On the image of the Galois representation associated to a non-CM Hida family

Abstract: Fix a prime $p > 2$. Let $\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{I})$ be the Galois representation coming from a non-CM irreducible component $\mathbb{I}$ of Hida's $p$-ordinary Hecke algebra. Assume the residual representation $\bar{\rho}$ is absolutely irreducible. Under a minor technical condition we identify a subring $\mathbb{I}_0$ of $\mathbb{I}$ containing $\mathbb{Z}_p[[T]]$ such that the image of $\rho$ is large with respect to $\mathbb{I}_0$. That is, $\text{Im} \rho$ contains $\text{ker}(\text{SL}_2(\mathbb{I}_0) \to \text{SL}_2(\mathbb{I}_0/\mathfrak{a}))$ for some non-zero $\mathbb{I}_0$-ideal $\mathfrak{a}$. This paper builds on recent work of Hida who showed that the image of such a Galois representation is large with respect to $\mathbb{Z}_p[[T]]$. Our result is an $\mathbb{I}$-adic analogue of the description of the image of the Galois representation attached to a non-CM classical modular form obtained by Ribet and Momose in the 1980s.