Nondegenerate curves of low genus over small finite fields

Wouter Castryck; John Voight

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arXiv · 0907.2060

Nondegenerate curves of low genus over small finite fields

Wouter Castryck·John Voight

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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 F_2 and one over F_3. Both of these curves have remarkable extremal properties concerning the number of rational points over various extension fields.

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Wouter Castryck, John Voight. 2009-07-12. Nondegenerate curves of low genus over small finite fields. https://arxiv.org/abs/0907.2060

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