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

Abstract

Fix a prime $p>2$. Let $\rho :Gal\left(\overline{\mathbb{Q}}∕\mathbb{Q}\right)\to {GL}_2\left(\mathbb{I}\right)$ 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 ${\mathbb{Z}}_p\left[\left[T\right]\right]$ such that the image of $\rho$ is large with respect to ${\mathbb{I}}_0$. That is, $Im\rho$ contains $ker\left({SL}_2\left({\mathbb{I}}_0\right)\to {SL}_2\left({\mathbb{I}}_0∕\mathfrak{a}\right)\right)$ for some nonzero ${\mathbb{I}}_0$-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 ${\mathbb{Z}}_p\left[\left[T\right]\right]$. 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.