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In Silverman's The arithmetic of elliptic curves, p. 201, theorem $7.1$ (Criterion of Neron-Ogg-Shafarevich), he applies the theorem "When $K$ is complete with respect to it's discrete value, then, $[E(K):E_0(K)]$ is finite" to deduce that $[E(K^{nr}):E_0(K^{nr})]$ is finite.

But $K^{nr}$ is not complete even in the simplest case $K＝\Bbb Q_p$.

This is why I need some modification of the proof. What kind of modification is needed here?

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    $\begingroup$ $K^{\mathrm{nr}}$ and its completion have the same residue field, so $E_0(K^{\mathrm{nr}}) = E_0(\hat{K}^{\mathrm{nr}}) \cap E(K^{nr})$. Hence $[E(K^{nr}) : E_0(K^{nr})] \le [ E(\hat{K}^{\mathrm{nr}}): E_0(\hat{K}^{\mathrm{nr}})]$. $\endgroup$
    – David Loeffler
    Sep 27, 2021 at 7:07
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    $\begingroup$ I think that the argument in Silverman goes through with $K^{\mathrm{nr}}$ replaced by its completion. This is not yet pointed out in the otherwise helpful Errata math.brown.edu/~jhs/AEC/AECErrata.pdf $\endgroup$
    – Chris Wuthrich
    Sep 27, 2021 at 8:40
  • $\begingroup$ This mistake has been fixed in the 2nd printing 2016 of the second edition (the pdf file can be found at link.springer.com/book/10.1007/978-0-387-09494-6). On page 201 you see "Let $K^\mathrm{nr}$ be the completion of the maximal unramified extension of $K$", so the mistake becomes an abuse of notation. $\endgroup$
    – WLOG
    Jan 1, 2022 at 2:11
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    $\begingroup$ But the completion of $K^{\mathrm{nr}}$ need not be perfect, while in Silverman elliptic curves are assumed to be defined over perfect fields. $\endgroup$
    – WLOG
    Feb 1, 2022 at 21:13

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