Revista Colombiana de Matemáticas Volumen 31 (1997), páginas 115–124

Zeta functions of singular curves over finite fields

W. A. Zúñiga-Galindo

Universidad Autónoma de Bucaramanga Bucaramanga, COLOMBIA

Abstract. Let X be a complete, geometrically irreducible, algebraic curve defined over a finite field 𝔽_{q} and let ζ(X, t) be its zeta function [Ser1]. If X is a singular curve, two other zeta functions exist. The first is the Dirichlet series Z(Ca(X), t) associated to the effective Cartier divisors on X; the second is the Dirichlet series Z(Div(X), t) associated to the effective divisors on X. In this paper we generalize F. K. Schmidt’s results on the rationality and functional equation of the zeta function ζ(X, t) of a non-singular curve to the functions Z(Ca(X), t) and Z(Div(X), t) by means of the singular Riemann-Roch theorem. Key words and phrases. Zeta functions, finite fields, singular curves, generalized Jacobians, compactified Jacobians. 1991 AMS Mathematics Subject Classification. Primary 14H25. Secondary 14H05.

In [G] Galkin introduced a zeta function for orders in global fields which coincides with Schmidt’s zeta function in the case of a non-singular curve but satisfies a functional equation only in the Gorenstein case. Green [Gr] obtained a functional equation by slightly modifying the definition of the zeta function of Galkin. However, this new zeta function is not uniquely determined by the curve. Finally, Stöhr [Sto] defines the zeta function Z(Div(X), t) which coincides with the zeta function of Galkin in the Gorenstein case and always satisfies a functional equation ([Sto, p. 133]). In [Z] we studied the local factors of the zeta function Z(Ca(X), t) using an adelic approach a lá Tate. As a consequence we obtained the rationality and a global functional equation of Z(Ca(X), t)([Z, thm. 5.3]). In this paper we generalize F. K. Schmidt’s results on the rationality and functional equation of the zeta function ζ(X, t) of a non-singular curve to the zeta functions Z(Ca(X), t) and Z(Div(X), t) using an approach based on the singular Riemann-Roch theorem. This allows us to find the residue at s = 0 of the zeta function Z(Ca(X), q^{−s}) and Z(Div(X), t) (see thms. 2.1, 2.2, 3.1).

1. Preliminaries

In this section we present the basic facts about singular curves and zeta functions (cf. e.g. [Ros1], [Ros2], [Ser2], [Sto]).

Let X be a complete, geometrically irreducible, algebraic curve defined over a field k. We denote by K = k(X) the function field of X over k, g will stand for its arithmetic genus, g̃ for its geometric genus, X̃ will be the normalization of X over k (also named the non-singular model of X) and π : X̃ ⟶ X will denote the normalization map. The regular surjective map π is birational. In particular, the function field of the smooth curve X̃ is K/k.

By a divisor of X we mean a coherent fractional ideal sheaf or, equivalently, a formal product

A = ∏_{P ∈X} A_{P} ,

where for each point P the P -component A_{P} (i.e. the stalk of A at P) is a non-zero fractional ideal of O_{P} and A_{P} = O_{P} for all but finitely many points. Given two divisors A and B, we define pointwise the product divisor AB and the quotient divisor A : B by

(AB)_{P} := A_{P} B_{P} ,

and

(A : B)_{P} := {z ∈ K | zB_{P} ⊆ A_{P} }.

We denote by Div(X) the set of divisors of X. A divisor A is called a locally principal or a Cartier divisor if each A_{P} is a principal fractional ideal. The Cartier divisors form a multiplicative group having the structure divisor

O := ∏_{P ∈X} O_{P} ,

as the identity. We denote by Ca(X) the group of the Cartier divisors. We define a partial order on Div(X) by

A ⩽ B ⟺ A_{P} ⊆ B_{P} for all P ∈ X.

A divisor A is called effective if A ≥ O. For our purposes, it is more convenient to work with the above ordering than with the usual one.

The degree of a divisor is uniquely defined by the following properties:

(i) deg(O) = 0.

(ii) deg(B) − deg(A) = ∑_{P ∈X} dim_{k}(B_{P} /A_{P}) whenever B ≥ A. We observe that in general deg(AB) ≠ deg(A) deg(B) (cf. [Ha, sect. 1]). However, if at least one of the divisors A or B is a Cartier divisor then the equality holds.

