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# Symmetric and alternating groups as monodromy groups of Riemann surfaces I : generic covers and covers with many branch points

Author: Robert M Guralnick; John Shareshian Providence, R.I. : American Mathematical Society, ©2007. Memoirs of the American Mathematical Society, no. 886. Print book : EnglishView all editions and formats 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.  Read more... (not yet rated) 0 with reviews - Be the first.

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Document Type: Book Robert M Guralnick; John Shareshian Find more information about: Robert M Guralnick John Shareshian 9780821839928 0821839926 125406381 vi, 128 pages : illustrations ; 26 cm. 1. Introduction and statement of main results -- 2. Notation and basic lemmas -- 3. Examples -- 4. Proving the main results on five or more branch points--Theorems 1.1.1 and 1.1.2 -- 5. Actions on $2$-sets--the proof of Theorem 4.0.30 6. -- Actions on $3$-sets--the proof of Theorem 4.0.31 -- 7. Nine or more branch points--the proof of Theorem 4.0.34 --8. Actions on cosets of some $2$-homogeneous and $3$-homogeneous groups -- 9. Actions on $3$-sets compared to actions on larger sets -- 10. A transposition and an $n$-cycle -- 11. Asymptotic behavior of $g_k(E)$ -- 12. An $n$-cycle--the proof of Theorem 1.2.1 -- 13. 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. Memoirs of the American Mathematical Society, no. 886. Robert M. Guralnick, John Shareshian ; with an appendix by R. Guralnick and J. Stafford.

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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  Read more...

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