MONODROMY EIGENVALUES ARE INDUCED BY POLES OF ZETA

FUNCTIONS – THE IRREDUCIBLE CURVE CASE

ANDRÁS NÉMETHI AND WILLEM VEYS

Abstract. The ‘monodromy conjecture’ for a hypersurface singularity f predicts that a pole of its topological (or related) zeta function induces one of its monodromy eigenvalues. However, in general quite few eigenvalues are obtained this way. The second author proposed to consider zeta functions associated to the hypersurface and to a differential form and raised the following question. Can one find a list of differential forms ω_{i} such that any pole of the zeta function of f and an ω_{i} induces a monodromy eigenvalue of f, and such that all monodromy eigenvalues of f are obtained this way? Here we provide an affirmative answer for an arbitrary irreducible curve singularity f.

1. Introduction

1.1. The ‘Monodromy Conjecture’ is one of the most fertilizing conjectures in singularity theory. It relates poles of Igusa/motivic/topological zeta functions to monodromy eigenvalues. For instance for a local analytic isolated singularity f : (ℂ^{n}, 0) → (ℂ, 0) it predicts that if s_{0} is a pole of the local topological zeta function of f then λ = exp(2πis_{0}) is an eigenvalue of the local monodromy operator acting on H^{∗}(F_{0}, ℂ), where F_{0} is the Milnor fiber of f. This conjecture was proved for n = 2 by Loeser (originally in the context of p-adic Igusa zeta functions) in [11]. There are by now various other partial results, e.g. [2, 3, 10, 12, 14, 17].

It is easy to see on explicit examples that in this way not all the eigenvalues of the monodromy operator are realized; actually quite few eigenvalues are obtained this way (in general). Hence it is natural to try to extend in some way this set of poles, such that the same procedure would yield all eigenvalues. A natural way to do this is using the local topological zeta functions associated with the original germ f and with a set of analytic differential n-forms ω living in (ℂ^{n}, 0).

We now first describe these zeta functions; they are defined in terms of an embedded resolution π of f^{−1}{0} ∪ div(ω). We denote by E_{i}, i ∈ S, the irreducible components of the inverse image π^{−1}(f^{−1}{0} ∪ div(ω)) and by N_{i} and ν_{i} − 1 the multiplicities of E_{i} in the divisor of π^{∗}f and π^{∗}ω, respectively. We put E_{I}^{∘} := (∩_{i∈I}E_{i}) \ (∪_{j∉I}E_{j}) for I ⊂ S. So the E_{I}^{∘} form a stratification of X in locally closed subsets.

2000 Mathematics Subject Classification. Primary 14B05, 32S40, 14H20. Secondary 32S05, 14H50.

Key words and phrases. Monodromy, Topological zeta function, Differential forms.

The first author is partially supported by OTKA grants, the second author by the Fund of Scientific Research - Flanders (G.0318.06).

Definition 1.1.1. The local topological zeta function of (f, ω) at 0 ∈ ℂ^{n} is

Z(f, ω; s) := ∑_{I⊂S} χ(E_{I}^{∘} ∩ π^{−1}\{0\}) ∏_{i∈I} \frac{1}{ν_{i} + sN_{i}} ,

where s is a variable.

This invariant was introduced by Denef and Loeser in [6] for ‘trivial ω’, i.e. for ω = dx_{1} ∧ ⋯ ∧ dx_{n}. Their original proof that this expression does not depend on the chosen resolution is by describing them as a kind of limit of p-adic Igusa zeta functions. Later they obtained the statement as a specialization of the intrinsically defined motivic zeta functions [7]. Another technique is applying the Weak Factorization Theorem [4], [19] to compare two different resolutions. For arbitrary ω one can proceed analogously.

It is natural and useful to study these invariants incorporating such a more general ω, see for example [2, 3, 16]. Note however that there one restricts to the situation where supp(div(ω)) ⊂ f^{−1}{0}. In the original context of p-adic Igusa zeta functions, see e.g. [11, III 3.5].

1.2. The second author in [18] proved for arbitrary n that any given monodromy eigenvalue of f at 0 is induced by a pole of a local topological zeta function of f and an adequate ω. In particular for n = 2 the result is the following.

Theorem 1.2.1. Assume that n = 2 and let λ be a monodromy eigenvalue of f at 0. Then there exists a differential 2-form ω on (ℂ^{2}, 0) such that Z(f, ω; s) has a pole s_{0} satisfying exp(2πis_{0}) = λ.

On the other hand, the zeta functions Z(f, ω; s) constructed in the theorem above will in general have other poles that don’t induce monodromy eigenvalues of f. The second author in [18] posed the problem to investigate the validity of the following statement:

Problem 1.2.2. Let f : (ℂ^{2}, 0) → (ℂ, 0) be a non-zero singularity germ. Find differential 2-forms ω_{1}, ⋯, ω_{k} on (ℂ^{2}, 0) such that

(1) if s_{0} is a pole of a zeta function Z(f, ω_{j}; s), then exp(2πis_{0}) is a monodromy eigenvalue of f at 0, and

(2) for each monodromy eigenvalue λ of f at 0, there is a differential form ω_{j} in the above list and a pole s_{0} of Z(f, ω_{j}; s) such that exp(2πis_{0}) = λ.

