Generalized Weierstrass semigroups and their Poincar\'e series

We investigate the structure of the generalized Weierstrass semigroups at several points on a curve defined over a finite field. We present a description of these semigroups that enables us to deduce properties concerned with the arithmetical structure of divisors supported on the specified points and their corresponding Riemann-Roch spaces. This characterization allows us to show that the Poincar\'e series associated with generalized Weierstrass semigroups carry essential information to describe entirely their respective semigroups.


INTRODUCTION
Let X be a projective, non-singular, geometrically irreducible algebraic curve over a finite field F; throughout the paper we will refer to this simply as a curve. Let F(X ) be the field of rational functions of X over F. For m ≥ 1 pairwise distinct F-rational points Q 1 , ..., Q m on X , we consider the set where div ∞ ( f ) stands for the pole divisor of the rational function f ; this is in fact an additive sub-semigroup of N m 0 which we will call the classical Weierstraß semigroup of X at the m-tuple Q = (Q 1 , ..., Q m ).
The classical Weierstraß semigroup at one point is a numerical semigroup which goes back to classical works of Riemann, Weierstraß or Hurwitz. An extension to the case of several points was introduced by Arbarello et al. [1]. The arithmetical properties involved in the special case of two points was extensively investigated by Kim [16] and Homma [14] (see also their joint work [15]), and references to the general case include Ballico and Kim [2], Matthews [17], Carvalho and Torres [9], or the survey by Carvalho and Kato [8]. Specifically, carrying on the studies of [14,16], Matthews [17] introduced the notion of generating set of H(Q) that allows constructing H(Q) from the knowledge of a finite number of elements in the semigroup. Since these papers were published, many works have arisen mainly pursuing explicit descriptions of Weierstraß semigroups of specific curves in order to be applied in the analysis of algebraic-geometric codes.
Motivated by a different interpretation of Weierstraß numerical semigroups, Delgado [10] provided a generalization to the classical approach of Weierstraß semigroups at several points on curves over algebraically closed fields; Beelen and Tutas [6] considered these semigroups in the case of curves over finite fields: writing v Q i for the valuation of F(X ) associated with Q i , they considered the set of tuples ρ Q ( f ) := (−v Q 1 ( f ), . . ., −v Q m ( f )), for f ∈ F(X ) × with poles only at the points Q 1 , . . . , Q m ; this turns out to be the additive sub-semigroup of Z m where R Q denotes the ring of functions of X that are regular outside {Q 1 , . . . , Q m }, which we call the generalized Weierstraß semigroup of X at Q.
It is not difficult to see the relation between the classical and the generalized Weierstraß semigroups, provided that #F ≥ m (cf. [6,Prop. 2]): An important notion in the study of H(Q) due to Delgado [10] is that of maximal elements (see Def. 2.4) which in particular allows him to explore symmetry properties of that semigroups. The reader is referred to [3,4,5,12,18] for further information.
On the other hand, by using the combinatorial notion of Poincaré series related to a multiindex filtration, the first author [18] introduced the Poincaré series associated with generalized Weierstraß semigroups for the cases m = 1, 2; there he showed that the knowledge of the series ensures information about the maximal elements of the generalized Weierstraß semigroup.
In this paper we study the generalized Weierstraß semigroups at several points and present a generating set in the sense of [17] in terms of the maximal elements. This is related to the characterization of the classical Weierstraß semigroups stated in [17], but also carries information on the corresponding Riemann-Roch spaces of every divisor supported on the specified points; recall here that any divisor D = ∑ m i=1 µ i Q i on X defines a F-vector space L (D)-called Riemann-Roch space of D-consisting of the rational functions f with poles only at the points with µ i ≥ 0 (and, furthermore, with the pole order of f at Q i ≤ µ i ), and if µ j < 0 having a zero at Q j or order ≥ µ j .
In particular, we prove that this set of maximal elements-as well as the generating set of H(Q)-is finitely determined. In addition, we extend some results of [18] on the corresponding Poincaré series and conclude that they afford enough information to recover entirely the semigroup, allowing us to interpret these series as a combinatorial invariant of the generalized Weierstraß semigroups.
