function α : A → Z ∪ {∞} is integrable, if α−1 (n) is measurable for all n ∈ Z, and if the sum X µX∞ (α−1 (n))L−n Xs n∈Z

cX . In this case, we define converges in M s Z X L−α dµX∞ := µX∞ (α−1 (n))L−n Xs A

n∈Z

Remark 3. — If α takes only a finite number of values, and if all its fibers are stable cylinders, then this integral is already defined in MXs . A crucial example of an integrable function is the following. Let h : Y∞ → X∞ be a morphism between stf t formal R-schemes, purely of dimension d. Let y be a point of Gr(Y∞ )\Gr(Sg(Y∞ )), defined over a perfect field extension F of k. We denote by ϕ : SpfRF → Y∞ the corresponding morphism of formal R-schemes. We define ordπ Jach (y), the order of the Jacobian ideal of h at y, as follows. From the natural morphism h∗ ΩdX∞ /R → ΩdY∞ /R , one deduces a morphism ³ ´ ³ ´ ϕ∗ h∗ ΩdX∞ /R /(torsion) → ϕ∗ ΩdY∞ /R /(torsion) It follows from the³structure theorem for finite-type modules over ´ ³ ´ principal domains that the image of ϕ∗ h∗ ΩdX∞ /R /(torsion) in L := ϕ∗ ΩdY∞ /R /(torsion) is either 0, in which case we set ordπ Jach (y) = ∞, or π n L for some unique n ∈ N, in which case we set ordπ Jach (y) = n. One defines then the map ordπ Jach : Gr(Y∞ )\Gr(Sg(Y∞ )) → N ∪ {∞} by y 7→ ordπ Jach (y)


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This is an integrable function, in the above sense (see [23]/Proposition 5.1.6 and [23]/Lemme 5.1.2). i Proposition 3. — Let X∞ be a stf t formal R-scheme. Let (X∞ )i∈I be a covering of X∞ by a finite number of open formal R-suschemes. If A ⊂ Gr(X∞ ) is a measurable subset of Gr(X∞ ), and α : A → Z ∪ {∞} is an integrable function, then we have, in cX , M s Z Z X L−α dµX∞ L−α dµX∞ = (−1)|J|−1 A

i ) A∩Gr(∩i∈J X∞

∅6=J⊂I

i Proof. — It is a direct consequence of [23]/Lemme 6.5.3, since Gr(X∞ ) is a cylinder.

Definition 8. — Let h : Y∞ → X∞ be a morphism between stf t formal R-schemes purely of dimension d. The wild locus of h is the subset Σh of Gr(Y∞ ) defined by Σh := Gr(Sg(Y∞ )) ∪ h−1 (Gr (Sg(X∞ ))) ∪ {y ∈ Gr(Y∞ ) | ordπ Jach (y) = ∞} The set Σh is measurable. We say that the morphism h is tame, if the measure of Σh is zero. Example 3. — Isomorphisms and admissible blow-ups are tame morphisms. Lemma 1 allows to generalize [23]/Lemme 7.1.3 (or [18]/Lemma 3.9.2) to : Lemma 3. — Let h : Y∞ → X∞ be a morphism between stf t formal R-schemes purely of dimension d. We assume that Y∞ is smooth over R. Let B be a cylinder in Gr(Y∞ ), and put A := h(B). Assume that ordπ Jach has constant value e < ∞ on B, 0 and that A ⊂ Gr(e ) (X∞ ) for some e0 ≥ 0. Then A is a cylinder and, if the restriction of h to B is injective, then there exists N ∈ N such that, ∀n ≥ N , the piecewise Xs morphism πn (B) → πn (A) induced by h is a piecewise trivial Xs -fibration with fiber AeXs Proof. — The result follows from [23]/Lemme 7.1.3 and Lemma 1, applied to h : Grn (Y∞ ) → Grn (X∞ ) → Xs . From Lemma 1 and 3, we obtain a relative “ change of variables formula ” (cf. [23]/Théorème 8.0.4). Theorem 2. — Let h : Y∞ → X∞ be a tame morphism between stf t formal Rschemes purely of dimension d. We assume that Yη is smooth over K. Let B and A be strongly measurable subsets in Gr(Y∞ ) and Gr(X∞ ), respectively. We assume that h induces a bijection between between B and A. Let α : A → Z ∪ {∞} be an integrable function. Then the application α ◦ h + ordπ Jach is integrable, and Z Z L−α dµX∞ = Lα◦h+ordπ Jach dµY∞ A