(A similar problem can be stated for arbitrary n.) As an illustration the case f = y^{q}−x^{p} with gcd(p, q) = 1 was treated: for this f the forms x^{i−1}y^{j−1}dx∧dy with 1 ≤ i ≤ p−1 and 1 ≤ j ≤ q − 1 satisfy (1) and (2). However, the ‘simplicity’ of this case can be misleading. When one constructs an ω_{j} as in (2) to induce a given λ, the zeta function Z(f, ω_{j}; s) will typically have other poles inducing roots of unity that are not eigenvalues.

1.3. As a motivation for this type of problem we should mention that the same one can be stated for the p-adic Igusa zeta function, which is a certain p-adic integral. For similar complex integrals there are (for arbitrary n) theorems of Barlet [5], Kashiwara [9] and Malgrange [13] relating poles to monodromy eigenvalues. We refer to the introduction of [18] for more details.

1.4. The main result of the present article is to give a positive answer to the above Problem in the case when f is an arbitrary (analytically) irreducible singularity.

2. Preliminaries

We fix the notation that we will use throughout the paper for the invariants of an irreducible complex plane curve singularity {f = 0}. We recall some well known facts about them, and we state an efficient compact formula for the topological zeta function of f and the relevant differential forms ω.

2.1. Let {f = 0} be an irreducible curve singularity with r Puiseux pairs (r ≥ 1). The Eisenbud-Neumann, or splice diagram Γ of f is the following:

[figure: a_{1}; a_{2}; a_{3}; a_{r−1}; a_{r}; p_{1}; p_{2}; p_{3}; p_{r−1}; p_{r}; …; E_{1}; E_{2}; E_{3}; E_{r−1}; E_{r}]

A way to look at it is as the dual graph G of the minimal embedded resolution π : (X, E) → (ℂ^{2}, 0) of {f = 0}, where the vertices of valence 2 are deleted. The vertices E_{1}, ⋯, E_{r} correspond to the so-called rupture irreducible components of the exceptional locus E, i.e. those intersecting at least three times other components. The arrow corresponds to the strict transform of {f = 0}. For details regarding splice diagrams, see [8].

The splice decorations satisfy a_{ℓ} ≥ 2 and p_{ℓ} ≥ 2. Their meaning is the following. They are (the absolute value of) the determinant of the self-intersection matrix of all exceptional components of π, appearing in G away from E_{ℓ} in the direction of a_{ℓ}, resp. p_{ℓ}. In fact, they can be recovered from the Newton pairs (p_{ℓ}, q_{ℓ})_{ℓ=1}^{r} of f by the following formula:

(2.1.1)  \{ \begin{matrix}a_{1} = q_{1} \\ a_{2} = q_{2} + a_{1}p_{1}p_{2} \\ a_{3} = q_{3} + a_{2}p_{2}p_{3} \\ ⋯ \\ a_{r} = q_{r} + a_{r−1}p_{r−1}p_{r}. \end{matrix}

Above q_{ℓ} ≥ 1 and gcd(p_{ℓ}, q_{ℓ}) = 1. Moreover, q_{ℓ} is also the absolute value of the determinant of the self-intersection matrix of all exceptional components on the chain strictly between E_{ℓ−1} and E_{ℓ}, where we take q_{ℓ} = 1 if there are no such components (i.e. if E_{ℓ−1} intersects E_{ℓ}). In particular, we have for all ℓ = 2, ⋯, r

(2.1.2)  a_{ℓ} > a_{1}p_{1}p_{2}^{2} ⋯ p_{ℓ−1}^{2}p_{ℓ}.

2.2. Let F_{0} denote the local Milnor fibre of f at 0. Then the characteristic polynomial P (f; t) of the local monodromy action on the first cohomology H^{1}(F_{0}, ℂ) is

Δ_{a_{1},p_{1}}(t^{p_{2}p_{3}···p_{r}}) · Δ_{a_{2},p_{2}}(t^{p_{3}···p_{r}}) · . . . · Δ_{a_{r−1},p_{r−1}}(t^{p_{r}}) · Δ_{a_{r},p_{r}}(t),

where we use the notation Δ_{c,d}(t) := \frac{(t^{cd}−1)(t−1)}{(t^{c}−1)(t^{d}−1)} ∈ ℤ[t].

2.3. For ℓ = 1, ⋯, r, let N_{ℓ} and ν_{ℓ} − 1 be the multiplicities of E_{ℓ} in the divisor of π^{∗}f and π^{∗}(dx ∧ dy), respectively.

Lemma 2.3.1.