Our paper is organized as follows. In Section 2 we collect some basic results on the generalized Weierstraß semigroups and their Poincaré series. Section 3 is devoted to the study of a generating set for the generalized Weierstraß semigroup which is finitely determined; rather than providing such a characterization, we show how these elements are appropriated to describe the Riemann-Roch spaces. In Section 4 we investigate the support of the Poincaré series and introduce the semigroup polynomial, which determines the Poincaré series by a functional equation. Finally, we present functional equations for the Poincaré series associated with H(Q) satisfying a symmetry condition.

PRELIMINARIES
Let X be a (projective, non-singular, geometrically irreducible, algebraic) curve of genus g defined over a finite field F with its function field F(X ). Denote by X (F) the set of F-rational points on X . For m ≥ 2, let Q 1 , . . . , Q m ∈ X (F) be pairwise distinct. In this section we summarize the main properties of the generalized Weierstraß semigroup H(Q) defined in the Introduction. We also recall the definition of Poincaré series associated with these algebraic-geometric structures due to [18].
2.1. Generalized Weierstraß semigroups. Consider the generalized Weierstraß semigroup H(Q) of X at Q. First we want to characterize the elements of H(Q) in terms of dimensions of Riemann-Roch spaces associated. For this purpose, let us fix some helpful notation: . . , α m ) ∈ Z m and a non-empty J I, we set ⋄ For any J ⊆ I, denote by 1 J the m-tuple whose the j-th coordinate 1 if j ∈ J and 0 otherwise; for instance, 1 I = 1 is the all 1 m-tuple, 1 / 0 = 0 is the all zero m-tuple, and e i = 1 {i} .
The following result was originally proved by Delgado [10, p. 629] the local expansion of f i at Q j as a Laurent series in the indeterminate t Q j . We claim that there exists But as #F ≥ m, this choice of (b 1 , . . ., b m ) ∈ F m can always be done.
We next recall the notion of maximality due to Delgado in [10]. As we shall see, this concept translates arithmetical properties of Riemann-Roch spaces L (α) into subsets of Z m related to α. Notice that when m = 2, every maximal element is also absolute maximal.
Remark 2.5. By a Z-valued formal series (or a formal distributions) in the variables t 1 , . . .,t m , we mean a formal expressions of type S(t) = ∑ α∈Z m s(α)t α for s(α) ∈ Z. When s(α) = 0 for all but finitely many α ∈ Z m , we refer to S(t) as a Laurent polynomial. The support of S(t) is the set {α ∈ Z m : s(α) = 0}. The set of Z-valued formal series has a polynomial-like Z-module structure in the sense that addition and scalar multiplication are performed like for polynomials. However, the usual multiplication rule can not be used since it does not make sense in general. There, with the usual multiplication rule, the set of Z-valued formal series is no longer a ring, but this operation gives us it is a module over the ring of Laurent polynomials over Z. For further details on these objects, we refer the reader to [13].
Observe that when m = 1, L(t) is just a formal power series. It codifies the elements of the Weierstraß numerical semigroup H(Q 1 ) = H(Q 1 ), since α ∈ H(Q 1 ) if and only if d(α) = 1. We thus have However, this does not remain true for several points, i.e., there are elements outside H(Q) appearing in the support of L(t); cf. the so-called gaps in [9]. It thus motivates a more convenient definition of formal series related to H(Q).
In order to make a precise description of Poincaré series P(t) associated with generalized Weierstraß semigroups, we consider some auxiliary formal series that shall enable us to explore some computational aspects of P(t). Let Writing Q(t) = ∑ α∈Z m q(α) · t α , its coefficients q(α) are given exactly by where P(I) denotes the power set of I.
For each i ∈ I, let us also consider the formal series Lemma 2.6. Let α ∈ Z m and i ∈ I. The formal series P i (t) does not depend on i.
By (2), we can deduce that Definition 2.7. The Poincaré series associated with H(Q) is defined as the multivariate formal series P i (t) in t 1 , . . . ,t m . It will be denoted by P(t).
Observe that, writing P(t) = ∑ α∈Z m p(α)t α , we have an expression for the coefficients The following result connects the formal series Q(t) and P(t). It is the analogue of that established in [11,Prop. 8] in our context, and their proofs run very closely with minor adjustments that for completeness we shall indicate below.
