## Second Tower of Function Fields by García and Stichtenoth

Let *b* = *q*^{2} be the square of a prime power *q*. In [1] García and Stichtenoth consider the tower *T*_{1} ⊆ *T*_{2} ⊆ ⋯ of global function fields over **F**_{b}, where *T*_{1} is the rational function field **F**_{b}(*x*_{1}) and *T*_{i} := *T*_{i−1}(*x*_{i}) for *i* = 2, 3,…, where *x*_{i} satisfies the equation

*x*

_{i}

^{q}+

*x*

_{i}= .

Let *Q*_{i} := *q*^{⌊i/2⌋} and *R*_{i} := *q*^{⌈i/2⌉}. Then it is shown in [1, Remark 3.8] that

*g*

_{i}:=

*g*(

*T*

_{i}/

**F**

_{b}) = (

*Q*

_{i}– 1)(

*R*

_{i}− 1).

Let *P*_{∞} denote the pole of *x*_{1} and *P*_{α} for *α* ∈ **F**_{b} the zero of *x*_{1} – *α* in *T*_{1} = **F**_{b}(*x*_{1}). Let Ω := {*α* ∈ **F**_{b} : *α*^{q} + *α* = 0} and Ω^{*} := Ω ∖ {0}. Then it is shown in [1, Lemma 3.3] that *P*_{∞} and *P*_{α} with *α* ∈ Ω^{*} are completely ramified in *T*_{i}/*T*_{1}, whereas *P*_{α} with *α* ∈ **F**_{b} ∖ Ω splits completely in every extention *T*_{i}/*T*_{1}. Therefore the number of **F**_{b}-rational points of *T*_{i} is

*N*

_{i}:=

*N*(

*T*

_{i}/

**F**

_{b}) = (

*q*

^{2}–

*q*)

*q*

^{i−1}+1 + (

*q*– 1) +

*s*

_{i},

where *s*_{i} denotes the number of rational places over *P*_{0} in *F*_{i}. Obviously, *s*_{1} = 1. For even *q* ≥ 4 the exact value of *s*_{i} is given in [2] as *s*_{2} = *q*, *s*_{3} = *q*^{2}, and *s*_{i} = 2*q*^{2}−*q* for *i* ≥ 4.

### Weierstrass Semigroup

The Weierstrass semigroup *H*_{i} of the unique place over *P*_{∞} in *T*_{i} is *H*_{1} = ℕ_{0} for *i* = 1 and

*H*

_{i}=

*qH*

_{i−1}∪{ℕ

_{0}+

*R*

_{i}(

*Q*

_{i}– 1)}

for *i* ≥ 2 [3].

### Optimality

We have *N*_{i}/*g*_{i} = *q*−1 [1, Theorem 3.1], thus this tower attains the Drinfelʹd-Vlăduţ bound [4] and is therefore asymptotically optimal.

### References

[1] | Arnaldo García and Henning Stichtenoth. On the asymptotic behaviour of some towers of function fields over finite fields. Journal of Number Theory, 61(2):248–273, December 1996.doi:10.1006/jnth.1996.0147 |

[2] | Ilia Aleshnikov, P. Vijay Kumar, Kenneth W. Shum, and Henning Stichtenoth. On the splitting of places in a tower of function fields meeting the Drinfeld-Vlăduţ bound. IEEE Transactions on Information Theory, 47(4):1613–1619, May 2001.doi:10.1109/18.923746 MR1830111 (2002c:94042) |

[3] | Ruud Pellikaan, Henning Stichtenoth, and Fernando Torres. Weierstrass semigroups in an asymptotically good tower of function fields. Finite Fields and Their Applications, 4(4):381–392, October 1998.doi:10.1006/ffta.1998.0217 MR1648573 (99g:11139) |

[4] | Sergei G. Vlăduţ and Vladimir Gershonovich Drinfelʹd. Number of points of an algebraic curve. Functional Analysis and its Applications, 17:53–54, 1983. |

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Second Tower of Function Fields by García and Stichtenoth.”
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