Generalized Weil’s Reciprocity Law and Multiplicativity Theorems

ANDRÁS NÉMETHI^{1}

Abstract: Fix a one-dimensional group variety G with Euler–characteristic χ(G) = 0, and a quasi–projective variety Y, both defined over C. For any f ∈ Hom(Y, G) and constructible sheaf F on Y, we construct an invariant c_{F}(f) ∈ G, which provides substantial information about the topology of the fiber–structure of f and the structure of F along the fibers of f. Moreover: c_{F} : Hom(Y, G) → G is a group homomorphism.

1. Introduction and the main result.

We fix a one–dimensional (abelian) group variety G and a quasi–projective variety Y, both defined over the complex numbers. We will use the multiplicative notation for the group structure μ : G × G → G of G. We would like to relate the topological fiber–structure of morphisms f : Y → G and the group structure of Hom(Y, G).

If f : Y → G is a morphism, and F is a constructible sheaf on Y, then the Euler–characteristic χ(t) of the cohomology groups H^{∗}(f^{−1}(t), F) is constant on a non–empty Zariski open set of G. This generic value is denoted by χ(t_{gen}). (If dim f(Y) = 0 then, by definition, χ(t_{gen}) = 0.) An element t ∈ G is called χ–critical value if Δχ(t) := χ(t_{gen}) − χ(t) ≠ 0. The set of χ–critical values is denoted by C(f, F). Our first invariant associated with f and F is

a(f, F) = ∏_{t∈C(f,F)} t^{Δχ(t)} ∈ G.

If C(f, F) = ∅ then, by definition, a(f, F) = e_{G}, the identity element of G.

If we have two morphisms f_{i} : Y → G (i = 1, 2), then we define f_{1} · f_{2} : Y → G by (f_{1} · f_{2})(y) = μ(f_{1}(y), f_{2}(y)). The proposed “multiplicativity problem” is the following: fix G, and study the existence of the multiplicativity property:

(M.P.) a(f_{1} · f_{2}, F) = a(f_{1}, F) · a(f_{2}, F) for arbitrary spaces Y and morphisms f_{i} (i = 1, 2); or, if (M.P.) does not hold, then characterize the correction term a(f_{1} · f_{2}) · a(f_{1})^{−1} · a(f_{2})^{−1}.

1Partially supported by NSF–grant No. DMS–9203482 and by the Netherlands Organisation for the Advancement of Scientific Research N.W.O.

1991 Mathematics Subject Classification: Primary 14F05; Secondary 14B05.

It is not very difficult to see that in order to have (M.P.) one needs the vanishing of the Euler–characteristic of G: χ(G) = 0. (Take, for example, Y = G, F = C_{G}, f_{1} = id_{G}, and f_{2} =the constant map g ↦ g_{0}, for a generic g_{0}.) With this topological restriction there are exactly two candidates for G: G = C^{∗} and G supported by an elliptic curve. Now, for G = C^{∗}, the multiplicativity property again does not hold (see belove), but the correction term can be determined: it is related to the monodromies around the “points at infinity” of the non–complete space C^{∗}. On the other hand, for elliptic curves the following holds:

Proposition 1. Let G be an elliptic curve with an arbitrary fixed group structure μ : G × G → G. Then multiplicativity property (M.P.) holds for arbitrary morphisms f_{i} : Y → G, (i = 1, 2), and constructible sheaf Fon Y.

We will give the proof in section 4.

In order to describe the correction term in the case G = C^{∗}, we need one more invariant. For this, we need more information about the sheaves {R^{k}f_{!}F}_{k}. It is well known that there exists a finite set Γ(f, F) of C^{∗} such that the restriction of ⊕_{k}R^{k}f_{!}F on its complement determines a flat bundle. Let R > r > 0 be two positive numbers such that for any t ∈ Γ(f, F) one has r < |t| < R. The monodromy M_{0}^{∗} ∈ Aut H^{∗}(f^{−1}(r), F) of the fibration over {t ∈ C^{∗} : |t| = r} defines the zeta–function ζ_{0}(λ) = ∏_{q≥0} det(1 − λM_{0}^{q})^{(−1)^{q}}. Similarly, the monodromy M_{∞}^{∗} of the fibration over {t ∈ C^{∗} : |t| = R} defines the zeta function ζ_{∞}(λ). Since the roots and the poles of ζ_{0} and ζ_{∞} are roots of unity, these rational functions can be written in the following form:

ζ_{0}(λ) = ∏_{N>0}(1 − λ^{N})^{α_{N}} , and ζ_{∞}(λ) = ∏_{M>0}(1 − λ^{M})^{β_{M}} .

Let b_{0}(f) = ∏_{N>0} N^{N·α_{N}} and b_{∞}(f) = ∏_{M>0}(−M)^{M·β_{M}}. Our second invariant associated with f and F is b(f, F) = b_{0}(f)/b_{∞}(f).

The main result of this paper is the following theorem:

Theorem. Let Y be a quasi–projective space and F a constructible sheaf on Y. For any two morphisms f_{i} : Y → C^{∗} (i = 1, 2) one has:

(M.T.) \frac{a(f_{1}, F) · a(f_{2}, F)}{a(f_{1} · f_{2}, F)} = \frac{b(f_{1}, F) · b(f_{2}, F)}{b(f_{1} · f_{2}, F)}.

This means that the invariant c(f, F) = a(f, F)/b(f, F) is multiplicative: c(f_{1}, F) · c(f_{2}, F) = c(f_{1} · f_{2}, F).