\{ \begin{matrix}N_{ℓ} = a_{ℓ}p_{ℓ}p_{ℓ+1} ⋯ p_{r} \\ ν_{ℓ} = p_{ℓ}ν_{ℓ−1} + q_{ℓ}, where ν_{0} = 1. \end{matrix}

Proof. The first identity is a well-known multiplicity formula of splice diagrams: N_{ℓ} is the product of all decorations along but not on the path connecting E_{ℓ} with the arrowhead [8, §10]. For the second part, consider for a moment the minimal resolution graph with deleted arrowhead, denoted by G′. Let {F_{k}}_{k} be the set of exceptional divisors/vertices, and δ_{k} the valency of F_{k} in G′. If K is the canonical cycle on G′ (or on X), and F := ∑_{k} F_{k}, then by the adjunction formula K + F = ∑_{k}(δ_{k} − 2)F_{k}^{∗}, where F_{k}^{∗} is the dual cycle of F_{k} with respect to the intersection form ( , ) associated with G′. Since F_{k} with δ_{k} ≠ 2 can be identified with vertices of Γ, one has

ν_{ℓ} = ∑_{k}(δ_{k} − 2)(F_{k}^{∗}, E_{ℓ}^{∗}).

The point is that for δ_{k} ≠ 2 the integer −(F_{k}^{∗}, E_{ℓ}^{∗}) can be read from Γ too, it is the product of all the decorations along but not on the path connecting F_{k} with E_{ℓ}, see [8, §10 and §20]. Hence ν_{ℓ} can be computed explicitly in G′, and the inductive formula also follows. □

For I = (j_{1}, i_{1}, i_{2}, ⋯, i_{r}) ∈ (ℤ_{>0})^{r+1} we will consider differential forms

ω_{I} := g^{j_{1}−1} ∏_{ℓ=1}^{r} g_{ℓ}^{i_{ℓ}−1}dx ∧ dy.

Here g_{ℓ} = 0 and g = 0 are the (reduced) equations of irreducible curves, whose strict transforms intersect exactly one exceptional component as indicated below (these strict transforms are represented by dashed arrows). We denote this graph by Γ(I).

[figure: a_{1}; a_{2}; a_{3}; a_{r−1}; a_{r}; (j_{1} − 1); (i_{1} − 1); (i_{2} − 1); (i_{3} − 1); (i_{r−1} − 1); (i_{r} − 1); p_{1}; p_{2}; p_{3}; p_{r−1}; p_{r}; …; E_{1}; E_{2}; E_{3}; E_{r−1}; E_{r}]

For ℓ ∈ {1, ⋯, r}, let ν_{ℓ,I} − 1 denote the multiplicity of E_{ℓ} in the divisor of π^{∗}(ω_{I}). Then by a similar multiplicity computation as in the proof of (2.3.1), we get:

ν_{ℓ,I} = ν_{ℓ} + p_{ℓ} ⋯ p_{2}p_{1}(j_{1} − 1) + p_{ℓ} ⋯ p_{2}a_{1}(i_{1} − 1)
+ p_{ℓ} ⋯ p_{3}a_{2}(i_{2} − 1) + p_{ℓ} ⋯ p_{4}a_{3}(i_{3} − 1) + ⋯
+ p_{ℓ}a_{ℓ−1}(i_{ℓ−1} − 1) + a_{ℓ}(i_{ℓ} − 1) + p_{ℓ}a_{ℓ}(i_{ℓ+1} − 1)
+ p_{ℓ+1}p_{ℓ}a_{ℓ}(i_{ℓ+2} − 1) + ⋯ + p_{r−1} ⋯ p_{ℓ}a_{ℓ}(i_{r} − 1).

2.4. Using [15] we can write a compact formula for Z(f, ω_{I}; s), associating only a term to each vertex E_{ℓ}, to the straight arrow and to the edges in the graph Γ(I).

Proposition 2.4.1.

Z(f, ω_{I}; s) =\frac{a_{1}}{j_{1}(ν_{1,I} + sN_{1})} + ∑_{ℓ=1}^{r} \frac{−1}{ν_{ℓ,I} + sN_{ℓ}} + ∑_{ℓ=2}^{r} \frac{q_{ℓ}}{(ν_{ℓ−1,I} + sN_{ℓ−1})(ν_{ℓ,I} + sN_{ℓ})}

+ \frac{1}{(ν_{r,I} + sN_{r})(1 + s)} + ∑_{ℓ=1}^{r} \frac{p_{ℓ}}{i_{ℓ}(ν_{ℓ,I} + sN_{ℓ})}

=\frac{1}{ν_{1,I} + sN_{1}} (−1 + \frac{a_{1}}{j_{1}} + \frac{p_{1}}{i_{1}} ) + ∑_{ℓ=2}^{r} \frac{1}{ν_{ℓ,I} + sN_{ℓ}} (−1 + \frac{q_{ℓ}}{ν_{ℓ−1,I} + sN_{ℓ−1}} + \frac{p_{ℓ}}{i_{ℓ}})

+ \frac{1}{(ν_{r,I} + sN_{r})(1 + s)}.

Proof. Essentially, this is an application of [15, Theorem 3.3]. In fact, that theorem is only treated in the context of the standard differential form dx ∧ dy (so for ω_{I} with I = (1, 1, ⋯, 1) ), but it is straightforward to verify that analogously the formula is valid for arbitrary ω_{I}, I = (j_{1}, i_{1}, ⋯, i_{r}). □

Corollary 2.4.2. Fix ℓ ∈ \{1, ⋯, r\}. Then −\frac{ν_{ℓ,I}}{N_{ℓ}} is a pole of order two of Z(f, ω_{I}; s) if and only if

\frac{ν_{ℓ,I}}{N_{ℓ}} = \frac{ν_{ℓ−1,I}}{N_{ℓ−1}} or \frac{ν_{ℓ,I}}{N_{ℓ}} = \frac{ν_{ℓ+1,I}}{N_{ℓ+1}} ,

where the first condition is empty for ℓ = 1, and we put (ν_{r+1,I}, N_{r+1}) := (1, 1).

