Explicit Galois Deformations over Imaginary Quadratic Fields

Abstract

Let $K$ be an imaginary quadratic field, and let $p$ be an odd prime that splits as $(p)=\pi\bar{\pi}$ in $K$. Let $G_K^\pi$ and $G_K^p$ denote the Galois groups of the maximal algebraic extensions of $K$ unramified outside $\pi$ and outside the primes above $p$, respectively. In this paper, we construct certain Iwasawa-theoretic quotients of $G_K^\pi$ and $G_K^p$ and study the universal deformation rings of the induced residual representations. The defining relations of these quotient deformation rings are described through characteristic elements of Iwasawa modules, and hence through Katz $p$-adic $L$-functions.