Appeared in: Finite Fields and their Applications,
vol. 4, pp. 381-392, 1998.
WEIERSTRASS SEMIGROUPS IN AN ASYMPTOTICALLY GOOD
TOWER OF FUNCTION FIELDS
RUUD PELLIKAAN, HENNING STICHTENOTH, AND FERNANDO TORRES
Abstract. 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
number of rational places of Fm /Fl
> 0.
m→∞
genus of Fm
lim
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(m) ) resp. L(rP (m) ), where G(m) is a
divisor (resp. P (m) is a rational place) of Fm . The latter space L(rP (m) ) 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 (m) . 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
q
zi+1
+ zi+1 = xq+1
, where xi = zi /xi−1 .
i
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:
xqi
q
T1 = Fq2 (x1 ) and Ti+1 = Ti (xi+1 ) with xi+1 + xi+1 = q−1
.
xi + 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)q m−1 . The genus g(Tm ) is computed by
using the theory of Artin-Schreier extensions, and one finds that
lim
m→∞
N (Tm )
= q − 1.
g(Tm )
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 2 > 25 {7}.
(m)
The element x1 ∈ T1 ⊆ Tm has in Tm a unique pole that we denote by P∞ , and in
(m)
(m)
fact P∞ is a rational place. Hence it is natural to consider the spaces L(rP∞ ) 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
(m)
(m)
semigroup of P∞ , that is H(P∞ ) = {i ∈ N0 | there is some f ∈ Tm having a
(m)
(m)
pole of order i at P∞ and no pole outside P∞ }. 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 = Fq 2
- the finite field of cardinality q 2 .
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)F0
- the zero divisor of an element x 6= 0 in F .
(x)F∞
- the pole divisor of x.
(x)F = (x)F0 − (x)F∞ - 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)F ≥ −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)F∞ = iP }.
WEIERSTRASS SEMIGROUPS
3
If E/F is a finite extension of F/K and A is a divisor of F/K;
conE
F (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 xqi+1 + xi+1 =
xqi
for i = 1, . . . , m − 1.
xiq−1 + 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.
(m)
(x1 )T∞m = q m−1 · P∞
(m)
with a place P∞ ∈ P(Tm ) of degree one.
iii) The genus g(Tm ) is

(q m/2 − 1)2
if m ≡ 0 mod 2

g(Tm ) =
m−1
 m+1
(q 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
 m
m
if m ≡ 0 mod 2,
 q −q2
cm =
m+1
 m
q − q 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
(m)
place P∞ ∈ 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
Since cm−1
Sm = q · Sm−1 ∪ {x ∈ N0 | x ≥ cm }.
≤ cm /q and cm ∈ q · Sm−1 , it follows that
cm
#Sm (cm ) = #Sm−1 ( ) .
q
Hence
cm + 1 − g̃m = #Sm (cm ) = #Sm−1 (
cm
cm
)=
+ 1 − g̃m−1 .
q
q
This gives the recursion formula
g̃m =
q−1
cm + g̃m−1 .
q
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
q−1
cm + g(Tm−1 ).
g̃m =
q
a) If m ≡ 0 mod 2 then we obtain from Definition 3.1 and Proposition 2.1
g̃m =
q−1 m
(q
q
m
m
− q 2 ) + (q 2 − 1)(q
m
= (q m − q 2 ) − (q m−1 − q
m
m−2
2
m−2
2
− 1)
m
) + (q m−1 − q 2 − q
m−2
2
+ 1)
m
= q m − 2q 2 + 1 = (q 2 − 1)2 = g(Tm ).
b) If m ≡ 1 mod 2, then
g̃m =
q−1 m
(q
q
= (q m − q
= qm − q
−q
m+1
2
m+1
2
) − (q m−1 − q
m+1
2
−q
) + (q
m−1
2
m−1
2
− 1)2
m−1
2
+ 1 = (q
) + (q m−1 − 2q
m+1
2
− 1)(q
m−1
2
m−1
2
+ 1)
− 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
.
= q−1
xi + 1
(m)
(m)
Recall that H(P∞ ) denotes the Weierstrass semigroup of the unique pole P∞ 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:
(m)
Theorem 3.1. H(P∞ ) = Sm .
