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Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I: Generic Covers and Covers with Many Branch Points: with an Appendix by R. Guralnick and R. Stafford
Robert M. Guralnick, University of Southern California, Los Angeles, CA, and John Shareshian, Washington University, St. Louis, MO
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Memoirs of the American Mathematical Society
2007; 128 pp; softcover
Volume: 189
ISBN-10: 0-8218-3992-6
ISBN-13: 978-0-8218-3992-8
List Price: US$68 Individual Members: US$40.80
Institutional Members: US\$54.40
Order Code: MEMO/189/886

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.

• Introduction and statement of main results
• Notation and basic lemmas
• Examples
• Proving the main results on five or more branch points--Theorems 1.1.1 and 1.1.2
• Actions on $$2$$-sets--the proof of Theorem 4.0.30
• Actions on $$3$$-sets--the proof of Theorem 4.0.31
• Nine or more branch points--the proof of Theorem 4.0.34
• Actions on cosets of some $$2$$-homogeneous and $$3$$-homogeneous groups
• Actions on $$3$$-sets compared to actions on larger sets
• A transposition and an $$n$$-cycle
• Asymptotic behavior of $$g_k(E)$$
• An $$n$$-cycle--the proof of Theorem 1.2.1
• Galois groups of trinomials--the proofs of Propositions 1.4.1 and 1.4.2 and Theorem 1.4.3
• Appendix A. Finding small genus examples by computer search--by R. Guralnick and R. Stafford
• Appendix. Bibliography