Cassels Pairing for Fine Selmer groups

Question

Let $S_n$ be the Selmer group of $E/K_n$ where $K_n/K$ is the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension of $K$ and $C_n$ be the torsion part of $S_n$. By Cassels pairing, we know that there is a non-degenerate skew symmetric Galois equivariant pairing $C_n \times C_n \rightarrow \mathbb{Q}_p/\mathbb{Z}_p$.

Now if $R_n$ is the fine Selmer group and $T_n$ is the torsion part of $R_n$, does the restriction of the Cassels pairing still give a non-degenerate skew symmetric Galois equivariant pairing $T_n \times T_n \rightarrow \mathbb{Q}_p/\mathbb{Z}_p$?

Answer 1

Flach has defined in [1] a Cassels-Tate pairing on very general Selmer groups, and the kernels of the pairing on both sides are the maximal divisible subgroups. But you need to be careful when applying his results to the fine Selmer group of an elliptic curve. In general, the pairing pairs the Selmer group attached to a given Selmer structure on a Galois module with the Selmer group attached to the dual Selmer structure on the dual module. It just so happens that in the case of elliptic curves and the usual Selmer groups, both the modules and the Selmer structures on them are self-dual.

The construction of Flach has been carefully explicated by Wuthrich in [2], where, in Section 6, he explicitly describes the dual of the fine Selmer group.

If you just naively use the classical Cassels-Tate pairing to pair the fine Selmer group with itself, then this will be degenerate (modulo the maximal divisible subgroup) in general. Indeed, Wuthrich gives an example of an elliptic curve over $\mathbb{Q}$ of rank $0$ whose fine $2$-Selmer group has size $2$, which implies that the Cassels-Tate pairing, when restricted to the fine $2$-Selmer group of that curve, must be $0$.

[1] M. Flach, A generalisation of the Cassels-Tate pairing, J. reine angew. Math. 412 (1990).

[2] C. Wuthrich, The fine Tate-Shafarevich group, Math. Proc. Cambridge Phil. Soc. 142 (2007).

You can google both articles to find freely available copies online.