Weierstrass Semigroups in an Asymptotically Good Tower of Function Fields

The Weierstrass semigroups of some places in an asymptotically good tower of function fields are computed. 0. Introduction A tower F1 ⊆ F2 ⊆ F3 ⊆ . . . of algebraic function fields over a finite field Fl is said to be asymptotically good if lim m→∞ number of rational places of Fm/Fl genus of Fm > 0. Recently an explicit description was obtained of several asymptotically good towers {1}, {2}. The motivation to consider these came from coding theory: such towers give rise to asymptotically good sequences of codes. Although the existence of good codes on or above the Tsfasman-Vladut-Zink bound was guaranteed {7} and even a polynomial construction was given {5}, the methods used (namely, modular curves) and the degree of the complexity of the construction were such that hardly any of the resulting codes were known explicitly. Now that asymptotically good towers (Fm)m≥1 of function fields are known explicitly, the next step would be to give an explicit description of the vector spaces L(G) resp. L(rP ), where G is a divisor (resp. P (m) is a rational place) of Fm. The latter space L(rP ) is the Fl-vector space of all rational functions in Fm that have no poles outside P (m) and pole order at most r at P . The first attempts have been made in this direction: these vector spaces were explicitly determined for the fields F1, F2 and F3, by {8}, and for F4 over F16 by {3}, in the tower F = (Fm)m≥1 over Fq2 which is given {1} by F1 = Fq2(x1) and Fi+1 = Fi(zi+1) with z i+1 + zi+1 = x q+1 i , where xi = zi/xi−1. In this paper we consider another tower T = (Tm)m≥1 over Fq2 ; this tower was introduced in {2} and seems to be easier to handle than the tower F above. It is defined as follows: T1 = Fq2(x1) and Ti+1 = Ti(xi+1) with xqi+1 + xi+1 = xqi x i + 1 . The first and third author were supported by grants of Deutsche Forschungsgemeinschaft DFG. . 1 2 R. PELLIKAAN, H. STICHTENOTH, AND F. TORRES By the form of the defining equations it is readily seen that N(Tm), the number of rational places of Tm, is at least (q 2 − q)qm−1. The genus g(Tm) is computed by using the theory of Artin-Schreier extensions, and one finds that lim m→∞ N(Tm) g(Tm) = q − 1. Hence the tower T is asymptotically good and in fact optimal {2}. This implies that geometric Goppa codes which are constructed by means of this tower T lie on or above the Tsfasman-Vladut-Zink bound, which is better than the GilbertVarshamov bound for all q > 25 {7}. The element x1 ∈ T1 ⊆ Tm has in Tm a unique pole that we denote by P (m) ∞ , and in fact P (m) ∞ is a rational place. Hence it is natural to consider the spaces L(rP (m) ∞ ) for all m and r. The main result of our paper is Theorem 3.1 where the dimension of all these spaces is determined. In other words, we describe explicitly the Weierstrass semigroup of P (m) ∞ , that is H(P (m) ∞ ) = {i ∈ N0 | there is some f ∈ Tm having a pole of order i at P (m) ∞ and no pole outside P (m) ∞ }. We remark that the minimum distance of some geometric Goppa codes is related to Weierstrass semigroups (see {4} and the references therein). 1. Preliminaries and Notation Throughout this paper, we will use the following notation: K = Fq2 the finite field of cardinality q. F an algebraic function field of one variable over K. g(F ) the genus of F/K. P(F ) the set of all places of F/K. (x)0 the zero divisor of an element x 6= 0 in F . (x)∞ the pole divisor of x. (x) = (x)0 − (x)∞ the principal divisor of x. supp A the support of the divisor A in F . deg A the degree of the divisor A. L(A) the K-vector space of all elements x ∈ F with (x) ≥ −A. H(P ) the Weierstrass semigroup of a place P ∈ P(F ), i.e. H(P ) = {i ∈ N | there is some x ∈ F with (x)∞ = iP}. WEIERSTRASS SEMIGROUPS 3 If E/F is a finite extension of F/K and A is a divisor of F/K; conF (A) the conorm of A in E/F . We will consider the following tower T = (Tm)m≥1 of of function fields Tm/K: Tm = K(x1, . . . , xm) with x q i+1 + xi+1 = xqi x i + 1 for i = 1, . . . ,m− 1. This tower was studied in {2}; we need some results from that paper: Proposition 1.1. i) For all m ≥ 2, the extension Tm/Tm−1 is a Galois extension of degree q. ii) The pole of x1 in T1 is totally ramified in Tm/T1, i.e. (x1) Tm ∞ = q m−1 · P (m) ∞ with a place P (m) ∞ ∈ P(Tm) of degree one. iii) The genus g(Tm) is g(Tm) =  (q − 1) if m ≡ 0 mod 2 (q m+1 2 − 1)(q 2 − 1) if m ≡ 1 mod 2 Proof. i), ii) see {2, Lemma 3.3}. iii) see {2, Remark 3.8}. 