Computer search for curves with many points among abelian covers of genus 2 curves. (English) Zbl 1317.11006

Rökaeus, Karl

Aubry, Yves (ed.) et al., Arithmetic, geometry, cryptography and coding theory. 13th conference on arithmetic, geometry, cryptography and coding theory, CIRM, Marseille, France, March 14–18, 2011 and Geocrypt 2011, Bastia, France, June 19–24, 2011. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-7572-8/pbk; 978-0-8218-9027-1/ebook). Contemporary Mathematics 574, 145-150 (2012).

Summary: Using class field theory one associates to each curve \(C\) over a finite field, and each subgroup \(G\) of its divisor class group, unramified abelian covers of \(C\) whose genus is determined by the index of \(G\). By listing class groups of curves of small genus one may get examples of curves with many points; we do this for all curves of genus 2 over the fields of cardinality 5,7,9,11,13 and 16, giving new entries for the tables of curves with many points [http://www.manYPoints.org].
For the entire collection see [Zbl 1248.11004].

MSC:

11-04	Software, source code, etc. for problems pertaining to number theory
11G20	Curves over finite and local fields
11R37	Class field theory
14G15	Finite ground fields in algebraic geometry