CM elliptic curves of rank 2 and nonvanishing of generalised Kato classes

Let $E_{/\mathbf{Q}}$ be a CM elliptic curve and $p>3$ a prime of good ordinary reduction for $E$. Assume $L(E,1)=0$ with sign $+1$ (so ${\rm ord}_{s=1}L(E,s)\geq 2$) and $E$ has a rational point of infinite order. We prove that if ${\rm Sel}(\mathbf{Q},V_pE)$ is $2$-dimensional, then a certain generalised Kato class $\kappa_p\in{\rm Sel}(\mathbf{Q},V_pE)$ is nonzero; and conversely, the nonvanishing of $\kappa_p$ implies that ${\rm Sel}(\mathbf{Q},V_pE)$ is $2$-dimensional. For non-CM elliptic curves, a similar result was proved in a joint work with M.-L. Hsieh. The proof in this paper is completely different, and should extend to other settings. Moreover, combined with work of Rubin, we can exhibit explicit bases for $2$-dimensional ${\rm Sel}(\mathbf{Q},V_pE)$.Comment: 20 pages, comments welcom

Variation of anticyclotomic Iwasawa invariants in Hida families

Building on the construction of big Heegner points in the quaternionic setting, and their relation to special values of Rankin-Selberg $L$-functions, we obtain anticyclotomic analogues of the results of Emerton-Pollack-Weston on the variation of Iwasawa invariants in Hida families. In particular, combined with the known cases of the anticyclotomic Iwasawa main conjecture in weight $2$, our results yield a proof of the main conjecture for $p$-ordinary newforms of higher weights and trivial nebentypus.Comment: Essentially final version, to appear in Algebra & Number Theor