2. The semigroups Sm A numerical semigroup is a subset S ⊆ N0 having the following properties: i) 0 ∈ S; ii) a, b ∈ S ⇒ a+ b ∈ S; iii) N0 \ S is finite. The numbers c ∈ N0 \ S are called gaps of S. As an example, consider an algebraic function field F/K and a place P ∈ P(F ) of degree one. Then H(P ), the Weierstrass semigroup of P , is a numerical semigroup, and the number of gaps of H(P ) is equal to the genus g(F ) (this is the Weierstrass gap theorem; see {6, p. 32}). In this Section we study certain numerical semigroups Sm ⊆ N0 which are defined recursively as follows. Definition 2.1. i) For m ≥ 1, let cm =  q − qm2 if m ≡ 0 mod 2, q − qm+1 2 if m ≡ 1 mod 2. ii) S1 = N0 and, for m ≥ 1, Sm+1 = q · Sm ∪ {x ∈ N0 | x ≥ cm+1}. 4 R. PELLIKAAN, H. STICHTENOTH, AND F. TORRES We will prove in Section 3 that Sm is in fact the Weierstrass semigroup of the place P (m) ∞ ∈ P(Tm). Recall that g(Tm) denotes the genus of the function field Tm/F. Proposition 2.2. The number of gaps of Sm is g(Tm). Proof. Let g̃m be the number of gaps of Sm. Define for a subset S ⊆ N0 and c ∈ N0 the set S(c) = {x ∈ S | x ≤ c}. The integer cm − 1 is the largest gap of Sm (for m ≥ 2). So if c ≥ cm, then g̃m = #(N0 \ Sm) = #({0, 1, . . . , c} \ Sm(c)). Therefore #Sm(c) = c+ 1− g̃m if c ≥ cm. If m > 1, then Sm = q · Sm−1 ∪ {x ∈ N0 | x ≥ cm}. Since cm−1 ≤ cm/q and cm ∈ q · Sm−1, it follows that #Sm(cm) = #Sm−1( cm q ) . Hence cm + 1− g̃m = #Sm(cm) = #Sm−1( cm q ) = cm q + 1− g̃m−1. This gives the recursion formula g̃m = q − 1 q cm + g̃m−1. Now we proceed by induction on m. If m = 1, then S1 = N0. So g̃1 = 0 = g(T1). Assume now that m > 1 and g̃m−1 = g(Tm−1) as induction hypothesis. Then g̃m = q − 1 q cm + g(Tm−1). a) If m ≡ 0 mod 2 then we obtain from Definition 3.1 and Proposition 2.1 g̃m = q−1 q (q − qm2 ) + (qm2 − 1)(q 2 − 1) = (q − qm2 )− (qm−1 − q 2 ) + (qm−1 − qm2 − q 2 + 1) = q − 2qm2 + 1 = (qm2 − 1) = g(Tm). b) If m ≡ 1 mod 2, then g̃m = q−1 q (q − qm+1 2 ) + (q 2 − 1) = (q − qm+1 2 )− (qm−1 − q 2 ) + (qm−1 − 2q 2 + 1) = q − qm+1 2 − q 2 + 1 = (qm+1 2 − 1)(q 2 − 1) = g(Tm). WEIERSTRASS SEMIGROUPS 5 3. The Main Result We consider again the tower of function fields T = (Tm)m≥1 over the field of constants K = Fq2 ; i.e., T1 = K(x1) and Ti+1 = Ti(xi+1) with xqi+1 + xi+1 = xqi x i + 1 . Recall that H(P (m) ∞ ) denotes the Weierstrass semigroup of the unique pole P (m) ∞ of x1 in Tm, and that the numerical semigroup Sm and the number cm are given by Definition 3.1. Our main result is the following: Theorem 3.1. H(P (m) ∞ ) = Sm. The proof will be given in this Section. Proposition 3.2. Suppose that for all m ≥ 1 there exists a divisor A of Tm with the following properties: i) A ≥ 0 and deg A = cm − g(Tm); ii) dimL(cmP (m) ∞ − A) = 1. Then we have H(P (m) ∞ ) = Sm, i.e., Theorem 3.1 holds. Proof. The assertion is trivial for m = 1, since H(P (1) ∞ ) = N0 = S1. We proceed by induction. Assume that m > 1 and that H(P (m−1) ∞ ) = Sm−1 holds, as induction hypothesis. We have from i) and ii) that deg (cmP (m) ∞ − A) = g(Tm) and dimL(cmP (m) ∞ − A) = 1. This means that cmP (m) ∞ −A is a non-special divisor of Tm (see {6, p.33}). Hence for any divisor B ≥ cmP (m) ∞ − A one has dimL(B) = deg B + 1− g(Tm). In particular we obtain for c ≥ cm + 1 dimL((c− 1)P (m) ∞ ) = c− g(Tm), dimL(cP (m) ∞ ) = c+ 1− g(Tm). So c is a non-gap of P (m) ∞ for all c > cm. Moreover, since P (m) ∞ is totally ramified in the extension Tm/Tm−1, q · Sm−1 = q ·H(P (m−1) ∞ ) ⊆ H(P (m) ∞ ). As cm ∈ q · Sm−1 we conclude that Sm = q · Sm−1 ∪ {x ∈ N0 | x ≥ cm} ⊆ H(P (m) ∞ ). 6 R. PELLIKAAN, H. STICHTENOTH, AND F. TORRES By Proposition 3.2 and the Weierstrass gap theorem, both semigroups Sm and H(P (m) ∞ ) have the same number of gaps, namely g(Tm). Hence H(P (m) ∞ ) = Sm. It remains to prove the existence of divisors A as in Proposition 3.2. The following elements πj ∈ Tm will play a crucial role. Definition 3.3. For 1 ≤ j ≤ m we define