cX . in M s

B

12


Proof. — One proves the assertion as [23]/Théorème 8.0.4, using Lemma 3 instead of [23]/Lemme 7.1.3. Remark 4. — 1. We consider the right hand side of the equality in Theorem 2 as cX , via the forgetful morphism of Abelian groups M cY → M cX , an element of M s s s induced by h (see Subsection 4.1). 2. As in [18]/Theorem 3.9.4, if hη is étale, if α takes only a finite number of values on A, and if the fibers of α are stable cylinders, then the function α◦h+ordπ Jach takes only a finite number of values on B, and its fibers are stable cylinders. In this case, the equality in Theorem 2 holds already in MXs . 5. Weak N´ eron models 5.1. Definitions and preliminary results. — Let Xη be a separated, quasicompact smooth rigid K-space. A weak Néron R-model for Xη is a smooth stf t formal scheme U∞ over Spf R, whose generic fiber is an open rigid subspace of Xη , and which has the property that the natural map U∞ (Rsh ) → Xη (K sh ) is bijective [5]. By this latter property, we mean that U∞ (R0 ) → Xη (K 0 ) is bijective, for any finite unramified extension K 0 of K, where R0 is the normalization of R in K 0 . Observe that this map is always injective, since U∞ is separated. Definition 9. — If X∞ is a generically smooth, stf t formal R-scheme, we say that U∞ → X∞ is a Néron R-smoothening for X∞ , if it has the following properties: 1. U∞ is a weak Néron R-model for Xη , 0 2. there exists a morphism X∞ → X∞ of stf t formal R-schemes, inducing an 0 isomorphism Xη → Xη on the generic fibers, such that U∞ → X∞ factors 0 . through an open immersion U∞ → X∞ In this case, we say that X∞ admits a Néron smoothening (NS) over R (or a R-NS). Remark 5. — Since U∞ is a weak Néron R-model for Xη , the open immersion 0 0 U∞ → X∞ will automatically be an isomorphism of U∞ onto Sm(X∞ ). Indeed, 0 U∞ is smooth over R, by definition, and any point on the special fiber of Sm(X∞ ) 0 0 0 lifts to an R -point on Sm(X∞ ), for some finite unramified extension R of R. The result in [5]/§3, Theorem 3.1 can be interpreted in our context as follows : Theorem 3. — If X∞ is a generically smooth, stf t formal R-scheme, then X∞ admits a Néron smoothening (NS) over R. Lemma 4. — If X∞ is a regular, generically smooth, stf t formal R-scheme, then Sm(X∞ ) is a Néron R-smoothening of X∞ . Proof. — It suffices to prove that the inclusion morphism Sm(X∞ ) → X∞ induces a bijection Sm(X∞ )(R0 ) → X∞ (R0 ), for any finite unramified extension K 0 of K, where R0 is the normalization of R in K 0 . One can use the same arguments as in [4]/§3, Proposition 2.


13

Remark 6. — Néron smoothenings can be empty. For instance, let K 0 be a separable, finite, ramified extension of K, and let R0 be the normalization of R in K 0 . Then ∅ → Spf R0 is a Néron R-smoothening. Weak Néron models have a natural interpretation in terms of Greenberg schemes (see [18]/Proposition 2.7.3) : Proposition 4. — Let X∞ be a generically smooth, stf t formal R-scheme, and let U∞ → X∞ be a Néron R-smoothening for X∞ . Then the morphism U∞ → X∞ induces a bijection between the points of Gr(U∞ ) and Gr(X∞ ). We state the following result for later use. Lemma 5. — Let X be a separated, flat R-scheme of finite type. Let K 0 be a finite extension of K, and denote by R0 the normalization of R in K 0 . The π-adic completion morphism HomR (Spec R0 , X) → HomR (Spf R0 , X∞ ) is a bijection. Proof. — Since Spec R0 is proper over Spec R, this is a corollary of Grothendieck’s Existence Theorem; see [10],(5.4.1). 5.2. Weak N´ eron models and resolution of singularities. — Definition 10. — Let X∞ be a generically smooth stf t formal scheme over R. A resolution (of singularities) of X∞ , is a composition of admissible formal blow-ups 0 0 h : X∞ → X∞ , such that X∞ is regular. Proposition 5. — Let X∞ be a generically smooth stf t formal scheme over R, 0 and suppose that X∞ allows a resolution X∞ → X∞ . Then the induced morphism 0 Sm(X∞ ) → X∞ is a R-NS of X∞ . Proof. — Straightforward from Lemma 4. If R has equicharacteristic 0, any algebraizable, generically smooth stf t formal R0 scheme X∞ admits a resolution X∞ → X∞ , by Hironaka’s resolution of singularities for excellent schemes in characteristic zero [15]. 5.3. Description of the normalization. — The results of this paragraph are contained in [17]/§10.4, where a complete discussion of Deligne-Mumford’s theorem on semi-stable reduction of curves is given. The statements that we present here, are essentially reformulations of the arguments of the proof of Proposition 10.4.6 in [17]. The reader can also consult [1]/§1.1. The second author thanks Q. Liu for very helpful discussions on these questions. Suppose that the residue field k of R is an algebraically closed field of characteristic zero. Let X be a separated, regular, flat R-scheme P of finite type, and of dimension 1, such that Xs is a strict normal crossing divisor i∈I di Ei . The aim of this subsection, is to describe the normalization of X(e), locally around Eio , where di |e.

