# Hyperelliptic functions and their automorphisms

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The study of algebraic curves, in particular the hyperelliptic case, can also benefit from applied field theory. These are curves for which a 2:1 map to a genus 0 curve exists, and they can be written as y^{2}=p(x) where p(x) is a complex polynomial of degree 2g+2 or 2g+1 with g the genus of the curve. As such they all have an automorphism of order 2, and removing this yields what is called the reduced group of the curve.

The reduced group is a finite subgroup of PGL_{2}(C). Due to a classical result by F. Klein, it is isomorphic to one of the following: C_{n}, D_{n}, A_{4}, S_{4}, A_{5}. T. Shaska and I have studied those spaces in the case when the reduced automorphism group of the curve is isomorphic to A_{5} [1], completing in this way the study of the possible groups that he initiated.

The moduli spaces of hyperelliptic curves with prescribed automorphism groups are parametrized by means of curve invariants. The cases of low genus are amenable to computation using functional decomposition and Gröbner bases in order to manipulate these invariants. For genus 3, we have classified the loci of genus 3 with an involution other than the hyperelliptic one and provided rational models [2]. A parametrization for these spaces is to be determined, and we believe that our computational techniques allow one to make some improvements in this direction.

## Related publications

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**Hyperelliptic curves with reduced automorphism group A**(with T. Shaska), Applicable Algebra In Engineering, Communication and Computing 18 no. 1 (Feb 2007), p. 3-20. ISSN 0938-1279. DOI 10.1007/s00200-006-0030-9._{5}**Hyperelliptic curves of genus 3 with prescribed automorphism group**(with J. Gutierrez and T. Shaska), Lect. Notes Ser. Comput. 13 (May 2005),*Computational Aspects of Algebraic Curves*, p. 109-123. World Sci. Publ., Hackensack, NJ. ISBN 981-256-459-4.