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Frobenius groups as monodromy groups. (English) Zbl 1195.14040

This short note studies actions of Frobenius groups on curves. By definition, a Frobenius group is a finite group \(G\) that has a proper non-trivial subgroup \(H\) such that \(H\cap H^g=\{1\}\) for all \(g\in G\setminus H\). Both \(H\) and \(N=(G\setminus \bigcup_{g\in G}H^g)\cup\{1\}\) are uniquely determined subgroups of \(G\), called Frobenius complement and Frobenius kernel.
The main result is Corollary 3.1 (which in the paper itself is referred to as ‘Theorem 3.1’): Let \(G\) be a Frobenius group and \(X\) a smooth projective curve over an algebraically closed field \(k\). Let \(G\) act faithfully as a group of automorphisms on \(X\) such that \(X/G\) is of genus zero, or, in other words, fix a \(G\)-Galois cover \(X\rightarrow\mathbb{P}^1\). Then the genus \(g(X)\) of \(X\) can be expressed as \[ g(X) = g(X/N) + g(X/H)|H|. \]
From this result, the author then deduces classifications of Frobenius group actions on curves in certain cases. The special case \(g(X/H)=0\) of the main result was proven already in [J. Flynn, Near-exceptionality over finite fields, PhD thesis, Berkeley, (2001)] and [R. M. Guralnick, Rational functions with monodromy group a Frobenius group, preprint, (2000)]. In this special case it follows that \(g(X)\leq1\).
The proof of the main result applies tools from representation theory to the action of \(G\) on the Tate module of \(X\). In the final section, the author considers a more general class of group actions on curves and again deduces some genus estimates.

MSC:

14H30 Coverings of curves, fundamental group
14H05 Algebraic functions and function fields in algebraic geometry
12F10 Separable extensions, Galois theory
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
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References:

[1] DOI: 10.1073/pnas.45.4.578 · Zbl 0086.25101 · doi:10.1073/pnas.45.4.578
[2] Guralnick, Monodromy Groups of Coverings of Curves, Galois Groups and Fundamental Groups pp 1– (2003) · Zbl 1071.20001
[3] Guralnick, Mem. Amer. Math. Soc. 162 (2003)
[4] Silverman, The Arithmetic of Elliptic Curves (1992) · Zbl 0585.14026
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