2.5. Notice that both P (f; t) and Z(f, ω_{I}; s) depend only on the diagram Γ(I), and not on the particular choice of the equations of f, g_{ℓ} and g. In particular, all the next arguments regarding these invariants depend merely on this graph.

3. The main theorem and its proof

3.1. The goal of the present article is to find a set I of indices I (or set of differential forms ω_{I}) such that

(1) if s_{0} is a pole of some Z(f, ω_{I}; s) ( I ∈ I), then exp(2πis_{0}) is an eigenvalue of monodromy of f, and

(2) if λ is an eigenvalue of monodromy of f, then there is such a form ω_{I} ( I ∈ I) and a pole s_{0} of Z(f, ω_{I}; s) such that exp(2πis_{0}) = λ.

The size of I is obstructed by both conditions (1) and (2). A larger set is obstructed more by (1), while if I is too small then it may not realize in (2) all the eigenvalues. Therefore, besides the existence of such a set, one may also ask how large/small it can be.

Theorem 3.1.1. Let f be an irreducible plane curve singularity with r Puiseux pairs as above. Then there exist an infinite set I, and a finite subset J of I of cardinality (a_{r} − 1)(p_{r} − 1), such that both sets I and J satisfy conditions (1) and (2) above.

Remark 3.1.2. (a) Statement (1) is obvious for the ‘trivial’ candidate pole s_{0} = −1 since 1 = exp(2πi(−1)) is always the (unique) monodromy eigenvalue on H^{0}(F_{0}, ℂ). So further we can concentrate on candidate poles (≠ −1) induced by the exceptional curves E_{1}, ⋯, E_{r}. Concerning (2) for λ = 1, we will see that at least one constructed ω_{I} really has s_{0} = −1 as pole.

(b) For r = 1 the cardinality of J is sharp (it is the smallest possible), provided that we deal with differential forms used in the present article. In fact, it is exactly the Milnor number μ(f) of the germ. Nevertheless, for large r, we believe that the size of J can be decreased even more (even with the present forms). It was not our goal to find the best minimal #J, but to find (for r > 1) a pair J ⊂ I with #J < μ(f) < #I in order to emphasize that the cardinality of an index-set satisfying the combinatorial conditions (1) and (2) is not really related with the Milnor number of f (at least not in this combinatorial way).

The proof of (3.1.1) will occupy the remaining part of the article.

3.2. The inductive step. The construction and the proof run over induction on the number of Puiseux pairs, i.e. sub-diagrams of Γ with less and less rupture vertices. In order to compare nicely the corresponding invariants, the diagram with one Puiseux pair will be (strangely enough) the graph Γ_{r,r} with one rupture vertex corresponding to E_{r} and with splice decorations (a_{r}, p_{r}). For any 1 ≤ k ≤ r we consider the diagram Γ_{k,r} with rupture vertices corresponding to {E_{ℓ}}_{k≤ℓ≤r} and splice decorations {(a_{ℓ}, p_{ℓ})}_{k≤ℓ≤r}. They codify the topological type of an irreducible germ f_{k,r}. Clearly, for k = 1 we recover Γ_{1,r} = Γ, f_{1,r} = f (up to equisingularity).

Similarly, for an index set I_{k,r} of indices I_{k,r} = (j_{k}, i_{k}, …, i_{r}) ∈ (ℤ_{>0})^{r−k+2} we consider the (topological/numerical type of the) form ω_{I_{k,r}} and the corresponding topological zeta function Z(f_{k,r}, ω_{I_{k,r}}; s).

In particular, the construction and the proof is based on a decreasing induction over k, and it starts with the diagram

[figure: a_{r}; (j_{r} − 1); (i_{r} − 1); p_{r}; E_{r}; Γ_{r,r}(I_{r,r}) :]

3.3. The construction of the sets J ⊂ I. The sets J_{k,r} ⊂ I_{k,r} will be constructed inductively as follows. We start with k = r:

I_{r,r} := {(j_{r}, i_{r}) ∈ (ℤ_{>0})^{2} : j_{r} ∉ {2a_{r}, 3a_{r}, …}, i_{r} ∉ {2p_{r}, 3p_{r}, …} },
J_{r,r} := {(j_{r}, i_{r}) : 0 < j_{r} < a_{r}, 0 < i_{r} < p_{r} }.

Assume that the sets J_{k+1,r} ⊂ I_{k+1,r} are already constructed for some k < r. Then one defines a map

ψ_{k} : I_{k+1,r} → (ℤ_{>0})^{r−k+2}

such that ψ_{k}((j_{k+1}, i_{k+1}, …, i_{r})) has the form (j_{k}, i_{k}, i_{k+1}, …, i_{r}) satisfying the additional identity

(3.3.1)  j_{k+1} = p_{k}j_{k} + a_{k}i_{k} − p_{k}a_{k}.

