Why is there "no analogue of $2i\pi$ in $\mathbf C_p$"?

algebraic-geometry number-theory integration homology-cohomology p-adic-number-theory

I think Colmez is referring to the fact that $H^0\bigl(G_{\mathbb Q_p},\mathbb C_p(1)\bigr) = 0.$

Any such invariant would be be a $p$-adic period for the cyclotomic character; but as Tate showed, this space of invariants vanishes, and so the $p$-adic cyclotomic character does not have a period in $\mathbb C_p$; one has to go to $B_{\mathrm{dR}}$ to get a period (the famous element $t$ of Fontaine).

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