Nondegenerate curves of low genus over small finite fields

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 extremal properties concerning the number of rational points over various extension fields. 1991 Mathematics Subject Classification. Primary 14H45, Secondary 14M25. Key words and phrases. nondegenerate curves, finite fields, Newton polytope. The first author is a postdoctoral fellow of FWO-Vlaanderen. He would like to thank Alessandra Rigato for some helpful comments on curves over finite fields having many or few rational points.

Counting points on smooth plane quartics

“…Remark 5.20 For primes p | D where f has good reduction we can compute the Cartier-Manin matrix directly from its definition, but we can more efficiently treat p * 4 (f ) by simply applying Algorithm 5.16 to the nondegenerate reduction of f modulo p. In our implementation we do the same for good primes p | * 4 (f ) greater than 3 by applying a random linear transformation to the reduction of f modulo p until we obtain a nondegenerate polynomial f ∈ F p [x 0 , x 1 , x 2 ] that defines an isomorphic curve. For p > 3 such a nondegenerate polynomial is guaranteed to exist by Proposition 3.2 of [7], and in practice we can find one quickly. Note that we have assumed f (x 0 , x 1 , x 2 ) = 0 is a model for X that is smooth a p, but if not, replace f modulo p with the reduction of a model for X that is smooth at p. Before describing our average polynomial-time algorithms to compute A p for p N coprime to D, we briefly recall some background material on remainder trees and forests.…”