For each non-zero rational function z ∈ K^{∗}, let div(z) be its principal divisor, i.e.,

div(z) := ∏_{P ∈X} z^{−1}O_{P} .

We denote by Prin (X) the subgroup of principal divisors of X.

Let

L(A) := ⋂_{P ∈X} A_{P} = {z ∈ K | div(z)A ≧ O}

be the k−vector space of global sections of A (also denoted by H^{0}(X, A)). We denote the dimension of the above k−vector space by ℓ(A) (also denoted by h^{0}(X, A)).

The Riemann-Roch theorem for function fields was generalized by Rosenlicht to curves with singularities (cf. e.g. [Ros1], [Sto]).

Theorem 1.1. (Riemann-Roch theorem for singular curves). Each divisor A of X satisfies

ℓ(A) = deg(A) + 1 − g + ℓ(C : A),

where C denotes the canonical divisor of X.

The local duality theorem also generalizes to singular curves (cf. e.g. [Sto, thm. 1.5])

Theorem 1.2. (Local duality). Let A, B be divisors of X such that A ≧ B. Then for each point P we have the k-isomorphism

(C_{P} : B_{P})/(C_{P} : A_{P}) ⟶_{≅} Hom_{k}(A_{P} /B_{P} , k).

As a consequence of the theorem of local duality, we get the reciprocity formula (cf. e.g. [Sto, sect. 1.7]).

Corollary 1.3. (Reciprocity formula). For each divisor A,

C : (C : A) = A.

From now on, we understand point to mean closed point. Let P be a point of X and O_{P} be the local ring of X at P. Let Q_{1}, Q_{2}, …, Q_{d} be the points of X̃ lying over P, i.e., π^{−1}(P) = {Q_{1}, …, Q_{d}}, and let O_{Q_{1}}, …, O_{Q_{d}} be the corresponding local rings at these points. Since the function fields of X and X̃ are the same and X̃ is a non-singular curve, the local rings O_{Q_{1}}, …, O_{Q_{d}} are valuation rings of K/k over O_{P}. The integral closure of O_{P} in K is Õ_{P} = ⋂_{Q∈π^{−1}(P)} O_{Q}.

The degree of singularity of X at P is defined as

δ_{P} = dim_{k}(Õ_{P} /O_{P}).

By Theorem. 1 in [Ros1], δ_{P} < ∞. The total degree of singularity of X is defined as

δ = ∑_{P ∈X} δ_{P} .

The degree of singularity δ_{P} remains invariant under completion. The total degree of singularity δ remains invariant under separable constant extensions (cf. [Ros1, thm. 12]).

We recall that the genus formula of a complete, geometrically irreducible, algebraic curve X is (see [H])

g = g̃ + ∑_{P ∈X} δ_{P} ,

where g̃ is the geometric genus of X.

The conductor ideal F_{P} of Õ_{P} in O_{P} is defined as

F_{P} = {x ∈ K | xÕ_{P} ⊆ O_{P} }.

This ideal is the largest common ideal of Õ_{P} and O_{P}. Furthermore, since δ_{P} < ∞, then F_{P} ≠ 0. On the other hand, Õ_{P} is a Dedekind domain with a finite number of maximal ideals. Thus, F_{P} is an Õ_{P} -principal ideal. The degree of the conductor ideal F_{P} is defined as

deg F_{P} := dim_{k}(Õ_{P} /F_{P}).

The degree of F_{P} is also invariant under completions and under separable constant extensions.

We say that a local ring O_{P} is Gorenstein if deg F_{P} = 2δ_{P}. An algebraic curve is called a Gorenstein curve if all its local rings are Gorenstein.

The dualizing sheaf ω_{X} is locally free of rank 1 if and only if X is a Gorenstein curve (cf. [A-K, chap. VIII, prop. 1.16]). We denote by C the canonical divisor of a complete, geometrically irreducible, algebraic curve X. Then C_{P} is a principal ideal if and only if ω_{X,P} is free of rank 1. Summarizing, we have the following result of Rosenlicht.

