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.

30 pages

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