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Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism

$$DR(V) \rightarrow H^1_e(K, V)$$

and state that (Bloch-Kato, Example 3.10.1) for a commutative formal lie group $G$ of finite height over $\mathcal{O}_K$, their exponential map $$T_p \otimes \mathbb{Q} \xrightarrow{exp_{BK}} DR(T_p \otimes \mathbb{Q})$$(for $T_p$ the $p$-adic tate module of G) agrees with the "exponential map in the classical sense"

$$\textrm{tan}(G_K) \xrightarrow{exp_{classical}} G(O_K) \otimes \mathbb {Q} $$

That is, a certain diagram commutes (Bloch-Kato, Example 3.10.1).

$\textbf{Question:}$ Can you give me a reference defining the exponential map for the formal lie group associated to an abelian variety $A$ over $K$?

If necessary, feel free to assume $A$ has semistable reduction over $K$. Is this contained in the second half of Serre's "Lie groups and Lie algebras", or Fontaine's "Groupes p-divisibles sur les corps locaux"?

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  • $\begingroup$ For elliptic curves it is in Silverman IV.5. $\endgroup$ Jul 9, 2014 at 15:52
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    $\begingroup$ I'd assume it's the inverse (which he mentions should exist on a small enough neighbourhood) to the logarithm in the sense of section 2 of Tate's article on p-divisible groups fhoermann.org/Tate%20-%20p-Divisible%20Groups.pdf $\endgroup$ Jul 9, 2014 at 16:02

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