Theorem 1.4. Let X be a complete, geometrically irreducible, algebraic curve defined over k. The curve X is Gorenstein if and only if its canonical divisor is a Cartier divisor.

We denote by Ca^{0}(X) the subgroup of degree zero divisors of Ca(X), by Pic (X) the quotient group Ca(X)/Prin (X) and by Pic^{0}(X) the quotient group

Ca^{0}(X)/Prin (X).

Let k̄ be an algebraic closure of k. Pic^{0}(X ⨂_{k} k̄) is the generalized Jacobian of X ⨂_{k} k̄ (see [Ser2], [Ros2]).If k is a finite field Pic^{0}(X̃) is the group of rational points of Pic^{0}(X ⨂_{k} k̄). The generalized Jacobian can be enlarged to a projective algebraic scheme, called the compactified Jacobian, which parametrizes the equivalence classes of zero degree divisors of X ⨂_{k} k̄. The rational points of the compactified Jacobian correspond bijectively to equivalence classes of zero degree divisors of X.

2. Rationality and functional equation of Z(Ca(X),t)

In this section we present a generalization of results of F. K. Schmidt on the rationality and on the functional equation of ζ(X, t) to the function Z(Ca(X), t) (cf. [Sti, chap. V]). The zeta function Z(Ca(X), t) is defined as follows:

Z(Ca(X), t) := ∑_{A≥O} t^{deg(A)},

where A runs through all Cartier divisors on X and t = q^{−s}, s ∈ ℂ, Re(s) > 0. We shall see later on that Z(Ca(X), t) converges analytically and uniformly on the semiplane Re(s) > 0. This zeta function decomposes formally into an Euler product

Z(Ca(X), t) = ∏_{P ∈X} Z(O_{P} , t) = ∏_{P ∈X} ( ∑_{I⊇O_{P}} t^{dim_{k}(I/O_{P})}) ,  (2.1)

and where I runs through all fractional principal ideals of O_{P}, such that I ⫆ O_{P}. By comparing product (2.1) with ζ(X̃, t) = Z(Ca(X̃), t), we see that it converges on the semiplane Re(s) > 0. On the other side, the correspondence I ⟶ I^{−1} is a bijection between the principal fractional ideals containing O_{P} and the principal ideals contained in O_{P}. Furthermore, this correspondence preserves degrees, therefore, we have that

Z(O_{P} , t) = ∑_{I⊆O_{P}} t^{dim_{k}(O_{P} /I)}.

We observe that if P is a smooth point of X, then Z(O_{P} , t) = (1 − t^{deg(P)})^{−1}, where deg(P) is the cardinality of the residue field of P.

Theorem 2.1. Let X be a complete, geometrically irreducible algebraic curve over a finite field k = 𝔽_{q}. Then the zeta function Z(Ca(X), t) is a rational function. More precisely,

Z(Ca(X), t) = \frac{L(Ca(X), t)}{(1 − t)(1 − qt)},

where L(Ca(X), t) ∈ ℤ[t] is a polynomial of degree at most 2g, and L(Ca(X), 1) = #Pic^{0}(X), i.e. L(Ca(X), 1) is the number of rational points of the generalized Jacobian of X ⨂_{k} k̄.

Proof. In order to generalize the proof of F.K. Schmidt we need to prove the following three claims.

Claim 1.

#Pic^{0}(X) < ∞.  (2.2)

Claim 2. There exists a Cartier divisor of degree 1. Claim 3. For any integer d, the number of divisor classes in Pic (X) of degree d is independent of d and is equal to the cardinality of Pic^{0}(X).

After this, we can follow the argument of F.K. Schmidt as in [Sti, chap. V]. It is important to note that in this argument it is irrelevant to know whether a canonical divisor is a Cartier divisor or not. We denote by [A] the linear equivalence class of a divisor A and by π^{∗}(A) the pullback of A.

To establish claim 1, we first observe that Pic^{0}(X̃) is a finite group (more precisely, the group of divisor classes of degree zero of K(cf. [Sti, chap. V]), so, it is sufficient to show that the kernel of the morphism

π_{0}^{∗} : Pic^{0}(X) ⟶ Pic^{0}(X̃)
π_{0}^{∗}([A]) ⟼ [π^{∗}(A)],

is a finite group. We note that [A] ∈ ker(π_{0}^{∗}) if and only if π^{∗}(A) is a principal divisor of X̃, i.e.