The multplicativity theorem has the following

Corollary. Let Y be a quasi–projective space and let F and F′ be two constructible sheafs on Y such that dim F_{Q} = dim F′_{Q} for any Q ∈ Y. If f : Y → C^{∗} is a morphism, then we define the relative b–invariant by Δb(f; F, F′) := b(f, F)/b(f, F′). Then Δb(·) = Δb(·; F, F′) is multiplicative; i.e. for any two morphisms f_{1} and f_{2} one has:

Δb(f_{1}) · Δb(f_{2}) = Δb(f_{1}f_{2}).

Proof. By a Meyer–Vietoris argument, χ(f^{−1}(t), F) = χ(f^{−1}(t), F′) for any t ∈ C^{∗}. Now apply the multiplicativity formula (M.T.).

A direct proof of the corollary can be extracted from the proof of the main theorem (section 3). □

The proof the main Theorem is contained in sections 2 and 3. The key point is a generalization of Weil’s reciprocity law. This is stated in section 2. (For Weil’s result, see, for example, [1], p. 242.)

We end this section with the following:

Problem: Find the analogue of the Multiplicativity Theorem for higher dimensional group varieties G !

Acknowledgement, bibliographical note. The result of Theorem in the special case of F = C_{Y} was first obtained by J. Denef in January 1991 by using arithmetical results of [3] (Théorème 3.2.1.1) and [4]. Directly afterwards F. Loeser noticed that the result (with the same restriction) also follows from Théorème 3.3.1 of [5], by considering the determinant of the Aomoto complex of f_{1}^{s_{1}}f_{2}^{s_{2}}, putting s_{1} = s_{2} = s and comparing with the determinant of the Aomoto complex of (f_{1}f_{2})^{s}. The work of Loeser–Sabbah uses the theory of D–modules. (The proof of J. Denef, respectively of F. Loeser are both unpublished.)

Since the theorem can be formulated in topological terms, they asked for a more natural proof, which is able to explain the “mystery” of the relation. This paper not only generalizes their result, but we hope that satisfies this demand as well. (The first version of this note was written in the autumn of 1993.)

The author wishes to thank Prof. Jan Denef for communicating to him the problem.

2. The generalized reciprocity law.

The main result of this section is a generalization of Weil’s reciprocity law. This result is the heart of the proof of the multiplicativity theorem. In fact, the multiplicativity theorem can be considered as a reciprocity law, as we will see later.

If f is a meromorphic function defined on a compact Riemann surface S, then supp(f) denotes its support, and ord_{x}(f) is the order of f at x.

The generalized Weil’s Reciprocity Law. Let f_{i} and g_{i} be meromorphic functions (i = 1, …, s) on the compact Riemann surface S so that there exists a finite set P = {x_{1}, …, x_{n}} with the following properties:

(a) supp(f_{i})∩supp(g_{i}) ⊂ P for each i = 1, …, s;

(b) there exist integers n_{1}, …, n_{s}, respective m_{1}, …, m_{s} with ∑_{i=1}^{s} n_{i}m_{i} = 0, and integers o_{f}(x_{j}) respective o_{g}(x_{j}) (j = 1, …, n) such that

\{ \begin{matrix}ord_{x_{j}}(f_{i}) = n_{i} · o_{f}(x_{j}) \\ ord_{x_{j}}(g_{i}) = m_{i} · o_{g}(x_{j}) & for any i and j. \end{matrix}

Then

∏_{p∈S\P} ∏_{i=1}^{s} f_{i}(p)^{ord_{p}(g_{i})} · ∏_{x∈P}[(∏_{i=1}^{s} f_{i}^{m_{i}})(x)]^{o_{g}(x)} =

∏_{p∈S\P} ∏_{i=1}^{s} g_{i}(p)^{ord_{p}(f_{i})} · ∏_{x∈P}[(∏_{i=1}^{s} g_{i}^{n_{i}})(x)]^{o_{f} (x)}.

The proof is elementary and similar to the proof of the Weil’s reciprocity law [1, p. 242], and it is left to the reader.

The main application of this reciprocity law is an (apparently) particular version of the multiplicativity theorem.

We will fix the homogeneous coordinates [u_{0} : u_{1} : u_{2}] in P^{2}(C). Let P (u_{0}, u_{1}, u_{2}) be an irreducible homogeneous polynomial. Let X ∈ P^{2} be its zero set Z(P). Let π : X̃ → X be the normalization of X.

The set A = X ∩ {u_{0}u_{1}u_{2} = 0} has a natural decomposition in the following subsets: A_{ij} = A_{ji} = X ∩ {u_{i} = u_{j} = 0} for 0 ≤ i < j ≤ 2; and A_{i} = X ∩ {u_{i} = 0} \ (∪_{j≠i}A_{ij}) for 0 ≤ i ≤ 2. Their invers image in Ã = π^{−1}(A) are Ã_{ij} = π^{−1}(A_{ij}) and Ã_{i} = π^{−1}(A_{i}). The space X^{∗} = X \ A can be considered as an affine space in C^{2} = P^{2} \ {u_{2} = 0}. Introduce the affine coordinates (x, y) in C^{2} by [x : y : 1] = [u_{0} : u_{1} : u_{2}] and denote p(x, y) = P (x, y, 1). Obviously Z(p) ∩ (C^{∗} × C^{∗}) = X^{∗}.

We assume that X^{∗} ≠ ∅, and X^{∗} is not of the form {x = constant} or {y = constant} or {xy = constant}.

