Contents · Section 1: pre-history in arithmetic problems about counting solutions modulo pm,...

AN INTRODUCTION TO p-ADIC AND MOTIVIC INTEGRATION, ZETA

FUNCTIONS AND INVARIANTS OF SINGULARITIES

JUAN VIU-SOS

Abstract. Motivic integration was introduced by Kontsevich to show that birationallyequivalent Calabi-Yau manifolds have the same Hodge numbers. To do so, he constructed acertain motivic measure on the arc space of a complex variety, taking values in a completionof the Grothendieck ring of algebraic varieties. Later, Denef and Loeser, together with theworks of Looijenga and Batyrev, developed in a series of articles a more complete theory ofthe subject, with applications in the study of varieties and singularities. In particular, theydeveloped a motivic zeta function, generalizing the usual (p-adic) Igusa zeta function andDenef-Loeser topological zeta function.

These notes are a basic introduction to geometric motivic integration, the precedentp-adic ideas associated with it, and the theory of the above zeta functions related to them.We focus in practical computations and ideas, providing examples and a recent formulaobtained by means of partial resolutions.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. Pre-history: counting Fp-points, p-adic integration and Igusa zeta function . . . . . . 4

1.1. A problem from Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2. An arithmetic-geometric approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3. Basics on p-adic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4. Affine p-adic integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5. The Igusa zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6. p-adic integration in manifolds and change of variables formula . . . . . . . . . . . . . . 111.7. Arithmetic vs topological: Milnor fiber and the Monodromy conjecture. . . . . . 18

2. Motivic integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1. The Grothendieck ring of varieties as universal additive invariant . . . . . . . . . . . 222.2. Basics on jet spaces and arc spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3. Motivic measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4. Motivic integral and change of variables formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5. Kontsevich Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.6. Useful computations using (embedded) resolutions of singularities . . . . . . . . . . . 37

3. Applications to singularities and the motivic zeta function . . . . . . . . . . . . . . . . . . . . . . 433.1. Batyrev’s new stringy invariants of mild singularities . . . . . . . . . . . . . . . . . . . . . . . . 433.2. The Motivic zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3. Quotient abelian singularities and embedded Q-resolutions . . . . . . . . . . . . . . . . . . 50

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2010 Mathematics Subject Classification. Primary: 14B05; Secondary: 14E18, 14G10, 11S80, 32S25, 32S45.Key words and phrases. p-adic integration, motivic integration, zeta functions, Monodromy conjecture,

resolution of singularities, quotient singularities, embedded Q-resolutions.Last update: July 12, 2020.

1

2 JUAN VIU-SOS

Introduction

Motivic integration ideas were originally developed by Kontsevich [Kon95] as a toolto study the relation between additive invariants of varieties, e.g. Betti numbers or Hodgenumbers, in the context of Mirror Symmetry in String Theory.

In such a context, the main objects of study are Calabi-Yau varieties, i.e. compact, complexalgebraic varieties admitting a non-vanishing form of maximal degree. Motivated by therelation between birationally equivalent Calabi-Yau varieties, Batyrev proves the followingresult.

Theorem (Batyrev’95, [Bat99a]). Let X and Y be two d-dimensional smooth Calabi-Yauvarieties. If X and Y are birationally equivalent, then they have the same Betti numbers, i.e.

bi(X) = dimCHi(X,C) = dimCH

i(Y,C) = bi(Y ), ∀i = 0, . . . , d.

The ingredients of Batyrev’s proof are:

• Hironaka’s desingularization theorem: creates a common smooth birational model ofX and Y .

• Reduction mod pm and Weil’s conjectures: strong results proven by Dwork, Grothen-dieck and Deligne about rationality, functional equations and relation with Bettinumbers for a zeta function associated with counting points in X(Fpm) for smoothvarieties.

• p-adic integration: the p-adic integers Zp ={∑

k≥0 akpk | ak ∈ {0, . . . , p− 1}

}encode

reductions mod pm of the varieties, and p-adic integration provides a way to relatedifferent p-adic volumes based on X(Fpm) by a “change of variables formula”.

• Comparison theorem between `-adic cohomology (` 6= p) and usual Betti numbers.

In a talk made at Orsay at the end of the same year, Kontsevich provided a more directapproach avoiding p-adic integration and Weil’s conjectures: he used arc spaces CJtK (“t-adic”spaces) instead of p-adic numbers as domain of integration, and constructed an integral withrespect to a measure based on a universal additive invariant of varieties, i.e. the Grothendieckring of complex varieties K0(VarC). This integration theory comes with a change of variablesformula relating integrals of different birational equivalent varieties, where the “jacobian” isexpressed in terms of the contact order of arcs along divisors. Motivic integration was born.

Using these ideas, he generalizes Batyrev’s Theorem as a direct consequence of the changeof variables formula.

Theorem (Kontsevich, Dec’95). Under the hypothesis above, X and Y have the sameHodge numbers, i.e.

hp,q(X) = dimCHq(X,ΩpX) = dimCH

q(Y,ΩpY ) = hp,q(Y ), ∀p, q = 0, . . . , d.

This theory was quickly developed for arbitrary (in particular singular) algebraic varietieson an algebraic closed field k, char k = 0, in a series of articles [DL98, DL99, DL02b, Loo02]by Denef and Loeser, as well as Looijenga. Also, Batyrev [Bat98, Bat99a] used motivicintegration ideas to produce new “stringy” invariants of singularities, generalizing additiveinvariants “twisted” by numerical values of the relative canonical divisor. Later, Cluckersand Loeser [CL08] gave a general framework for motivic integration based on model theory.

Focusing on singularities, one of the main applications of this theory is the study of certainzeta functions associated with a polynomial f : the (p-adic) Igusa zeta function ZIgusa(f ; s)

p-ADIC AND MOTIVIC INTEGRATION, ZETA FUNCTIONS AND SINGULARITIES 3

and the Denef-Loeser topological zeta function Ztop(f ; s). The function ZIgusa(f ; s) is definedas a p-adic parametric integral of a polynomial f with coefficients in a p-adic field [Igu74], andZtop(f ; s) is a rational function constructed in terms of the numerical data associated with an

embedded resolution of Vf = {f = 0} ⊂ Cd, with f being now a complex polynomial [DL92].In particular, Ztop(f ; s) is obtained as certain limit of ZIgusa(f ; s).

The interesting information is contained in the poles of these zeta functions. It is believedthat the behavior of the poles is controlled by the topology of the Minor fiber: each poleshould correspond to an eigenvalue of the monodromy of the Milnor fiber at some point ofVf . This is known as the Monodromy conjecture.

It turns out that both ZIgusa(f ; s) and Ztop(f ; s) are specializations of the motivic zetafunction Zmot(f ; s), defined as a parametric motivic integral [DL98]. This gives a moregeometric way to understand and manipulate both functions, as well as opens the door tobetter understand the relation with the topology of the Milnor fiber.

Kontsevich’s original theory (sometimes called “naive” motivic integration) admits moresophisticated extensions, as the equivariant or monodromic motivic integration, as wellas the relative motivic integration with respect to X-varieties. The first one was studiedin [DL99, Loo02], and they derive the local motivic Milnor fiber Sf,0, which is an object lyingin a ring similar to the one constructed from K0(VarC) but also codifying the monodromyaction. This object is a “motivic incarnation” of the Milnor fiber Ff,0, in the sense thatthey both have same cohomological invariants, e.g. the Hodge-Steenbrink spectrum, Eulercharacteristic, etc. In the second case, developed in [Alu07, dFLNU07], the authors constructmotivic versions of the Chern-MacPherson classes for singular varieties using a relativeGrothendieck of X-varieties {V → X} ∈ K0(VarX).

Recently, a version of motivic integration in real algebraic geometry was developed byseveral authors [KP03, Fic05, Cam16, Fic17, Cam17, CFKP19], with some applications tothe study of blow-analytic equivalence, or to Lipschitz inverse map theorems of germs ofarc-analytic homeomorphisms.

A first version of these notes was produced as support material of a mini-course about thissubject for graduate students which I lectured at ICMC-USP. The notes were completed aftera second mini-course at IMPA as part of the Thematic Program on Singularity Theory 2020.

The main purpose of these notes is to give a geometric introduction of motivic integrationin complex geometry and the related zeta functions for graduated students, focusing onconcrete examples and practical computations. One of the main goals is that the readercould feel comfortable once she/he deals with more advanced techniques and papers based onmotivic integration and zeta functions. In order to do that, we present a panorama of thebasic theory, organized as follows:

• Section 1: pre-history in arithmetic problems about counting solutions modulo pm,p-adic integration and the Igusa zeta function, Monodromy conjecture and the Denef-Loeser topological zeta function.

• Section 2: construction of “naive” motivic integration and its specializations, changeof variables formula, proof of Kontsevich’s theorem, formulas from normal crossingsand embedded resolutions.

• Section 3: first applications on singularities, Bartyrev’s stringy invariants and themotivic zeta function, tecnhiques using quotient singularities and formulas fromembedded Q-resolutions.

We stress the use of resolution of singularities to obtain formulas, detailing one of the classicproofs of such a formula for the motivic integral. In addition, we introduce a recent formulaobtained in [LMVV19], which computes motivic integrals and zeta functions from “partial

4 JUAN VIU-SOS

resolutions” involving orbifolds and is very practical in order to make computations.

The content of these notes is mainly produced following [Pop] (for a complete introductionof the relation between p-adic integration, topology of algebraic varieties, Weil and Igusazeta functions and the basic theory of motivic integration), the detailed introductions [Vey06,Cra04] (for a more general theory on motivic integration and stringy invariants) and also [Bli11,Nic10]. More complete and good surveys on the motivic subject can be found in [Loo02]and [DL01]. Recently, Chambert-Loir, Nicaise and Sebag published a book on motivicintegration [CLNS18] which covers a huge part of the current theory.

Several exercises appear among these notes, the hardest ones are placed at the end of thefirst and last section.

Acknowledgments. The author would like to thank Edwin León-Cardenal, JorgeMart́ın-Morales and José I. Cogolludo-Agust́ın for their valuable comments andsuggestions which helped to improve these notes. Also, to Farid Tari for proposing me togive a first mini-course on this subject at ICMC-USP in São Carlos. Finally, to MarceloSaia for giving me the opportunity to present a second one at the Thematic Program onSingularity Theory 2020 held at IMPA. The author was supported by a PNPD/CAPES grantand by a postdoctoral grant #2016/14580-7 by Fundação de Amparo à Pesquisa do Estadode São Paulo (FAPESP).

1. Pre-history: counting Fp-points, p-adic integration and Igusa zeta function

The theory of p-adic integration and zeta functions is a rich area which involves analysisover totally disconnected fields, theory of local zeta functions, arithmetic problems,... and hasmultiple interactions which real and complex analysis, theory of partial differential equations,algebraic geometry as well as singularity theory. See [Meu16, LZ19] for recent surveys onp-adic local zeta functions and related problems, as well as many bibliography references. Foran extensive introduction, see Igusa’s book [Igu00], and also Denef’s report [Den91].

1.1. A problem from Number Theory.Let f ∈ Z[X1, . . . , Xd] and fix p an arbitrary prime number. We want to investigate the

number of solutions of f modulo a power pm, i.e.

m ≥ 0 : Nm(f) := #{x ∈ (Z/pmZ)d | f(x) ≡ 0 mod pm}with the convention N0(f) = 1.

Example 1.1.

(1) f0(x) = x: we have x ≡ 0 mod pm if and only if x = 0 ∈ Z/pmZ, then Nm(f0) = 1for all m ≥ 0.

(2) f1(x) = x2: so x2 ≡ 0 mod pm. Studying it for different values of m, we see that:

• p|x2 ⇔ p|x, then N1(f1) = 1.• p2|x2 ⇔ p|x, then N2(f1) = p.• p3|x2 ⇔ p2|x, then N3(f1) = p.

