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Asymptotically good towers of function fields with small \(p\)-rank. (English) Zbl 1490.11111

Let \({\mathbb F}_q\) be a finite field of characteristic \(p\). For a function field \(F\) over \({\mathbb F}_q\), \(g(F)\) denotes the genus of \(F\) and \(N(F)\) the number of rational places in \(F\). A tower of function fields is an infinite sequence \({\mathcal F}=\big(F_i\big)_{i\geq 0}\) of function fields \(F_i\) over \({\mathbb F}_q\) such that \(F_0\subseteq F_1\subseteq F_2\subseteq\ldots \), \(F_{i+1}/F_i\) is separable for all \(i\) and \(g(F_i)\xrightarrow[i\to\infty]{}\infty\). Call \(\lambda({\mathcal F}) :=\lim\limits_{i\to\infty} N(F_i)/g(F_i)\) the {\em limit} of the tower. We have \(0\leq \lambda({\mathcal F})\leq \sqrt q-1\). The tower \({\mathcal F}\) is called {\em asymptotically good} if \(\lambda({\mathcal F})>0\). Let \(\sigma({\mathcal F}):=\liminf\limits_{i\to\infty}s(F_i)/g(F_i)\) be the asymptotic \(p\)-rank of \({\mathcal F}\), where \(s(F_i)\) is the \(p\)-rank of \(F_i\). We have \(0\leq \sigma({\mathcal F}) \leq 1\).
The aim of this paper is to construct asymptotically good towers \({\mathcal F}\) with \(\sigma({\mathcal F})\) as small as possible. In Section 4 the authors construct asymptotically good towers over quadratic fields \({\mathbb F}_q\), that is \(q\) is a square, whose asymptotic \(p\)-rank is small.
Let \({\mathcal G}:=(G_i)_{i\geq 0}\) be the optimal tower introduced by A. Garcia and H. Stichtenoth [J. Number Theory 61, No. 2, 248–273 (1996; Zbl 0893.11047)]. That is: \(G_1:= {\mathbb F}_q(x_1)\) is a rational function field, \(G_0:= {\mathbb F}_q(x_0)\) with \(x_0=x_1^l+x_1\) and for \(i\geq 1\), \(G_{i+1}=G_i(x_{i+1})\) with \(x_{i+1}^l+x_{i+1}=\frac{x_i^l}{ x_i^{l-1}+1}\), where \(q=l^2\). Let \(E:=G_0(y)={\mathbb F}_q (x_0,y)\) with \(y^m=x_0\). Let \({\mathcal E}=E\cdot {\mathcal G}=\big(E_i\big)_{i\geq 0}\) be the composite of the function field \(E\) and the tower \({\mathcal G}\). The main result of the paper is that \(\lambda({\mathcal E})=(l-1)\frac{\gcd(l+1,m)}{m}\) and \(\sigma({\mathcal E})=\frac 1m\).
It is also shown that for any \(\varepsilon >0\), there exists a constant \(B>0\), depending on \(q\), and an asymptotically good tower \({\mathcal F}=\big(F_i\big)_{i\geq 0}\) over \({ \mathbb F}_q\) such that \(\sigma({\mathcal F})<\varepsilon\) and \(|\mathrm{Aut}(F_i)|\geq B\cdot g(F_i)\) for all \(i\geq 0\). That is, there exist function fields over \({\mathbb F}_q\) having large genus, which have simultaneously many rational points, many automorphisms and small \(p\)-rank.

MSC:

11R58 Arithmetic theory of algebraic function fields
11G20 Curves over finite and local fields
14G50 Applications to coding theory and cryptography of arithmetic geometry
14H05 Algebraic functions and function fields in algebraic geometry

Citations:

Zbl 0893.11047
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Full Text: DOI

References:

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