Question

In Conrad and Brinon's notes http://math.stanford.edu/~conrad/papers/notes.pdf, two uniformizers of $B_{dR}$ are produced: one is $\xi := [\tilde{p}]-p$ (bottom of p.58), where $\tilde{p} = (p, p^{1/p}, p^{1/p^2}, \ldots)$ is some choice of compatible $p^r$ roots of unity, and the other is $t := \log(\epsilon)$, where $\epsilon$ is some basis of the Tate module of $\mathbb{C}_K$.

Furthermore, we know that $B_{dR}^+ / (t) \simeq \mathbb{C}_K$. Therefore, there should be some $a_1 \in \mathbb{C}_K$ such that $\xi = a_1t + ...$. My question is: how can we describe more explicitly what this $a_1$ is?

Answer 1

What you're asking for is a description of $$a_1 = \theta(\frac{[\tilde{p}]-p}{t}).$$ It is an element of $C_p$ and you can't really "write it down explicitly". What you can do is let $G_{Qp}$ act on it and see what happens. If $g \in G_{Qp}$ then $g(\tilde{p})=\tilde{p} \cdot \epsilon^{c(g)}$ where $c(\cdot)$ is the Kummer cocycle and $g(t)=\chi(g) t$ where $\chi(\cdot)$ is the cyclotomic character. This should imply that $$g(a_1) = \theta(\frac{[\tilde{p}][\epsilon^{c(g)}]-p}{\chi(g) t}) = \frac{a_1}{\chi(g)} + p \frac{c(g)}{\chi(g)}.$$ Now that you see this formula, you should recognize it. Let $V$ be the semistable extension of $Q_p$ by $Q_p(1)$. It has Hodge-Tate weights $0$ and $1$ and $a_1$ is basically the period corresponding to weight $0$, ie the period that tells you that $(V \otimes_{Qp} C_p)^{G_{Qp}} \neq 0$.