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Weierstrass points on Kummer extensions. (English) Zbl 1420.14070
Let \(F/K(x)\) be a Kummer extension of algebraic function fields of one variable, where \(K\) is an algebraically closed field of characteristic \(p\geq 0\). For a fully ramified place \(P\) of the extension \(F/K(x)\), the authors characterize the Weierstrass gaps at \(P\). This result is parallel to C. Towse’s [Trans. Am. Math. Soc. 348, No. 8, 3355–3378 (1996; Zbl 0877.14025)]. As an application, if \(F\) is maximal over a finite field of order \(q^2\) given by \(y^m=f(x)\), under mild conditions, they show that \(m\) is a divisor of \(q+1\). This was known already for a class of function fields; see e.g. A. Garcia and S. Tafazolian [J. Pure Appl. Algebra 212, No. 11, 2513–2521 (2008; Zbl 1186.11034)].
MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
11R58 Arithmetic theory of algebraic function fields
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