SYMMETRIC AND ALTERNATING GROUPS AS MONODROMY GROUPS OF RIEMANN SURFACES by Robert Guralnick, John Shareshian ( 9780821839928 )

Item Details

Title:

SYMMETRIC AND ALTERNATING GROUPS AS MONODROMY GROUPS OF RIEMANN SURFACES
GENERIC COVERS AND COVERS WITH MANY BRANCH POINTS - WITH AN APPENDIX BY R. GURALNICK AND R. STAFFORD

Volume:

v. 1

By:

Robert Guralnick, John Shareshian

Format:

Paperback

List price:	£64.00
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ISBN 10:

0821839926

ISBN 13:

9780821839928

Publisher:

AMERICAN MATHEMATICAL SOCIETY

Pub. date:

15 July, 2007

Edition:

Illustrated edition

Series:

Memoirs of the American Mathematical Society No. 189

Pages:

128

Description:

Considers indecomposable degree $n$ covers of Riemann surfaces with monodromy group an alternating or symmetric group of degree $d$. The authors 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 $n =d(d-1)/2$.

Synopsis:

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.

Publication:

Imprint:

American Mathematical Society

Returns:

Returnable

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