Polynomials with $\operatorname{PSL}(2)$ monodromy | Annals of Mathematics

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∈k[X]∖k[Xp] 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 PSL2(q), subject to a certain ramification hypothesis. As a consequence, we describe all polynomials f∈k[X] such that 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∈k[X] such that 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.

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数学年刊(Annals of Mathematics)

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