In this talk we look at polynomials having the property that all compositional iterates are irreducible, which we call dynamical irreducible. After surveying some previous results (mostly over finite fields), we will concentrate on the question of the dynamical irreducibility of integer polynomials being preserved in reduction modulo primes. More precisely, for a class of integer polynomials $f$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions, he set of primes $p$ such that $f$ is dynamical irreducible modulo $p$ is of relative density zero. The proof of this result relies on a combination of analytic (the square sieve) and diophantine (finiteness of solutions to certain hyperelliptic equations) tools, which we will briefly describe.[-]

Given a finite connected undirected graph $X$, its fundamental group plays the role of the absolute Galois group of $X$. The familiar Galois theory holds in this setting. In this talk we shall discuss graph theoretical counter parts of several important theorems for number fields. Topics include
(a) Determination, up to equivalence, of unramified normal covers of $X$ of given degree,
(b) Criteria for Sunada equivalence,
(c) Chebotarev density theorem.
This is a joint work with Hau-Wen Huang.[-]

The number $F(h)$ of imaginary quadratic fields with class number $h$ is of classical interest: Gauss' class number problem asks for a determination of those fields counted by $F(h)$. The unconditional computation of $F(h)$ for $h \le 100$ was completed by M. Watkins, and K. Soundararajan has more recently made conjectures about the order of magnitude of $F(h)$ as $h \to \infty$ and determined its average order.
For odd $h$ we refine Soundararajan's conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number $F(G)$ of imaginary quadratic fields with class group isomorphic to a given finite abelian group $G$.
Using Watkins' tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance $(\mathbb{Z}/3\mathbb{Z})^3$ does not). This observation is explained in part by the Cohen-Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine the Cohen-Lenstra heuristics with a refinement of Soundararajan's conjecture to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of “missing” class groups, for families of $p$-groups as $p$ tends to infinity. For instance, it appears that no groups of the form $(\mathbb{Z}/p\mathbb{Z})^3$ and $p$ prime occurs as a class group of a quadratic imaginary field.
Conditionally on the Generalized Riemann Hypothesis, we extend Watkins' data, tabulating $F(h)$ for odd $h \le 10^6$ and $F(G)$ for $G$ a $p$-group of odd order with $|G| \le 10^6$. (To do this, we examine the class numbers of all negative prime fundamental discriminants $-q$, for $q \le 1.1881 \cdot 10^{15}.$) The numerical evidence matches quite well with our conjectures.
This is joint work with S. Holmin, N. Jones, C. McLeman, and K. Petersen.[-]