The pair (j_{k}, i_{k}) is defined by the following principle:

Case A: j_{k+1} ≢ 0 (mod p_{k}) and j_{k+1} ≢ 0 (mod a_{k}).

Note that this is equivalent to ‘i_{k} ≢ 0 (mod p_{k}) and j_{k} ≢ 0 (mod a_{k})’. Since gcd(p_{k}, a_{k}) = 1 the numbers p_{k}x + a_{k}y for x, y ∈ ℤ_{>0} fill up all integers larger than p_{k}a_{k}. So, for a given j_{k+1} ≥ 1 there exist i_{k} ≥ 1 and j_{k} ≥ 1 satisfying (3.3.1). We take such a pair (j_{k}, i_{k}).

Case B: j_{k+1} ≢ 0 (mod p_{k}) and j_{k+1} ≡ 0 (mod a_{k}).

Take j_{k} = a_{k} and i_{k} = \frac{j_{k+1}}{a_{k}} .

Case C: j_{k+1} ≡ 0 (mod p_{k}).

Take j_{k} = \frac{j_{k+1}}{p_{k}} and i_{k} = p_{k}.

Definition 3.3.2. Set I_{k,r} := ψ_{k}(I_{k+1,r}) and J_{k,r} := ψ_{k}(J_{k+1,r}) for all k ∈ {1, …, r − 1}.

It is easy to verify that ψ_{k} is injective, hence #J_{k,r} = #J_{r,r} = (a_{r} − 1)(p_{r} − 1). In the main theorem we will take I := I_{1,r} and J := J_{1,r}.

3.4. The proof of (3.1.1) in the case of one Puiseux pair. (This is essentially Proposition 2.7 in [18].) Consider the diagram Γ_{r,r}(I_{r,r}) from (3.2), where I_{r,r} = (j_{r}, i_{r}) ∈ I_{r,r} as in (3.3).

By (2.3.1) we have N_{r} = a_{r}p_{r} and ν_{r,I} = ν_{r} + p_{r}(j_{r} − 1) + a_{r}(i_{r} − 1) = p_{r}j_{r} + a_{r}i_{r}. Using the formula of (2.4.1) we obtain

\begin{matrix}Z(f_{r,r}, ω_{I_{r,r}}; s) = & \frac{1}{(p_{r}j_{r} + a_{r}i_{r}) + sa_{r}p_{r}} · ( \frac{a_{r}}{j_{r}} + \frac{p_{r}}{i_{r}} + \frac{1}{1 + s} − 1 ) \\ = & \frac{(p_{r}j_{r} + a_{r}i_{r}) + (p_{r}j_{r} + a_{r}i_{r} − j_{r}i_{r})s}{j_{r}i_{r}(p_{r}j_{r} + a_{r}i_{r} + sa_{r}p_{r})(1 + s)}. \end{matrix}

First notice that −1 is always a pole, hence it can be paired with the trivial eigenvalue of the monodromy acting on H^{0}(F_{0}, ℂ). The candidate pole s_{0} := −\frac{p_{r}j_{r}+a_{r}i_{r}}{a_{r}p_{r}} = −(\frac{j_{r}}{a_{r}} + \frac{i_{r}}{p_{r}}) cannot be a pole of order 2, since one easily verifies that s_{0} ≠ −1. It is a pole (of order 1) if and only if p_{r}j_{r} + a_{r}i_{r} − j_{r}i_{r} ≠ a_{r}p_{r}, i.e. if and only if i_{r} ≠ p_{r} and j_{r} ≠ a_{r}. Since in this case

P (f_{r,r}; t) = Δ_{a_{r},p_{r}}(t) = \frac{(t^{a_{r}p_{r}} − 1)(t − 1)}{(t^{a_{r}} − 1)(t^{p_{r}} − 1)},

the statements (1) and (2) are clearly satisfied for both J_{r,r} and I_{r,r}.

3.5. Inductive properties of the candidate poles. In order to simplify the notations, we will consider the case k = 1 (but evidently, the same inductive formulae hold for arbitrary k when one connects the steps k + 1 with k). Moreover, we write I′ := I_{2,r} = (j_{2}, i_{2}, …, i_{r}) and I := ψ_{1}(I′) = (j_{1}, i_{1}, …, i_{r}). Then ν_{ℓ,I} and ν_{ℓ} (for any 1 ≤ ℓ ≤ r) are computed from Γ_{1,r}(I), while their analogues ν_{ℓ,I′} and ν′_{ℓ} (for any 2 ≤ ℓ ≤ r) from Γ_{2,r}(I′).

In fact, with the substitution j_{2} = p_{1}j_{1} + a_{1}i_{1} − p_{1}a_{1}, one has the expressions

\{ \begin{matrix}ν_{1,I} & = p_{1}j_{1} + a_{1}i_{1} + a_{1}p_{1}(i_{2} − 1) + a_{1}p_{1}p_{2}[i_{3} − 1 + p_{3}(i_{4} − 1) + ⋯] \\ & ν_{2,I} = p_{2}j_{2} + a_{2}i_{2} + a_{2}p_{2}[i_{3} − 1 + p_{3}(i_{4} − 1) + ⋯], \end{matrix}

where the terms between square brackets are the same. Therefore

(3.5.1)  a_{2}ν_{1,I} − a_{1}p_{1}ν_{2,I} = j_{2}(a_{2} − a_{1}p_{1}p_{2}) = j_{2}q_{2}.