14


regular, flat R-scheme of finite type, such Definition 11. — Let X be a separated, P that Xs is a strict normal crossings divisor i∈I di Ei . For each i ∈ I, we define a Ei ] → X(e). ei (e) as the inverse image of Ei under the normalization map X(e) variety E ] e ◦ (e) as E ◦ (e) ×X(e) X(e). Likewise, we define a E ◦ -variety E i

i

i

Let e be an integer such that di |e, for some i ∈ I. Let ∆ be an irreducible ei (e). The description in [17]/§10.4 shows that the morphism E e ◦ (e) → component of E i ◦ ◦ Ei is an étale cover of Ei of degree di , and does not depend on the integer e (see the proof of [17], Prop. 10.4.6, as well as [17], Remark 10.4.8). So : Lemma 6. — Let XK be a separated, smooth curve over K, and X be a separated, regular, flat R-model of XK such that Xs is a strict normal crossings divisor P 0 0 i∈I di Ei . Then, for e, e ∈ N such that di |e and di |e , e ◦ (e) ∼ e ◦ (e0 ) E =E i i as Ei◦ (e)-schemes. Proof. — As we noted above, for every integer e ≥ 1 such that di |e, the canonical e ◦ (di ) is an isomorphism over E ◦ . In particular, if e0 ≥ 1 is e ◦ (e) → E morphism E i i i e ◦ (e) → E e ◦ (e0 ). another integer such that di |e0 , we deduce a canonical isomorphism E i i Thanks to this lemma, we can omit the integer e from the notation, and we’ll e ◦ instead of E e ◦ (e). simply write E i i Remark 7. — Let X be obtained by base change from a regular, irreducible, flat k[t]-variety Y , X := SpecB, with B of the form A/(ud − tb), where A is a smooth ˜ o /E o is given exk[t]-algebra and u is a local parameter and b ∈ A× . Then the cover E i i d plicitely by SpecBs [W ]/(W − b), where Bs = B/(t) (see proof of [17], Prop. 10.4.6). In particular, if Y is a complex, irreducible, smooth C-variety, and f : Y → A1C = Spec C[t] is a flat morphism of schemes, such that f −1 (0) ⊂ Y is a strict normal P ˜ ◦ /E ◦ , associated to the crossing divisor i∈I di Ei , then, for any i ∈ I, our cover E i i C[[t]]-variety X = Y ×C[t] Spec C[[t]], coincides with the one constructed in [8],(2.3). Note that the C[[t]]-variety X is isomorphic to the closed subscheme V (f − t) ⊂ Y ×C Spec C[[t]]. Proposition 6. — Let XK be a separated, smooth curve over K, and X be a separated, regular, flat R-model of XK such that Xs is a strict normal crossings divisor P i∈I di Ei . Then ] s = td |e E e◦ Sm(X(e)) `

`

ei (e) is equal to di /gcd(e, di ) and Proof. — The multiplicity of a component ∆ of E ◦ e every point in Ei (e) is smooth, when di |e. Besides, if di |e and if xi is a point of ∆ lying above an intersection point of Ei with another component Ej , then xi is not smooth (an explicit description can be found in the proof of [17], Prop. 10.4.6). So the result follows from these facts.


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5.4. Weak N´ eron models and ramification. — The notion of weak Néron model is stable under unramified base change: let K 0 be a finite unramified extension of K, denote by R0 the normalization of R in K 0 , and let X∞ be any generically smooth, stf t formal R-scheme. Then U∞ → X∞ is a Néron R-smoothening for X∞ , iff b R R0 → X∞ × b R R0 is a Néron R0 -smoothening for X∞ × b R R0 . U∞ × However, it is not stable under ramified base change. Example 4. — Consider the formal R-scheme X∞ := SpfR{x, y}/(xy − π), and the base change induced by the ring extension R(2) := R[T ]/(T 2 − π). Denote by O the closed point (x = 0, y = 0) on the special fiber Xs . Since X∞ is regular, its smooth locus Sm(X∞ ) = X∞ \ {O} is a Néron Rsmoothening. While no R-points of X∞ pass through the point O, there are R(2)b R R(2) points of X∞ passing through O, e.g. the point (T, T ). Hence, Sm(X∞ )× b b is not a weak Néron model for Xη ×K K(2), since the image of Sm(X∞ )×R R(2) → b K K(2) does not contain the point (T, T ). Xη × It is natural to ask how weak Néron models behave under ramified base change. In this paper, we will only consider the case where the residue field k of R is an algebraically closed field of characteristic zero. Moreover, we restrict to algebrizable stf t formal R-schemes of dimension one, i.e. to π-adic completions of curves over R. The general case will be studied in [21]. Lemma 7. — Let X be a separated, regular R-scheme of finite type, such that the P special fiber Xs is a strict normal crossing divisor i∈I di Ei . Let e ≥ 1 be an integer. Let K 0 be a finite and unramified extension of K(e), and denote by R0 the normalization of R(e) in K 0 . Let ψ : Spf R0 → X∞ be a morphism of formal R-schemes, and denote by ψ(0) the image of the closed point of Spf R0 under ψ. If J is the unique subset of I with ψ(0) ∈ EJo , then e is J-linear. Proof. — By Lemma 5, the morphism ψ is obtained by completion from a morphism ψ : Spec R0 → X. Let K 0 be the field of fractions of R0 , and denote by vK 0 the unique discrete valuation on K 0 with vK 0 (π) = e. The maximal ideal of the completed local Q d ring of X at ψ(0) ∈ EJo contains elements xj , j ∈ J, with π = u j∈J xj j , with u a unit. Hence, X e = vK 0 (π) = dj vK 0 (xj (ψ)). j∈J

Lemma 8. — Let X be a separated, regular, generically smooth R-scheme of finite type, P of pure relative dimension m, such that Xs is a strict normal crossing divisor i∈I di Ni . Let e ≥ 1 be an integer. If e is not Xs -linear (Definition 1), then ] ∞ → (X(e)) ^ Sm(X(e)) ∞ is a Néron R(e)-smoothening for X∞ (e).