For any point Q ∈ X, in the local ring O_{Q}(P^{2}) the germ of P has a prime–decomposition P = ∏_{j∈J(Q)} P_{j}. The space–germs Z_{j} = {P_{j} = 0} give the local irreducible components of the germ of Z(P) at Q. Corresponding to this decomposition, the inverse image π^{−1}(Q) contains exactly J(Q) points π^{−1}(Q) = ∪_{j∈J(Q)}Q̃_{j}. We will use the notation t_{j} for a local coordinate at the smooth points Q̃_{j}. If f is a meromorphic function on X then the composition f ∘ π is denoted by f̃. It is a meromorphic function on X̃.

The following lemma will be useful in multiplicity and order computations: Lemma 1. Let Q ∈ X be a point in X ∩ C^{2} and φ : (C^{2}, Q) → (C, 0) be the germ of an analytic function which does not vanish on any of the space–germs Z_{j}. Let t_{j} be a local coordinate at Q̃_{j}. Then:

(a) ord_{t_{j}}Jac(p_{j}, φ)˜ = μ(p_{j}) + i_{Q}(φ, p_{j}) − 1 for any j;

(b) ord_{t_{j}}Jac(p, φ)˜ = μ(p_{j}) + i_{Q}(p_{j}, p/p_{j}) + i_{Q}(φ, p_{j}) − 1 for any j.

Here Jac(ψ, ϕ) denotes the Jacobian ψ_{x}ϕ_{y} − ψ_{y}ϕ_{x} of ψ and ϕ (ψ_{x} and ψ_{y} are the partial derivatives), p_{j}(x, y) = P_{j}(x, y, 1), μ(p_{j}) is the Milnor number of the germ p_{j}, and i_{Q}(·, ·) denotes the intersection multiplicity at Q. Proof. Consider a good representative φ̄ : (X , Q) → (D_{2}, 0) of φ such that Z_{j} ∩ ∂X ⊂ φ̄^{−1}(∂D_{2}) and φ̄|Z_{j} ∩ X : (Z_{j} ∩ X , Q) → (D_{2}, 0) is a good representative for the restriction of φ on Z_{j}. For ϵ sufficiently small consider the Milnor fiber F_{j} = {p_{j} = ϵ} ∩ X. Now, ord_{t_{j}}Jac(p_{j}, φ)˜ = i_{Q}(p_{j}, Jac(p_{j}, φ)) = ∑_{T} i_{T} (p_{j} − ϵ, Jac(p_{j}, φ)), where the sum is over the intersection points T of the fiber F_{j} with Z(Jac(p_{j}, φ)). In these intersection points T, the fiber φ^{−1}(φ(T)) ∩ X is smooth. These points T are exactly those points where φ^{−1}(φ(T)) ∩ X intersects F_{j} tangentially. Therefore, the restriction φ̄ : F_{j} → D_{2} on F_{j} is a branched covering of degree i = i_{Q}(φ, p_{j}), with branch points of index i_{T} + 1 = i_{T} (p_{j} − ϵ, Jac(p_{j}, φ)) + 1 corresponding to the points T.

By an Euler–characteristic argument, we obtain: 1 − μ(p_{j}) = χ(F_{j}) = i · χ(D_{2}) − ∑_{T} i_{T} which gives (a).

(b) For the second part, write p = p_{j} · q_{j}. Then the result follows from (a) and the relations Jac(p, φ) = q_{j} · Jac(p_{j}, φ) + p_{j} · Jac(q_{j}, φ) and ord_{t_{j}} q̃_{j} = i_{Q}(p_{j}, q_{j}). □

Let X ⊂ P^{2} as above and Q̃_{j} ∈ π^{−1}(Q) for j ∈ J(Q). The map–germ π : (X̃, Q̃_{j}) → (Z_{j}, Q) can be written in the form t_{j} ↦ [x(t_{j}) : y(t_{j}) : 1] = [t_{j}^{N_{j}}a_{j}(t_{j}) : t_{j}^{M_{j}}b_{j}(t_{j}) : 1], where N_{j}, M_{j} are integers, and a_{j}(t_{j}), b_{j}(t_{j}) are invertible germs at Q̃_{j}.

Define the set P_{i} = {Q̃_{j} ∈ Ã_{i2} : N_{j}+M_{j} = 0} for i = 0, 1. Let P = P_{0}∪P_{1}.

Lemma 2. Consider the following meromorphic functions on X̃:

f_{1} = x̃, f_{2} = ỹ, g_{1} = (\frac{yp_{y}}{yp_{y} − xp_{x}})˜ , g_{2} = (\frac{xp_{x}}{xp_{x} − yp_{y}})˜ .

Then the Riemann surface S = X̃, the set P ⊂ X̃ and the functions f_{i}, g_{i} (i = 1, 2) satisfy the hypothesis of the Generalized Reciprocity Law. More precisely:

(a) supp(f_{i}) ∩ supp(g_{i}) ⊂ P for i = 1, 2; and

(b) ord_{x}(g_{1}) = ord_{x}(g_{2}) and ord_{x}(f_{1}) = −ord_{x}(f_{2}) for any x ∈ P.

Therefore, the reciprocity law

(∗)  ∏_{x∈S\P} f_{1}(x)^{ord_{x}g_{1}}f_{2}(x)^{ord_{x}g_{2}} ∏_{x∈P}[(f_{1}f_{2})(x)]^{ord_{x}g_{1}} =

∏_{x∈S\P} g_{1}(x)^{ord_{x}f_{1}}g_{2}(x)^{ord_{x}f_{2}} ∏_{x∈P}[ \frac{g_{1}}{g_{2}} (x)]^{ord_{x}f_{1}}

can be applied for these functions.

