Cyclicity of adjoint Selmer groups and fundamental units

Abstract

We study the universal minimal ordinary Galois deformation $\rho_{\mathbb{T}}$ of an induced representation $\operatorname{Ind}_F^{\mathbb{Q}} \varphi$ from a real quadratic field $F$ with values in $\mathrm{GL}_2(\mathbb{T})$. By Taylor–Wiles, the universal ring $\mathbb{T}$ is isomorphic to a local ring of a Hecke algebra. Combining an idea of Cho–Vatsal [CV03] with a modified Taylor–Wiles patching argument in [H17], under mild assumptions, we show that the Pontryagin dual of the adjoint Selmer group of $\rho_{\mathbb{T}}$ is canonically isomorphic to $\mathbb{T}/(L)$ for a non-zero divisor $L \in \mathbb{T}$ which is a generator of the different $\mathfrak{d}_{\mathbb{T}/\Lambda}$ of $\mathbb{T}$ over the weight Iwasawa algebra $\Lambda=W[[T]]$ inside $\mathbb{T}$. Moreover, defining $\langle\varepsilon\rangle := (1+T)^{\log_p(\varepsilon)/\log_p(1 + p)}$ for a fundamental unit $\varepsilon$ of the real quadratic field $F$, we show that the adjoint Selmer group of $\operatorname{Ind}_F^\mathbb{Q}\Phi$ for the (minimal) universal character $\Phi$ deforming $\varphi$ is isomorphic to $\Lambda/(\langle\varepsilon\rangle - 1)$ as $\Lambda$-modules.