Exercise 9.2.5

Proof.

By Exercise 4 , we obtain (9.12):

We must prove that every period $(f,{\lambda }_j),j=1,\dots ,d,$ is of the form $(f,{\lambda }_j)=\sigma (\eta )=\sigma ((f,\lambda ))$, where $\sigma \in \mathrm{Gal}(\mathbb{Q}({\zeta }_p)∕L_{f^{\prime }})$.

Write $[i]={[\lambda ]}^{-1}[{\lambda }_j]$. As $[{\lambda }_j]\in [\lambda ]H_{f^{\prime }}$, then $[i]={[\lambda ]}^{-1}[{\lambda }_j]\in H_{f^{\prime }}$.

Since $[{\lambda }_j]=[\mathit{i\lambda }]$,

Let $\sigma \in \mathrm{Gal}(\mathbb{Q}({\zeta }_p)∕L_{f^{\prime }})$ be defined by $\sigma ({\zeta }_p)={\zeta }_p^i$, where $[i]\in H_{f^{\prime }}$, so by Lemma 9.2.4(c),

Every $(f,{\lambda }_j),j=1,\dots ,d,$ is a conjugate of $(f,\lambda )$ over $L_{f^{\prime }}$. □