The contributions from the sets X̃^{∗} = π^{−1}(X^{∗}), Ã_{ij} and Ã_{i}, in the left, respectively in the right hand side of the reciprocity law are listed in the next table:

contributions in the  contributions in the
left hand side of (*)  right hand side of (*)
X̃^{∗}  ∏_{Q∈X^{∗}} \frac{x_{Q}^{i_{Q}(x−x_{Q},p)−1}·y_{Q}^{i_{Q}(y−y_{Q},p)−1}}{(x_{Q}y_{Q})^{i_{Q}(xy−x_{Q}y_{Q},p)−1}} 1
Ã_{01}  1  ∏_{j} \frac{N_{j}^{N_{j}} ·M_{j}^{M_{j}}}{(N_{j}+M_{j})^{N_{j} +M_{j}}}
Ã_{0}  ∏_{Q∈A_{0}} y_{Q}^{i_{Q}(y−y_{Q},p)}  1
Ã_{1}  ∏_{Q∈A_{1}} x_{Q}^{i_{Q}(x−x_{Q},p)}  1
Ã_{2}  1  ∏_{Q∈A_{2}} ∏_{j∈J(Q)} \frac{1}{4^{N_{j}}}

Ã_{02}

(i) N_{j} ≠ 0 ≠ N_{j} + M_{j} 1  ∏_{j} \frac{N_{j}^{N_{j}} ·M_{j}^{M_{j}}}{(N_{j}+M_{j})^{N_{j} +M_{j}}}
(ii) N_{j} = 0  ∏_{j} a_{j}(0)^{ord_{t_{j}} [a_{j}(t_{j})−a_{j}(0)]}  1
(iii) N_{j} + M_{j} = 0  ∏_{j}[a_{j}(0)b_{j}(0)]^{−ord_{t_{j}} [a_{j}b_{j}−a_{j}(0)b_{j}(0)]} ∏_{j}(−1)^{N_{j}}

Ã_{12}

(i) M_{j} ≠ 0 ≠ N_{j} + M_{j} 1  ∏_{j} \frac{N_{j}^{N_{j}} ·M_{j}^{M_{j}}}{(N_{j}+M_{j})^{N_{j} +M_{j}}}
(ii) M_{j} = 0  ∏_{j} b_{j}(0)^{ord_{t_{j}} [b_{j}(t_{j})−b_{j}(0)]}  1
(iii) N_{j} + M_{j} = 0  ∏_{j}[a_{j}(0)b_{j}(0)]^{−ord_{t_{j}} [a_{j}b_{j}−a_{j}(0)b_{j}(0)]} ∏_{j}(−1)^{M_{j}}

In the table (x_{Q}, y_{Q}) are the coordinates of the point Q; and the product ∏_{j} is over the index set J(Q), where the point Q is clear from the context. The points in Ã_{i2} (or equivalently, the local irreducible components of Z(P) at A_{i2}) are separated into three groups, which have geometrically different behaviour. In fact, they correspond to those χ–critical values which come from the bad behaviour of the functions at infinity.

Proof. Let R be one of the sets X̃^{∗}, Ã_{ij} or Ã_{i}. The conditions (a) and (b) are equivalent to the set of conditions:

(a_{R}) supp(f_{i}) ∩ supp(g_{i}) ∩ R ⊂ P ∩ R, (i = 1, 2);

(b_{R}) the identities of (b) are satisfied for any point x ∈ P ∩ R, where R = X̃^{∗}, Ã_{ij} and Ã_{i}.

The case R = X̃^{∗}.

Since x̃ and ỹ are invertible holomorphic functions on X̃^{∗}, the conditions (a_{R}) and (b_{R}) are satisfied. This also shows that the contribution in the right hand side is trivial. The support of g_{i} in X̃^{∗} has the following geometric interpretation. Fix a point Q ∈ X^{∗} and let φ be one of the functions x, y or xy. Then Jac(φ, p)(Q) = 0 if and only if the tangent line at Q of the smooth curve {φ = φ(Q)} is in the Zariski tangent space of Z(p) at Q. Let Q̃_{j} ∈ (supp(g_{1}) ∪ supp(g_{2})) ∩ X̃^{∗}. Then by lemma 1 one has: ord_{t_{j}}g_{1} = i_{Q}(x − x_{Q}, p_{j}) − i_{Q}(xy − x_{Q}y_{Q}, p_{j}) and symmetrically for g_{2}. This gives the result from the table.

The case R = Ã_{02}.

In this case M_{j} < 0 and M_{j} < N_{j}. Notice that g_{1} + g_{2} = 1. The identity g_{1} = 1/(1 + \frac{xy′}{yx′}) implies that

(#) g_{1}(t_{j}) = \frac{N_{j}a_{j}b_{j} + t_{j}b_{j}a′_{j}}{(N_{j} + M_{j})a_{j}b_{j} + t_{j}(a_{j}b_{j})′} .