The proof will be given in this Section.
Proposition 3.2. Suppose that for all m ≥ 1 there exists a divisor A(m) of Tm with
the following properties:
i) A(m) ≥ 0 and deg A(m) = cm − g(Tm );
(m)
ii) dim L(cm P∞ − A(m) ) = 1.
Then we have
(m)
H(P∞
) = Sm ,
i.e., Theorem 3.1 holds.
Proof. The assertion is trivial for m = 1, since
(1)
) = N0 = S1 .
H(P∞
We proceed by induction. Assume that m > 1 and that
(m−1)
H(P∞
) = Sm−1
holds, as induction hypothesis. We have from i) and ii) that
(m)
(m)
deg (cm P∞
− A(m) ) = g(Tm ) and dim L(cm P∞
− A(m) ) = 1.
(m)
This means that cm P∞ − A(m) is a non-special divisor of Tm (see {6, p.33}). Hence
(m)
for any divisor B ≥ cm P∞ − A(m) one has
dim L(B) = deg B + 1 − g(Tm ).
In particular we obtain for c ≥ cm + 1
(m)
dim L((c − 1)P∞
) = c − g(Tm ),
(m)
) = c + 1 − g(Tm ).
dim L(cP∞
(m)
(m)
So c is a non-gap of P∞ for all c > cm . Moreover, since P∞ is totally ramified in
the extension Tm /Tm−1 ,
(m−1)
(m)
q · Sm−1 = q · H(P∞
) ⊆ H(P∞
).
As cm ∈ q · Sm−1 we conclude that
(m)
Sm = q · Sm−1 ∪ {x ∈ N0 | x ≥ cm } ⊆ H(P∞
).
6
R. PELLIKAAN, H. STICHTENOTH, AND F. TORRES
By Proposition 3.2 and the Weierstrass gap theorem, both semigroups Sm and
(m)
H(P∞ ) have the same number of gaps, namely g(Tm ). Hence
(m)
H(P∞
) = Sm .
It remains to prove the existence of divisors A(m) as in Proposition 3.2. The
following elements πj ∈ Tm will play a crucial role.
Definition 3.3. For 1 ≤ j ≤ m we define
j
Y
πj =
(xiq−1 + 1)
i=1
and
(m)
Zj
= {P ∈ P(Tm ) | P is a zero of xq−1
+ 1, f or some i ∈ {1, . . . , j}}.
i
i) Let 1 ≤ j ≤ m. Then the principal divisor of πj in Tm is given
Lemma 3.4.
by
(m)
(πj )Tm = Bj
(m)
where Bj
(m)
− (q m − q m−j )P∞
,
≥ 0 is a divisor of Tm with
(m)
(m)
supp (Bj ) = Zj
.
ii) Let 1 ≤ j ≤ m − 1 and 0 ≤ e ≤ q − 1. Then the principal divisor of πj xej+1
in Tm is given by
(m)
(m)
(πj xej+1 )Tm = Cj,e − (q m − q m−j + eq m−j−1 )P∞
,
(m)
where Cj,e ≥ 0 is a divisor of Tm with
(m)
(m)
supp (Cj,e ) ⊇ Zj
.
Proof. i) We proceed by induction on m. The case m = 1 is trivial. Now let m ≥ 2
and assume that the assertions hold for m − 1 and all j = 1, . . . , m − 1.
a) j ≤ m − 1. Then πj ∈ Tm−1 ⊆ Tm , and by the induction hypothesis,
(m−1)
(πj )Tm−1 = Bj
(m−1)
Observing that P∞
is totally ramified in Tm /Tm−1 , we obtain
(m)
− (q m−1 − q m−1−j ) · qP∞
(m)
− (q m − q m−j )P∞ ,
(πj )Tm = Bj
= Bj
(m)
(m−1)
.
− (q m−1 − q m−1−j )P∞
(m)
(m)
(m−1)
where Bj is the conorm of Bj
in Tm /Tm−1 . Note that the places of
(m)
(m−1)
Zj are exactly those of Tm lying above Zj
.
