Tower of Function Fields by Bezerra and García

Let b = q2 be a square of a prime power q. In [1] Bezerra and García consider the tower F1 ⊆ F2 ⊆ ⋯ of global function fields over Fb, where F1 is the rational function field Fb(x1) and Fi+1 := Fi(xi+1) for i = 1, 2,…, where xi+1 satisfies the equation

$\displaystyle {\frac{{x_{i+1}−1}}{{x_{i+1}^{q}}}}$ = $\displaystyle {\frac{{x_{i}^{q}−1}}{{x_{i}}}}$.

Then it is shown in [1, Lemma 4] that

gi := g(Fi/Fb) = $\displaystyle {\frac{{1}}{{q−1}}}$⋅\begin{displaymath}\begin{cases}\left(q^{n/2}−1\right)^{2} & \textrm{if $i$\ i… …\left(q^{(n+1)/2}−1\right) & \textrm{if $i$\ is odd}\end{cases}\end{displaymath}

and that

Ni := N(Fi/Fb) ≥ qi

for all i ≥ 1.

Optimality

It is easy to see that

$\displaystyle \lim_{{i\to\infty}}^{}$Ni/gi ≥ q−1,

therefore this tower attains the Drinfelʹd-Vlăduţ bound [2].

References

[1]Juscelino Bezerra and Arnaldo García.
A tower with non-Galois steps which attains the Drinfeld-Vladut bound.
Journal of Number Theory, 106(1):142–154, May 2004.
doi:10.1016/j.jnt.2003.11.004
[2]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. “Tower of Function Fields by Bezerra and García.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_FBezerraGarciaTower.html

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