Nondegenerate curves of low genus over small finite fields

2009/07/12 by Wouter Castryck, Castryck, Wouter, John Voight +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.0907.2060

8 pages; uses pstricks

arxiv created 2009/07/12 · arxiv updated 2009/12/01

Abstract

In a previous paper, we proved that over a finite field k of sufficiently large cardinality, all curves of genus at most 3 over k can be modeled by a bivariate Laurent polynomial that is nondegenerate with respect to its Newton polytope. In this paper, we prove that there are exactly two curves of genus at most 3 over a finite field that are not nondegenerate, one over F₂ and one over F₃. Both of these curves have remarkable extremal properties concerning the number of rational points over various extension fields.

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