On Boston's Unramified Conjecture for $\GL_2$ and McLeman's $(3,3)$-Conjecture

Abstract

Let $p$ be an odd prime number. Based on recent work of Zhang, we prove that for a finite set $S$ of primes of $\Q$ containing $\infty$ but not $p$, any continuous odd representation $G_{\Q,S} \to \GL_2(A)$ over a complete Noetherian local ring $A$ with finite residue field of characteristic $p$ has finite image, where $G_{\Q,S}$ denotes the Galois group of the maximal extension of $\Q$ unramified outside $S$. This proves the two-dimensional odd case of Boston's strengthening of the unramified Fontaine--Mazur conjecture over $\Q$. Furthermore, we show that for an imaginary quadratic field $K$, any continuous conjugate self-dual two-dimensional $p$-adic representation of its pro-$p$ Galois group $G_{K,S}(p)$ has finite image, provided the primes in $S$ satisfy a modest condition. As an application, we resolve the sufficiency direction of McLeman's $(3,3)$-conjecture on $p$-class field towers for $p>3$. Namely, we prove that if $K$ is an imaginary quadratic field with $p$-class rank two and the Galois group $G_{K,\varnothing}(p)$ of the maximal unramified $p$-extension of $K$ has Zassenhaus type $(3,3)$, then the $p$-class field tower of $K$ is finite. For $p=3$, the results of Ahlqvist and Pink give the same finiteness conclusion in ten of the thirteen possible cases for the fourth Zassenhaus quotient.