Proposition 2.8. The Poincaré series associated with H(Q) satisfies the following functional equation Proof. According to the definition of P(t), it is sufficient to prove that p i (α)− p i (α −1) = q(α) for some i ∈ I. First notice that for any reordering {i 1 , . . . , i m−1 } of I\{i} and any β ∈ Z m we can write By (2), we have and, by taking (4), we obtain which equals to we thus obtain which completes the proof by noticing that P(I) = P(I\{i})∪{J ∪{i} : J ∈ P(I\{i})} in Eq. (1).
In [18,Sec. 3.2], Moyano-Fernández specializes the two point case Q = (Q 1 , Q 2 ) and obtains that the related Poincaré series is expressed simply as (5) where M (Q) is the set of maximal elements of H(Q). In Section 4 we shall investigate the support of the Poincaré series in the general case.

GENERALIZED WEIERSTRASS SEMIGROUPS AT SEVERAL POINTS
Throughout this section X will be a curve over a finite field F and Q = (Q 1 , ..., Q m ) will be an m-tuple of pairwise distinct F-rational points on X . Assume that #F ≥ m ≥ 2. We will provide a generating set of generalized Weierstraß semigroups at several points which extends the description of Matthews [17] to these objects. We furthermore generalize some results of Beelen and Tutas [6] to prove the finiteness determination of the generators.
It follows from [6, Prop. 14 (i)] that which yields the equality these sets in (6) coincide exactly with the set of maximal elements M (Q) of H(Q) (cf. Definition 2.4). Furthermore, adapting ideas from [16,14], the maximal elements M (Q) of H(Q) provide a generating set for H(Q) in the sense that The next example elucidates the role of the absolute maximal elements in the description of a generalized Weierstraß semigroup at a pair of points.
Example 3.1. This example illustrates the maximal elements of a generalized Weierstraß semigroup of the Hermitian curve given by the affine equation x 4 = y 3 + y over F 9 at pair (Q, P), where Q is the point at infinity and P = (0 : 0 : 1). In Figure 1, the maximal elements are represented by the circles "•", whereas the remaining elements of H(Q, P) are represented by the filled circles "•".
In the remaining of this subsection, we investigate the relationship between maximal elements and generating sets for the generalized Weierstraß semigroups at several points, taking account the characterization from Eq. (7) in the case of two points.
The following proposition states equivalences concerning the absolute maximal property.
Proposition 3.2. Let α ∈ H(Q). Then the following statements are equivalent: Proof. Let us first prove that (i) implies (ii). Since α ∈ H(Q), we get {α} ⊆ ∇ m i (α) for all i ∈ I. If β ∈ ∇ m j (α) with β = α for some j ∈ I, then there exists a subset J I containing j such that β ∈ ∇ J (α), contradicting the hypothesis. Since (ii) immediately implies (iii), let us assume (iii). To prove (iv) it is sufficient to show that ℓ(α − 1 J c ) = ℓ(α − 1 J c ∪{i} ) for every subset J I containing i, which is equivalent to ∇ m i (α − 1 J c ) = / 0 by Proposition 2.1 (2). But, since which gives us a contradiction. This completes the proof. We next prove a characterization of the generalized Weierstraß semigroups through least upper bounds of their absolute maximal elements. This description is analogous to that afforded by Matthews in [17] for classical Weierstraß semigroups at several points. Proof. To begin with, observe that lub(β 1 , β 2 ) ∈ H(Q) for β 1 , β 2 ∈ Γ(Q) ⊆ H(Q), since it is always possible to find (b 1 , b 2 ) ∈ F 2 outside the union of at most m one-dimensional linear spaces thanks to #F ≥ m; therefore we may set h : . This argument may be extended to show that lub(β 1 , . . . , β m ) ∈ H(Q) for β 1 , . . . , β m ∈ Γ(Q).
