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Abelian subgroups with a normal cyclic decomposition December 11, 2012

Posted by bossudenotredame in Automorphism Group, Automorphisms of curves, Function fields, Galois Theory.
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And I’m not that dumb, I mean normal in the bigger group. This strange criteria becomes the only way I can construct some of these coverings for the attack. I wish I could get rid of this strange condition but for now I can’t think of an alternative algorithm. Maybe my brain is rotting.

Just to make the problem interesting I have seen C_3\times C_3 with a non-normal decomposition and at the same time in the same group and same subgroup with the normal decomposition.

But beside having this condition, exhausting the possibility that a group has one of these subgroup (with the fixed field of the Abelian subgroup being rational otherwise $latex  (e)$ would have done it) is another problem.

The golden key is that if it’s normal then this is not the problem. suppose C_i is a part of a decomposition and normal. then if in a new decomposition

C_i \subset eq D_1x...xD_n and C_i is empty. C_i, it should be that C_i \times D_1x...xD_n/C_i so gcg^{-1}gdg^{-1} and because the first part is normal the second part have to be, otherwise will be traped in C_i and contradiction.

This opens the way for a greedy algorithm. If you take off a normal part you are not going to harm the problem, so let take the biggest cyclic part that we can.

Get the Automorphism group, spit out the curve equation. September 2, 2012

Posted by bossudenotredame in Automorphism Group.
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I’m writing a program in Sage, that take an Automorphism group, and “tries” to spit out the curve equation. At this stage I only use “Über die Automorphismengruppen von algebraischen Funktionenkörpern” results. This means that I look at the centre of the Automorphism group G and check if I can write G as an extension of one of a cyclic subgroup of the centre. If I find such a cyclic subgroup then I’m the winner.

This For now, but I might make it more sophisticated later. This is good for my PhD and this is also good for the whole world. I think we lack such a tools.