π^{∗}(A)_{Q} = z^{−1}O_{Q} = A_{P} O_{Q},  (2.3)

for some z ∈ K^{∗} and every point Q of X̃ lying over P. Therefore

F_{P} ⫅ zA_{P} ⫅ Õ_{P} .

Since deg F_{P} < ∞, the above relation implies that the kernel of π_{0}^{∗} contains only a finite number of linear equivalence classes. The equivalence class of an effective Cartier divisor A contains \frac{q^{ℓ(A)−1}}{q−1} linearly equivalent Cartier divisors. Therefore the kernel of π_{0}^{∗} is a finite group.

The second claim can be reduced to the non-singular case using the fact that the morphism π^{∗} : Ca(X) ⟶ Ca(X̃) is surjective and preserves degrees. The preservation of the degree of a Cartier divisor under π^{∗} follows from the fact that dim_{k} O_{P} /hO_{P} = dim_{k} Õ_{P} /hÕ_{P}, for every non-zero rational function h of X. Thus, there exists a Cartier divisor of degree 1. We observe that claim 1 and claim 2 imply that the number of effective Cartier divisors with a given degree is finite. Using the same argument as in the nonsingular case, we prove that Z(Ca(X), t) converges absolutely and uniformly on the semiplane Re(s) > 0 (cf. [Sti, chap. V, prop. 1.6]). The last claim follows from claim 2 and the fact that deg(AB) = deg(A) + deg(B), for any two Cartier divisors A and B (see [Ha, sect. 1]). □ ✓

Corollary 2.2. The quotient \frac{Z(O_{P} ,t)}{Z(Õ_{P} ,t)} is a rational function of t.

Proof. By taking a partial resolution of singularities of X, we may assume that P is the only singular point of X. Then

\frac{Z(Ca(X), t)}{Z(Ca(X̃), t)} = ∏_{Q∈π^{−1}(P)} (1 − t^{deg(Q)}) Z(O_{P} , t) = \frac{Z(O_{P} , t)}{Z(Õ_{P} , t)}.

The result thus follows from the previous theorem. □ ✓

Theorem 2.3. Let X be a complete, geometrically irreducible, algebraic curve defined over a finite field k = 𝔽_{q}. Then the zeta function Z(Ca(X), t) satisfies the functional equation

Z(Ca(X), t) = q^{g−1}t^{2g−2}Z(Ca(X), \frac{1}{qt})  (2.3)

if and only if X is a Gorenstein curve.

Proof. If X is a Gorenstein curve, the argument of Schmidt for the non-singular case, the reciprocity (Corollary 1.2), and the observations made in the proof of Lemma 2.1 imply the functional equation (2.3) (see [Sti, chap. V, prop. 1.13]). Conversely, if the zeta function Z(Ca(X), t) satisfies the functional equation (2.3) and g ≧ 1, then the Riemann-Roch theorem and (2.2) imply that

Z(Ca(X), t) = Z_{1}(X, t) + Z_{2}(X, t)

with

Z_{1}(X, t) = \frac{1}{q − 1} ∑_{0≤deg[A]≤2g−2} q^{ℓ(A)}t^{deg(A)} = \frac{1}{q − 1} ∑_{j=0}^{2g−2} a_{j}t^{j}  (2.4)

and

Z_{2}(X, t) = \frac{1}{q − 1} ∑_{deg[A]≧2g−1} q^{ℓ(A)}t^{deg(A)} − \frac{1}{q − 1} ∑_{deg[A]≧0} t^{deg(A)}

= \frac{1}{q − 1} (#Pic^{0}(X)q^{g}t^{2g−1} \frac{1}{1 − qt} − \frac{#Pic^{0}(X)}{1 − t}) .  (2.5)

By hypothesis, Z(Ca(X), t) satisfies the functional equation (2.3). A direct verification shows that the function Z_{2}(X, t) also satisfies the functional equation (2.3). Therefore, Z_{1}(X, t) satisfies the functional equation (2.3). This implies that

a_{2g−2−j} = a_{j}q^{g−1−j}, j = 0, 1, …, 2g − 2.

