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In explicit examples that I have seen worked out, it appears that when the fine Selmer group is finite in the cyclotomic extension it is in fact trivial.

Is there any reason to expect that this should always be the case? Can someone point me to examples where the fine Selmer group is finite but non-trivial?

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  • $\begingroup$ Apparently, I hadn't looked hard enough and there are examples in Wuthrich's thesis of non-trivial finite fine Selmer groups! $\endgroup$ – debanjana Jun 10 at 15:34

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