By Golasinski M., et al. (eds.)

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**Example text**

5 iii). 5. 1 repeatedly) it follows that for any subextension of degree p in the refined pyramid, the different exponents are either 2p − 2 or 0. This holds in particular along the left edge of the pyramid which represents the tower W1 . 6 now follows. 37 A. Garcia and H. 7. Let q = 2 be a square, and let W1 = (F0 , F1 , F2 , . ) be the tower over Fq which is recursively defined by the equation Y + Y = X /(X −1 + 1). Then we have: γ(W1 ) = , ν(W1 ) = 2 − and λ(W1 ) = − 1. In particular, the tower W1 attains the Drinfeld-Vladut bound and it is therefore asymptotically optimal over Fq .

12] V. G. Drinfeld and S. G. Vladut, “The number of points of an algebraic curve”, Funktsional. Anal. i Prilozhen, Vol. 17, 68-69 (1983). [Funct. Anal. , Vol. ] [13] I. Duursma, B. Poonen and M. Zieve, “Everywhere ramified towers of global function fields”, Proceedings of the 7th International Conference on Finite Fields and Applications Fq7, Toulouse (eds. G. ), LNCS, Vol.

The right hand side of the last equality above takes the value 1 at the place P∞ , and since the exponent m = (q − 1)/( − 1) is the norm exponent of Fq /F 28 Towers of Function Fields we conclude that P∞ splits completely in the extension F1 /F0 . Let Q∞ be a m place of F1 above P∞ . Then we have from the equation xm 1 = (x0 + 1) − 1 that vQ∞ (x1 ) = vQ∞ (x0 ) = −1. Since F2 = F1 x2 x1 + 1 and x2 x1 + 1 m =1− 1 x1 + 1 m , it follows as above that the place Q∞ splits completely in the extension F2 /F1 .