Polynomials with PSL(2) monodromy

TL;DR

This paper classifies certain polynomials over fields of positive characteristic with PSL(2) monodromy, describing their structure, decomposability, and exceptionality, and provides a comprehensive understanding of their Galois groups and ramification properties.

Contribution

It determines all polynomials with PSL(2) monodromy under specific conditions and characterizes their decomposability and exceptionality over fields of positive characteristic.

Findings

Classified polynomials with PSL(2) monodromy in characteristic p.

Described conditions for polynomial decomposability over extensions.

Identified all exceptional indecomposable polynomials of non-p-power degree.

Abstract

Let K be a field of characteristic p>0, and let q be a power of p. We determine all polynomials f in K[t]\K[t^p] of degree q(q-1)/2 such that the Galois group of f(t)-u over K(u) has a transitive normal subgroup isomorphic to PSL_2(q), subject to a certain ramification hypothesis. As a consequence, we describe all polynomials f in K[t], of degree not a power of p, such that f is functionally indecomposable over K but f decomposes over an extension of K. Moreover, except for one ramification setup (which is treated in the companion paper arxiv:0707.1837), we describe all indecomposable polynomials f in K[t] of non-p-power degree which are exceptional, in the sense that x-y is the only absolutely irreducible factor of f(x)-f(y) which lies in K[x,y]. It is known that, when K is finite, a polynomial f is exceptional if and only if it induces a bijection on infinitely many finite extensions of K.