• p4|x2 ⇔ p2|x, then N4(f1) = p2.• p5|x2 ⇔ p3|x, then N5(f1) = p2.• It is easy to show (Exercise):N2k(f1) = N2k+1(f1) = pk.

(3) f2(x, y) = y−x2: Fixing an arbitrary x ∈ Z/pmZ, y is uniquely determined by y ≡ x2mod pm. Then, Nm(f2) = pm.


(4) f3(x, y) = xy: Exercise: Nm(f3) = (m+ 1)pm −mpm−1.(5) f4(x, y) = y

2 − x3: we list the first values. We have that N1(f4) = p and Nm(f4) =pm−1F (p) as follows:

• 2 ≤ m ≤ 5: F (p) = 2p− 1.• 6 ≤ m ≤ 7: F (p) = p2 + p− 1.• 8 ≤ m ≤ 11: F (p) = 2p2 − 1.

• 12 ≤ m ≤ 13: F (p) = p3 + p2 − 1.• 14 ≤ m ≤ 17: F (p) = 2p3 − 1.

etc.

Note that, if we look at the complex sets Vi = {fi = 0}, V2 is smooth, V3 has a simplenodal singularity and V4 is an ordinary cusp. In fact, the behavior of Nm(f) turns out to be“more complicated” precisely when {f = 0} ⊂ Cd has singularities.

1.2. An arithmetic-geometric approach.Consider the associated Poincaré power series

Q(f ;T ) :=∑m≥0Nm(f)Tm ∈ ZJT K, (1)

codifying the number of solutions modulo pm. Coming back to the previous examples:

Example 1.2.

(1) Q(f0;T ) = 1 + T + T2 + · · · = 1

1− T.

(2) Q(f1;T ) = 1 + T + pT2 + pT 3 + · · · = (1 + T )(1 + pT 2 + p2T 4 + · · · ) = 1 + T

1− pT 2.

(3) Q(f2;T ) =1

1− pT.

(4) We claim Q(f4;T ) =1− (p− p2)T 2 − p6T 6

(1− p7T 6)(1− pT ).

Exercise 1.3. Compute Q(f3;T ) and Q(g;T ) where g = xN11 · · ·x

Ndd and N1, . . . , Nd ≥ 1.

In fact, Nm(f) has a regular behavior, as it was conjectured by Borewicz and Shafare-vich and then proved by Igusa.

Theorem 1.4 (Igusa’75, [Igu74]). Q(f ;T ) is a rational function, i.e. Q(f ;T ) ∈ Q(T ).

We are going to prove the previous theorem at the end of Section 1.6. The main ideas ofIgusa’s proof are:

• Take T = p−s and express Q(f, p−s) in terms of a p-adic integral∫Zp|f |sp |dx| (the

Igusa zeta function).

• An embedded resolution of singularities of {f = 0} ⊂ Cd.• A change of variables formula for p-adic integrals.

Remark 1.5. In fact, Q(f ;T ) can be computed from an embedded resolution of {f = 0} ⊂ Cd.

1.3. Basics on p-adic numbers.

6 JUAN VIU-SOS

1.3.1. Construction. Fix a prime number p. The p-adics give a analytic way to deal withproblems of polynomials in Z/pmZ, since any root modulo pm can be lifted in a p-adic root(Hensel’s Lemma).

Definition 1.6. Let 0 6= x ∈ Q. Consider the unique presentation x = pm · ab

, where m ∈ Z,a/b irreducible and both p - a and p - b. We define in Q:

• The order ordp : Q→ Z t {∞} by

ordp(x) :=

{m if x 6= 0

+∞ if x = 0 .

• The (p-adic) absolute value:

|x|p :={p− ordp(x) if x 6= 0

0 if x = 0.

The idea here is that numbers which are divisible by large powers of p are considered small.

Exercise 1.7. Prove that the map | · |p : Q→ R≥0 is a non-archimedean absolute value (orultrametric) on Q, i.e. for any x, y ∈ Q:

(1) |x|p ≥ 0, and |x|p = 0 if and only if x = 0.

(2) |xy|p = |x|p · |y|p.

(3) |x+ y|p ≤ max{|x|p , |y|p

}.

We define a topology on Q induced by the distance d(x, y) := |x− y|p, for any x, y ∈ Q.

Theorem 1.8 (Ostrowski). Let ‖ · ‖ be a non-trivial absolute value on Q. Then ‖ · ‖ isequivalent either to the usual | · |, or to a p-adic | · |p.

Definition 1.9. The field of p-adic numbers Qp is defined as the completion of (Q, | · |p),i.e. the set of equivalence classes of Cauchy sequences with respect to | · |p.

Remark 1.10.

(1) We have an embedding Q ↪→ Qp, identifying any x ∈ Q with the constant sequence(x, x, x, x, x, . . .).

(2) As a consequence, charQp = 0.

(3) Every x ∈ Qp could be standardly represented by an unique “Laurent series expansionin base p”, i.e.

x =∑

k≥ordp(x)

akpk

with aordp(x) 6= 0 and ak ∈ {0, . . . , p− 1} for any k.

Exercise 1.11. Prove that the following identities:

−1 = (p− 1) + (p− 1)p+ (p− 1)p2 + · · · and 11− p

= 1 + p+ p2 + · · · ,

hold in Qp.


1.3.2. Topology. As Qp is a normed space, we can consider the unit disk space, which is alsoa subring of Qp.

Definition 1.12. The ring of p-adic integers is defined as

Zp :={x ∈ Qp | |x|p ≤ 1

},

or, equivalently, the set of x = akpk + ak+1p

k+1 + · · · ∈ Qp such that ak = 0 for any k < 0.

Proposition 1.13.

(1) Qp is a totally disconnected locally compact topological space.

(2) Zp is open and closed in Qp, moreover Zp is compact.

(3) Zp is a local ring, with maximal ideal pZp ={x ∈ Zp | |x|p < 1

}and residue field

Zp/pZp ' Fp.

As a consequence, we have a disjoint union decomposition by equivalence classes:

Zp = pZp t (1 + pZp) t · · · t (p− 1 + pZp) .(4) Qp has as basis of open and closed neighborhoods given by elements of the form

a+ pmZp ={x ∈ Qp

∣∣∣ |x− a|p ≤ p−m} ,for any a ∈ Qp and m ∈ N.

Z77Z7 1 + 7Z7

2 + 7Z7

3 + 7Z74 + 7Z7

5 + 7Z7

6 + 7Z7

72Z7

1 + 72Z7

Figure 1. Topology of Z7, as the union of translates of 7mZ7, for m ≥ 0.

Remark 1.14.

(1) There is an algebraic way to construct the p-adic numbers: for any m,n ∈ N, m ≥ n,consider the natural projections

πmn : Z/pm+1Z −→ Z/pn+1Z

given by the natural reduction modulo pn+1. We have thus an inverse system{(Z/pm+1Z

)m∈N , (π

mn )m,n∈N

}.

8 JUAN VIU-SOS

The p-adic integers are expressed as the inverse limit :

Zp = lim←−m

(Z/pm+1Z

).

Then, Qp is simply the field of fractions of Zp. Moreover, using the representations

we deduce that Qp '(pN)−1

Z×p , i.e. the localization of Z×p =

{x ∈ Qp | |x|p = 1

},

the units of Zp, by the multiplicative system given by powers of p.

(2) The previous p-adic construction can be extended to any discrete valuation ring(R,m) via completions in the m-adic topology (see for example [Pop]). Then, takingK a finite extension of Qp and its corresponding integral closure OK of Zp in K, allthe following theory in this section can be generalized.

1.4. Affine p-adic integration.

1.4.1. Haar measure. We use the basis of neighborhoods described in Proposition 1.13 todefine a Borel measure in Qp.

Definition 1.15. Let G be a topological group. A Haar measure defined on G is a Borelmeasure µ : G→ C satisfying:

(1) µ(gE) = µ(E), for any g ∈ G and for any Borel-measurable E ⊂ G.(2) µ(U) > 0 for any open set U ⊂ G.(3) µ(K) < +∞ for any compact K ⊂ G.

Proposition 1.16. Any abelian locally compact topological group G admits an unique Haarmeasure up to scalars.

Definition 1.17. The normalized Haar measure µ : Qp → R is defined by

µ(a+ pmZp) =1

pm,

for any a ∈ Qp and m ∈ N.

In particular, µ is invariant by translation and µ(Zp) = 1.

Exercise 1.18. Verify that µ(Z×p ) = 1− p−1.

1.4.2. p-adic cylinders. Consider Qdp with the product topology and the natural projection

πm : Zdp →

(Z/pm+1Z

)d, m ≥ 0. We introduce an elementary concept which will play a

central role in the construction of the motivic measure.

Definition 1.19. A subset C ⊂ Zdp is called a cylinder if C = π−1m (πm(C)), for some m ≥ 0.

Remark 1.20. Any pm+1Zdp = π−1m (0) = π

−1m

(πm(p

m+1Zdp))

is a cylinder.

The measure of a cylinder can be computed using the following “silly” property, considering

the cardinal |πm(C)| of the basis πm(C) ⊂(Z/pm+1Z

)d.


Proposition 1.21. For any cylinder C ⊂ Zdp, the sequence(|πm(C)|pd(m+1)

)m≥0

is constant for m� 0, and its limit is equal to µ(C). Moreover, if we choose m0 > 0 suchthat C = π−1m0(πm0(C)), then for any m ≥ m0, we have

|πm(C)|pd(m+1)

=|πm0(C)|pd(m0+1)

= µ(C).

Proof. Remember that |πm(C)| is a finite set. For m ≥ m0, C can be written as

C =⊔

a∈πm(C)

a+(pm+1Zp

)dBy invariance of the Haar measure,

µ(C) =∑

a∈πm(C)

µ((pm+1Zp

)d)= |πm(C)| ·

1

pd(m+1).

Now, considering the projection πmm0 :(Z/pm+1Z

)d → (Z/pm0+1Z)d, it is easy to see that∣∣∣(πmm0)−1 (a)∣∣∣ = pd(m−m0) for any a ∈ (Z/pm0+1Z)d. Since πm0 = πmm0 ◦ πm, we have|πm(C)| =

∣∣πm (π−1m0(πm0(C)))∣∣ = ∣∣∣(πmm0)−1 (πm0(C))∣∣∣ = |πm0(C)| · pd(m−m0),and the result holds. �

1.4.3. Integration over Znp . Let F : Qp → C be a measurable function. Assume that theimage Im(F ) is a countable subset and take A ⊂ Qp a measurable set. For any c ∈ Im(F ),consider the level sets of F in A:

AF (c) := {x ∈ A | F (x) = c} .

Then, ∫AF (x)dµ =

∑c∈Im(F )

∫AF (c)

F (x)dµ =∑

c∈Im(F )

µ(AF (c)) · c.

We are interested in functions of the form F (x) = |f(x)|sp, where s ∈ C such that Re(s) > 0.Note that, in this case:∫

A|f(x)|sp dµ =

∫Ap− ordp(f(x))sdµ =

∑n≥0

µ {x ∈ A | ordp(f(x)) = m} · p−ms.

Example 1.22. For N ≥ 0, consider ∫Zp

∣∣xN ∣∣sp

dµ.

Taking F (x) =∣∣xN ∣∣s

p= p− ordp(x)Ns, the image is countable and only depends on the order of

x. In fact, for any m ≥ 0,

AF(p−mNs

)= pmZp \ pm+1Zp.

10 JUAN VIU-SOS

which gives a partition of Zp. Thus,∫Zp

∣∣xN ∣∣sp

dµ =∑m≥0

µ(pmZp \ pm+1Zp

)· p−mNs =

∑m≥0

(p−m − p−(m+1)

)· p−mNs

=∑m≥0

(1− p−1

)· p−m(Ns+1) =

(1− p−1

)∑m≥0

(p−(Ns+1)

)m=(1− p−1

) 11− p−(Ns+1)

=p− 1

p− p−Ns.