Using this identity as inductive step, we obtain for all ℓ ∈ {1, …, r − 1}:

(3.5.2)  a_{ℓ+1}ν_{ℓ,I} − a_{ℓ}p_{ℓ}ν_{ℓ+1,I} = j_{ℓ+1}q_{ℓ+1}.

Notice also that N_{ℓ} for ℓ ≥ 2 computed from Γ_{1,r} or from Γ_{2,r} is the same.

We list now several facts regarding the candidate poles −\frac{ν_{ℓ,I}}{N_{ℓ}} (ℓ ≥ 1).

Lemma 3.5.3.

\frac{ν_{1,I}}{N_{1}} > \frac{ν_{2,I}}{N_{2}} > ⋯ > \frac{ν_{r,I}}{N_{r}} .

In particular, −\frac{ν_{1,I}}{N_{1}} is not a pole of order 2 of Z(f_{1,r}, ω_{I}; s).

Proof. For the inequalities use (3.5.2), then apply (2.4.2). □

We now compute the contribution R_{1} of E_{1} to the residue of −\frac{ν_{1,I}}{N_{1}} as candidate pole of order 1 of Z(f_{1,r}, ω_{I}; s). Namely, by (2.4.1),

N_{1}R_{1} = −1 + \frac{a_{1}}{j_{1}} + \frac{p_{1}}{i_{1}} + \frac{q_{2}}{ν_{2,I} − \frac{ν_{1,I}}{N_{1}} N_{2}} = −1 + \frac{a_{1}}{j_{1}} + \frac{p_{1}}{i_{1}} + \frac{a_{1}p_{1}q_{2}}{a_{1}p_{1}ν_{2,I} − a_{2}ν_{1,I}} .

Note that this is the residue, unless \frac{ν_{1,I}}{N_{1}} = 1.

Lemma 3.5.4. R_{1} = 0 if and only if j_{1} = a_{1} or i_{1} = p_{1}.

Proof. Use identity (3.5.1). □

Lemma 3.5.5. For ℓ ≥ 2 one has:

(i) ν_{ℓ,I} = ν_{ℓ,I′}, and

(ii) s_{0} = −\frac{ν_{ℓ,I}}{N_{ℓ}} = −\frac{ν_{ℓ,I′}}{N_{ℓ}} is a pole of Z(f_{1,r}, ω_{I}; s) if and only if it is a pole of Z(f_{2,r}, ω_{I′}; s).

Proof. By the general formula

\{ \begin{matrix}ν_{ℓ,I′} & = ν′_{ℓ} + p_{ℓ} ⋯ p_{2}(j_{2} − 1) + [p_{ℓ} ⋯ p_{3}a_{2}(i_{2} − 1) + ⋯] \\ ν_{ℓ,I} & = ν_{ℓ} + p_{ℓ} ⋯ p_{2}p_{1}(j_{1} − 1) + p_{ℓ} ⋯ p_{2}a_{1}(i_{1} − 1) + [p_{ℓ} ⋯ p_{3}a_{2}(i_{2} − 1) + ⋯] \end{matrix}

where the terms between square brackets are the same. Also, one easily verifies by induction on ℓ that ν′_{ℓ} = ν_{ℓ} + p_{ℓ} ⋯ p_{2}(p_{1} − 1)(a_{1} − 1). Hence (i) follows.

For (ii) notice that for ℓ ≥ 3 the relevant parts of the two zeta functions are exactly the same. For ℓ = 2 again most relevant parts are the same; the difference is the contribution \frac{q_{2}}{(ν_{1,I}+sN_{1})(ν_{2,I}+sN_{2})} for Z(f_{1,r}, ω_{I}; s) versus the contribution \frac{a_{2}}{j_{2}(ν_{2,I}+sN_{2})} for Z(f_{2,r}, ω_{I′}; s). In order to check the claim about s_{0} being a pole we must verify that \frac{q_{2}}{ν_{1,I}+s_{0}N_{1}} = \frac{a_{2}}{j_{2}} . This is equivalent to q_{2}j_{2} = a_{2}(ν_{1,I} − \frac{ν_{2,I}}{N_{2}} N_{1}) = a_{2}ν_{1,I} − a_{1}p_{1}ν_{2,I}, which is precisely (3.5.1). □

Before we start the proof, we need another property of the set J = J_{1,r}.

Lemma 3.5.6. The set J contains a subset J′ such that the set of possible values {ν_{1,I} : I ∈ J′} contains all the integers a_{1}p_{1} ⋯ p_{r−1} + 1, …, a_{1}p_{1} ⋯ p_{r−1} + N_{1}.

Proof. We construct the desired J′ = J′_{1,r} inductively as follows. Set

J′_{r,r} = {(j_{r}, i_{r}) : 1 ≤ i_{r} ≤ p_{r} − 1, j_{r} = αp_{2} ⋯ p_{r−1} for some 1 ≤ α ≤ N_{1}}.

