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Abstract:We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P (x) with L regular over any PAC field P of characteristic zero. A special case of this implies that G is a Galois group over F p (x) for almost all primes p. §0. INTRODUCTIONMany attempts have been made to realize finite groups as Galois groups of extensions of Q(x) that are regular over Q (see the end of this introduction for definitions). We call this the "regular inverse Galois problem." We show that to each finite group G with trivial center and integer r ≥ 3 there is canonically associated an algebraic variety, H in r (G), defined over Q (usually reducible) satisfying the following. It is convenient to introduce the following terminology: A group G is called regular over a field k if G is isomorphic to the Galois group of an extension of k(x) that is regular over k. The above has the following immediate corollary for P(seudo)A(lgebraically)C(losed) fields P of characteristic 0: Every finite group is regular over P (Theorem 2). Another corollary (which can be viewed as a special case of the previous one) is that every finite group is regular over the finite prime field F p for almost all primes p (Corollary 2).In §6 we derive an addendum to our main result that is crucial for the preprint [FrVo]. In that paper we prove a long-standing conjecture on Hilbertian PAC-fields P (in the case char(P ) = 0): Every finite embedding problem over P is solvable. For countable P this, combined with a result of Iwasawa, implies that the absolute Galois group of P is ω-free. That is, G(P /P ) is a free profinite group of countably infinite rank, denotedF ω . By a result of [FrJ, 2], every countable Hilbertian field k of characteristic 0 has a Galois extension P with the following properties: P is Hilbertian and PAC, and G(P/k) ∼ = ∞ n=2 S n (where S n is the symmetric group of degree n). From the above, G(k/P ) = G(P /P ) ∼ =Fω, and we get the exact sequenceModuli spaces for branched covers of P 1 were already considered by Hurwitz [Hur] in the special case of simple branching (where the Galois group is S n ). Fulton [Fu] showed -still in the case of simple branching -that the analytic moduli spaces studied by Hurwitz are the sets of complex points of certain schemes. Fried [Fr,1] studied more generally moduli spaces for covers of P 1 with an arbitrary given monodromy group G ⊂ S n and with a fixed number of branch points. The new moduli spaces H in r (G) studied in the present paper are coverings of those previous ones, parametrizing equivalence classes of pairs (χ, h) where χ is a Galois cover of P 1 with r branch points and h is an isomorphism between G and the automorphism group of the cover χ. The extra data of the isomorphism h associated to the points of H in r (G) ensures that a Qrational point corresponds to a cover of P 1 that can be defined over Q such that also all ...

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Abstract:We use the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz's conjecture (1966). An exceptional polynomial f over a finite field F q is a polynomial that is a permutation polynomial on infinitely many finite extensions of F q . Carlitz's conjecture says f must be of odd degree (if q is odd). Indeed, excluding characteristic 2 and 3, arithmetic monodromy groups of exceptional polynomials must be affine groups.We don't, however, know which affine groups appear as the geometric monodromy group of exceptional polynomials. Thus, there remain unsolved problems. Riemann's existence theorem in positive characteristic will surely play a role in their solution. We have, however, completely classified the exceptional polynomials of degree equal to the characteristic. This solves a problem from Dickson's thesis (1896). Further, we generalize Dickson's problem to include a description of all known exceptional polynomials.Finally: The methods allow us to consider covers X → P 1 that generalize the notion of exceptional polynomials. These covers have this property: Over each F q t point of P 1 there is exactly one F q t point of X for infinitely many t. Thus X has a rare diophantine property when X has genus greater than 0. It has exactly q t + 1 points in F q t for infinitely many t. This gives exceptional covers a special place in the theory of counting rational points on curves over finite fields explicitly. Corollary 14.2 holds also for an indecomposable exceptional cover having (at least) one totally ramified place over a rational point of the base. Its arithmetic monodromy group is an affine group.• Supported by NSA grant MDA 14776 and BSF grant 87-00038.• 2 First author supported by the Institute for Advanced Studies in Jerusalem and IFR Grant #90/91-15.• 3 Supported by NSF grant DMS 91011407. Dedication:To the contributions of John Thompson to the classification of finite simple groups; and to the memory of Daniel Gorenstein and the success of his project to complete the classification.