Exercise 1.23. Prove that

∫Zdp

∣∣∣xN11 · · ·xNdd ∣∣∣sp

dµ =

d∏i=1

p− 1p− p−Nis

.

(Hint: factorize the sums coming from different xi)

1.5. The Igusa zeta function.We introduce the first of the zeta functions, studied by Igusa in order to determine the

numbers Nm(f).

Definition 1.24. Let f ∈ Zp[X1, . . . , Xd] and let s ∈ C. The (local) Igusa zeta function off is given by

ZIgusa(f ; s) :=

∫Zdp

|f(x)|sp dµ.

Remark 1.25. ZIgusa(f ; s) is holomorphic for any s ∈ {z ∈ C | Re(z) > 0}.

We can recover the power series Q(f ;T ) in (1) from this zeta function as follows.

Proposition 1.26. We have the following relation

ZIgusa(f ; s) = Q

(f,

1

ps+d

)(1− ps) + ps.

Before proving the relation above, we notice the following relation between level sets of|f |p and the numbers Nm(f) previously defined.

Lemma 1.27. The set Vm ={x ∈ Zdp | |f(x)|p ≤ p−m

}is a cylinder in Zdp. Moreover,

µ(Vm) = Nm(f) · p−dm.

Proof. We can rewrite Vm ={x ∈ Zdp | ordp f(x) ≥ m

}. Note that V0 = Z

dp and the formula

holds in this case. For anym ≥ 1, we have πm−1(Vm) ={x ∈ (Z/pmZ)d | f(x) = 0 mod pm

}.

Moreover, the set Vm = (π−1m−1 ◦ πm−1)(Vm) is a cylinder, and by Proposition 1.21, it follows

µ(Vm) =|πm−1(Vm)|

pdm=Nm(f)pdm

.

�


Proof of Proposition 1.26. For any m ≥ 0, the level sets of |f(x)|sp can be expressed asVm \ Vm+1. Thus

ZIgusa(f ; s) =∑m≥0

µ(Vm \ Vm+1) · p−ms =∑m≥0

(Nm(f)pdm

− Nm+1(f)pd(m+1)

)· p−ms

=∑m≥0

Nm(f)pm(d+s)

−∑m≥0

Nm+1(f)pd(m+1)+ms

=∑m≥0Nm(f)

(1

pd+s

)m−∑m≥1Nm(f)

(1

pm(d+s)−s

)

= Q

(f,

1

ps+d

)− ps

(Q

(f,

1

ps+d

)− 1).

�

Remark 1.28. Taking the substitution T = p−s in ZIgusa(f ; s), the above relation can berewritten as

Q(f ; p−dT ) =T · ZIgusa(f ; s)− 1

T − 1.

Example 1.29. For f(x) = xN , we obtain

Q(f ;T ) =pT · p−1

p−pNTN − 1pT − 1

=−1 + (p− 1)T + pN−1TN

(1− pN−1TN )(pT − 1).

Exercise 1.30. Obtain Q(f ;T ) in the same way for f(x) = xN11 · · ·xNdd .

We have a way of computing the series Q(f ; s) for p-adic integration! But it should benoticed that the previous integrals are easy to compute since f was a monomial and wehave a formula for the norm of a product, but as soon as f involves sums, the computationsbecome harder and complicated. For example, try to compute∫

Z2p

|xy(x− y)|sp dµ.

Solution: we can use resolution of singularities to transform f in a function which locallyseems like a monomial! Thus, we need to introduce integration in Qp-analytic manifolds andprove a change of variables formula.

1.6. p-adic integration in manifolds and change of variables formula.

1.6.1. Integration in Qp-analytic manifolds. The notion of manifolds, analytic functions andits integration theory over Qp are quite natural. However, this theory has some advantagesover the p-adics, since closed balls are both open and compact sets.

Definition 1.31.

(1) For any open U ⊂ Qdp, a function f : U → Qp is called Qp-analytic map if for anyx ∈ U , there exists a neighborhood V ⊂ U such that f|V is given by a convergentpower series.

(2) We call f = (f1, . . . , fd) : U → Qdp a Qp-analytic map if any fi is Qp-analytic.

(3) A Qp-analytic manifold of dimension d is a Hausdorff topological space X together

with an atlas (Ui, ϕi)i∈I in Qdp and such that any change of charts ϕj ◦ ϕ−1i is by-

analytic, i, j ∈ I.

Remark 1.32.

12 JUAN VIU-SOS

(1) A Qp-analytic manifold is a locally compact, totally disconnected topological space.

(2) Every open U ⊂ Qdp is a Qp-analytic manifold. In particular, U = Zp is a compactQp-analytic manifold.

Example 1.33. Consider the p-adic projective line P1(Qp) ={

[u : v] | (u, v) ∼ λ(u′, v′), λ ∈ Q×p}

.

We can see that P1(Qp) is covered by two disjoint compact open sets:

U ={

[u : v] | v 6= 0, |u/v|p ≤ 1}

and V ={

[u : v] | u 6= 0, |v/u|p < 1},

since we have two bianalytic maps ϕU : U → Zp, ϕU [u : v] = u/v, and ϕV : V → pZp,ϕV [u : v] = v/u. Note that pZp is homeomorphic to Zp.

Example 1.34. Let π : Bl0(Q2p)→ Q2p be the blow-up at the origin of the Qp-affine plane,

i.e. the 2-dimensional Qp-analytic manifold

Bl0(Q2p) :=

{((x, y), [u : v]) ∈ Q2p × P1(Qp) | xv − yv = 0

},

with π being the projection on Q2p. By using the (disjoint) charts of P1(Qp) = U t V , we get

two disjoint charts of Bl0(Q2p) = Ũ t Ṽ , i.e.

Ũ = Bl0(Q2p) ∩

(Q2p × U

)and Ṽ = Bl0(Q

2p) ∩

(Q2p × V

).

Both are bianalytic to Q2p, and map to the base Q2p via the applications

ϕ1 : Ũ −→ Q2p(s1, t1) 7−→ (s1, s1t1)

andϕ2 : Ṽ −→ Q2p

(s2, t2) 7−→ (s2t2, t2)

Denote by E = π−1(0) the exceptional divisor, note that Bl0(Q2p) \E

π' Q2p \O, i.e. ϕ1|{s1 6=0}and ϕ2|{t2 6=0} are bianalytic maps. For a Qp-analytic submanifold X ⊂ Q2p, we define its stricttransform, denoted by X̃, as the Zariski closure of π−1(X \ 0).

[−1 : 1]

[1 : 1][0 : 1]

[1 : 0]

E

C̃π

x− y = 0x+ y = 0

x = 0

y = 0

C

Figure 2. Local blow-up of C = {x2 + y2(y − 1) = 0} at the origin in R2.

Take Y = π−1(Z2p). We can express it in two disjoint compact charts Y = YU ∪ YV using

Ũ and Ṽ . Looking locally:

ϕ−11(Z2p)

={

(s1, t1) ∈ Q2p | |s1|p ≤ 1, |s1|p · |t1|p ≤ 1},

and analogously for ϕ−12 . We obtain a partition of Z2p with respect to the “metric diagonal”{

|x|p = |y|p}

by setting the two compact polydisks:

YU ={

(s1, t1) | |s1|p ≤ 1, |t1|p ≤ 1}

and YV ={

(s2, t2) | |s2|p < 1, |t2|p ≤ 1}.


And we have:

π(YU ) ={

(x, y) | |y|p ≤ |x|p ≤ 1}

and π(YV ) ={

(x, y) | |x|p < |y|p ≤ 1}.

Proposition 1.35. Every compact Qp-analytic manifold of dimension d is bianalytic to a

finite disjoint union of copies of Zdp (so, open-compact).

Remark 1.36. We can define k-differential forms ω ∈ Ωk(X) = Γ(∧k T ∗X) in X in the usual

way, such that for any open U ⊂ X with coordinates x1, . . . , xd:

ω|U =∑

1≤i1

14 JUAN VIU-SOS

polydisks

YU ={

(s1, t1) | |s1|p ≤ 1, |t1|p ≤ 1}

and YV ={

(s2, t2) | |s2|p < 1, |t2|p ≤ 1}.

By partitioning Z2p = π(YU ) t π(YV ), we have∫Z2p

|f(x, y)|sp |dx ∧ dy|p =∫YU

|π∗f(x, y)|sp |π∗(dx ∧ dy)|p︸︷︷︸

IU

+

∫YV

|π∗f(x, y)|sp |π∗(dx ∧ dy)|p︸︷︷︸

IV

As IU is defined over the first chart π|YU = ϕ1 : (s1, t1)→ (s1, s1t1), we have:

IU =

∫|s1|p≤1, |t1|p≤1

∣∣∣sa+b+c1 tb1(1− t1)c∣∣∣sp|s1ds1 ∧ dt1|p

=

(∫|s1|p≤1

|s1|(a+b+c)s+1p |ds1|p

)·

(∫|t1|p≤1

∣∣∣tb1(1− t1)c∣∣∣sp|dt1|p

)

=p− 1

p− p−((a+b+c)s+1)

∫|t1|p≤1

∣∣∣tb1(1− t1)c∣∣∣sp|dt1|p

We have already seen the first integral in Example 1.22. For the second one, we need tounderstand how the non-monomial part changes in the poly-disk of integration. Rememberthat we can decompose Zp =

⊔p−1k=0(k + pZp), thus{

|t1|p ≤ 1}

=

p−1⊔k=0

Tk, where Tk =

{|t1 − k|p ≤

1

p

}and now ∫

|t1|p≤1

∣∣∣tb1(1− t1)c∣∣∣sp|dt1|p =

p−1∑k=0

∫Tk

∣∣∣tb1(1− t1)c∣∣∣sp|dt1|p

With the previous decomposition, every integral can be computed as a “monomial integral”.

Note that, in T0 ={|t1|p ≤ 1/p

}, we show that |t− 1|p = 1 (using for example the p-adic

representation), and then:∫T0

∣∣∣tb1(1− t1)c∣∣∣sp|dt1|p =

∫|t1|p≤1/p

|t1|bsp |dt1|p =(p− 1)p−bs

p− p−bs

In the same way, in T1 ={|t1 − 1|p ≤ 1/p

}, we have |t1|p = |(t1 − 1) + 1|p = 1 and∫

T1

∣∣∣tb1(1− t1)c∣∣∣sp|dt1|p =

∫|t1−1|p≤1/p

|t1 − 1|csp |dt1|p =(p− 1)p−cs

p− p−cs

For k ≥ 2, it is easy to see that∣∣tb1(t1 − 1)∣∣p = 1 in Tk, then∫

Tk

∣∣∣tb1(1− t1)c∣∣∣sp|dt1|p = µ(Tk) = µ(pZp) =

1

p.

Regrouping the above integrals, we have:

IU =p− 1

p− p−((a+b+c)s+1)·(

(p− 1)p−bs

p− p−bs+

(p− 1)p−cs

p− p−cs+p− 2p

)Following similar arguments, we can compute IV only by monomial computations and getI = IU + IV (Exercise).


Remark 1.42. Geometrically, we are resolving the singularities of f(x, y) = 0, i.e. constructinga birational model by blowing-up the origin in Z2p where π

∗f is locally monomial. AscharQp = 0, Hironaka’s Theorem on resolution of singularities applies and we can alwaysobtain this model in any dimension. Note there exist explicit algorithmic resolutions ofsingularities (see Villamayor’s work in [Vil89]).

In the case of Qp-manifolds, the resolution of singularities can be formulated as follows.

Theorem 1.43 (Qp-analytic embedded resolution of singularities). Let f ∈Qp[X1, . . . , Xd] be non-constant. Then there exist a Qp-analytic manifold X, dimX = d, a

proper surjective Qp-analytic map π : Y → Qdp which is an isomorphism outside a set ofmeasure zero, and finitely many submanifolds E0, . . . , Er ⊂ X, codimEi = 1, such that:

(1)∑r

i=0Ei is a simple normal divisor.