As a consequence, the generalized Weierstraß semigroups are entirely determined by their absolute maximal elements. In what follows, we present the outcomes of this property which seem-at first glance-to justify why this characterization is appropriated to the study of these objects. To be precise, we analyze the relationship between the Riemann-Roch spaces of divisors supported on subsets of {Q 1 , . . ., Q m } and the absolute maximal elements of generalized Weierstraß semigroups at Q.
For i ∈ I, define on Γ(α) the relation Notice that ≡ i is an equivalence relation on Γ(α). Denote by Γ(α)/≡ i the set of equivalence classes [β ] i for β ∈ Γ(α). In our next theorem, we formulate a characterization of the dimensions ℓ(α) in terms of absolute maximal elements in Γ(α).
Proof. We first observe that the conditions imposed on β ∈ Γ(Q) by β ≤ α and |β |≥ 0 imply that Γ(α) is finite. Note also that the dimension ℓ(α) is precisely the number of proper inclusions in the filtration It is equivalent to the existence of an absolute maximal element β ∈ H(Q) with β i = α i − j and β ≤ α − je i ≤ α, which, according to Remark 3.3, is a minimal element with respect to ≤ in the set {γ ∈ H(Q) : γ i = α i − j}.
for a choice of representative classes, and denote, by convenient abuse of notation, ρ −1 . . , ℓ(α). Refining the connection established above, we next describe how the absolute maximal elements of H(Q) carry intrinsic information on R Q as a F-vector space.
Corollary 3.6. Let α ∈ Z m . The set ρ −1 Q (Γ(α)/≡ i ) is a basis for the Riemann-Roch space L (α). In particular, the ring of functions of X that are regular outside Q 1 , . . ., Q m is spanned by ρ −1 Q (Γ(Q)) as an infinite-dimensional F-vector space.

3.2.
Determining generating sets. Although the generalized Weierstraß semigroups have the description as in Theorem 3.4, the computation of all absolute maximal elements is not an easy task since the set formed by them is infinite. In this way, we would like to know whether the set Γ(Q) of absolute maximal elements can be finitely determined. For this purpose, let us start this discussion with the case m = 2. Here the characterization of elements of M (Q) as in (6) is essential since by [6,Prop. 14 (vii)], the function σ 2 has the periodical property (8) where a is the smallest positive integer t such that (t, −t) ∈ H(Q). Hence, it is possible to determine M (Q) from the knowledge of both a and the finite set For the general case we can consider a similar approach: for i = 1, . . ., m − 1, let a i be the smallest positive integer t such that t(Q i − Q i+1 ) is a principal divisor on X , and denote by η i ∈ Z m the m-tuple whose j-th coordinate is Notice that the existence of these a i 's is guaranteed by the finiteness of the divisor class group. Consider the region As noticed in [6,Sec. 2], the functions f ∈ R × Q are such that the support of div( f ) is a subset of the set {Q 1 , . . ., Q m } and their image under ρ Q form a lattice in the hyperplane {α ∈ R m : |α|= 0}. Denote by Θ(Q) its sublattice generated by the elements η 1 , . . . , η m−1 .
Similarly to the properties (8) and (9) satisfied by pairs, we can establish the following general statement on maximal and absolute maximal elements in generalized Weierstraß semigroups.
We close this section by exploring the consequences of Theorem 3.7 for the Riemann-Roch spaces associated with divisors whose support is contained in {Q 1 , . . ., Q m }. To this end, let us define for α, α ′ ∈ Z m the relation (11) α ≡ α ′ if and only if α − α ′ ∈ Θ(Q).
Observe that the foregoing relation is an equivalence relation in Z m because Θ(Q) is a lattice in {α ∈ R m : |α|= 0}. Writing [α] for the equivalence class of α for ≡, we can state the following property of dimensions in these equivalence classes.
Since Γ(Q) ∩ C is finite, let β 1 , . . ., β c be its elements. For i = 1, . . ., c, let f i ∈ R Q be a function such that ρ Q ( f i ) = β i . Analogously, let g i ∈ R Q be a function such that ρ Q (g i ) = η i for i = 1, . . ., m − 1. We can thus state the following concerning such functions.