WEIERSTRASS SEMIGROUPS
7
b) j = m. The field Hm = K(x2 , . . . , xn ) is isomorphic to Tm−1 , and we write
m
Y
πm = (xq−1
+
1)
·
ρ
with
ρ
=
(xq−1
+ 1) ∈ Hm−1 .
1
i
i=2
By induction hypothesis, the principal divisor of ρ in Hm−1 is
,
(ρ)Hm−1 = C − (q m−1 − 1)Q(m−1)
∞
(m−1)
where Q∞
∈ P(Hm−1 ) is the unique pole of x2 in Hm−1 and C ≥ 0 is a
divisor of Hm−1 whose support is the set of all zeroes of xq−1
+ 1, . . . , xq−1
2
m +1
(m−1)
in Hm−1 . By {2, Lemma 3.2 and 3.3}, Q∞
splits in Tm /Hm−1 as follows:
X
(m)
+
Q(m) .
) = P∞
conTHmm−1 (Q(m−1)
∞
(m)
Q(m) ∈Z1
Hence we obtain
(m−1)
+ 1)Tm + conTHmm−1 (C − (q m−1 − 1)Q∞
(πm )Tm = (xq−1
1
= q m−1
X
)
(m)
Q(m) − (q − 1)q m−1 P∞
(m)
Q(m) ∈Z1
X
(m)
+ conTHmm−1 C − (q m−1 − 1)(P∞
+
Q(m) )
(m)
Q(m) ∈Z1
X
= conTHmm−1 C +
(m)
Q(m) − (q m − 1)P∞
.
(m)
Q(m) ∈Z1
Note that the support of the divisor
X
Tm
(m)
Bm
= conH
C+
m−1
Q(m)
(m)
Q(m) ∈Z1
(m)
is Zm , as claimed.
The proof of ii) is similar; we leave it to the reader.
Definition 3.5. For 1 ≤ j ≤ m, let
(m)
Aj
X
=
P.
(m)
P ∈Zj
(m)
(m)
Remark. πj ∈ L((q m − q m−j )P∞ − Aj ). This follows immediately from Lemma
3.4.
Proposition 3.6. For 1 ≤ j ≤ m,
(m)
(m)
L((q m − q m−j )P∞
− Aj ) =< πj > ;
(m)
(m)
i.e., the space L((q m − q m−j )P∞ − Aj ) is one-dimensional.
8
R. PELLIKAAN, H. STICHTENOTH, AND F. TORRES
(1)
(1)
Proof. The assertion is trivial for m = 1 since deg ((q − 1)P∞ − A1 ) = 0. Suppose
now that m ≥ 2 and that the Proposition holds for m − 1. We have to show that
(m)
(m)
any z ∈ L((q m − q m−j )P∞ − Aj ) can be written as z = α · πj , with α ∈ K.
a) 1 ≤ j ≤ m − 1. Observe that
(m−1)
(m)
(m−1)
(m)
− Aj
− Aj ) ∩ Tm−1 = L((q m−1 − q m−1−j )P∞
L((q m − q m−j )P∞
(m)
).
(m−1)
This follows easily from the definition of the divisors Aj and Aj
and the
(m)
fact that P∞ is totally ramified in Tm /Tm−1 . The latter space is generated
by πj , by induction hypothesis, hence we have
(m)
(m)
− Aj ) ∩ Tm−1 =< πj > .
L((q m − q m−j )P∞
(m)
(m)
Assume now that L((q m − q m−j )P∞ − Aj ) 6=< πj > . Then there exists
(m)
(m)
an element in L((q m − q m−j )P∞ − Aj ) \ Tm−1 , and we choose such an
(m)
element z of minimal pole order at P∞ , say v (m) (z) = −r (where v (m)
(m)
denotes the discrete valuation of Tm corresponding to the place P∞ ). Let
(m)
(m)
(m)
(m)
σ ∈ Gal (Tm /Tm−1 ). Then σP∞ = P∞ and σAj = Aj , so σz ∈
(m)
(m)
(m)
L((q m − q m−j )P∞ − Aj ) and v (m) (σz) = −r. The place P∞ has degree
one, hence there is some α ∈ K × such that
v (m) (σz − αz) > −r.
