eBook ISBN:  9781470404901 
Product Code:  MEMO/189/886.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 
eBook ISBN:  9781470404901 
Product Code:  MEMO/189/886.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 189; 2007; 128 ppMSC: Primary 14; 20
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 \(n =d(d1)/2\).
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 \(n=d(d1)/2\). 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.

Table of Contents

Chapters

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 2sets — the proof of Theorem 4.0.30

6. Actions on 3sets — 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 2homogeneous and 3homogeneous groups

9. Actions on 3sets 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


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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 \(n =d(d1)/2\).
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 \(n=d(d1)/2\). 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.

Chapters

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 2sets — the proof of Theorem 4.0.30

6. Actions on 3sets — 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 2homogeneous and 3homogeneous groups

9. Actions on 3sets 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