In this paper, we solve a function field analogue of classical problems in analytic number theory, concerning the autocorrelations of divisor functions, in the limit of a large finite field.

We prove the following theorem for a finitely generated field K: Let M be a Galois extension of K which is not separably closed. Then M is not PAC over K.

The notion of ‘Pseudo Algebraically Closed (PAC) extensions’ is a generalization of the classical notion of PAC fields. It was originally motivated by Hilbert’s tenth problem, and recently had new… (More)

We extend Haran’s Diamond Theorem to closed subgroups of a finitely generated free profinite group. This gives an affirmative answer to Problem 25.4.9 in [FrJ].

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite… (More)

We analyze the rel leaves of the Arnoux-Yoccoz translation surfaces. We show that for any genus g ě 3, the leaf is dense in the connected component of the stratum Hpg ́1,g ́1q to which it belongs,… (More)

The main goal of this work is to answer a question of D` ebes and Haran by relaxing the condition for Hilbertianity. Namely we prove that for a field K to be Hilbertian it suffices that K has the… (More)

We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions… (More)

Given a power q of a prime number p and “nice” polynomials f1, . . . , fr ∈ Fq[T, X] with r = 1 if p = 2, we establish an asymptotic formula for the number of pairs (a1, a2) ∈ Fq such that f1(T, a1T… (More)