Since N_{1}p_{2} ⋯ p_{r−1} = a_{1}p_{1}p_{2}^{2} ⋯ p_{r−1}^{2}p_{r} ≤ a_{r} − 1, cf. (2.1.2), J′_{r,r} ⊂ J_{r,r}. Define inductively J′_{k,r} := ψ_{k}(J′_{k+1,r}) for all 1 ≤ k < r.

Take (j_{r}, i_{r}) ∈ J′_{r,r}. Then ψ_{r−1}((j_{r}, i_{r})) = (j_{r}/p_{r−1}, p_{r−1}, i_{r}), and inductively

ψ_{2} ∘ ⋯ ∘ ψ_{r−1}((j_{r}, i_{r})) = (α, p_{2}, ⋯, p_{r−1}, i_{r}).

Therefore, any I ∈ J′_{1,r} has the form (j_{1}, i_{1}, p_{2}, ⋯, p_{r−1}, i_{r}) with α = p_{1}j_{1} + a_{1}i_{1} − p_{1}a_{1}, cf. (3.3.1). Hence, we have

ν_{1,I} =p_{1}j_{1} + a_{1}i_{1} + a_{1}p_{1}(i_{2} − 1) + a_{1}p_{1}p_{2}(i_{3} − 1) + a_{1}p_{1}p_{2}p_{3}(i_{4} − 1)
+ ⋯ + a_{1}p_{1}p_{2} ⋯ p_{r−2}(i_{r−1} − 1) + a_{1}p_{1}p_{2} ⋯ p_{r−1}(i_{r} − 1)
=α + a_{1}p_{1}[p_{2} + p_{2}(p_{3} − 1) + p_{2}p_{3}(p_{4} − 1)
+ ⋯ + p_{2}p_{3} ⋯ p_{r−2}(p_{r−1} − 1) + p_{2}p_{3} ⋯ p_{r−1}(i_{r} − 1)]
=α + (a_{1}p_{1}p_{2} ⋯ p_{r−1})i_{r}.

Since 1 ≤ α ≤ a_{1}p_{1}p_{2} ⋯ p_{r} and 1 ≤ i_{r} ≤ p_{r} − 1 are arbitrary, these ν_{1,I} contain all the integers 1 + a_{1}p_{1} ⋯ p_{r−1}, …, a_{1}p_{1} ⋯ p_{r} + a_{1}p_{1} ⋯ p_{r−1}(p_{r} − 1). Moreover

a_{1}p_{1} ⋯ p_{r} + a_{1}p_{1} ⋯ p_{r−1}(p_{r} − 1) ≥ a_{1}p_{1} ⋯ p_{r} + a_{1}p_{1} ⋯ p_{r−1} = N_{1} + a_{1}p_{1} ⋯ p_{r−1}.

This ends the proof. □

3.6. Proof of Theorem (3.1.1). We will use induction starting with the situation (3.4). Assume that the theorem is already verified for Γ_{2,r}(I_{2,r}) and Γ_{2,r}(J_{2,r}); we will check it in the next paragraphs for (1, r). Clearly, it is enough to verify statement (1) of (3.1.1) only for I and (2) only for J.

(1) Take I ∈ I. If s_{0} = −\frac{ν_{ℓ,I}}{N_{ℓ}} for some ℓ ≥ 2 is a pole of Z(f_{1,r}, ω_{I}; s), by (3.5.5)(ii) it is also a pole of Z(f_{2,r}, ω_{I′}; s) for some I′ ∈ I_{2,r}. Then the induction hypothesis says that exp(2πis_{0}) is a root of P (f_{2,r}; t) = Δ_{a_{2},p_{2}}(t^{p_{3}···p_{r}}) · . . . · Δ_{a_{r},p_{r}}(t), and hence of P (f_{1,r}; t). Suppose now that s_{0} = −\frac{ν_{1,I}}{N_{1}} is a pole of Z(f_{1,r}, ω_{I}; s). By (3.1.2)(a) we can assume that s_{0} ≠ −1. Therefore R_{1} in (3.5.4) is the residue, hence j_{1} ≠ a_{1} and i_{1} ≠ p_{1}; i.e. Case A occurs. In other words, j_{1} ≢ 0 (mod a_{1}) and i_{1} ≢ 0 (mod p_{1}). But then, since s_{0} = −\frac{p_{1}j_{1}+a_{1}i_{1}−a_{1}p_{1}(···)}{a_{1}p_{1}p_{2}···p_{r}}, we have that exp(2πis_{0}) is a root of Δ_{a_{1},p_{1}}(t^{p_{2}p_{3}···p_{r}}).

(2) Take a root λ of P (f_{1,r}; t). If it is a root of Δ_{a_{2},p_{2}}(t^{p_{3}···p_{r}}) · . . . · Δ_{a_{r},p_{r}}(t) = P (f_{2,r}t), the induction hypothesis yields a pole s_{0} of Z(f_{2,r}, ω_{I′}; s) for some I′ ∈ J_{2,r}, hence by (3.5.5)(ii) also of Z(f_{1,r}, ω_{I}; s) (I ∈ J), such that exp(2πis_{0}) = λ.

