Robert M. Guralnick and Michael E. Zieve:
Polynomials with PSL(2) monodromy,
Annals of Math. 172 (2010), 1315–1359.

(Both the published version and the arXiv version are available online.)

Let k be a field of characteristic p≥0, and let g be a polynomial over k which is indecomposable (not a composition of lower-degree polynomials over k). Guralnick and Saxl showed that, if g decomposes over an extension of k, then the degree of g is either a power of  p or 21 or 55. We begin by determining all possibilities with the latter degrees. It turns out that there exists such a polynomial of degree 21 if and only if  p=7 and k contains nonsquares, in which case the polynomials can be presented explicitly. A similar conclusion holds for degree 55.

A polynomial g over Fq is called exceptional if the map ag(a) induces a bijection on Fqn for infinitely many n. The composition of two polynomials is exceptional if and only if both polynomials are exceptional, so it suffices to study the indecomposable exceptional polynomials. Letting  p denote the characteristic of  FqFried, Guralnick and Saxl showed that any indecomposable exceptional polynomial has degree either prime or a power of  p, except perhaps when  p ≤ 3 in which case they could not rule out the possibility that the degree is  pr(pr-1)/2  with r > 1 odd. Soon afterward, Müller, Cohen–Matthews, and Lenstra–Zieve produced indecomposable exceptional polynomials having all degrees allowed by the Fried–Guralnick–Saxl result. We determine all indecomposable exceptional polynomials of non-prime power degree, except for one ramification possibility which we treat in a companion paper with Joel Rosenberg. The conclusion is that, except for the new family of polynomials discovered in the latter paper, all such indecomposable exceptional polynomials are twists of the examples discovered previously.

We solve these two problems as consequences of a more general problem. Let k be a field of characteristic p>0, let q be a power of  p, and let t be transcendental over k. We determine all polynomials g in k[X] of degree q(q-1)/2 for which g(X)-t has simple roots and its Galois group over k(t) has a transitive normal subgroup isomorphic to PSL(2,q). These include the polynomials discussed in the previous two paragraphs. More generally, Guralnick and Saxl have produced a list of groups satisfying the known conditions necessary for occurring as the Galois group of g(X)-t over K(t), where g is a polynomial over an algebraically closed field K. Our result determines all such polynomials for one of the main families of groups on the Guralnick–Saxl list; this complements work of Abhyankar, who has exhibited polynomials realizing some of the other families of groups on the list.

Our strategy for proving these results is as follows. Let g be a polynomial of degree q(q-1)/2 over an algebraically closed field K of characteristic p, where q is a power of  p. Let Ω be the splitting field of g(X)-t over K(t), and suppose that G := Gal(Ω/K(t)) normalizes PSL(2,q). Then G is contained in the automorphism group PΓL(2,q) of PSL(2,q). We use the classification of subgroups of PΓL(2,q), together with Hilbert's different formula, to determine all possibilities for the inertia and higher ramification groups in Ω/K(t) that are consistent with K(x) having genus zero. For each such possibility, we determine all candidates for Ω: this is the most difficult step, and involves various ingredients. We then determine the automorphism group of each of these (infinitely many) candidates for Ω, and use invariant theory to compute the subfields of Ω corresponding to K(t) and K(x), and finally to compute the polynomial g. This solves the geometric part of our problem; to solve the original arithmetic problems, we use the factorizations of g(X)-g(Y) determined previously by Cohen–Matthews and Zieve.

Our results should be viewed in the context of Abhyankar's conjecture (the Raynaud/Harbater theorem). Given a smooth projective irreducible curve C over an algebraically closed field of characteristic p, and a finite set S of points on C, this conjecture describes the groups occurring as Galois groups of covers of C whose branch locus is contained in S. However, this work does not address the problem of determining the dimension of the moduli space of covers having prescribed Galois group and prescribed ramification data, let alone the problem of constructing the covers explicitly. The p=0 case of Abhyankar's conjecture follows from Riemann's existence theorem; however, even in that situation, the proof is nonconstructive. We have constructed all polynomial covers of  P1 realizing certain permutation representations of PSL(2,q). In future work we hope to extend this to other classes of groups, which will provide data towards a conjecture on the dimension of the moduli space of covers having prescribed Galois group and prescribed ramification.