I am trying to understand Weierstrass subdomains of $\Spm\DeclareMathOperator\QP{\mathbb{Q}_p}\QP$.
Recall that a Weierstrass algebra of an affinoid space $\Spm A$, where $A$ is a Banach algebra with norm $|\cdot|$, associated to an $n$-tuple $(a_1,...,a_n)\in A^n$ is the relative Tate algebra:
$$A\langle X_1,...,X_n\rangle/(X_1-a_1, ..., X_n - a_n)$$
where $A\langle X\rangle = \sum_{n\ge 0}a_nX^n$, where $|a_n|\longrightarrow 0$ as $n\longrightarrow 0$. More generally one has to work with multi-indices.
The Weierstrass algebra associated to an $n$-tuple $(a_1,...,a_n)$ is nothing but $A\langle a_1,...,a_n\rangle$.
The natural map $A \longrightarrow A\langle a_1,...,a_n\rangle$ induces a map on the max spectra in the opposite direction, $M(A\langle a_1,...,a_n\rangle)\longrightarrow M(A)$, and in Conrad's "several approaches to ...", he says that the image of $M(A\langle a_1,...,a_n\rangle)$ inside $M(A)$ consists of points $x\in M(A)$ for which $|a_i(x)|\le 1$ for all $1\le i\le n$.
Consider the case $A = \QP$, and the Weierstrass algebra associated to the $1$-tuple $1/p$, i.e. $\QP\langle 1/p\rangle$. One could show very easily that as abstract rings,
$$\QP\langle 1/p\rangle \cong \sum_{n=-\infty}^{\infty}a_np^n,\quad a_n\in [0,p-1].$$
This ring is a field, hence $M(\QP\langle 1/p\rangle) = \{0\}$. Similarly $M(\QP) = \{0\}$, so the map on spectra sends $\{0\} \in M(\QP\langle 1/p\rangle)$ to $\{0\}\in M(\QP)$, and by Conrad's claim, it should be the case that $|1/p(0)| \le 1$, but the value of $1/p$ in the residue field $\QP/(0)$ is $1/p$, and the Banach norm on $\QP$ is the $p$-adic norm, so that the statement of Conrad implies that $|1/p|_p \le 1$, while clearly $|1/p|_p > 1$.
What am I missing?