Case (i): Assume that N_{j} ≠ 0 and N_{j} + M_{j} ≠ 0. Then (#) implies that Q̃_{j} ∉ supp(g_{i}) (i = 1, 2) and the contribution follows. Case (ii): Assume that N_{j} = 0. Then (#) shows that ord_{t_{j}}g_{1} > 0. Since g_{2} = 1 − g_{1} we obtain that ord_{t_{j}}g_{2} = 0. On the other hand ord_{t_{j}} x̃ = 0 (by assumtion). Therefore Q̃_{j} ∉ supp(f_{i}) ∩ supp(g_{i}) (i = 1, 2), i.e. the conditions are satisfied. Since g_{2}(Q̃_{j}) = 1 and ordx̃ = 0, this case has no contribution to the left hand side. The right hand side is: ∏_{j} a_{j}(0)^{ord_{t_{j}} g_{1}}. But ord_{t_{j}}g_{1} = 1 + ord_{t_{j}}a′_{j}(t_{j}) = ord_{t_{j}}[a_{j}(t_{j}) − a_{j}(0)]. Case (iii): Assume that N_{j} + M_{j} = 0.

These local irreducible components correspond exactly to the set P_{0}. In particular condition (a) is satisfied but we have to verify (b). The identity ord_{t_{j}}f_{1} + ord_{t_{j}}f_{2} = 0 is assured by the assumtion. The identity (#) gives that ord_{t_{j}}g_{1} < 0. Since g_{1} + g_{2} = 1 we obtain that ord_{t_{j}}g_{1} = ord_{t_{j}}g_{2}, hence (b) is satisfied too.

The contribution in the right hand side is ∏_{j}[−\frac{yp_{y}}{xp_{x}} (Q̃_{j})]^{N_{j}}. But [−\frac{yp_{y}}{xp_{x}}](Q̃_{j}) = \frac{yx′}{xy′} (Q̃_{j}) = \frac{N_{j}}{M_{j}} = −1.

The contribution in the left hand side is ∏_{j}[a_{j}(0)b_{j}(0)]^{ord_{t_{j}} g_{1}}. But, by (#), ord_{t_{j}}g_{1} = −[1 + ord(a_{j}b_{j})′] = −ord[(a_{j}b_{j})(t_{j}) − (a_{j}b_{j})(0)].

In the other cases the (case-by-case) verification is similar (or even simpler) to the above cases, and it is left to the reader. □

The next proposition relates the reciprocity law with the multiplicativity theorem.

Let Y = Z(p) be an irreducible curve in C^{∗}×C^{∗} as above. Any morphism f : Y → C^{∗} defines two invariants a(f) = a(f, C_{Y}) and b(f) = b(f, C_{Y}), where C_{Y} is the constant sheaf on Y (cf. with the definitions in the introduction).

Proposition 2. The expression

\frac{a(x)a(y)}{a(xy)} ( respectively \frac{b(x)b(y)}{b(xy)})

is equal to the left (respectively to the right) hand side of the Generalized Reciprocity Law (*), applied in Lemma 2 for the meromorphic functions f_{1}, f_{2}, g_{1} and g_{2}.

In particular, the Reciprocity Law implies the Multiplicativity Theorem in the following case: Y is an irreducible curve in C^{∗} × C^{∗}, f_{1}(x, y) = x, f_{2}(x, y) = y, and F is the constant sheaf C_{Y}.

Proof. If f : Y → C^{∗} is a morphism, we denote the set of critical values {f(y) : y ∈ Y satisfies df(y) = 0} by Σ(f).

As a first step, we we will find the χ–critical set C(x) and the invariant a(x).

It is not hard to verify that the “points at infinity” contained in the union A_{0} ∪ A_{2} ∪ A_{01} ∪ A_{12} have no contribution in C(x).

A point Q ∈ X^{∗} is χ–critical if and only if it is critical. In this case x_{Q} will be a χ–critical value of the function x, and the contribution in a(x) is x_{Q}^{i_{Q}(x−x_{Q},p)−1}.

Any point “at infinity” Q ∈ A_{1} gives a χ–critical value x_{Q} and a contribution equal to x_{Q}^{i_{Q}(x−x_{Q},p)} in a(x).

The last and more interesting case is given by the points Q̃_{j} ∈ Ã_{02}. A point Q̃_{j} is a “bad point at infinity” if the tangent cone of the local irreducible component Z_{j} at [0 : 1 : 0] is not u_{0} = 0 or u_{2} = 0. Consider the parametrization t_{j} ↦ [x(t_{j}) : y(t_{j}) : 1] of Z_{j} similarly as above. The line which joins [0 : 1 : 0] and [u_{0} : u_{1} : u_{2}](t_{j}) is u_{0} − t_{j}^{N_{j}}a_{j}(t_{j})u_{2} = 0. The tangent cone is given by the limit equation lim_{t_{j}→0}[u_{0} − t_{j}^{N_{j}}a_{j}(t_{j})u_{2}] = 0. If N_{j} > 0 then the tangent cone is u_{0} = 0; if N_{j} < 0 then it is u_{2} = 0. If N_{j} = 0 then its equation is u_{0} − a_{j}(0)u_{2} = 0, in affine coordinates x = a_{j}(0). This means that the vertical line x = a_{j}(0) is an asymtote of Z(p) and a_{j}(0) is a χ–critical value.

Consider the set of intersection points of Z_{j} and the line u_{0} − (a_{j}(0) + ϵ)u_{2} = 0 which are in a small neighbourhood of [0 : 1 : 0] and are lying in X^{∗}. Their number is the number of the local, non–zero solutions of the equation a_{j}(t_{j}) − a_{j}(0) = ϵ; in particular, the jump in Euler–characteristic is ord_{t_{j}}[a_{j}(t_{j}) − a_{j}(0)].