(2) div(π∗f) =∑r

i=0NiEi, for some N1, . . . , Nr ∈ Z≥0.(3) div(Jac(π)) = div(π∗(dx1 ∧ · · · ∧ dxd)) =

∑ri=0(νi − 1)Ei, for some ν1, . . . , νr ∈ Z≥1.

Moreover, π : Y → Qdp can be constructed as a composition of successive blowing-ups oversmooth centers.

Remark 1.44. In terms of equations, the above assure that in suitable local coordinatesy = (y1 . . . , yd) around any point a ∈ X, we have

π∗f = yNi11 · · · y

Nikk · u(y), u(a) 6= 0,

and

π∗(dx1 ∧ · · · ∧ dxd) = yνi1−11 · · · y

νik−1k · v(y) · dy1 ∧ · · · ∧ dyd, v(a) 6= 0,

for some 1 ≤ k ≤ d. The sequence {(Ni, νi)}ri=0 is called the numerical data of the resolution(or discrepancies).

Example 1.45. Consider the cusp C : y2 − x3 = 0. After successive blow-ups at the origin(see Figure 3), we obtain three exceptional divisors E1, E2, E3 with

div(π∗f) = Ĉ + 2E1 + 3E2 + 6E3 and div(Jacπ) = E1 + 2E2 + 4E3,

where Ei ' P1 and Ĉ ' A1.

•

C

•

ĈE1 : (2, 2)

•

Ĉ

E1 : (2, 2)

E2 : (3, 3)

×• •

ĈE1 : (2, 2) E2 : (3, 3)

E3 : (6, 5)

Figure 3. Embedded resolution of the cusp by successive blowing-ups.

Theorem 1.46. ZIgusa(f ; s) is a rational function on p−s. Moreover, if π : X → Qdp is an

embedded resolution of {f = 0} with numerical data {(Ni, νi)}ri=0, then

ZIgusa(f ; s) =P (p−s)(

1− p−(N0s+ν0))· · ·(1− p−(Nrs+νr)

) ,where P ∈ Z[1/p][T ].

16 JUAN VIU-SOS

Proof. We study the restriction of the embedded resolution π : Y → Zp. Recall thatY = π−1(Zp) is compact and it can be covered by a finite disjoint compact-open charts{Ui}i ∈ I, where locally

π∗f = yNi11 · · · y

Nikk · u(y), u(0) 6= 0,

and

π∗(dx1 ∧ · · · ∧ dxd) = yνi1−11 · · · y

νik−1k · v(y) · dy1 ∧ · · · ∧ dyd, v(0) 6= 0,

for any y ∈ Ui and some 1 ≤ k ≤ d. Also, {(Ni1 , νi1), . . . , (Nik , νik)} is contained in thenumerical data of π. Using the previous decomposition and the change of variables,

ZIgusa(f ; s) =∑i

∫Ui

|y1|Ni1s+νi1−1p · · · |yk|Niks+νik−1p |u(y)|sp |v(y)|p · |dy1 ∧ · · · ∧ dyd|p︸︷︷︸

IUi

.

Fact: |u(y)|p and |v(y)|p are locally constant. We can assume that they are constant inUi, say |u(y)|p = p−a and |v(y)|p = p−b, for any y ∈ Ui. Each of the Ui is identified with apolydisk Pi =

{|yj |p ≤ p

−mj | j = 1, . . . , d}

for some m1, . . . ,mj ∈ Z≥0, then

IUi = p−as−b ·

∫Pi

|y1|Ni1s+νi1−1p · · · |yk|Niks+νik−1p · |dy1 ∧ · · · ∧ dyd|p

= p−as−b ·∫|y1|p≤p−m1

|y1|Ni1s+νi1−1p dy1 · · · · ·∫|yk|p≤p−mk

|yk|Niks+νik−1p dyd

= p−as−b ·(p− 1p

)d· p

−m1(Ni1s+νi1 )

1− p−m1(Ni1s+νi1 )· · · p

−mk(Niks+νik )

1− p−mk(Niks+νik ).

Since {(Ni1 , νi1), . . . , (Nik , νik)} ⊂ {(Ni, νi)}i=0r , then it is clear that {−νi/Ni}i∈I are polescandidates for ZIgusa(f ; s). It remains to check that the numerator is a polynomial over p

−s

with coefficients on Z[1/p], since the number a above could be negative. In fact, we knowthat (f ◦ π)(Pi) ⊂ Zp and this implies that |π∗f |p ≤ 1 on Pi. This is equivalent to

1 ≥ |y1|Ni1p · · · |yk|Nikp · |u(y)|p = p

−a−m1Ni1−···−mkNik ⇐⇒ a+m1Ni1 + · · ·+mkNik ≥ 0.

Thus, p−s appears with a non-negative power in the numerator of IUi . �

Corollary 1.47. The power series Q(f ;T ) is rational.

Remark 1.48.

(1) In the previous theorem, we have deduced that the poles of ZIgusa(f ; s) are of theform

s0 = −ν

N+

2πik

Nlog p,

for k ∈ Z, where the −ν/N are contained in the set{− ν1N1

, . . . ,− νrNr

}⊂ Q


data {(Ni, νi)}ri=0. Here, it suffices to know that the previous condition is satisfiedfor almost any p. Consider the stratification defined by

E◦I :=⋂i∈I

Ei \⋃j 6∈I

Ej ,

for any I ⊂ {0, . . . , r} (see Figure 5). Then, for p� 0,

ZIgusa(f ; s) = p−d

∑I⊂{1,...,r}

|E◦I (Fp)|∏i∈I

(p− 1)p−(Nis+νi)

1− p−(Nis+νi), (2)

where |E◦I (Fp)| is the number of Fp-rational points in E◦I reduced modulo p.(3) Another practical way to compute ZIgusa(f ; s) without involving resolutions of singu-

larities is Igusa’s Stationary Phase Formula, which can be found in [LZ19, Propo. 6.1].

The above expression allows to explore the computation of ZIgusa(f ; s) and the completedetermination of its poles form the combinatorics of resolutions, i.e. the possible exceptionaldivisors and associated numerical data appearing for fixed f . In particular, the determinationof poles is involved in the study of an astonishing problem on Singularity Theory, which isdiscussed in the next section.

Exercise 1.49. Using Denef’s formula above, compute ZIgusa(fi; s) for the polynomialsf1, . . . , f4 appearing in Examples 1.1 and 1.2, and verify the rational expressions describingQ(fi; s).

Remark 1.50 (Final remarks).

(1) We can extend the previous notion of measure of cylinders to varieties in the p-adics.Let X be a smooth d-dimensional subvariety of Znp , defined algebraically. We canprove that the sequence (

|πm(X)|pd(m+1)

)m≥0

is constant for m� 0, and it is called the volume µ(X) of X.(2) (Oesterlé [Oes82]) Now, consider X singular. One defines its volume as

µ(X) := limε→0

µ(X \ Tε(Xsing)) ∈ R

where Tε(Xsing) is a tubular neighborhood of radius ε > 0. Then, in fact

µ(X) = limn→∞

|πm(X)|pd(m+1)

(3) One can prove (Exercise) that if X is a d-dimensional smooth variety defined overZp and it admits a nowhere vanishing d-form ω, then∫

X(Zp)dµω =

|X(Fp)|pd

=|X(Fp)||Fdp|

∈ Z[1/p].

This is in fact one of the main points of Batyrev’s proof: from the above relationtogether with the change of variables between integrals, he concludes that twobirationally equivalent smooth Calabi-Yau varieties have the same Weil zeta function.

18 JUAN VIU-SOS

1.7. Arithmetic vs topological: Milnor fiber and the Monodromy conjecture.The poles of the ZIgusa(f ; s) are quite mysterious and it was suspected by Igusa that they

are connected with the topology of the complex variety Vf = {f = 0} ⊂ Cd, in particularwith some aspects of the Milnor fibration of f [Mil68, Lê79]. A beautiful survey about thissubject and related problems is [Sea19]; see also [Bud12].

C2

Vf = f−1(0)

Ff,x0

•

Bε(x0)

Bε(x0) ∩ f−1(Dη)

•• t

CDη

f

Figure 4. Milnor fiber at the origin.

Consider a polynomial mapping f : Cd → C, and fix a point x0 ∈ Vf (which is possiblysingular and no necessarily isolated). The Milnor fibration of f at x0 is the C∞-locally trivialfibration given by the restriction

f| : Bε(x0) ∩ f−1(D∗η) −→ D∗η,

where Bε(x0) is the open ball of radius ε around x0, and D∗η = {z ∈ C | |z| < η} is the open

punctured disk, with 0 < η � ε� 1 small enough.The Milnor fiber of f at x0 is any fiber Ff,x0 := f

−1| (t) of the previous fibration, for

t ∈ D∗η. Milnor showed that the class of diffeomorphism of Ff,x0 does not depend on t, andeach lifting of a small loop around 0 ∈ Dη induces a well-defined diffeomorphism (up toisotopy) Ff,x0 → Ff,x0 , called the geometric monodromy transformation of the Milnor fiber.The corresponding linear action in the cohomology Tx0 : H

•(Ff,x0 ;C) → H•(Ff,x0 ;C) iscalled the complex algebraic monodromy action of the Milnor fiber. It is well known thatHq(Ff,x0 ;C) = 0 for any q ≥ d.

Conjecture 1.51 (Igusa p-adic Monodromy conjecture). Let s0 be a pole of ZIgusa(f ; s)for almost all primes p. Then exp(2πiRe(s0)) is an eigenvalue of the monodromy action Tx0on some level Hq(Ff,x0 ;C) at some point x0 ∈ Vf .

Remark 1.52 (A’Campo formula). The set of eigenvalues of the monodromy action Tx0is composed with roots of unity and is closed by conjugation. One can study it using themonodromy zeta function of f at x0, which is the alternating product of the characteristicpolynomials of Tx0 at each level H

q(Fx0 ;C), i.e.

ζf,x0(t) :=∏q≥0

det (Id−t · Tx0 | Hq(Fx0 ;C))(−1)q .


For any embedded resolution of singularities h : X → Cd of Vf , A’Campo proved [A’C75]the following useful formula:

ζf,x0(t) =∏i∈I

(1− tNi

)χ(Ěi∩h−1(x0)), (3)

where div(h∗f) =∑

i∈I NiEi, and Ěi := Ei \(⋃

j 6=iEj

). Note that the collection (Ěi)i∈I is

defined such that for any q ∈ Ěi, h∗f is locally of the form xN = 0, with N constant alongthe stratum.

Example 1.53. In the case of the cusp f(x, y) = y2−x3, we already know that s0 = −1,−5/6are the only two real parts of poles of ZIgusa(f ; s). From the resolution in Example 1.45,

ζf,0(t) =(1− t2)(1− t3)

1− t6=

1− tt2 − t+ 1

,

since Ei ' P1 for exceptional divisors, so χ(Ei \ {k pts}) = 2 − k. Thus, t0 = 1, eiπ/3 is azero and a pole of ζf,0(t), respectively.

Using a particular completion over the p-adics for almost all p, Denef and Loeser [DL92]define a specialization of the Igusa zeta function for complex polynomials f , “taking thelimit p→ 1” in ZIgusa(f ; s). In [Den91, Sec. 4.3] the author gives a short explanation of theaforementioned limit for the definition of the topological zeta function.

Let h : X → Cd be an embedded resolution of singularities of Vf = {f = 0}, with numericaldata {(Ni, νi)}ri=0, and consider the stratification {E◦I }I⊂{0,...,r} as in Remark 1.48.

Definition 1.54. The topological zeta function of f is defined by

Ztop(f ; s) :=∑

I⊂{0,...,r}

χ(E◦I )∏i∈I

1

Nis+ νi∈ Q(s).

If f : (Cn, 0) → (C, 0) is a germ of a function at the origin, we define Ztop,0(f ; s) the localtopological zeta function of f at 0 by taking χ

(E◦I ∩ h−1(0)

)instead of χ(E◦I ) in the above

expression.

