Symmetric and Alternating Groups As Monodromy Groups of Riemann Surfaces 1: Generic Covers and Covers With Many Branch Points: Generic Covers and ... an Appendix by R. Guralnick and R. Stafford - Brossura

Guralnick, Robert M.; Shareshian, John

Brossura

ISBN 10: 0821839926 ISBN 13: 9780821839928

Casa editrice: Amer Mathematical Society, 2007

Riassunto

The authors consider indecomposable degree $n$ covers of Riemann surfaces with monodromy group an alternating or symmetric group of degree $d$. They show that if the cover has five or more branch points then the genus grows rapidly with $n$ unless either $d = n$ or the curves have genus zero, there are precisely five branch points and $n =d(d-1)/2$. Similarly, if there is a totally ramified point, then without restriction on the number of branch points the genus grows rapidly with $n$ unless either $d=n$ or the curves have genus zero and $n=d(d-1)/2$. One consequence of these results is that if $f:X \rightarrow \mathbb{P 1$ is indecomposable of degree $n$ with $X$ the generic Riemann surface of genus $g \ge 4$, then the monodromy group is $S n$ or $A n$ (and both can occur for $n$ sufficiently large). The authors also show if that if $f(x)$ is an indecomposable rational function of degree $n$ branched at $9$ or more points, then its monodromy group is $A n$ or $S n$. Finally, they answer a question of Elkies by showing that the curve parameterizing extensions of a number field given by an irreducible trinomial with Galois group $H$ has large genus unless $H=A n$ or $S n$ or $n$ is very small.

Su questo libro

Product Description:

Book by Guralnick Robert M Shareshian John

Su questa edizione

• Editore: Amer Mathematical Society
• Data di pubblicazione: 2007
• ISBN 10: 0821839926
• ISBN 13: 9780821839928
• Rilegatura: Copertina flessibile
• Numero di pagine: 128