Relating two genus 0 problems of John Thompson:

Excluding a precise list of groups like alternating, symmetric, cyclic and dihedral, from 1st year algebra, we expect there are only finitely many monodromy groups of primitive genus 0 covers. Denote this nearly proven genus 0 problem as Prob 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 Prob ₁. We pose uniformization problems for j-line covers in two cases. First: From the three 0-sporadic examples of Prob ₂. Second: From finite collections of genus 0 curves with aspects of Prob ₁.