Relating Two Genus 0 Problems of John Thompson

Abstract

Abstract. Excluding a precise list of groups like alternating, symmetric, cyclic and dihedral, from 1st year algebra ( §7.2.3), we expect there are only finitely many monodromy groups of primitive genus 0 covers. Denote this nearly proven genus 0 problem as Problem g=0 2 . We call the exceptional groups 0-sporadic. Example: Finitely many Chevalley groups are 0-sporadic. A proven result: Among polynomial 0-sporadic groups, precisely three produce covers falling in nontrivial reduced families. Each (miraculously) defines one natural genus 0 Q cover of the j-line. The latest Nielsen class techniques apply to these dessins d'enfant to see their subtle arithmetic and interesting cusps.John Thompson earlier considered another genus 0 problem: To find θ-functions uniformizing certain genus 0 (near) modular curves. We call this Problem g=01 . We pose uniformization problems for j-line covers in two cases. First: From the three 0-sporadic examples of Problem g=0 2 . Second: From finite collections of genus 0 curves with aspects of Problem g=01 .

A curve algebraically but not rationally uniformized by radicals

“…One is therefore led to study the Galois groups of primitive coverings X → P 1 with X general, and of genus zero coverings. There have been many developments on the genus zero problem: see [5][6][7]9,11]. As for the primitive generic case, in a series of papers [12][13][14][15] it is proven that, for any fixed g 4, the Galois group of a primitive covering f : X → P 1 , with X general in M g , is either A n or S n , with n > (g + 1)/2, a statement that strengthen Zariski's result.…”