Since r was chosen to be minimal, we conclude that σz − αz ∈ Tm−1 , so
σz − αz = β · πj with β ∈ K. But
v (m) (σz − αz) > −r ≥ −(q m − q m−j ) = v (m) (πj ),
so β = 0 and
σz = αz (with α ∈ K × ).
The order of Gal (Tm /Tm−1 ) is q, so σ q is the identity and
z = σ q z = αq z.
As αq = 1 ⇒ α = 1 it follows that σz = z for all σ ∈ Gal (Tm /Tm−1 ),
therefore z ∈ Tm−1 . This is a contradiction because z ∈ L((q m − q m−j )P∞ −
(m)
Aj ) \ Tm−1 .
(m)
(m)
b) j = m. We know from a) that dim L((q m − q)P∞ − Am−1 ) = 1, so
(m)
(m)
dim L((q m − 1)P∞
− Am−1 ) ≤ q.
(m)
(m)
The elements πm−1 · xem (0 ≤ e ≤ q − 1) are in L((q m − 1)P∞ − Am−1 ), by
Lemma 3.4 ii), and they are linearly independent. Since
(m)
(m)
m
(m)
L((q m − 1)P∞
− A(m)
m ) ⊆ L((q − 1)P∞ − Am−1 ),
(m)
(m)
any element y ∈ L((q m − 1)P∞ − Am ) can be written as
y = πm−1 · h(xm )
WEIERSTRASS SEMIGROUPS
9
(m)
with a polynomial h(xm ) ∈ K[xm ] of degree ≤ q − 1. The divisor Am
contains all zeroes of xq−1
m + 1 in Tm , and these places are not zeroes of πm−1 .
q−1
So h(xm ) = γ · (xm + 1) with γ ∈ K and therefore
y = πm−1 · γ · (xq−1
m + 1) = γ · πm ∈< πm > .
Lemma 3.7. Let 1 ≤ j ≤ m/2. Then
(m)
deg Aj
= q j − 1.
(m)
Proof. Let Ai = {P ∈ P(Tm ) | P is a zero of xiq−1 + 1}. It follows from {2, Lemma
3.6} that for 1 ≤ i ≤ m/2,
X
deg (
P ) = (q − 1)q i−1 .
(m)
P ∈Ai
Since
(m)
Aj
=
j
X
X
P,
i=1 P ∈A(m)
i
we obtain
deg
(m)
Aj
=
j
X
(q − 1)q i−1 = q j − 1.
i=1
Definition 3.8. We define a divisor A(m) of Tm as follows: A(1) = 0 and, for m ≥ 2,
 m
f or m ≡ 0 mod 2
 2
(m)
with j =
A(m) = Aj
 m−1
f or m ≡ 1 mod 2.
2
By Proposition 3.2, the proof of Theorem 3.1 will be finished when we prove the
following Lemma:
Lemma 3.9.
i) deg A(m) = cm − g(Tm ).
(m)
ii) dim L(cm P∞ − A(m) ) = 1.
Proof. For m = 1, all assertions are obvious since c1 = g(T1 ) = 0 and A(1) = 0. Now
let m ≥ 2.
a) m ≡ 0 mod 2. Then
cm = q m − q m/2 and gm = (q m/2 − 1)2
(see Definition 2.1 and Proposition 1.1). Hence
cm − gm = q m/2 − 1 = deg A(m) ,
by Lemma 3.7. On the other hand, we have
(m)
(m)
(m)
− A(m) ) = L((q m − q m/2 )P∞
− Am/2 ) = < πm/2 > ,
L(cm P∞
by Proposition 3.6.
10
R. PELLIKAAN, H. STICHTENOTH, AND F. TORRES
b) m ≡ 1 mod 2. The proof is similar.
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Department of Mathematics and Computing Science, Eindhoven University of
Technology, 5600 MB Eindhoven, The Netherlands
E-mail address: ruudp@win.tue.nl
Universität GH Essen, FB 6 Mathematik und Informatik, D-45117 Essen, Germany
E-mail address: stichtenoth@uni-essen.de
E-mail address: fernando.torres@uni-essen.de