Yuan Liu

We consider the distribution of the Galois groups $\operatorname{Gal}(K^{\operatorname{un}}/K)$ of maximal unramified extensions as $K$ ranges over $\Gamma$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We prove two properties of $\operatorname{Gal}(K^{\operatorname{un}}/K)$ coming from number theory, which we use as motivation to build a probability distribution on profinite groups with these properties. In Part I, we build such a distribution as a limit of distributions on $n$-generated profinite groups. In Part II, we prove as $q\rightarrow\infty$, agreement of $\operatorname{Gal}(K^{\operatorname{un}}/K)$ as $K$ varies over totally real $\Gamma$-extensions of $\mathbb{F}_q(t)$ with our distribution from Part I, in the moments that are relatively prime to $q(q-1)|\Gamma|$. In particular, we prove for every finite group $\Gamma$, in the $q\rightarrow\infty$ limit, the prime-to-$q(q-1)|\Gamma|$-moments of the distribution of class groups of totally real $\Gamma$-extensions of $\mathbb{F}_q(t)$ agree with the prediction of the Cohen--Lenstra--Martinet heuristics.

Let $k$ be an algebraically closed field of characteristic $p$ and let $X$ the projective line over $k$ with three points removed. We investigate which finite groups $G$ can arise as the monodromy group of finite \'{e}tale covers of $X$ that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime $p\ge 5$, there are families of tamely ramified covers with monodromy the symmetric group $S_n$ or alternating group $A_n$ for infinitely many $n$. These covers come from the moduli spaces of elliptic curves with $PSL_2(\mathbb{F}_\ell)$-structure, and the analysis uses work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder, about Markoff triples modulo $\ell$.

We say a closed point $x$ on a curve $C$ is sporadic if $C$ has only finitely many closed points of degree at most $\operatorname{deg}(x)$ and that $x$ is isolated if it is not in a family of effective degree $d$ divisors parametrized by $\mathbb{P}^1$ or a positive rank abelian variety (see Section 4 for more precise definitions and a proof that sporadic points are isolated). Motivated by well-known classification problems concerning rational torsion of elliptic curves, we study sporadic and isolated points on the modular curves $X_1(N)$. In particular, we show that any non-cuspidal non-CM sporadic, respectively isolated, point $x \in X_1(N)$ maps down to a sporadic, respectively isolated, point on a modular curve $X_1(d)$, where $d$ is bounded by a constant depending only on $j(x)$. Conditionally, we show that $d$ is bounded by a constant depending only on the degree of $\mathbb{Q}(j(x))$, so in particular there are only finitely many $j$-invariants of bounded degree that give rise to sporadic or isolated points.