The Quarterly Journal of Mathematics Advance Access published online on May 28, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap015
GROUP ACTION ON GENUS 7 CURVES AND THEIR WEIERSTRASS POINTS
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA
E-mail: rezo{at}msn.com
Received 15 September 2004;
revised 2 September 2008
 |
Abstract |
|---|
In this work, we generalize the theory of elliptic modular functions, to the case of genus 7. We investigate the equations of all algebraic curves of genus 7, their automorphism groups and their link to modern algebraic geometry and the theory of hyperelliptic curves. We discuss the cyclic covers of any curve of genus 7, the local structure of the moduli space at the corresponding Weierstrass points for each curve. We show that the largest finite group acting as the full automorphism group of a hyperelliptic curve of genus 7 has order 64 and we find its equation. We then obtain all the 3g – 3 = 18 hyperelliptic curves of genus 7 and their full automorphism groups. We discover that there are merely three other finite groups of the order >64 acting on some non-hyperelliptic curves of genus 7. We also obtain the equations of the non-hyperelliptic curves.
Present address: 331 S. Negley Avenue, 5, Pittsburgh, PA 15232-1119, USA.