关于Hida族的Frobenius半单性
On Frobenius semisimplicity in Hida families
摘要：
设$p\geq 5$是素数，$\ell\neq p$是不分$p$ -普通Hida族的驯服水平的素数.我们证明了Frobenius元素在$\ell$上对几乎所有算术专业化的伽罗瓦表示的作用是半单的和非标量的。若$k_f$表示尖点形式$f(z)= \sum_{n\geq 1} a_\ell(f) e^{2πi n z}$的权，则Ramanujan猜想所预言的不等式$$|a_\ell(f) | \leq 2 \ell^{(k_f-1)/2},$$，对几乎所有的族$f$都是严格不等式.
英文摘要：
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 $f(z)= \sum_{n\geq 1} a_\ell(f) e^{2πi n z}$, 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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