Therefore,

C(x) = Σ(x : X^{∗} → C^{∗}) ∪ A_{1} ∪_{j∈J(A_{02}),N_{j}=0} {a_{j}(0)},

and:

a(x) = ∏_{q∈X^{∗}} x_{Q}^{i_{Q}(x−x_{Q},p)−1} · ∏_{Q∈A_{1}} x_{Q}^{i_{Q}(x−x_{Q},p)} · ∏_{j∈J(A_{02}),N_{j}=0}[a_{j}(0)]^{ord[a_{j}−a_{j}(0)]}.

We have symmetric identities for C(y) and a(y).

In the case of the function xy the χ–critical values are provided by the critical points in X^{∗} and maybe by some “critical point at infinity” contained in Ã_{i2} (i = 0, 1). Corresponding to the point Q̃_{j} ∈ Ã_{02}, we have to compute the number of local, non–zero solutions of the equation t_{j}^{N_{j}+M_{j}}a_{j}(t_{j})b_{j}(t_{j}) = s ∈ C^{∗}. This number is zero for any s if N_{j} + M_{j} ≠ 0. If N_{j} + M_{j} = 0, then for s = a_{j}(0)b_{j}(0) we have a jump of ord[a_{j}b_{j} − a_{j}(0)b_{j}(0)]. Therefore

C(xy) = Σ(xy) ∪_{j∈J(A_{i2}),N_{j}+M_{j}=0} {a_{j}(0)b_{j}(0)},

and

a(xy) = ∏_{Q∈X^{∗}}[x_{Q}y_{Q}]^{i_{Q}(xy−x_{Q}y_{Q},p)−1}· ∏_{j∈J(A_{i2}),N_{j}+M_{j}=0}[a_{j}(0)b_{j}(0)]^{ord[a_{j}b_{j}−a_{j}(0)b_{j}(0)]}.

Now using the table it is easy to verify that a(x)a(y)/a(xy) is exactly the left hand side of the reciprocity law.

The expression b(x)b(y)/b(xy) depends only on the points at infinity X − X^{∗}.

If j ∈ J(A_{01}), then Z_{j} contributes with 1 − λ^{N_{j}} in ζ_{0}(x), with 1 − λ^{M_{j}} in ζ_{0}(y), with 1 − λ^{N_{j}+M_{j}} in ζ_{0}(xy), and with 1 in ζ_{∞}(x), ζ_{∞}(y) and ζ_{∞}(xy). So, the contribution in b(x)b(y)/b(xy) is N_{j}^{N_{j}}M_{j}^{M_{j}}/(N_{j} + M_{j})^{N_{j}+M_{j}}, exactly as indicated in the table.

The points in A_{0} have no contribution in ζ_{∞}(x), ζ_{0}(y), ζ_{∞}(y) and ζ_{∞}(xy); and they have the same contributions in ζ_{0}(x) and ζ_{0}(xy).

Similarly, in all other cases the result can be verified case by case. For this, the following remark is useful. Look at the equation t_{j}^{N_{j}}a_{j} = re^{2πiα}. If N_{j} > 0, then this gives 1 − λ^{N_{j}} in ζ_{0}(x) and 1 in ζ_{∞}. If N_{j} < 0 then the contributions are inverse. If N_{j} = 0 then both contributions are trivial. □

3. The proof of the main theorem and examples.

1. Consider the map ϕ = (f_{1}, f_{2}) : Y → C^{∗} ×C^{∗}. Then f_{i} = pr_{i} ∘ϕ (i = 1, 2), where x = pr_{1} and y = pr_{2} are the projections. Let G^{•} = Rϕ_{!}F. Then the definition of the invariants a and b can be extended to the case of complexes, and we have the identities a(g, G^{•}) = a(g ∘ ϕ, F) and b(g, G^{•}) = b(g ∘ ϕ, F) for any function g : C^{∗} × C^{∗}. Therefore, we can assume that Y = C^{∗} × C^{∗} and f_{1} = x and f_{2} = y. Moreover, by the additivity of the invariants a and b, the complex G^{•} can be replaced by only one constructible sheaf, which in the sequel will be denoted again by F. We will use the following notations:

A(F) = \frac{a(x, F)a(y, F)}{a(xy, F)} and B(F) = \frac{b(x, F)b(y, F)}{b(xy, F)}.

2. Assume that suppF is zero–dimensional. Then a(x)a(y)/a(xy) = b(x) = b(y) = b(xy) = 1, hence the result follows.

3. Assume that there exists an irreducible curve X ↪^{i} P^{2} such that F = i_{∗}(C_{X})|Y. If X^{∗} := X ∩Y is not of the form {g = constant}, where g = x, y or xy, then the result follows by section 2. If X^{∗} is of the exceptional type {g = cons.} mentioned above, then the result follows by an easy verification.

4. Assume that suppF is one–dimensional. Let X_{1}^{∗}∪⋯∪X_{i}^{∗} be the irreducible decomposition of suppF. Using the second step (and the additivity of our invariants), we can assume that the restriction of F on the intersection points of the irreducible components is zero. Now, it is easy to verify that a(f, F) = ∏_{j} a(f, F|X_{j}^{∗}) (and similarly for b) for any f. Therefore we can assume that X^{∗} = suppF is an irreducible curve.

Now, again using step 2, we can assume that Q ↦ dim_{C} F_{Q} is constant on X^{∗}. Denote this common dimension by d. It is easy to verify that a(f, F) = a(f, C_{X^{∗}})^{d}. In particular

A(F) = [ A(C_{X^{∗}}) ]^{d}.

If we establish a similar relation for the B–invariant, then this case follows from step 3.