Remark 1.55.

(1) Ztop(f ; s) does not depend on the chosen resolution (see [DL92]), but it is worthnoticing that a priori it is not defined intrinsically.

(2) Despite of the name, Ztop(f ; s) is an analytical invariant of f , but not topological : acounterexample is given in [ACLM02a].

(3) When f is non-degenerate with respect to its Newton polyhedron Γ(f), there existcombinatorial formulas in terms of Γ(f) for ζf,0(t), ZIgusa(f ; s) and Ztop(f ; s), as wellas their local versions (see [Var76, Loe90, DH01]). These formulas are implementedin Maple [HL00] and Sagemath [VS12].

Using this zeta function, one can get a local version of the Monodromy conjecture forgerms of holomorphic functions f : (Cn, 0)→ (C, 0).

Conjecture 1.56 (Local Monodromy conjecture). Let s0 be a pole of Ztop,0(f ; s).

Then exp(2πis0) is an eigenvalue of the monodromy action Tx0 on some level Hq(Ff,x0 ;C)

at some point x0 ∈ Vf .

Remark 1.57.

20 JUAN VIU-SOS

(1) We should consider the monodromy action Tx0 in points x0 ∈ Vf others than theorigin, since Ztop,0 also encodes information of the nearby strata of the singular locus

around 0. In the isolated case, it is clear that it suffices to look at the eigenvalues ofthe monodromy action at the origin.

(2) However, Denef proved in [Den93, Lemma 4.6] that if λ is an eigenvalue of Tx0 atx0 ∈ Vf , then there exist x1 ∈ Vf (arbitrarily close to x0) such that λ appears aszero or pole of ζf,x1(t). Thus, from the practical point of view, to determine all theeigenvalues is equivalent to compute all the possible monodromy zeta functions!

The classical strategy to try to prove (or disprove) the Monodromy conjecture for aparticular hypersurface Vf ⊂ Cd is as follows. First, compute an explicit embedded resolutionof singularities of Vf , or instead, study the possible combinatorial properties appearing ina subfamily of those embedded resolutions. Then, the determinantion of the possible polesof ZIgusa(f ; s) or Ztop,0(f ; s) from the resolution, excluding as much as possible “fake poles”

appearing from the numerical data. Finally, compute all possible monodromy zeta functionsusing A’Campo formula.

Following the previous strategy, the Monodromy conjecture is proved for some families ofsingularities e.g.:

• d = 2 [Loe88, Rod04b] (the existence of a minimal embedded resolution of singularitiesin dimension 2 is one of the key points in this setting),

• d = 3 and f homogeneous [RV01, ACLM02b],• superisolated surface singularities [ACLM02b],• f is a product of linear forms (hyperplane arrangements) [BMT11],• f is quasi-ordinary [ACLM05],• f with non-degenerate surface singularities with respect to the Newton polyhe-

dron [BV16, Loe90].

It is worth noticing that if we want to study the possible poles using different resolutionsof Vf or other birational maps, then in principle the behavior of ZIgusa(f ; s) would be betterthan Ztop,0(f ; s) since we can study these transformations via the changes of variables formula.

On the other hand, the expressions appearing in Ztop,0(f ; s) are much easier to study, as well

as the determination of “fake poles”. The Weak Factorization theorem [AKMW02, W lo03]becomes the principal tool to study different birational maps in this case. A combinationof both points of view was used e.g. by Veys in [Vey97] for determining zeta functions ofcurves from partial resolutions.

As we will see in the following sections, the use of motivic integration will give a generalframework for the study of all these zeta functions, their relations via birational maps and away to understand some of the connections with other topological and analytical aspects ofvarieties.

Remark 1.58 (Strong Monodromy conjecture). There exists a stronger version of theMonodromy conjecture involving the roots of a polynomial associated to the meromorphiccontinuation of local zeta functions. The b-function or Bernstein-Sato polynomial associatedto f ∈ C[x1, . . . , xd] is the monic polynomial bf (s) ∈ C[s] of smaller degree verifying thatthere exists a differential operator P ∈ C[s, x1, . . . , xd, ∂x1 , . . . , ∂xd ] such that

P · fs+1 = bf (s)fs,

for any s ∈ Z. It is know [Kas77] that the roots of bf (s) are negative rational numbers.By the above functional equation and using integration by parts, one sees that the polesof ZIgusa(f ; s) lie in {λ− j | bf (λ) = 0, j ∈ Z≥0}. The Strong Monodromy conjecture says


that if s0 is a pole of ZIgusa(f ; s) then Re(s0) is a root of bf (s). The same is stated forZtop(f ; s) and the respective local versions. This statement is in fact stronger in the sensethat implies the Monodromy conjecture, since it was proved by Malgrange and Kashiwarain [Mal83, Kas83] that any root λ of bf (s) gives an eigenvalue exp(2πiλ) of Tx0 at somex0 ∈ Vf . See [Den91, Meu16] for further details.

Exercise 1.59. Compute Ztop,0(fi; s) and verify that the (local) Monodromy conjecture

holds for f1(x, y) = y2−x3, f2(x, y) = xm+ym (m ≥ 2), f3(x, y, z) = (x2 +y3)(x2y2 +x6 +y6)

and f4(x, y, z) = y2 − xz.

Exercise 1.60. Study Ztop,x0(g; s) and ζg,x0(t) for g(x, y, z) = z2 − x3y2 over the origin

and nearby points. Is the Monodromy conjecture verified in this case? (How Vf looks intransverse sections to the singular set around 0?).

22 JUAN VIU-SOS

2. Motivic integration

Based on p-adic integration, Kontsevich constructed an integral using the arc space L(X)of CJtK-points of a variety. However, one cannot construct a real-valued measure on L(X)similar to the p-adic case, since C((t)) is not locally compact. Kontsevich’s key idea was todefine a measure using additive invariants of complex varieties, avoiding Weil’s conjecturesand relaying directly those invariants by a change of variables formula coming from morphismbetween varieties.

The following table summarizes the similarities and differences between p-adic integrationin last section and the motivic one presented in the following.

p-adic integral (geometric) motivic integral

Functions to study f ∈ Z[x1, . . . , xd]f regular over

a complex variety X

Arithmetic vs.geometric

sols. of f = 0 over the ringZ/pm+1Z ' Zp/pm+1Zp, i.e. an

d-tuple with coord.a0 + a1p+ · · ·+ ampm(ai ∈ {0, . . . , p− 1})

sols. of f = 0 over the ringC[t]/

(tm+1

)' CJtK/

(tm+1

), i.e.

an d-tuple with coord.a0 + a1t+ · · ·+ amtm

(ai ∈ C)

Domains ofintegration

Zdp = lim←−

(Z/pm+1Z

)d(liftings of all sols. mod pm+1

with coord.∑∞

k≥0 akpk)

L(X) = lim←−Lm(X)

(arcs: liftings of all sols. modtm+1 with coord.

∑∞k≥0 akt

k)

Algebra ofmeasurable sets

CylindersCylinders/stable

sets/semi-algebraic sets of L(X)

Value ring of themeasure

Z[1/p]M̂C, a localized and completed

universal ring of additiveinvariants of complex varieties

Interesting class ofintegrablefunctions

Order of cancellation of f overthe p-adic integers

Contact order of an arc along adivisor

Operations Change of variables/Fubini Change of variables

2.1. The Grothendieck ring of varieties as universal additive invariant.Denote by VarC the category of complex algebraic varieties. It is worth noticing that VarC

is a small category, i.e. the class of objects Obj(VarC) forms a set (this follows from the factthat we can identify each algebraic variety with its structural sheaf of rings).

Definition 2.1. Let R be a ring. A map λ : Obj(VarC)→ R is an additive invariant if forany X,Y varieties:

(1) If X and Y are isomorphic, then λ(X) = λ(Y ).

(2) For any (Zariski) closed subset F ⊂ X, we have λ(X) = λ(X \ F ) + λ(F ).(3) λ(X × Y ) = λ(X) · λ(Y ).

Example 2.2. The following are examples of additive invariants:


(1) Euler characteristic: χ(X) =∑i≥0

(−1)i dim Hi(X,Q) =∑i≥0

(−1)ibi(X).

It is worth noticing that χ(·) is additive when X is a complex algebraic variety. In gene-ral, is the compactly supported Euler characteristic χc(X) =

∑i≥0(−1)i dim H

ic(X,Q)

which verifies additivity relations. Fox example, S1 = R t {pt} and χ(S1) = 0 butχ(R) = χ(pt) = 1. However, χc(R) = −1.

(2) Virtual Hodge-Deligne polynomial: To a smooth projective variety X, one canassociate

HX(u, v) :=∑p,q≥0

(−1)p+qhp,q(X)upvq ∈ Z[u, v],

where hp,q(X) = dim Hp,q(X,Q) are the Hodge numbers. In the same way, onecan associate a virtual Poincaré polynomial PX(t) =

∑i≥0(−1)ibi(X)ti ∈ Z[t].

Note that HX(t, t) = PX(t) and χ(X) = PX(1).

(3) Counting points: Assume X is defined over Q and fix p prime, then the mapN p(X) = |X(Fp)| is an additive invariant over complex varieties defined over Q.

Any of the previous additive invariants can be considered as the realization of a universaladditive invariant of complex algebraic varieties.

Definition 2.3. The Grothendieck ring of varieties (K0(VarC),+, ·) is generated by theclasses [X], where

[X] = [Y ], if X,Y ∈ Obj(VarC) are isomorphic,

and relations:

• for any (Zariski) closed subset F ⊂ X, we have: [X] = [X \ F ] + [F ],• [X × Y ] = [X] · [Y ].

The unit elements for addition and multiplication are 0 := [∅] and 1 := [pt], respectively.Denote by L := [A1C] the Lefschetz motive.

Example 2.4.

(1) [Cn] = Ln and [C∗] = [C \ {pt}] = [C]− [pt] = L− 1.(2) [CP1] = [C t {∞}] = L+ 1. In fact, we know that CPn = Cn t CPn−1, n ≥ 1, thus

[CPn] = [Cn] + [Cn−1] + · · ·+ [C] + [pt] = Ln + Ln−1 + · · ·+ L+ 1.(3) Let Pm be a pencil of m affine lines at the origin O. Then

[Pm] = [Pm \ O] + [O] = m[C∗] + 1 = m(L− 1) + 1 = mL− (m− 1).(4) Take the ordinary cusp C : y2 − x3 = 0 in C2. Using the parametrization ϕ : C→ C,

ϕ(t) = (t2, t3), which restrings into an isomorphism ϕ|C∗ , we have:

[C] = [C \ O] + [O] = [C∗] + 1 = L.

Note that ϕ is a bijection of points, but not an isomorphism. However, [C] = [C].(5) A subset C ⊂ X is called constructible if it is the finite disjoint union of locally closed

subsets of X. In fact, we can write C =⊔k Uk and this well-defines:

[C] :=∑k

[Uk].

Exercise: Prove that [C] does not depend on the chosen decomposition of C.

24 JUAN VIU-SOS

Remark 2.5. The above examples give the idea that K0(VarC) is a scissors ring : there areelements A 6' B, but verifying [A] = [B] after cutting-and-pasting operations. In fact, anylocally trivial fibration in this setting is trivial in K0(VarC), as it is shown in the next result.

Proposition 2.6. The product on K0(VarC) extends to Zariski locally trivial fibrations: if

Fi↪→ X p→ B verifies that for any x ∈ B there is a Zariski open x ∈ U ⊂ B such that

p−1(U) ' F × U , then [X] = [F ] · [B].

Proof. It follows from induction over dimB and operations on K0(VarC) (Exercise).�

Remark 2.7. It is easy to prove, using successive stratifications on singular sets, that{[X] | X smooth and projective} is a set of generators of K0(VarC). In fact, Bittner [Bit04]proved that K0(VarC) can be described equivalently by the previous set of generators, subjectto the following relation: if Y ⊂ X is smooth and projective, and π : BlY (X) → X is theblow-up of X along Y , with exceptional divisor E = π−1(Y ), then

[X]− [Y ] = [BlY (X)]− [E].