16


] ∞ is the open formal subscheme Proof. — By Proposition 6, we know that Sm(X(e)) ◦ ] ˜ of X(e)∞ with underlying space td` |e E` . Let K 0 be a finite unramfied extension of K(e), and denote by R0 the normalization ] → X(e) ] induces a of R(e) in K 0 . We’ll show that the open immersion Sm(X(e)) bijection 0 ] ] 0) Sm(X(e))(R ) → X(e)(R ] → X(e) is proper, and is an isomorphism on The normalization morphism X(e) ] → X(e)η . By the valuative the generic fibers, hence induces an isomorphism X(e) η ] 0 ) → X(e)(R0 ) is a bijection. criterion for properness, the map X(e)(R Hence, it suffices to prove the following statement: if ψ is an element of X(e)(R0 ), then the image ψ(0) of the closed point of Spec R0 is contained in some Eò , with d` | e. This follows from Lemma 7, and the fact that e is not Xs -linear. Lemma 9. — Fix an integer e > 0. Let X be a separated, regular, flat R-scheme of finite type, such that the special fiber Xs is a strict normal crossing divisor E = P (j) : X (j+1) → X (j) , i∈I di Ei . There exists a composition of admissible blow-ups π j = 0, . . . , r − 1, such that – X (0) = X, – the special fiber of X (j) is a strict normal crossing divisor E (j) =

kj X

(j)

(j)

di Ei ,

i=1 (j)

– π (j) is the formal blow-up with center EJ (j) , for some subset J (j) of {1, . . . , kj }, with |J (j) | > 1, – e is not E (r) -linear. Proof. — We’ll prove the following, stronger, property: there exist a composition of π (j) as in the statement of the lemma, such that, for each subset J ⊂ {1, . . . , kr }, (r) with |J| > 1, and with (E (r) )oJ non-empty, there exists a j in J with dj > e. For each integer a > 0, we denote by J (a, E) the set of subsets J ⊂ I, with |J| > 1, with EJo 6= ∅, and with dj ≤ a for all j ∈ J. Let a0 be the smallest integer with J (a0 , E) non-empty. We will proceed by induction on the cardinality of J (a0 , E). 0 Fix a subset J ⊂ I, with |J| > 1, such that J belongs to J (a0 , E). Let π : X∞ → X∞ be the formal blow up with center EJ , denote the exceptional component by E00 , 0 0 and the strict transform of Ei by E i , for each i in I. The special P P fiber of X∞ is a 0 0 strict normal crossing divisor E = i∈It{0} di Ei , where d0 = j∈J dj . By minimality of a0 , one of the dj , j ∈ J, equals a0 , so d00 > a0 , and J 0 ⊂ I ∪ {0} does not belong to J (a0 , E 0 ), if 0 ∈ J 0 . Since (E 0 )oJ is empty, J ∈ / J (a0 , E 0 ). Hence, 0 0 it suffices to observe that for each J ⊂ I, J ∈ / J (a0 , E) implies J 0 ∈ / J (a0 , E 0 ). We can summarize the previous results as follows :


17

regular, generically smooth, flat R-scheme Proposition 7. — Let X be a separated, P of finite type, such that Xs = d N is a strict normal crossing divisor. Let i i i∈I e ≥ 1 be an integer. Then there exists a flat, regular R-model X 0 , deduced from X by a finite sequence of admissible blow-ups (i.e. blow-ups with center contained in 0 (e) ) → X 0 (e) ^ the special fiber), such that Sm(X eron R(e)-smoothening. In ∞ is a N´ ∞

