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arXiv · 1801.03070

On Frobenius semisimplicity in Hida families

Jyoti Prakash Saha
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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.

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Jyoti Prakash Saha. 2018-01-09. On Frobenius semisimplicity in Hida families. https://arxiv.org/abs/1801.03070

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