M{o}bius cancellation on polynomial sequences and the quadratic Bateman-Horn conjecture over function fields

We establish cancellation in short sums of certain special trace functions over $mathbb{F}_q[u]$ below the P{o}lya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the $mathbb{F}_q[u]$-analog of Chowlas conjecture on cancellation in M{o}bius sums over polynomial sequences, and of the Bateman-Horn conjecture in degree $2$, for some values of $q$. A final application is to sums of trace functions over primes in $mathbb{F}_q[u]$.

The Hanna Neumann conjecture for Demushkin Groups

We confirm the Hanna Neumann conjecture for topologically finitely generated closed subgroups $U$ and $W$ of a nonsolvable Demushkin group $G$. Namely, we show that begin{equation*} sum_{g in U backslash G/W} bar d(U cap gWg^{-1}) leq bar d(U) bar d(W) end{equation*} where $bar d(K) = max{d(K) - 1, 0}$ and $d(K)$ is the least cardinality of a topological generating set for the group $K$.

Balanced presentations for fundamental groups of curves over finite fields

We show that the algebraic fundamental group of a smooth projective curve over a finite field admits a finite topological presentation where the number of relations does not exceed the number of generators.

On the Chowla and twin primes conjectures over $mathbb F_q[T]$

Using geometric methods, we improve on the function field version of the Burgess bound, and show that, when restricted to certain special subspaces, the M{o}bius function over $mathbb F_q[T]$ can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to $1$ for the M{o}bius function in arithmetic progressions, and resolve Chowlas $k$-point correlation conjecture with large uniformity in the shifts. Using a function field variant of a result by Fouvry-Michel on exponential sums involving the M{o}bius function, we obtain a level of distribution beyond $1/2$ for irreducible polynomials, and establish the twin prime conjecture in a quantitative form. All these results hold for finite fields satisfying a simple condition.

Irreducibility of polynomials with a large gap

We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N > operatorname{deg}(c) + operatorname{deg}(d)$ such that the polynomial $$f_N(x) = x^N c(x^{-1}) + d(x)$$ is irreducible over $mathbb Q$, where $c, d in mathbb Z[x]$ are polynomials with nonzero constant terms and satisfying suitable conditions. As an application, we show that $x^N - k x^2 + 1$ is irreducible for all $N ge 5$ and $k in {3, 4, ldots, 24} setminus {9, 16}$. We also give a complete description of the factorization of polynomials of the form $x^N + k x^{N-1} pm (l x + 1)$ with $k, l in mathbb Z$, $k eq l$.

Virtual retraction and Howsons theorem in pro-$p$ groups

We show that for every finitely generated closed subgroup $K$ of a non-solvable Demushkin group $G$, there exists an open subgroup $U$ of $G$ containing $K$, and a continuous homomorphism $tau colon U to K$ satisfying $tau(k) = k$ for every $k in K$. We prove that the intersection of a pair of finitely generated closed subgroups of a Demushkin group is finitely generated (giving an explicit bound on the number of generators). Furthermore, we show that these properties of Demushkin groups are preserved under free pro-$p$ products, and deduce that Howsons theorem holds for the Sylow subgroups of the absolute Galois group of a number field. Finally, we confirm two conjectures of Ribes, thus classifying the finitely generated pro-$p$ M. Hall groups.

Gradients of sequences of subgroups in a direct product

For a sequence ${U_n}_{n = 1}^infty$ of finite index subgroups of a direct product $G = A times B$ of finitely generated groups, we show that $$lim_{n to infty} frac{min{|X| : langle X rangle = U_n}}{[G : U_n]} = 0$$ once $[A : A cap U_n], [B : B cap U_n] to infty$ as $n to infty$. Our proof relies on the classification of finite simple groups. For $A,B$ that are finitely presented we show that $$ lim_{n to infty} frac{log |mathrm{Torsion}(U_n^{mathrm{ab}})|}{[G : U_n]} = 0. $$

Schreiers formula for Prosupersolvable groups

We classify the finitely generated prosupersolvable groups that satisfy Schreiers formula for the number of generators of open subgroups.

An axiomatizable profinite group with infinitely many open subgroups of index 2

We show that a profinite group with the same first-order theory as the direct product over all odd primes $p$ of the dihedral group of order $2p$, is necessarily isomorphic to this direct product.