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In Conrad and Brinon's notes, 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?

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May be I miss something but both are uniformizers and both reduce to $0$ modulo $\mathrm{ker}(\theta)$. So the $a_1$ you write is $0$. All you can say is that they differ multiplicatively by some $u\in (B_{dR}^+)^\times$: observe that the action of Galois on both is different, and so is the action of $\varphi$ (although this $\varphi$ acts on them only as elements of $B_{cris}^+$) so this might be quite non-trivial a unit.... – Filippo Alberto Edoardo Dec 21 '12 at 2:16
$a_1$ is not the reduction of the uniformizers mod $ker(\theta)$, but the reduction of the ratio mod $ker(\theta)$, which as you note is nonzero. – Will Sawin Dec 21 '12 at 4:19

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 $(V \otimes_{Qp} C_p)^{G_{Qp}} \neq 0$.

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Thank you for this discussion. In fact, you've reconstructed my original application, which is to describe the periods of the Hodge-Tate isomorphism for a Tate curve. I was just trying to discuss an example of how $B_{dR}$ lets you analyze this sort of question. I think I've now figured it out to my satisfaction. – Tony Dec 23 '12 at 13:59

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