The invariant B(F) is a product of local contributions corresponding to the points at infinity X − X^{∗}. We make the verification explicit in the case of Q = [0 : 0 : 1]. In all the other points the verification is similar, and is left to the reader. Let Z_{j} be a local component as above, and consider the space–germ (Z_{j}, Q). Then F is flat on Z_{j}^{∗} := Z_{j} − Q (in a neighbourhood of Q), hence it is determined by a representation ρ_{j} : Z = π_{1}(Z_{j}^{∗}) → AutC^{d}, i.e. by ρ_{j}(1) = A ∈ AutC^{d}. Then this point contributes in ζ_{0}(x, F) with det(I − λ^{N_{j}}A), in ζ_{0}(y, F) with det(I − λ^{M_{j}}A), and in ζ_{0}(xy, F) with det(I − λ^{N_{j}+M_{j}}A) (and it has no contribution in ζ_{∞}’s).

Write det(I − λA) = c · ∏_{k}(I − λ^{k})^{α_{k}}. Then

\frac{det(I − λ^{N_{j}}A) det(I − λ^{M_{j}}A)}{det(I − λ^{N_{j}+M_{j}}A)} = c · ∏_{k} \frac{(1 − λ^{kN_{j}})^{α_{k}}(1 − λ^{kM_{j}})^{α_{k}}}{(1 − λ^{k(N_{j}+M_{j})})^{α_{k}}} .

Its contribution in B(F) is

∏_{k}[ \frac{(kN_{j})^{kN_{j}}(kM_{j})^{kM_{j}}}{[k(N_{j} + M_{j})]^{k(N_{j}+M_{j})}} ]^{α_{k}} = [ \frac{N_{j}^{N_{j}}M_{j}^{M_{j}}}{(N_{j} + M_{j})^{N_{j}+M_{j}}} ]^{∑ kα_{k}}.

But ∑_{k} kα_{k} = d, therefore B(F) = B(C_{X^{∗}})^{d}. This shows that the case when suppF is one–dimensional (even if the corresponding representation is very complicated) can be reduced to the constant sheaf case. But this case is proved in the third step.

5. Consider now an arbitrary constructible sheaf on Y = C^{∗} × C^{∗}. Then there is a stratification of Y such that F is flat on each stratum. Using step 2 and step 4, we can assume that F restricted to any zero and one dimensional stratum is zero. In particular, there exists a curve X^{∗} (maybe not irreducible) such that F|X^{∗} = 0 and F|(Y − X^{∗}) is a flat bundle. Let d be the dimension of its fiber.

Now, by a Mayer–Vietoris argument: A(F) = A(i_{∗}C_{X^{∗}})^{−d}. We wish to establish a similar relation: B(F) = B(i_{∗}C_{X^{∗}})^{−d}. Since a factor of type (1 − λ)^{α} in a zeta–function has no contribution in the b–invariants, therefore by a Mayer–Vietoris argument, B(F) can be localized in the neighbourhood of the points X −X^{∗} (where X is the closure of X^{∗}). But in these points, any zeta function is a product of some zeta functions which are associated with constructible sheafs with one dimensional support ([6]; see the precise argument belove). Therefore, by similar computation as in the one–dimensional case (step 4), we obtain the wanted relation.

We present the explicit argument in the case when Q = [0 : 0 : 1]. Assume that Q ∈ X − X^{∗}. The local situation is the following. We have a constructible sheaf F defined in (C^{2}, Q) with discriminant X. This means that F on the local complement of the curve germ X is flat. Moreover (by step 4), we can assume that F|X = 0. Consider the embedded resolution ρ : (U, E) → (C^{2}, Q) of the curve–germ X. Let X′ be the strict transform of X ∪{xy = 0}. For each exceptional divisor E_{i} construct a smooth transversal curve S_{i} in U, and let C_{i} = ρ(S_{i}). Let E_{i}^{0} = E_{i} −(∪_{j≠i}E_{i} ∪X′). Let f = x, y or xy. We are interested in the zeta function ζ_{0}(f, F) of the monodromy M_{0}^{∗} defined on H^{∗}(f^{−1}(t) ∩ [(C^{2}, Q) − {xy = 0}], F), where f^{−1}(t) is the local Milnor fiber. Then for any of the functions f = x, y or xy one has the following “reduction formula” [6]:

ζ_{0}(f, F) = ∏_{E_{i}} ζ(f|C_{i}, F|C_{i})^{χ(E_{i}^{0})}.

Since the zeta functions can be computed as a product of zeta functions associated with “one–dimensional” objects, the same computation works as in step 4.

This ends the proof of the multiplicativity theorem. □ Example 1. Let X^{∗} = {xy − 1 − x^{2} = 0} ⊂ C^{∗} × C^{∗}, and F = C_{X^{∗}}. Then C(x) = {−i, +i} and a(x) = −b(x) = 1. The invariants of y are: C(y) = {−2, 2}, a(y) = −4, ζ_{0}(y) = ζ_{∞}(y) = (1−λ)^{2}, hence b(y) = 1. On the other hand, C(xy) = {1}, a(xy) = 1, ζ_{0}(xy) = (1−λ)^{2}, but ζ_{∞}(xy) = 1−λ^{2}. Therefore b(xy) = (−2)^{−2} = 1/4.

