Michael D. Fried

The inverse Galois problem and rational points on moduli spaces.

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 Fp(x) for almost all primes p.

Schur covers and Carlitz's conjecture

AbstractWe use the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz’s conjecture (1966). An exceptional polynomialf over a finite field

The embedding problem over a Hilbertian PAC-field

{\mathbb{F}}_q

Galois stratification over Frobenius fields

is a polynomial that is a permutation polynomial on infinitely many finite extensions of

Realizing alternating groups as monodromy groups of genus one covers

{\mathbb{F}}_q

Alternating groups and moduli space lifting invariants

. Carlitz’s conjecture saysf must be of odd degree (ifq 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 metric 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 coversX→