Symmetric and Alternating Groups As Monodromy Groups of Riemann Surfaces, Volume 1

Generic Covers and Covers with Many Branch Points - with an Appendix by R. Guralnick and R. Stafford

Robert Guralnick, John Shareshian (Autoren)

Buch | Softcover

128 Seiten

2007 | illustrated Edition
American Mathematical Society (Verlag)
9780821839928 (ISBN)

Infos

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 $n =d(d-1)/2$.

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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.

Inhalt

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.

Produkt-Details

• Erscheint lt. Verlag: 1.9.2007
• Reihe/Serie: Memoirs of the American Mathematical Society
• Zusatzinfo: illustrations
• Verlagsort: Providence
• Sprache: englisch
• Gewicht: 267 g
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• ISBN-13: 9780821839928 / 9780821839928
• Zustand: Neuware