More generally, when $E(K_m)$ is stable as $m$ increases for an extension equence $K_0<K_1<K_2<\cdots<K_m<\cdots$ ? In the case, is $\mathrm{Sel}(E/K_m)$ stable as $m\rightarrow \infty$?
1
-
1$\begingroup$ I assume that Sel$(E/K)$ is the group that fits into a short exact sequence between $E(K)$ and Sha$(E/K)$. Otherwise you need to fix your $n$ in $n$-Selmer group (or your isogeny). So your question is equivalent to asking about the growth of the Tate-Shafarevich group in the tower. Basically anything can happen. Iwasaw theory gives examples of exploding growth as well as frequent stabilisation. $\endgroup$– Chris WuthrichCommented Mar 3, 2015 at 20:03
Add a comment
|