The above result uses the Weak Factorization theorem.

Theorem 2.8 (Universal property). For any additive invariant λ : Obj(VarC) → R,there exists a unique λ̃ : K0(VarC)→ R such that the following diagram commutes:

Obj(VarC) R

K0(VarC)

λ

[ · ]λ̃

Corollary 2.9. There exist well-defined ring morphisms

Z χ([X]) = χ(X)

Z[t] P ([X]) = PX(t)

K0(VarC)

Z[u, v] H([X]) = HX(u, v)

Z N p([X]) = N p(X)

χ

P

H

N p

Remark 2.10.

(1) The idea is that the class [X] is the “most general” way to “count points” or “measurethe size of a variety”. Note the analogy

p =∣∣∣A1Fp∣∣∣←→ L = [A1C].

(2) In general, a class [X] ∈ K0(VarC) cannot be expressed as a polynomial in Z[L]. LetCg be a smooth projective curve of genus g > 0. Since Cg is a compact Riemann surface,it is known that the Hodge-Deligne polynomial of Cg is HCg(u, v) = 1− gu− gv + uv.However H(Li) = (uv)i for every i ∈ N.

(3) The ring K0(VarC) is quite mysterious and complicated to deal with in general. Inthe last years, it has been an object of research, in fact:• (Pooner’02): K0(VarC) is not a domain, i.e. it contains zero divisors [Poo02].


• (Borisov’14): L is a zero divisor [Bor18] (but not a nilpotent element sinceχ(Ln) = 1, for any n ≥ 0).• Larsen-Lunts conjecture’03 [LL14]: If [X] = [Y ], then X and Y admit

a decomposition into isomorphic locally closed subvarieties. This is True fordimX ≤ 1 (Liu-Sebag’10 [LS10]) but False in general (Borisov’14 [Bor18]) .

2.2. Basics on jet spaces and arc spaces.Let X be a complex algebraic variety.

Definition 2.11. Assume that X is affine, i.e. X ⊂ Cd and that X = {f1 = · · · = fk = 0},where fi ∈ C[x1, . . . , xd].

• The m-jet space of X, denoted by Lm(X), is the algebraic variety over C[t]/(tm+1)defined by

Lm(X) ={γm = (x1(t), . . . , xd(t)) ∈

(C[t]/(tm+1)

)d ∣∣∣ f1(γm) = · · · = fk(γm) = 0}Note that the previous equalities are established modulo tm+1.

• The (formal) arc space of X, denoted by L(X), is the algebraic variety over CJtKdefined by

L(X) ={γ = (x1(t), . . . , xd(t)) ∈ CJtKd

∣∣ f1(γ) = · · · = fk(γ) = 0}For any m-jet γm (resp. arc γ), we said that γm(0) (resp. γ(0)) is its origin on X.

Remark 2.12.

(1) Lm(X) and L(X) are complex algebraic varieties via the identifications:

{pts of Lm(X) with coord. in C} ={

pts of Xwith coord. in C[t]/(tm+1)}

{pts of L(X) with coord. in C} = {pts of Xwith coord. in CJtK} .Note that L(X) is infinite dimensional in general.

(2) The natural projections modulo tn+1

C[t]/(tm+1) −→ C[t]/(tn+1) and CJtK −→ CJtK/(tn+1) ' C[t]/(tn+1),for every n ≤ m, induces natural truncation maps between arc and jets spaces

πmn : Lm(X)→ Ln(X) and πn : L(X)→ Ln(X).

Note that πmn = πkn ◦ πmk and πn = πkn ◦ πk, for every n ≤ k ≤ m.

Example 2.13. Let X = Cd:

Lm(Cd) ={(a

(1)0 + a

(1)1 t+ · · ·+ a

(1)m t

m, . . . , a(d)0 + a

(d)1 t+ · · ·+ a

(d)m t

m) ∣∣∣ a(i)j ∈ C}

' Cd(m+1).

Remark 2.14. For X ⊂ Cd, looking at the coefficients in Lm(X), we can identify the variety

with a subvariety of Cd(m+1) ' Cd×m+1)· · · ×Cd such that the truncation maps πmn , n ≤ m,

are induced by projections πmn : Cd(m+1) → Cd(n+1) on the first d(n+ 1) components.

Example 2.15.Let X =

{y2 − x3 = 0

}be the ordinary cusp:

• L0(X) ={

(a0, b0) ∈ C2 | b20 − a30 = 0}

= X.

26 JUAN VIU-SOS

• L1(X) ={

(a0 + a1t, b0 + b1t) ∈(C[t]/(t2)

)2 ∣∣ (b0 + b1t)2 − (a0 + a1t)3 = 0 mod t2}={

(a0 + a1t, b0 + b1t) ∈(C[t]/(t2)

)2 ∣∣ b20 − a30 = 0 and 2b0b1 − 3a20a1 = 0}.Taking coefficients, we can see the map π10 : L1(X) → L0(X) = X induced by theprojection C4 → C2 : (a0, b0, a1, b1) 7→ (a0, b0). Note that:

– The fiber at (0, 0) is the whole (a1, b1)-plane W = {(0, 0, a1, b1)} ' C2, which isthe tangent space of X at the origin T(0,0)X ' C2.

– For (a0, b0) 6= (0, 0), the fiber correspond to a line La0,b0 passing thought Pa0,b0 =(a0, b0, 0, 0) with equation (2b0)b1 − (3a20)a1 = 0. Note that La0,b0 ⊂ Pa0,b0 +W ,corresponds to the tangent line of X at (a0, b0) and T(a0,b0)X ' C.

Resuming, L1(X) is the tangent bundle TX and π10 : TX → X is the naturalprojection.

• In the same way, L2(X) can be seen as a variety in C6 given by the equations b20 − a30 = 0

2b0b1 − 3a20a1 = 0b21 + 2b0b2 − (3a0a21 + 3a20a2) = 0

Note that the fiber of π20 at the origin is the plane {(0, 0, a1, 0, a2, b2)} ' C3, but itsimage by π21 is the line {a0 = b0 = b1 = 0} ⊂ C4. We deduce that π21 : L2(X)→ L1(X)is not surjective.

However, we can prove that the maps πm+1m : Lm+1(X)→ Lm(X) are surjective above thenon-singular part of X = L0(X), moreover, they are fibrations of fiber C.

Remark 2.16. The previous spaces can be defined for any variety X:

Lm(X) = Hom(SpecC[t]/(tm+1), X

).

Then, we have the affine truncation morphisms πm+1m : Lm+1(X)→ Lm(X) induced by thenatural truncation C[t]/(tm+2) � C[t]/(tm+1), and we define the arc space as an inverse limit

L(X) = lim←−Lm(X) = Hom (SpecCJtK, X) .

Any morphism between varieties ϕ : Y → X induces well-defined morphisms:

ϕm : Lm(Y )→ Lm(X) and ϕ∞ : L(Y )→ L(X).

As SpecCJtK = {(0), (t)}, the map SpecCJtK → X define two points: the image of theclosed one ϕ

((t))

(the origin) and the image of the generic one ϕ((0)).

Determining the geometry of the varieties Lm(X) is in general a hard problem. Nevertheless,there exists several practical results about how “complicated” are the fibers of the mapsπmn : Lm → Ln.

Proposition 2.17. Let X be a d-dimensional complex variety. Then:

(1) L0(X) = X and L1(X) = TX.

(2) If X is smooth, then πmn is a Zariski locally trivial fibration with fiber Cd(m−n), for any

n ≤ m. In particular, πmn and πm are surjections and Lm(X) is smooth of dimensiond(m+ 1).

(3) For any γm ∈ Lm(X), the fiber(πm+1m

)−1(γm) is either empty or isomorphic to Tx0X,

where γm(0) = x0.

(4) Assume that X is irreducible and consider Xreg = X \ Xsing. Then the closure of(πm0 )

−1 (Xreg) is an irreducible component of Lm(X) of dimension d(m+ 1).


Remark 2.18.

(1) If X is singular, then TX is not locally trivial. Also, we have shown in Example 2.15that πmn is not surjective.

(2) In [Mus01], Mustaţă obtained several results relating the geometry of the spacesLm(X) with the one of X for the locally complete intersection case. In particular,he obtains that Lm(X) is irreducible for any m > 0 if and only if X has rationalsingularities. It turns out that the he original proof of these results is based onthe use of motivic integration! Other results in this direction, as well as formulasfor the log-terminal threshold involving the dimensions of jet spaces, were obtainede.g. in [EM04, EMY03, ELM04].

Example 2.19 (Lm(X) vs πm(L(X))). We know that πm(L(X)) ⊂ Lm(X) but in thesingular case not any m-jet can be lifted on an arc of X. Take X = {xy = 0} the ordinarynode, we have

L(X) ={

(x(t), y(t)) ∈ CJtK2 | x(t)y(t) = 0}.

If we take generic arcs x(t) =∑

i≥0 aiti and y(t) =

∑j≥0 bjt

j , we have that (x(t), y(t)) ∈ L(X)if and only if for any k ≥ 0:

a0bk + a1bk−1 + · · ·+ ak−1b1 + akb0 = 0 (Jk)

Note that, for any m ≥ 0:

Lm(X) ={

(x(t), y(t)) ∈(C[t]/(tm+1)

)2 | x(t)y(t) ≡ 0 mod tm+1}={

(a0, b0, . . . , am, bm) ∈ C2(m+1)∣∣∣ (Jk) is verified for any k = 0 . . . ,m} ,

since x(t) · y(t) is a generic polynomial of degree t2m. In particular, if we look for L1(X), weshow that

L1(X) = {(a0 + a1t, b0 + b1t) | a0b0 = 0, a0b1 + a1b0 = 0}= {a0 = a1 = 0}︸︷︷︸

V2,0

∪{a0 = b0 = 0}︸︷︷︸V1,1

∪{b0 = b1 = 0}︸︷︷︸V0,2

.

Thus L1(X) has three irreducible components isomorphic to C2. What happens withπ1(L(X))? If we study the spaces Wl,h = Vl,h ∩ π1(L(X)):

• A 1-jet is in W2,0 if it is of the form ϕ1 = (0, b0 + b1t), verifying that there exists andarc ϕ̃ = (a2t

2 + a3t3 + . . . , b2t

2 + b3t3 + . . .) ∈ CJtK2 such that ϕ1 + ϕ̃ verifies (Jk) for

any k ≥ 0. Note that this is automatic for k = 0, 1. Now, (J2) and (J3) are equivalentto

a2b0 = 0 and a2b1 + a3b0 = 0,

respectively. Taking ϕ̃ = 0 for any b0, b1 ∈ C, it is easy to see that (Jk) is verified forany k. Then W2,0 = V2,0 and W0,2 = V0,2, by symmetry.

• For W1,1, we study the lifts of (a1t, b1t) in L(X). In this case, (J2) is equivalent toa1b1 = 0, thus W1,1 ⊂W2,0 ∩W0,2. In fact, taking again ϕ̃ = 0 as lifted part, we seethat W1,1 = V1,1 ∩ {a1b1 = 0}.

We deduce that π1(L(X)) is a projection over only two of the irreducible components ofL1(X).

In general, we can prove (Exercise) that Lm(X) =⋃

l+h=m+1

Vl,h where

Vl,h = {a0 = · · · = al−1 = 0, b0 = · · · = bh−1 = 0} ' Cm+1,

28 JUAN VIU-SOS

and those are exactly the irreducible components of Lm(X). Moreover, it can be shown thatπm(L(X)) = Vm+1,0 ∪ V0,m+1. We deduce that

[Lm(X)] = (m+ 2)Lm+1 − (m+ 1)Lm and [πm(L(X))] = 2Lm+1 − 1.

