Polynomials with $\operatorname{PSL}(2)$ monodromy

Abstract

Let $k$ be a field of characteristic $p>0$, let $q$ be a power of $p$, and let $u$ be transcendental over $k$. We determine all polynomials $f\in k[X]\setminus k[X^p]$ of degree $q(q-1)/2$ for which the Galois group of $f(X)-u$ over $k(u)$ has a transitive normal subgroup isomorphic to $\operatorname{PSL}_2(q)$, subject to a certain ramification hypothesis. As a consequence, we describe all polynomials $f\in k[X]$ such that $\operatorname{deg}(f)$ is not a power of $p$ and $f$ is functionally indecomposable over $k$ but $f$ decomposes over an extension of $k$. Moreover, except for one ramification configuration (which is handled in a companion paper with Rosenberg), we describe all indecomposable polynomials $f\in k[X]$ such that $\operatorname{deg}(f)$ is not a power of $p$ and $f$ is 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 $k$.