0 (e) ) is a weak N´ ^ b K K(e). particular, Sm(X eron model for Xη × ∞

6. Relative motivic Serre invariants Using motivic integration on formal schemes [23] and the theory of weak Néron models [5], Loeser and the second author showed in [18] how one can associate a notion of volume to a smooth separated quasi-compact rigid space Xη over K, and a gauge form ω on Xη , in analogy with the classical p-adic case [24]. This volume is an element of the localized Grothendieck ring Mk , and its residue class in Mk /(L − 1) = K0 (V ark )/(L − 1) depends only on Xη and not on ω. This residue class is called the motivic Serre invariant of Xη , and denoted by S(Xη ). Using the constructions in §4, we will generalize these definitions to the relative setting. Our main result is the following. Theorem 4. — Let X∞ be a generically smooth, stf t formal R-scheme, purely of dimension d. If U∞ → X∞ and V∞ → X∞ are two Néron R-smoothenings of X∞ , then [Vs ] = [Us ] in K0 (V arXs )/IXs . Definition 12 (relative motivic Serre invariant). — Let X∞ be a generically smooth, stf t formal R-scheme, purely of dimension d. Let U∞ → X∞ be any Néron R-smoothening of X∞ . We define the relative motivic Serre invariant S(X∞ ) associated to X∞ , by S(X∞ ) := [Us ] ∈ K0 (V arXs )/IXs By Theorem 4 above, this definition does not depend on the choice of Néron Rsmoothening U∞ → X∞ . For the proof of Theorem 4, we need some additional ingredients. Let X∞ be a generically smooth, stf t formal R-scheme, purely of dimension d. Consider a differential d-form ω ˜ in ΩdX∞ /R (X∞ ). Let x be a point of Gr(X∞ )\Gr(Sg(X∞ )) defined over some perfect field extension F of k. We denote by ϕ : Spf(RF ) ³ → X∞ the correspond´ ing morphism of formal R-schemes. The submodule of L := ϕ∗ ΩdX∞ /R /(torsion) generated by ϕ∗ ω ˜ is either 0, in which case we set (ordπ (˜ ω )) (x) = 0, or of the form π n L for some integer n ∈ N, in which case we set (ordπ (˜ ω )) (x) = 0. We obtain an integrable map ordπ (˜ ω ) : Gr(X∞ )\Gr(Sg(X∞ )) → N ∪ {∞} (see [18]/TheoremDefinition 4.1.2). Now assume X∞ is affine, and take ω ∈ ΩdXη /R (Xη ). Using the canonical isomorphism ΩdXη /R (Xη ) ' ΩdX∞ /R (X∞ )×R K (see [3]/Proposition 1.5), we write ω = π −n ω ˜,

18


with ω ˜ ∈ ΩdX∞ /R , and we define ordπ,X∞ (ω) : Gr(X∞ )\Gr(Sg(X∞ )) → N ∪ {∞} by (ordπ,X∞ (ω)) (x) = ordπ (˜ ω )(x) − n Clearly, this definition does not depend on the choice of ω ˜ . Since the construction is local, we can drop the assumption that X∞ is affine, and define ordπ,X∞ (ω) for any generically smooth, stf t formal R-scheme X∞ , purely of dimension d. 0 Lemma 10. — Let X∞ → X∞ be a morphism of generically smooth, stf t formal R-schemes, purely of dimension d, inducing an isomorphism Xη0 → Xη on the generic fibers, and let ω be a d-form on Xη . Then Z Z 0 (ω) cX 0 L−ordπ,X∞ ∈ M dµX∞ L−ordπ,X∞ (ω) dµX∞ = s 0 )\Gr(Sg(X 0 )) Gr(X∞ ∞

Gr(X∞ )\Gr(Sg(X∞ ))

Proof. — One can copy the proof of [18]/Theorem-Definition 4.1.2, using Theorem 2 instead of [18]/Theorem 3.9.4. Moreover, if ω is a gauge form on Xη , the proofs of [18]/Theorem-Definition 4.1.2 and [18]/Theorems 4.5.1, show that the integral Z Z |ω| := L−ordπ,X∞ (ω) dµX∞ X∞

Gr(X∞ )\Gr(Sg(X∞ ))

is already defined in MXs . Lemma 11. — Let X∞ be a smooth stf t formal R-scheme, and let ω be a gauge i )i∈I be the connected components of X∞ , with generic points form ¡on X¢η . Let (X∞ i ηi ∈ X ∞ . The function ordπ,X∞ (ω) takes exactly one value on each connected s i component X∞ of X∞ . If we denote this value by ordηi (ω), then Z X i L−ordπ,X∞ (ω) dµX∞ = L−d [πn,X∞ (B)∩Grn (X∞ )]L−ordηi (ω) in MXs B\Gr(Sg(X∞ ))

i∈I

for any n-cylinder B ⊂ Gr(X∞ ) of G(X∞ ). Proof. — This lemma is a relative analogue of [18]/Proposition 4.3.1. We give a proof from a slightly different point of view. By the Maximum Principle (see [3]/Lemma 5.4 or [5]/Lemma 4.1, 4.2, 4.3), we see that the function ordπ,X∞ (ω) takes exactly one i value on each connected component X∞ of X∞ . We denote this value by ordηi (ω), for all i ∈ I. The definition of the motivic integral implies the result. Lemma 11 implies that Z X∞

|ω| = [Us ] ∈ K0 (V arXs )/IXs

where U∞ → X∞ is a R-NS of X∞ . Indeed, by the change of variables formula, we can assume that U∞ → X∞ is an open immersion. Since U∞ is a weak Néron R-model, the inclusion map Gr(U∞ ) → Gr(X∞ ) is an isomorphism.