Corollary 3.9. The ring R Q of functions of X that are regular outside Q 1 , . . . , Q m is spanned as an F-vector space by the set of functions Proof. According to Corollary 3.6, ρ −1 Q (Γ(Q)) spans R Q as a F-vector space. Since Γ(Q) = (Γ(Q) ∩ C ) + Θ(Q) by Theorem 3.7, we have the assertion.

POINCARÉ SERIES AND THEIR FUNCTIONAL EQUATIONS
This section is devoted to the study of the support of Poincaré series associated with generalized Weierstraß semigroups and their functional equations. In particular, we extend some results of [18]. As in the previous section, let X be a curve over a finite field F and let Q = (Q 1 , . . . , Q m ) be an m-tuple of pairwise distinct F-rational points on X . We also assume that #F ≥ m ≥ 2.

Poincaré series as an invariant. Eq. (2) provides an expression for the coefficients of P(t).
In the proposition to be proved, we use that expression to state a characterization of likely elements in the support of P(t), extending somehow the formula (5) to the case of several points. This result is the version of [19,Prop. 3.8] for our Poincaré series associated with generalized Weierstraß semigroups; we shall point its proof out for the sake of clarity.
Proof. By Proposition 2.1 (2) we have d i (β ) = 0 if and only if ∇ m i (β ) = / 0; thus for any i, j ∈ I with i = j and β ∈ Z m , we get Hence, for any i ∈ I and any reordering {i 1 , . . . , i m−1 } of I\{i}, we have (12) 0 To prove (a), we observe that d i (α) = 0 for some i ∈ I by Proposition 2.1. Then, by (12) we deduce that each sum d i (α − 1 + 1 J + e i ) of the coefficients p i (α) in (2) vanishes and therefore p i (α) = 0, which gives (a) by Proposition 2.8. Now, assuming α ∈ H(Q)\M (Q), then ∇ i (α) = / 0 for some i ∈ I, which implies d i (α − 1 + e i ) = 1 by the equality ∇ i (α) = ∇ m i (α − 1 + e i ) together with Proposition 2.1 (2). Therefore, all other following terms in the inequalities (12) are equal to 1 and so Notice that in this form, P(t) agrees exactly with that afforded by (5) for the two point case. Furthermore, by item (c), the absolute maximal elements of H(Q) do appear in the support of P(t).
We next show a general expression for the Poincaré series which allows us to describe them from a certain Laurent polynomial and the generators of Θ(Q).
Theorem 4.2. Let P * (t) be the multivariate Laurent polynomial

Then the Poincaré series P(t) associated with H(Q) satisfies
Proof. After writing P(t) as in (14), the proof follows by Theorem 3.7 provided that we show the equality p(α) = p(α + η) for any α ∈ M (Q) ∩ C and η ∈ Θ(Q). Since the coefficient p(α) is as in (2)  We now conclude the current section by proving a result which summarizes the main properties of Poincaré series associated with a Weierstraß semigroup. Here we introduce an equivalent notion of symmetry to that in [18] which will enable us to extend the results there. The next lemma generalizes the result stated in [10, p. 629] to non-algebraically closed fields. Although their proof follows similar lines, some adaptations will be necessary. Proof. By Proposition 2.1 (2) and the Riemann-Roch theorem, we have the following equivalences: where K ′ is a canonical divisor on X . Therefore, By the assumption #F ≥ m, we can proceed exactly as in the proof of Proposition 2.1 to choose an appropriate element . Consequently, the canonical divisor K = div( f ) + K ′ satisfies v Q i (K) = α i − 1 and K − D(α − 1) ≥ 0, which is the desired conclusion.
In [18,Prop. 6], the Poincaré series associated with symmetric generalized Weierstraß semigroups at two points are shown to satisfy the functional equation (16) P(t) = t σ P(t −1 ), where σ is as in Proposition 4.7 (3). We can now formulate a generalization of the functional equations (16) to multivariable Poincaré series of generalized Weierstraß semigroups at several points.  Hence, using the equality above together with Eq. (14) for P(t), Remark 4.8 (2) implies that In particular, we can also derive an functional equation for Q(t) as follows. Proof. Let α ∈ Z m . By Proposition 2.8 and the relation p(α) = (−1) m p(σ − α) in the proof of Theorem 4.10, we get which proves the assertion.