On the other hand, a_{0} = 1. Thus, a_{2g−2} = q^{g−1}. Since g ≥ 1, a divisor class [C] with deg(C) = 2g − 2 and ℓ(C) = g must appear in the sum in (2.4). These properties characterize the canonical class. Hence, by theorem 1.4, X is Gorenstein. In the case g = 0, the genus formula implies δ = 0, so that X is a non-singular curve. Thus, in this case, also X is Gorenstein. □ ✓

As a consequence of the functional equation (2.3), the degree of the polynomial L(Ca(X), t) is 2g.

Corollary 2.4. The zeta function Z(O_{P} , t) satisfies the functional equation

\frac{Z(O_{P} , t)}{Z(Õ_{P} , t)} = q^{δ_{P}} t^{2δ_{P}} \frac{Z(O_{P} , \frac{1}{qt})}{Z(Õ_{P} , \frac{1}{qt})}

if and only if O_{P} is Gorenstein.

3. Rationality and functional equation of Z(Div(X),t)

In this section, we study the rationality and functional equation of the zeta function Z(Div(X), t) associated to the set of divisors Div(X). This zeta function is defined as follows:

Z(Div(X), t) = ∑_{A≥O} t^{deg(A)},

where A runs through all effective divisors on X and t = q^{−s}, s ∈ ℂ, Re(s) > 0. This zeta function decomposes formally into the following Euler product

Z(Div(X), t) = ∏_{P ∈X} Z(O_{P} , t) = ∏_{P ∈X} ( ∑_{I⊇O_{P}} t^{dim_{k}(I/O_{P})}) ,

where I runs trough all O_{P} -fractional ideals containing O.

Theorem 3.1. Let X be a complete, geometrically irreducible algebraic curve over a finite field k = 𝔽_{q}. Then the zeta function Z(Ca(X), t) is a rational function. More precisely,

Z(Ca(X), t) = \frac{L(Div(X), t)}{(1 − t)(1 − qt)},

where L(Div(X), t) ∈ ℤ[t] is a polynomial of degree at most 2g, and L(Ca(X), 1) = #Cl^{0}(X), the number of classes of divisors of degree zero, i.e. L(Div(X), 1) is the number of rational points of the compactified Jacobian of X ⨂_{k} k̄.

Proof. The proof is similar to the proof of theorem 2.1. In order to prove that

#Cl^{0}(X) = Div^{0}(X)/P rin(X) < ∞,

we observe that Pic^{0}(X) acts on Cl^{0}(X) by multiplication and the quotient set is isomorphic to

{ ∏_{P ∈X_{sing}} A_{P} | A_{P} is an ideal in O_{P} and F_{P} ⊆ A_{P} ⊆ Õ_{P} }.  (3.1)

Since dim_{k}(Õ_{P} /F_{P}) < ∞ and #Pic^{0}(X) < ∞, we conclude from (3.1) that #Cl^{0}(X) < ∞. Now we can follow the proof of theorem 2.1. □ ✓

The zeta function Z(Div(X), t) satisfies the functional equation (2.3). The proof of this fact follows from the singular Riemann-Roch theorem and the reciprocity, by a similar reasoning to that of Schmidt (see [Sti, chap. V]). Stöhr defined Z(Div(X), t) and proved that Z(Div(X), t) satisfies the functional equation (2.4) (cf. [Sto, p. 133]). The local factors of Z(Div(X), t) are rational functions (as in Corollary 2.2) and satisfy a functional equation (as in Corollary 2.4). The functional equation (2.3) implies that the degree of the polynomial L(Div(X), t) is 2g.

Acknowledgments

The author wishes to thank the support of the following institutions CNPq of Brazil, COLCIENCIAS of Colombia, Academia Colombiana de Ciencias Exactas, Físicas y Naturales. The author also thanks to Prof. Karl-Otto Stöhr for all the clarifying conversations about this topic.

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(Recibido en noviembre de 1996; revisado en febrero de 1998)

Universidad Autónoma de Bucaramanga Laboratorio de Cómputo Especializado A.A. 1642, Bucaramanga, COLOMBIA

e-mail: wzuniga@bumanga.unab.edu.co