Karl Rökaeus: New curves with many points over small finite fields

We use class field theory to search for curves with many rational points over the finite fields of cardinality <=5. The maximum number of points on a curve of fixed genus over GF(q) is bounded from above by a theorem of Weil, but this bound is rarely attained, and in most cases it is unknown how many points such a curve can have. Interest in this question started in the 1980's, and by now the best known results for genus up to 50 are collected on http://manYPoints.org.

Let q<=5. We did a computer aided search that goes through abelian covers of each curve of genus 2 over GF(q). This gave a number of improvements of the old records. For example, over GF(2) we settled the question of how many points a curve of genus 17 can have; we also found new record curves of genus 45, 46 and 48. In some cases this search was exhaustive for this type of extension of genus up to 50.