No, there are no such examples known. In fact, with the current technology, the two questions are more or less equally hard. That's because for any given prime $p$, you can, in principle, establish finiteness of $Ш(E/\mathbb{Q})[p^\infty]$ algorithmically , see M. Stoll, E. Fby performing $p^n$-descent for higher and higher $n$, until the upper bound on the rank of $Ш(E/\mathbb{Q})[p^n]$ stabilises. SchaeferOf course, How to do we cannot prove a p-descent on an elliptic curvepriori that this would ever happen, Trans. Amer. Math. Soc. 356 (2004), 1209–1231but in practice, and the references therein. So if you knew finiteness of $p$-primary parts of sha outside a finite set of primes, you would run your computer to do $p^n$-descent for a provably finite amount of time and the remaining primes, until you establish finiteness for this finite set, too.
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No, there are no such examples known. In fact, with the current technology, the two questions are more or less equally hard. That's because for any given prime $p$, you can, in principle, establish finiteness of $Ш(E/\mathbb{Q})[p^\infty]$ algorithmically, see M. Stoll, E. F. Schaefer, How to do a p-descent on an elliptic curve, Trans. Amer. Math. Soc. 356 (2004), 1209–1231, and the references therein. So if you knew finiteness of $p$-primary parts of sha outside a finite set of primes, you would run your computer for a provably finite amount of time and establish finiteness for this finite set, too. |
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