The study of πm(L(X)) was already considered by Nash [Nas95] and it is in general avery difficult problem. For a more detailed good-survey in arc and n-jet spaces, see [dF16]. Itshould be noticed that πm(L(X)) is a constructible set in Lm(X), since Greenberg [Gre66]proved that there exists a constant c > 0 such the image of πm is equal to the one of π

cmm , for

any m ≥ 0.

2.3. Motivic measure.Looking at the p-adic case, we look for defining a measure µ normalized over L(C), i.e.

µ(L(C)) = 1, and with a formula of the type

“ µ(C) =|πmC||(AdC)m+1|

”,

when m � 0 and for any “cylinder” C ⊂ L(X), in the spirit of Proposition 1.21. In thissetting, the “number of points” does not make sense anymore and we are going to reflect thisinvariant by the class in the Grothendieck ring.

Thus, in order to give a well-defined framework for expressions of the type [X]/Ln, we denoteby MC the localized ring K0(VarC)[L−1]. Remember that the map K0(VarC)→MC is notinjective, since L is know to be a zero divisor, but any additive invariant λ : K0(VarC)→ Rsuch that λ(L) 6= 0 extends to a mapMC → R[λ(L)−1]. In particular, there exist well-definedring morphisms

Z χ(L) = 1

Z[t, t−1] P (L) = t2

MC since

Z[u, v, (uv)−1] H(L) = uv

Z[1/p] N p(L) = p

χ

PH

N p

2.3.1. Construction of the measure. We start generalizing naturally the notion of cylinder.

Definition 2.20. A subset A ⊂ L(X) is called a cylinder (or constructible) if A = π−1m (Cm)for some m ∈ Z≥0 and some constructible set Cm ⊂ Lm(X).

Remark 2.21.

(1) Morally, a cylinder is a set of arcs A ' Cm×C∞ or A ' πm(A)×C∞. The collection ofcylinders in L(X) forms a boolean algebra of sets, i.e. finite unions and complementsof cylinders are cylinders, as well as the empty set and L(X) = π−10 (X).

(2) If we have a partition by constructible sets X =⊔ki=0Wi, then L(X) =

⊔ki=0 π

−10 (Wi).

Proposition 2.22. Assume X is smooth and let A = π−1m0(Cm0) be a cylinder in L(X). Then

[πm(A)]

Ld(m+1)∈MC

is constant for any m ≥ m0.


Proof. Since X is smooth, the maps πmm0 : Lm(X) = πm(L(X)) −→ Lm0(X) = πm0(L(X))are locally trivial fibrations with fiber Cd(m−m0). Thus [πm(A)] = [Cm0 ] · [Cd(m−m0)] =[Cm0 ] · Ld(m−m0) and the result holds. �

Remark 2.23. For A = L(X) with X smooth d-dimensional, note that the previous expressionis simply [X]L−d since L(X) = π−10 (X).

For general varieties, the previous stabilizations are not assured.

Definition 2.24. We call A ⊂ L(X) stable if for some m0 ∈ Z≥0, we have:(1) πm0(A) is constructible and A = π

−1m0 (πm0(A)),

(2) for any m ≥ m0, the projection πm+1(A) → πm(A) is a piecewise trivial fibrationwith constant fiber Cd, i.e. there exists a finite partition of πm(A) into locallyclosed sets S such that any of them admits a open covering S =

⋃k Uk verifying

(πm+1m )−1(Uk) ' Uk × Cd, for any k.

Lemma 2.25 (Denef-Loeser, [DL99]). If A ⊂ L(X) is a cylinder and A ∩ L(Xsing) = ∅,then A is stable.

The above implies that for any stable A, the limit limm→∞[πm(A)]

Ld(m+1)exists in MC. This

defines an additive invariant with respect to finite unions and intersections

µ̃L(X) : {Stable subsets of L(X)} −→MC,

which is called the naive motivic measure. However, in general the stable subsets do notform an algebra of sets, as L(X) is not stable for general X. Take X = {xy = 0}, fromExample 2.19, we see that the sequence

[πm(L(X))]Ld(m+1)

=2Lm+1 − 1Lm+1

= 2− 1Lm+1

does not stabilize. We are going to define a measure µL(X), extending µ̃L(X)

2.3.2. Completion of MC. As in the p-adic case, we will consider a completion by a normfor which the values L−m are small. For this, Looijenga [Loo02] introduces the notion ofvirtual dimension.

Definition 2.26. An element τ ∈ K0(VarC) is called d-dimensional if there is a finiteexpression

τ =∑i

ai[Xi],

with ai ∈ Z and Xi ∈ Obj(VarC) such that d = maxi{dimXi}, and if there is no suchexpression with all dimXi ≤ d− 1. We set that the dimension of [∅] is −∞.

The above can be extended to elements in MC setting dim(L−1

)= −1.

Proposition 2.27. The virtual dimension map dim :MC → Z ∪ {−∞} is well-defined andsatisfies, for any τ, τ ′ ∈ K0(VarC):

(1) dim (τ · τ ′) ≤ dim τ + dim τ ′.(2) dim (τ + τ ′) ≤ max{dim τ,dim τ ′}, with equality if dim τ 6= dim τ ′.

Exercise 2.28. LetA,B ⊂ L(X) be stable subsets. Show that, ifA ⊂ B then dim µ̃L(X)(A) ≤dim µ̃L(X)(B).

30 JUAN VIU-SOS

We construct the completion with respect the ascending filtration defined by the virtualdimension over MC:

· · · ⊂ Fm−1MC ⊂ FmMC ⊂ Fm+1MC ⊂ · · ·given by the subgroups

FmMC = {τ ∈MC | dim τ ≤ −m} =〈

[X]

Li

∣∣∣∣ X ∈ Obj(VarC), dimX − i ≤ −m〉 .Note that FmMC · FnMC ⊂ Fm+nMC.

Definition 2.29. We define the ring

M̂C = lim←−m

MC/FmMC,

i.e. the completion of MC with respect to the filtration F•MC.

Remark 2.30.

(1) By definition, a sequence(

[Xk]

Lik

)k∈N

converges to zero in M̂C if and only if

dimXk − ik −→k→∞

−∞.

(2) As described by Batyrev [Bat98], the ring M̂C is the completion with respect thenorm

δ : MC −→ R≥0τ 7−→ δ(τ) = edim τ

setting δ(∅) = 0. This norm is non-archimidean, i.e. for any τ, τ ′ ∈MC:(a) δ(τ) = 0 if and only if τ = 0 = [∅] in MC.(b) δ(τ · τ ′) ≤ δ(τ) · δ(τ ′).(c) δ(τ + τ ′) ≤ max{δ(τ), δ(τ ′)}.

It is worth noticing that this norm is not known to be multiplicative, i.e. if thecondition (2) is in fact an equality, since MC could not be a domain.

Exercise 2.31.

(1) Show that a sum∑∞

i=0 τi, with τi ∈MC, converges in M̂C if and only if τi → 0.(2) Fix N ∈ Z, show that

∞∑i=0

L−Ni =1

1− L−N

in M̂C.

Example 2.32. Revisiting the sequence associated with X = {xy = 0}, we see that the limitexists in M̂C and

limm→∞

[πm(L(X))]Ld(m+1)

= limm→∞

(2− 1

Lm+1

)= 2.

Theorem 2.33 (Denef-Loeser, [DL99]). For any cylinder A ⊂ L(X), the limit

µL(X)(A) = limm→∞

[πm(A)]

Ld(m+1)

exists in M̂C. Moreover, the map

µL(X) : {Cylinders of L(X)} −→ M̂CA 7−→ µL(X)(A)


is a σ-additive measure, i.e. for any family {Ak}k∈N of pairwise disjoint cylinders, we have

µL(X)

⊔k≥0

Ak

= ∑k≥0

µL(X)(Ak).

The measure µL(X) is called the motivic measure on L(X). When there is not confusion,we denote the measure above simply by µ = µL(X).

As a consequence of the above theory on measurable sets, we obtain that the arc spaces ofsubvarieties of X are “negligible” in L(X).

Proposition 2.34 (Denef-Loeser, [DL99]). Let Z ⊂ X be a proper closed subvariety ofX. Then µL(X)(L(Z)) = 0.

Remark 2.35.

(1) Other authors (for example [Bat98, Vey06]) consider another normalization of themotivic measure with an extra factor Ld, such that µ(L(X)) = [X]. In particular,µ(Ld) = Ld.

(2) It is not known whether the natural map MC → M̂C is injective or not. It is easy tosee that

ker(MC −→ M̂C

)=⋂m∈ZFmMC.

However, the Hodge polynomial H : MC → Z[u, v, (uv)−1] factors on the imageMC ⊂ M̂C, i.e. H(τ) = 0 for any τ ∈

⋂m≥0FmMC, since

degH([Xm]L

−im) = 2 dimXm − 2im ≤ −2m −→ −∞where [Xm]L

−im ∈ FmMC for any m ∈ Z. As a consequence, the specializations Pand χ factor too.

(3) The main point to deal with the singular case is to express L(X)\L(Xsing) as a unionof “complements in L(X) of smaller and smaller tubular neighborhoods around thesingular locus”, constructed in terms of the “contact levels” of arcs with respect toXsing. More precisely,

L(X) \ L(Xsing) =⋃e≥0L(e)(X),

where L(e)(X) = L(X) \π−1e (Le(Xsing)). As it turns out, A∩L(e)(X) is stable at anylevel m0 ≥ e for any cyinder A ⊂ L(X), and one can associate a measure to A takingthe limit in M̂C when e→∞. Compare this with Oesterlé’s result in the case ofp-adics (Remark 1.50).

(4) Denef-Loeser and Batyrev extend the motivic measure for a more general classof measurable sets. In particular: (C[t]-)semi-algebraic sets of L(X), defined by finiteboolean combinations of conditions involving polynomials, lower coefficients of arcsand (in)equalities using the order ordt of arcs (see [DL99] for more details).

2.4. Motivic integral and change of variables formula.

32 JUAN VIU-SOS

We can now define a motivic integral naturally generalizing the previous p-adic integral,and based on the same principle: an integral of a function with countable image values canbe expressed as a sum of level values.

Definition 2.36. Let A ⊂ L(X) be a cylinder and let α : A→ Z∪ {∞} be a map such thatthe fibers α−1(m) = {x ∈ A | α(x) = m} are cylinders. We define the motivic integral of αover A as the expression∫

AL−αdµL(X) :=

∑m∈Z

µL(X)(α−1(m)

)L−m

in M̂C, whenever the right-hand side converges. In this case, L−α is called integrable.

Note that any map α bounded from bellow gives an integrable function. We can produce“simple” examples such as “characteristic functions”: let C be a finite collection of disjointcylinders, then

α =∑C∈C

aC1C

gives an integrable function for any aC ∈ Z.

Example 2.37. For X smooth of dimension d and α ≡ 0,∫L(X)

L−0dµ = µ(L(X)) = [X]L−d.

We are going to integrate with respect to cylinders expressed using the order of an arc

ordt : CJtK −→ Z≥0 ∪ {∞}γ 7−→ ordt(γ)

defined as

ordt(γ) = sup {e ∈ Z≥0 | γ(t) ∈ (te)} ,

or equivalently, if there exists a unit ϕ(t) ∈ CJtK× such that γ(t) = tordt(γ)ϕ(t). Note that

ordt(γ) =∞ ⇐⇒ γ = 0ordt(γ) = 0 ⇐⇒ γ(0) 6= 0, i.e. γ is a unit.

Example 2.38. Let X = C and m ∈ Z≥0. Consider the set Am = {γ ∈ L(C) | ordt(γ) =m}. Then Am is a cylinder because it can be written as Am = π−1m (Cm) where Cm ={γ ∈ Lm(C) | γ(t) = amtm, for some am ∈ C∗}. Therefore

µ(Am) = [Cm]L−(m+1) = (L− 1)L−(m+1), ∀m ≥ 0.

Remark 2.39. For any cylinders C1 ⊂ Cd1 and C2 ⊂ Cd2 , we have

µL(Cd1+d2 )(C1 × C2) = µL(Cd1 )(C1) · µL(Cd2 )(C2).