Notice the importance of the sign in the definition of b_{∞}. This sign correction can be explained as follows. It is clear that the theorem is a residue–type result, in particular in a local parametrization of type t_{j} ↦ x(t_{j}) = t_{j}^{N_{j}}a_{j}(t_{j}) the order ord_{t_{j}}x = N_{j} is crucial. On the other hand, the zeta function recovers only 1 − λ^{|N_{j}|}, i.e. only the absolute value |N_{j}| of the order. Therefore, in case N_{j} < 0, we have to make a sign correction if we want to recover the good object, (i.e. the order) from the zeta function ζ_{∞}. Example 2. Let X^{∗} = {x^{3} = (y − u^{3})^{2}} ⊂ C^{∗} × C^{∗} and F as above. Here u ∈ C^{∗} is a free parameter. Then a(x) = u^{6}, a(y) = u^{9} and a(xy) = \frac{3^{3}2^{2}}{5^{5}} u^{15}. (It is instructive to make this verification). The zeta functions are: ζ_{0}(x) = 1 − λ^{2}, ζ_{0}(y) = (1 − λ)^{3}, ζ_{0}(xy) = (1 − λ)^{3}(1 − λ^{2}), ζ_{∞}(x) = 1 − λ^{2}, ζ_{∞}(y) = 1 − λ^{3}, and ζ_{∞}(xy) = 1 − λ^{5}. Notice that the final results A = 5^{5}/2^{2}3^{3} = B come out in two completely different ways. (This gives a very mysterious flavour of the formula.)

Notice also that if we have some parameters in the definition of the functions f_{i} or in the definition of the sheaf (see the above example) then the invariants a(f_{i}, F) can depend strongly on the parameters. But, by our theorem, A(F) has to be discrete and all the variations with respect to the parameters are cancelled. The author knows no other argument which would imply this fact.

4. The proof of Proposition 1.

Let G be a smooth elliptic curve and fix a point P_{0} ∈ G on G. This defines a group structure on (G, P_{0}) with P_{0} as its neutral element [2, p. 321]. In this section it is more convenient to use the additive notation ⊕ for the group structure: P ⊕ Q = R if and only if P + Q = R + P_{0} in Div(G).

We have to prove that a(f_{1} ⊕ f_{2}, F) = a(f_{1}, F) ⊕ a(f_{2}, F).

By similar arguments as in section 3, we can assume that Y ⊂ G × G is a projective irreducible curve, F = C_{Y}, and f_{1} (respectively f_{2}) is induced by the first (respectively, the second) projection G × G → G. Let π : S → Y be the normalization map of Y, and K_{S}, respectively K_{G} the canonical divisor of S, respectively of G.

Let f : Y → G be a finite morphism. Since for arbitrary two points Q_{1} and Q_{2} one has: Q_{1} ∼ Q_{2} in Div(G) if and only if Q_{1} = Q_{2}, the invariant a(f) = ⊕_{t∈C(f)}Δχ(t) · t is determined by the bifurcation divisor B_{f} := ∑_{t∈C(f)} Δχ(t) · t ∈ Div(G): a(f) is the unique point in G with the property

(1) a(f) ∼ B_{f} − (deg B_{f} − 1)P_{0} in Div(G).

Let Sing(Y) = {P_{1}, …, P_{s}} be the (reduced) singular locus of Y, and let (r_{i} + 1) be the number of local irreducible components of Y at P_{i} (i = 1, …, s). Define the divisor S(Y) ∈ Div(Y) by ∑_{i=1}^{s} r_{i}P_{i}. Then, if f̃ = f ∘ π then, by [2, p. 301], the ramification divisor Rf̃ of f̃is Rf̃ ∼ K_{S}−f̃^{∗}K_{G} ∼ K_{S} (because K_{G} ∼ 0).

Consider the morphism f̃_{∗} : Div(S) → Div(G) defined by f̃_{∗}(∑ n_{i}Q_{i}) = ∑ n_{i}f̃(Q_{i}), and similarly f_{∗}(S(Y)) := ∑_{i=1}^{s} r_{i}f(P_{i}). Then:

B_{f} ∼ f̃_{∗}(Rf̃) + f_{∗}(S(Y)) ∼ f̃_{∗}(K_{S}) + f_{∗}(S(Y)) in Div(G).

By a calculation: B_{f_{1}⊕f_{2}} ∼ B_{f_{1}} +B_{f_{2}} −(deg K_{S} +∑_{i=1}^{s} r_{i})P_{0}, and deg B_{f_{1}⊕f_{2}} = deg B_{f_{1}} = deg B_{f_{2}} = deg K_{S} + ∑_{i=1}^{s} r_{i}. Now, from (1), one has: a(f_{2} ⊕ f_{2}) + P_{0} ∼ a(f_{1}) + a(f_{2}), hence a(f_{1} ⊕ f_{2}) = a(f_{1}) ⊕ a(f_{2}).

References

[1] P. Griffiths and J. Harris: Principles of Algebraic Geometry, A Wiley– Interscience Publication, 1978.

[2] R. Hartshorne: Algebraic Geometry, Graduate Text in Mathematics 52, Springer–Verlag, 1977.

[3] G. Laumon: Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Pub. Math. I. H. E. S., 65, 131-210, 1987.

[4] F. Loeser: Déterminants et Faisceaux de Kummer, unpublished.

[5] F. Loeser and C. Sabbah.: Equations aux différences finies et déterminants d’intégrales de fonctions multiformes, Comm. Math. Helvetici. 66, 458-503, 1991.

[6] A. Némethi.: The zeta function of singularities, Journal of Algebraic Geometry, 2, 1-23, 1993.

Institute of the Romanian Academy, Bucharest, and The Ohio State University, Columbus, 231 West 18th Avenue, OH 43210, USA E-mail: nemethi@math.ohio-state.edu