On Frobenius semisimplicity in Hida families

TL;DR

This paper proves that Frobenius actions on Galois representations in Hida families are mostly semisimple and non-scalar, and confirms the Ramanujan bound as strict for almost all forms in the family.

Contribution

It establishes Frobenius semisimplicity and non-scalar action for almost all specializations in Hida families, and verifies the strict Ramanujan bound for these forms.

Findings

Frobenius actions are semisimple and non-scalar for almost all specializations.

The Ramanujan bound is strictly satisfied for almost all forms.

Provides new insight into the structure of Galois representations in Hida families.

Abstract

Let $p\geq 5$ be a prime and $\ell\neq p$ 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 $f(z)= \sum_{n\geq 1} a_\ell(f) e^{2\pi i n z}$, then the inequality $$|a_\ell(f) | \leq 2 \ell^{(k_f-1)/2},$$ predicted by the Ramanujan conjecture, is a strict inequality for almost all members $f$ of the family.