Global Construction of General Exceptional Covers with motivation for applications to encoding

The paper [FGS] uses the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz’s conjecture (1966). We consider only separable polynomials; their derivative is nonzero. Then, f ∈ Fq [x] is exceptional if it acts as a permutation map on infinitely many finite extensions of the finite field Fq , q = pa for some prime p. Carlitz’s conjecture says f must be of odd degree (if p is odd). The main theorem of [FGS; Theorem 14.1] restricts the list of possible geometric monodromy groups of exceptional indecomposable polynomials (§1.1): either p = 2 or 3 or these must be affine groups. The proof of Carlitz’s conjecture motivates considering general exceptional covers of nonsingular projective algebraic curves. For historical reasons we sometimes call these Schur covers [Fr2]. Suppose φ : X → P1 is an exceptional cover over Fq . Then, for some integer s, there is a unique x ∈ X(Fqt ) over each z ∈ P(Fqt ) for each integer t with (t, s) = 1. In particular, |X(Fqt )| = qt+1 when (t, s) = 1. We include a complete proof that exceptionality is equivalent to a statement about the geometric/arithmetic monodromy pair of the cover. Theorem 2.5 shows all geometric/arithmetic monodromy pairs satisfying necessary conditions (§1.1–§1.2) derive from covers over Fp for all suitably large primes p. Other topics: (i) How modular curve points over finite fields explicitly produce rational function exceptional covers of prime degree (Corollary 3.5). (ii) How fiber products produce abundant general exceptional covers