This property is again a consequence of the fact that the truncation map πm,n : Lm(Cd)→Ln(Cd), for m ≥ n, is a locally trivial fibration with fiber Cd(m−n). The combination ofthis property together with Example 2.38 will help us simplify some calculations in explicitexamples.


2.4.1. The Contact order of an arc along a divisor. Let D be an effective Cartier divisor on asmooth variety X, i.e. a subvariety of X which is locally given by one equation. One definesthe function

ordD : L(X) −→ Z≥0 ∪ {∞}γ 7−→ ordt fD(γ)

where fD is a local equation of D around the origin π0(γ) = γ(0) ∈ X, and

ordt fD(γ) = ordt(fD ◦ γ)(t).

Remark 2.40.

(1) Note that

ordD(γ) =∞ ⇐⇒ γ ∈ L(Dred)ordD(γ) = 0 ⇐⇒ π0(γ) 6∈ Dred

In fact, we can prove that ord−1D (∞) = L(Dred) is not a cylinder, but it is a µL(X)-measurable set of measure 0.

(2) If we write D =∑s

i=1NiDi as a linear combination of prime divisors, then locally

hD decomposes as a product fD =∏si=1 f

NiDi

of defining equations for Di, hence

ordD =s∑i=1

Ni ordDi

(3) The previous construction can be generalized for a sheaf of ideals I on X. We define:

ordI : L(X) −→ Z≥0 ∪ {∞}γ 7−→ min {ordt g(γ) | g ∈ I(U), U open affine cover γ(0) ∈ U}

What about the m-th contact locus of D, ord−1D (m), for m ∈ Z≥0 ? Assume that X = Cd,it is easy to see as in the p-adic case that, for m ≥ 1:

{γ ∈ L(X) | ordD(γ) ≥ m} = {γ ∈ L(X) | ordt(fD ◦ γ) ≥ m}= {γ ∈ L(X) | (fD ◦ γ)(t) ≡ 0 mod tm}= (πm−1)

−1(Lm−1(D))

The above remains true for a general X using coordinate open covers. Thus,

{γ ∈ L(X) | ordD(γ) = m} = (πm−1)−1(Lm−1(D)) \ (πm)−1(Lm(D)),

which is clearly a cylinder. Then, if X is smooth:

µ(ord−1D (m)

)=

[Lm−1(D)]Ldm

− [Lm(D)]Ld(m+1)

.

Note that ord−1D (0) = π−10 (X \D). Also, µ

(ord−1D (0)

)= L−d[X \D] whenever X is smooth.

Example 2.41. Let X = C and D = div(f) ⊂ C be the divisor associated with the functionf(x) = x. We have that Lm(D) = {0} ⊂ Lm(X) for any m ≥ 0 and that

µ(ord−1D (m)) =1

Lm− 1Lm+1

In this way, we obtain:∫L(C)

L− ordDdµ =∑m≥0

(1

Lm− 1Lm+1

)· 1Lm

=

(1− 1

L

)∑m≥0

1

L2m

=L− 1L· 1

1− L−2=

L− 1L− L−1

=L

L+ 1.

34 JUAN VIU-SOS

Example 2.42. Let X = C and D be the divisor associated with the function f(x) = xN ,for N ≥ 0, representing “the origin with multiplicity N”. Note that, for any γ ∈ CJtK oforder k ≥ 0, i.e. γ(t) = tkϕ(t) with ϕ ∈ CJtK×,

(f ◦ γ)(t) = tkNϕ(t)N .

Thus, N | ordD(γ) for any γ ∈ L(C), hence ord−1D (m) = ∅ for any m 6= kN . One can compute,using Example 2.38:

µ {γ ∈ L(C) | ordD(γ) = kN} = µ {γ ∈ L(C) | ordt(γ) = k} =L− 1Lk+1

,

for any k ≥ 0. Then∫L(C)

L− ordDdµ =∑k≥0

µ {γ ∈ L(C) | ordD(γ) = kN} ·1

LkN

= L−1(L− 1)∑k≥0

1

Lk(N+1)=

L− 1L− L−N

.

(Compare this result with the p-adic one obtained for

∫Zp

|x|Np dµ.)

Exercise 2.43. Compute the motivic integral associated with D : xN1yN2 = 0 over C2,where N1, N2 ≥ 1.

Exercise 2.44. Let X be a d-dimensional smooth variety and D0 a smooth divisor. Showthat, if D = ND0 for N ≥ 1, then∫

L(X)L− ordDdµ = L−d

([X \D0] + [D0] ·

L− 1LN+1 − 1

).

(Hint: Since D is smooth, Lm(D) is a locally trivial fibration over D with fiber C(d−1)m)

The motivic integral can be defined with respect to a constructible set W ⊂ X in a naturalway: ∫

L(X)WL− ordDdµL(X),

setting L(X)W = π−10 (W ). In particular, there is a local version of the motivic integrals forgerms of functions f : (X,x0)→ (C, 0), given by∫

L(X)x0L− ordDdµL(X),

where D is locally defined by f .

2.4.2. The change of variables formula. We will continue with more calculations later. First,let introduce the main tool on motivic integration theory: the change of variables. This toolinvolves the ordinary and relative canonical divisors of varieties and, more generally, theconcept of Jacobian ideal sheaf.

Let h : Y → X be a proper birational map, with Y smooth and d = dimY = dimX.Recall that the exceptional locus of h is the maximal subvariety of Y where h is not anisomorphism.

First, the properness condition in h implies that there is an induced bijection of arcs fromL(Y ) to L(X) away from those contained in the exceptional locus. More precisely, if E ⊂ Y


is the exceptional locus of h and F = h(E), we have a bijection

L(Y ) \ L(E) L(X) \ L(F ).h∞

This is due to the valuative criterion of properness [Har77, Thm. II.4.3].

Now, consider the sheaf of Kähler differential d-forms ΩdX =∧d Ω1X , constructed as

follows. The sheaf Ω1X is the unique one such that for any U ⊂ X with sections OX(U) =C[x1, . . . , xd]/(f1, . . . , fr), we have

Ω1X(U) = OX(U) ·〈dx1, . . . ,dxd〉(df1, . . . ,dfr)

.

In this case, note that Ω2X(U) is then aOX(U)-module generated by {dxi ∧ dxj | 1 ≤ i < j ≤ d},and with relations given by the ideal

(df1 ∧ dxk, . . . , dfr ∧ dxk | 1 ≤ k ≤ d

). Local presenta-

tions of the sheaf ΩdX can be computed inductively. Since Y is smooth, then ΩdY is the usual

sheaf of regular differential d-forms.We are going to define the ideal sheaf Jac(h) associated with h as follows:

• If X is smooth, then ΩdX is locally generated by one element. In this way, we candefine the canonical divisor KX as the divisor div(ω) for a non-zero top rationaldifferential form ω ∈ Γ(ΩdX , X). In this case, Jac(h) is the discrepancy divisor (orrelative canonical divisor) between Y and X, denoted by Kh (see Section 3.1.1).Recall that Kh is defined as KY − h∗KX , where the representatives KY and KX arechosen such that h∗KY = KX , i.e. Kh is locally generated by the ordinary Jacobiandeterminant with respect to local coordinates.

• For singular X, ΩdX is not necessarily locally generated by one element, but wecan still compare h∗(ΩdX) with Ω

dY . Taking locally a generator ωY of Ω

dY , then any

h∗(ω) ∈ h∗(ΩdX) can be written as h∗(ω) = gω · ωY , for some regular function gω.Then Jac(h) is defined as the ideal sheaf which is (locally) generated by those gω.

Both Kh and Jac(h) are supported over the exceptional locus of h.

Example 2.45. Let C : y2 − x3 = 0 be the ordinary cusp in C2, and consider the properbirational map h : C→ C given by the parametrization h(u) = (u2, u3). Taking the coordinatering OC = C[x, y]/(y2 − x3), it is easy to see that

Ω1C = OC ·〈dx, dy〉

(2ydy − 3x2dx).

Thus, the module h∗Ω1C is generated by udu, so it is principal in Ω1C and Jac(h) = 〈u〉.

Example 2.46. Let X be smooth d-dimensional. In a blowing-up π : Y = BlZ(X) → Xover a smooth center Z ⊂ X of codimension c, the relative canonical divisor is given byKπ = (c− 1)E, where E = π∗Y is the exceptional divisor. We compute∫

L(Y )L− ordKπdµ =

∫L(Y )

L− ord(c−1)Edµ

= L−d(

[Y \ E] + [E] · L− 1Lc − 1

)= L−d

([Y \ E] + [E]

[Pc−1]

)= L−d ([X \ Z] + [Z])

= L−d[X],

since Y \ E ' X \ Z by definition and also E is a Pc−1-bundle over Z.

36 JUAN VIU-SOS

The above gives the idea of how the change or variables should work.

Theorem 2.47 (Denef-Loeser, [DL99]). Let h : Y → X be a proper birational morphismbetween algebraic varieties X and Y , where Y is not singular. Let A ⊂ L(X) be a cylinderand α : A→ Z ∪ {∞} such that L−α is integrable on A. Then,∫

AL−αdµL(X) =

∫h−1(A)

L−(α◦h)−ordJac(h)dµL(Y ).

In particular, when both X and Y are smooth varieties and α = ordD for an effective divisorD, the above takes the form∫

AL− ordDdµL(X) =

∫h−1(A)

L− ordh∗D+KhdµL(Y ),

where h∗D is the pull-back of D and Kh = KY − h∗KX is the relative canonical divisor.

Remark 2.48. Accessible proofs of the change of variables formula for the smooth case can befound in [Cra04, Pop]. The above result is generalized in [DL02b] for both possibly singularX and Y .

2.5. Kontsevich Theorem.As a consequence of the change of variables formula, we are going to prove in an almost

straightforward way some strong results on additive invariants, starting with the originalleitmotiv of motivic integration: Kontsevich’s Theorem on Hodge numbers of Calabi-Yauvarieties.

Recall that a Calabi-Yau variety X is a smooth, projective algebraic variety, admittinga nowhere vanishing regular differential form of maximal degree. An alternative way toformulate this last condition is to say that the canonical divisor KX is trivial.

Theorem 2.49 (Kontsevich, [Kon95]). Let X and Y be two birationally equivalent Calabi-

Yau manifolds. Then [X] = [Y ] in M̂C, in particular they have the same Hodge numbers.

Proof. By Hironaka’s desingularization Theorem [Hir64], since X and Y are birationallyequivalent, there exists a compact smooth variety Z and birational morphisms hX : Z → Xand hX : Z → Y . Note that KhX = KZ − h∗XKX = KZ = KZ − h∗YKY = KhY because ofthe Calabi-Yau hypothesis. Also, µ(L(X)) = [X]L−d and µ(L(Y )) = [Y ]L−d since both Xand Y are smooth. Using the change of variables formula

L−d[X] = µ(L(X)) =∫L(X)

1dµL(X) =

∫L(Z)

L− ordKhX dµL(Z)

=

∫L(Z)

L− ordKhY dµL(Z) =

∫L(Y )

1dµL(Y ) = µ(L(Y )) = L−d[Y ].

Thus, [X] = [Y ]. We showed that the Hodge polynomial H factorizes over the image of MCon M̂C, hence HX(u, v) = HY (u, v), which implies that both X and Y have the same Hodgenumbers. �

Following the proof above, Kontsevich Theorem can be generalized in a straightforwardmanner for K-equivalent varieties.

Definition 2.50. Two compact algebraic varieties are K-equivalent if there exists a compactsmooth variety Z and birational morphisms hX : Z → X and hX : Z → Y such thath∗XKX = h

∗YKY .

p-ADIC AND MOTIVIC INTEGRATION, ZETA FUNCTIONS

Date post:	28-Jan-2021
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Contents · Section 1: pre-history in arithmetic problems about counting solutions modulo pm,...

Documents