Galois representations ramified at one prime via relative deformation theory

Abstract

Let $p$ be an odd prime and let $\mathbf{G}/\mathbb Z$ be a split connected reductive group with $\operatorname{dim} Z(\mathbf{G})\leq1$. Assume that a split maximal torus of $\mathbf{G}$ admits a cocharacter whose pairing with every simple root is odd, and impose an explicit root-theoretic condition on the reduction of $\mathrm{Lie}(\mathbf{G}^{\mathrm{der}})$ modulo $p$. We construct infinitely many continuous representations $ρ:G_{\{p\}}\longrightarrow\mathbf{G}(\mathbb{Z}_p)$ which are unramified at every finite prime different from $p$ and whose images contain a principal congruence subgroup of $\mathbf G^{\mathrm{der}}(\mathbb{Z}_p)$. When the center has dimension one, the representations may be chosen to have open image in $\mathbf{G}(\mathbb{Z}_p)$. The construction continues the author's earlier work on $\mathrm{GL}_n$-valued representations ramified at one prime, but replaces the residual unobstructedness used there by a relative lifting argument. For $\mathbf{G}=\mathrm{GL}_n$ all the required root-theoretic conditions are automatic for every odd prime. We therefore obtain, for every odd $p$ and every $n>1$, infinitely many representations $G_{\{p\}}\to\mathrm{GL}_n(\mathbb{Z}_p)$ with open image. In particular, this removes the weak Vandiver-type hypothesis occurring in the author's previous work, as well as the restriction $p\geq7$ in that construction.