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An asymptotically good tower of curves over the field with eight elements. (English) Zbl 1062.11037

Given an infinite tower \(C_\bullet\) of curves \(\cdots\rightarrow C_i\rightarrow C_{i-1}\rightarrow \cdots\rightarrow C_0\) over the finite field \({\mathbb F}_q\), with \(C_i/C_{i-1}\) separable of degree \(>1\) and \(g(C_i)\rightarrow \infty\) as \(i\rightarrow \infty\), put \[ \lambda(C_\bullet)=\lim_{i\rightarrow\infty} \#C_i({\mathbb F}_q)/g(C_i). \] T. Zink [Degeneration of Shimura surfaces and a problem in coding theory. In: Fundamentals of computation theory, Proc. 5th Int. Conf., Cottbus/Ger. 1985, Lect. Notes Comput. Sci. 199, 503–511 (1985; Zbl 0581.94014)] used Shimura surfaces to a construct a tower \(C_\bullet\) of curves over \({\mathbb F}_{p^3}\), for \(p\) a prime, such that \(\lambda(C_\bullet)\geq 2(p^2-1)/(p+2)\). A. Garcia, H. Stichtenoth, and M. Thomas [Finite Fields Appl. 3, No. 3, 257–274 (1997; Zbl 0946.11029)] constructed an explicit tower \(C_\bullet\) of Kummer curves defined over \({\mathbb F}_{p^m}\), where \(m\geq 2\), by the simple recursive equation \(x_i^m+(x_{i-1}+1)^m=1\) such that \(\lambda(C_\bullet)\geq 2/(q-2)\). The authors present an explicit tower \(C_\bullet\) of Artin-Schreier curves defined over \({\mathbb F}_8\) by the simple recursive equation \(x_i^2+x_i=x_{i-1}+1+(1/x_{i-1})\), and they show that \(\lambda(C_\bullet)=3/2\).
The key, as usual, is to determine the ramification divisor of \(C_i\) over \(C_{i-1}\). The number of points of \(C_{i-1}\) that are totally ramified in the cover \(C_i/C_{i-1}\) breaks up into two cases depending on whether \(i\) is even or odd.
An interesting open problem is to see if this explicit tower is related to the modular construction of Zink.

MSC:

11G20 Curves over finite and local fields
11R58 Arithmetic theory of algebraic function fields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
14H25 Arithmetic ground fields for curves
14G50 Applications to coding theory and cryptography of arithmetic geometry
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