In particular,

19

Z X∞

|ω| = [Us ] ∈ K0 (V arXs )/IXs

only depends on X∞ , and not on the gauge form ω. This value is exactly the relative motivic Serre invariant S(X∞ ) (Definition 12). Remark 8. — A gauge form on Xη does not always exist. But the smoothness of Xη implies that it does exist locally. So, by the additivity of motivic integrals (see Proposition 3), we can still define an element Z |ω| X∞

in K0 (V arXs )/IXs , associated to a family ω := (ωi )i∈Γ of local gauge forms ωi ∈ ΩdX i /R (Xηi ), where (Xηi )i∈Γ is an admissible (finite) covering of Xη by admissible η

(quasi-compact) open rigid subspaces Xηi of Xη . Proof of Theorem 4. — Let us prove Theorem 4. All the definitions being local, it follows from Proposition 3 that we can assume that Xη admits a global gauge form. The remarks above imply that Z |ω| = [Us ] ∈ K0 (V arXs )/IXs X∞

for every R-NS U∞ → X∞ . Lemma 12. — Let Xη a K-rigid analytic space, quasi-compact, separated and smooth over K, purely of dimension d. Let X∞ be a R-formal model of Xη . Via the forgetful morphism F or : K0 (V arXs )/IXs → K0 (V ark )/(L − 1), we have F or(S(X∞ )) = S(Xη ) in K0 (V ark )/(L − 1). Proof. — Take a R-NS U∞ → X∞ . Both members are equal to the class [Us ] in K0 (V ark )/(L − 1). If Y∞ is another model for Xη , dominating X∞ , the forgetful morphism K0 (V arYs )/IYs → K0 (V arXs )/IXs maps S(Y∞ ) to S(X∞ ). So the data of the Serre invariants computed over all Rmodels of Xη form a projective system. Definition 13. — The relative Serre invariant Srel (Xη ) is the class 0 lim S(X∞ ) ← 0 X∞

in K0 (V arXη )/IXη := lim K0 (V arXs0 )/IXs0 , ← 0 X∞

0 X∞

where runs over the projective system of formal R-models for Xη , ordered by admissible blow-up.

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Remark 9. — This construction depends only on Xη . Compare to the construction of the Zariski-Riemann space of a rigid variety in [11]. 7. Motivic Serre invariants, ramification and curves 7.1. Motivation. — One motivation of this work (and more generally of [21] or [19]) is the study of the motivic zeta function, the motivic Milnor fiber, and the Monodromy Conjecture; see [20] for the general philosophy of our project. In this final section, we will link the Serre invariants of the completion of a C[t]-curve and its ramifications, to the motivic zeta function associated to this curve. Let us recall some notations and results of [8]. Let Y be a smooth irreducible complex variety, and let f : Y → A1C = Spec C[t] be a dominant C-morphism of schemes. We define Yn,1 as the constructible subset of Ln (Y ) given by © ª Yn,1 := x ∈ Ln (Y ) | f (x) = tn mod tn+1 Denef and Loeser’s motivic zeta function, associated to f , is the Poincaré series X [Yn,1 ]T n ∈ K0 (V arX0 ) Zf (T ) := n≥1

It reflects some crucial properties of the critical points of f , see, for instance, [6]. We will show how this Poincaré series can be understood in terms of the motivic Serre invariant of the nearby fiber of f (i.e. the generic fiber of the t-adic completion of Y /C[t]). Our main result is Theorem 7. 7.2. Motivic Serre invariants of curves. — We assume that k is an algebraically closed field of characteristic zero, unless explicitly stated otherwise. Let X be a separated, flat, generically smooth R-scheme of finite type, such that Xη has pure dimesion 1. In order to simplify notations, we write S(X) instead of S(X∞ ) to denote the motivic Serre invariant associated to the π-adic completion X∞ of X. Definition 14. — An embedded reslution of singularities for the pair (X, Xs ), is a finite composition of blow-ups h : X 0 → X, with centers inPthe special fiber, such that X 0 is regular, and Xs0 is a strict normal crossing divisor i∈I di Ei0 . Since k has characteristic zero, the ring R is excellent, so we can always find an embedded resolution for (X, Xs ), by [15]. Proposition 8. — Let h : X 0 → X be an embedded resolution of singularities for P (X, Xs ), such that Xs0 is a strict normal crossing divisor i∈I di Ei0 . Then X [(Ei0 )◦ ] S(X) = i∈I, di =1

in K0 (V arXs )/IXs . In particular, the right hand side does not depend on the choice of the resolution h.


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Proof. — The composition 0 0 Sm(X∞ ) → X∞ → X∞

is a Néron R-smoothening for X∞ , by Proposition 5. It is clear that 0 )s = ti∈I,di =1 (Ei0 )o Sm(X∞

7.3. Motivic Serre invariants and ramification. — Proposition 9. — Let k be a field of characteristic p ≥ 0. Let R0 be a finite, totally ramified extension of R. Let X∞ be a flat, stft formal R-model of a smooth rigid analytic K-space Xη . We assume that Sm(X∞ ) is a NS of X∞ over R. Then there i+1 i exists a finite sequence of admissible blow-ups hi : X∞ → X∞ , 0 ≤ i ≤ n − 1, with center C(i) and exceptional divisor E(i), such that : 0 ˆ R R0 = X∞ × 1. X∞ n 0 2. Sm(X∞ ) is a NS of X∞ over R0 Moreover, ˆ R R0 ) − S(X∞ ) = ([Sg(Xs )] − [Sg(Xsn )]) + S(X∞ ×

n X

([E(i)] − [C(i)])

i=0

and, in particular, the right hand side does not depend on the choice of such sequence of blow-ups. Proof. — The existence of such a sequence of blow-ups is just a reformulation of Theorem 3.1 of [5]. We have the following diagram / ...

