Abstract
In this paper, explicit equations for algebraic curves with genus 4, 5, and 10 already studied in characteristic zero, are analyzed in positive characteristic p. We show that these curves have an interesting behaviour on the number of their rational places. Namely, they are either maximal or minimal over the finite field with p2 elements for infinitely many p's. The key tool is the investigation of their Jacobian decomposition. Lists of small p's for which maximality holds are provided. In some cases we also describe the automorphism group of the curve.
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