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**Example text**

In many interesting cases (see Section 4 and Section 5), the polynomial f (X, Y ) guarantees a very special ramification behaviour at the base of the pyramid, which implies immediately that the equation f (X, Y ) = 0 indeed defines a recursive tower. 3. 24 Towers of Function Fields This picture means: the place P0 of Fq (x0 ) is ramified in the extension Fq (x0 , x1 ) over Fq (x0 ) with ramification index e = m. Hence there is just one place Q1 of Fq (x0 , x1 ) lying above P0 , and this place Q1 is totally ramified over P0 .

Stichtenoth 25 Performing such a fractional linear transformation can sometimes transform the defining equation to a nicer form, or it can make it easier to describe the ramification locus or the splitting locus of the tower. 10 and hence they have a finite ramification locus. Moreover they have a non-empty splitting locus Z(T ). 1. 10. 10. Proof. 10). 5). The defining equations that we will consider in this section do give rise to towers of function fields T = (F0 , F1 , F2 , . 15). 1 The Tower T1 Consider the tower T1 over the field F4 with four elements, which is given recursively by the equation Y 3 = X 3 /(X 2 + X + 1).

All places (x0 = α) with (α − 1) + +1 = −1 and α = 0 split completely in the tower W5 . In particular we have that the splitting rate satisfies ν(W5 ) ≥ 2 + . It is much more difficult to determine the genus γ(W5 ) of the tower W5 . For the proof of the following result we refer to [8]. 20. Let q = 3 where is any prime power, and let W5 be the tower over Fq which is recursively defined by the equation Y −Y −1 = 1 − X + X −( −1) . 21. For any cubic prime power q = A( 3 ) ≥ 2( 3 2( − 1) . +2 2 one has − 1) .