n X∞

/ X0 ∞

/ X1 ∞

² X∞ We have [Xs0 ] = [Xs ], and [Xsi ] − [C(i)] = [Xsi+1 ] − [E(i)] in K0 (V arXs ), for 0 ≤ i ≤ n − 1. Hence, n n X X [Xs ] − [C(i)] = [Xsn ] − [E(i)] i=0

i=0

The result follows from the definition of the Serre invariant. Indeed, we deduce from the hypotheses that, in K0 (V arXs )/IXs , 0 n [S(X∞ )] = [Sm(X∞ )] = [Xsn ] − [Sg(Xsn )]

and [S(X∞ )] = [Sm(X∞ )] = [Xs ] − [Sg(Xs )]

22


Lemma 13. — Let X be a separated, regular, flat R-scheme of finite Ptype, of pure relative dimension 1, such that Xs is a strict normal crossing divisor i∈I di Ei . Let h : X 0 → X be the blow-up with center {xi,j } := Ei ∩ Ej , for some pair i 6= j in I. Denote by E00 the exceptional divisor of h. Then e00 )◦ ] = 0 [(E in K0 (V arXs )/IXs . Proof. — Choose an integer e > 0 with (di +dj )|e. Let D be an irreducible component e 0 (e). The morphism of normalization induces a finite morpism D → E 0 ' P1 , of E 0 0 k which is étale outside {y1 , y2 } := E00 \ (E00 )◦ . The Hurwitz formula applied to the morphism D → E00 yields 2 − 2g(D) = d(2 − 2g(E00 )) − 2d + k1 + k2 f0 0 (e) ×E 0 yi , where d is the degree of this morphism, and ki the number of points on E 0 0 for i = 1, 2. Since g(D) ≥ 0 and g(E0 ) = 0, 2 ≥ 2 − 2g(D) = k1 + k2 ≥ 2 We conclude that k1 = k2 = 1 and that g(D) = 0. This result implies that D ' P1k , e 0 )◦ is isomorphic to the k-torus Gm,k . In particular, since the structural and that (E 0 e 0 )◦ → Xs factors through xi,j → Xs , we see that morphism (E 0

e00 )◦ ] = [Gm,k ×k xi,j ] ∈ K0 (V arX ) [(E s The lemma follows from the fact that [Gm,k ×k xi,j ] = 0 in K0 (V arXs )/IXs . Indeed, we have the relations [[Gm,k ×k xi,j ]] = [Gm,Xs ][xi,j ] = (LXs − [Xs ])[xi,j ] = 0 in K0 (V arXs )/IXs . Theorem 5. — Let k be an algebraically closed field of characteristic zero. Let X be a separated, flat, generically smooth R-scheme of finite type, of pure relative dimension 1. Take an embedded resolution of singularities X 0 → X for (X, Xs ). The special P 0 fiber Xs is a strict normal crossing divisor i∈I di Ei . For any integer e > 0, X ei◦ ] S(X∞ (e)) = [E i, di |e

in K0 (V arXs )/IXs . Proof. — We may suppose X 0 = X. If e is not Xs -linear, it follows from Lemma 8 and Proposition 6 that X ] )] = S(X∞ (e)) = ei◦ ] [Sm(X(e) [E ∞ i, di |e

Now assume that e is Xs -linear. By Lemma 9, we can find a composition of admissible blow-ups π (j) : X (j+1) → X (j) , j = 0, . . . , r − 1, such that – X (0) = X,


23

– the special fiber of X (j) is a strict normal crossing divisor E (j) =

kj X

(j)

(j)

di Ei ,

i=1 (j)

– π (j) is the formal blow-up with center EJ (j) , for some subset J (j) of {1, . . . , kj }, with |J (j) | = 2, – e is not E (r) -linear. By Lemma 8 and Proposition 6, S(X∞ (e)) =

X

(r)

e )◦ ] [(E i

(r)

i, di |e

Lemma 13 implies that

X (r)

i, di |e

(r)

e )◦ ] = [(E i

X

ei◦ ] [E

i, di |e

7.4. The rationality of the generating power series. — Let k be an algebraically closed field of characteristic zero. Let X∞ be an stf t formal R-scheme, with smooth generic fiber Xη . For each n ∈ N, we consider the element S(X∞ (n)) ∈ ˆ R (R(n)), where R(n) := R[T ]/(T n −π)). K0 (V arXs )/IXs (recall that X∞ (n) := X∞ × We encode these data in a generating power series. Definition 15 (Serre Poincar´ e series). — The Serre Poincaré series associated to a generically smooth, stf t formal R-scheme X∞ , is the generating series X S(X∞ ; T ) := S(X∞ (n))T n ∈ (K0 (V arXs )/IXs ) [[T ]] n≥1

