On Frobenius semisimplicity in Hida families

Abstract

Let $p\ge 5$ be a prime and $\ell \ne p$ be a prime not dividing the tame level of a $p$-ordinary Hida family. We prove that the actions of the Frobenius element at $\ell$ on the Galois representations attached to almost all arithmetic specializations are semisimple and non-scalar. If $k_f$ denotes the weight of a cusp form $f(z)={\sum }_{n\ge 1}{a}_{\ell }(f)e^{2\pi inz}$, then the inequality $$|a_{\ell }(f)|\le 2{\ell }^{(k_f-1)/2},$$ predicted by the Ramanujan conjecture, is a strict inequality for almost all members $f$ of the family.