Suppose on the other hand that λ is a root of Δ_{a_{1},p_{1}}(t^{p_{2}···p_{r}}). By (3.5.6), for some I ∈ J we get ν_{1,I} with exp(2πi(−\frac{ν_{1,I}}{N_{1}} )) = λ. Since λ ≠ 1, we get that ν_{1,I}/N_{1} ≠ 1. Also, for such a λ necessarily j_{1} ≢ 0 (mod a_{1}) and i_{1} ≢ 0 (mod p_{1}), and so s_{0} = −\frac{ν_{1,I}}{N_{1}} is really a pole (of order 1) of Z(f_{1,r}, ω_{I}; s), since the contribution R_{1} of (3.5.4) is really the residue of s_{0}.

Finally, assume that λ = 1, the eigenvalue of the monodromy on H^{0}(F_{0}, ℂ). Then take α = i_{r} = 1 in the construction from the proof of (3.5.6). Then by the computation of that proof, we have ν_{1,I} = 1 + a_{1}p_{1} … p_{r−1} < N_{1}. Hence, ν_{ℓ,I}/N_{ℓ} < 1 for all ℓ, and s = −1 is indeed a pole of Z(f_{1,r}, ω_{I}; s) for this choice of I.

This finishes the proof of Theorem (3.1.1).

3.7. Example. We describe the case of two Puiseux pairs more explicitly. A standard equation for f is then

f = (y^{a_{1}} + x^{p_{1}})^{p_{2}} + x^{u}y^{v}

where ua_{1} +vp_{1} = a_{2}. The occurring g, g_{1}, g_{2} in the forms ω_{I}, I = (j_{1}, i_{1}, i_{2}), can be chosen in the following easy way:

ω_{I} = y^{j_{1}−1}x^{i_{1}−1}(y^{a_{1}} + x^{p_{1}})^{i_{2}−1}dx ∧ dy .

By the construction in (3.3) the sets I in J are as follows. The number i_{2} can take all values satisfying 0 < i_{2} < p_{2}. The couples (j_{1}, i_{1}) can be either

i_{1} = p_{1} and 0 < j_{1} ≤ [\frac{a_{2}−1}{p_{1}}] (Case C),

j_{1} = a_{1} and 0 < i_{1} ≤ [\frac{a_{2}−1}{a_{1}}] but excluding here the multiples of p_{1} (Case B), or the values from Case A. In this last case it is in general not so clear how to describe really explicitly (j_{1}, i_{1}) in terms of a_{1}, p_{1} and a_{2}.

We mention a particular situation; suppose that a_{1} = p_{1} + 1. Then we can take i_{1} = \{\frac{j_{2}−1}{p_{1}}\}p_{1} + 1 and j_{1} = [\frac{j_{2}−1}{p_{1}}] + p_{1} − \{\frac{j_{2}−1}{p_{1}}\}p_{1}, where j_{2} takes all values 0 < j_{2} < a_{2}.

3.8. Remark. Theorem (3.1.1) is also valid for the motivic or Hodge zeta functions associated to f and the forms ω_{I}, and, in the context of p-adic f, for its associated Igusa zeta functions.

Condition (2) for these zeta functions follows from the result for topological zeta functions, since the latter is a specialization of the former ones. (We refer to [18] for more details.) Concerning condition (1) we must be sure that in our case a pole of the motivic/Hodge/Igusa zeta function also yields a pole of the topological one, and this is true. The only non-obvious statement here is the ‘if’ part of the analogue of Lemma (3.5.4), and this is easily verified, for instance using [15].

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[11] Loeser, F.: Fonctions d’Igusa p–adiques et polynômes de Bernstein, Amer. J. Math. 110, 1988, 1–22.

[12] Loeser, F.: Fonctions d’Igusa p–adiques, polynômes de Bernstein, et polyèdres de Newton, J. Reine Angew. Math. 412, 1990, 75–96.

[13] Malgrange, B.: Polynômes de Bernstein–Sato et cohomologie évanescente, Astérisque 101/102, 1983, 243–267.

[14] Veys, W.: Poles of Igusa’s local zeta function and monodromy, Bull. Soc. Math. France 121, 1993, 545–598.

[15] Veys, W.: Zeta functions for curves and log canonical models, Proc. London Math. Soc., 74, 1997, 360–378.

[16] Veys, W.: Zeta functions and ‘Kontsevich invariants’ on singular varieties, Canadian J. Math. 53, 2001, 834–865.

[17] Veys, W.: Vanishing of principal value integrals on surfaces, J. Reine Angew. Math. 598, 2006, 139-158.

[18] Veys, W.: Monodromy eigenvalues and zeta functions with differential forms, Adv. Math. 213, 2007, no. 1, 341–357.

[19] Włodarczyk, J.: Combinatorial structures on toroidal varieties and a proof of the weak factorization theorem, Invent. Math. 154, 2003, 223–331.

Rényi Institute of Mathematics, 1053 Budapest, Reáltanoda u. 13–15, Hungary

E-mail address: nemethi@renyi.hu
URL: http://www.renyi.hu/~nemethi

K.U.Leuven, Departement Wiskunde, Celestijnenlaan 200B, B–3001 Leuven, Belgium

E-mail address: wim.veys@wis.kuleuven.be
URL: http://www.wis.kuleuven.be/algebra/veys.htm