It is natural to ask whether this series is rational (see [19] or [21], for a result in the general case). Theorem 6. — Suppose that k is an algebraically closed field of characteristic zero. Let X be a separated, flat, generically smooth R-scheme of finite type, of pure relative dimension one, and denote by X∞ its π-adic completion. The generating series S(X∞ ; T ) is rational, and, moreover, £ ¤ S(X∞ ; T ) ∈ (K0 (V arXs )/IXs ) T, (1 − T i )−1 i∈N More precisely, if X 0 → X is an embedded P resolution of singularities for (X, Xs ), and Xs is a strict normal crossing divisor i∈I di Ei , then S(X∞ ; T ) =

di X e◦ ] T [E ∈ (K0 (V arXs )/IXs ) [[T ]] i 1 − T di i∈I

24


Proof. — We may suppose that X = X 0 . By Theorem 5, we have   X X ei◦ ] T n  [E S(X∞ ; T ) = n≥1

di |n

in (K0 (V arXs )/IXs )[[T ]]. Regrouping terms, we obtain X X ei◦ ] S(X∞ ; T ) = [E T qdi i∈I

=

X

q≥1

ei◦ ] [E

i∈I

T di 1 − T di

Theorem 7. — Let Y be a smooth, irreducible complex surface, and let f : Y → A1C = Spec C[t] be a flat C-morphism of schemes. Denote by X∞ the t-adic completion of Y ×C[t] C[[t]]. Its generic fiber Xη is a quasi-compact, separated and smooth rigid variety over K, and S(X∞ ; T ) = Zf (T ) in (K0 (V arX0 )/IX0 ) [[T ]]. Proof. — This follows immediately from Remark 7, Theorem 6, and the explicit formula for Zf (T ) in [8], Theorem 2.4. References [1] A. Abbes, “Réduction semi-stable des courbes”, in J.-B. Bost (ed.) et al., Courbes semistables et groupe fondamental en géométrie algébrique. Actes des conférences données au CIRM ` a Luminy, France, 30 novembre au 4 décembre 1998, Prog. Math. 187, 2000, 59-110. [2] S. Bosch, W. L¨ utkebohmert, Formal and rigid geometry I: rigid spaces, Math. Ann., 295:2, 1990, 291–317. [3] S. Bosch, W. L¨ utkebohmert, M. Raynaud Formal and rigid geometry III: the relative maximum principle, Math. Ann. 302, 1995, 1–29. [4] S. Bosch, W. L¨ utkebohmert, and M. Raynaud. Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete 21, 1990. [5] S. Bosch, K. Schl¨ oter, Néron models in the setting of formal and rigid geometry, Math. Ann. 301:2, 1995, 339–362. [6] J. Denef and F. Loeser. Motivic Igusa zeta functions J. Algebraic Geom. 7, 1998, 505–537. [7] J. Denef and F. Loeser. Germs of arcs on singular algebraic varieties and motivic integration Invent. Math. 135:1, 1999, 201–232. [8] J. Denef and F. Loeser. Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41:5, 2002, 1031–1040. ´ ements de géométrie algébrique, I,, Publ. Math., [9] A. Grothendieck, J. Dieudonné, El´ ´ Inst. Hautes Etud. Sci. 4 ,1960, 5–228.


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´ ements de géométrie algébrique, III,, Publ. Math., [10] A. Grothendieck, J. Dieudonné, El´ ´ Inst. Hautes Etud. Sci. 11, 1961, 5–167. [11] K. Fujiwara. Theory of tubular neighborhood in étale topology. Duke Math. J., 80:1, 1995, 15–57. [12] J. Giraud, Résolution des singularités (d’après Heisuke Hironaka) (French) [Resolution of singularities (after Heisuke Hironaka)] Séminaire Bourbaki 10, Exp. No. 320, Paris: Soc. Math. France, 1995, 101–113. [13] M.J. Greenberg, Rational points in Henselian discrete valuation rings, Inst. Hautes ´ Etudes Sci. Publ. Math. 31, 1966, 59–64. [14] M.J. Greenberg, Schemata over local rings, Ann. of Math. 73:2, 1961, 624–648. [15] H. Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. of Math., 79:2, 1964, 109–326. [16] G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal embeddings, Lecture Notes in Mathematics 339, 1973. [17] Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics 6, Oxford: Oxford University Press, 2002. [18] F. Loeser and J. Sebag, Motivic integration on smooth rigid varieties and invariants of degenerations, Duke Math. J. 119, 2003, 315–344. [19] J. Nicaise and J. Sebag, Invariant de Serre et fibre de Milnor analytique, C. R. Acad. Sci. Paris, Ser. I 341:1, 2005, 21–24. [20] J. Nicaise and J. Sebag, Rigid geometry and the monodromy conjecture, to appear in the Conference Proceedings of Singularités, CIRM, Luminy, 24 January-25 February [21] J. Nicaise and J. Sebag, The Serre invariant, ramification, and the analytic Milnor fiber, in preparation. [22] N. Schappacher, Some remarks on a theorem of M.J. Greenberg, Proceedings of the Queen’s Number Theory Conference, (Kingston, Ont., 1979), Queen’s Papers in Pure and Appl. Math. 54, Queen’s Univ., 1980, 101–113 [23] J. Sebag, Intégration motivique sur les schémas formels, Bull. Soc. Math. France 132:1, 2004, 1–54. [24] J.-P. Serre. Classification des variétés analytiques p-adiques compactes. Topology, 3, 1965, 409–412.

Johannes Nicaise, Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium • E-mail : [email protected] Julien Sebag, Universit´ e Bordeaux 1, IMB, Laboratoire A2X, 351 cours de la lib´ eration, 33405 Talence cedex, France • E-mail : [email protected]