Bulletin of the
Korean Mathematical Society BKMS

Let $E$ be an elliptic curve over $\mathbb Q$. Using Iwasawa theory, we give what seems to be the first general upper bound for the order of vanishing of the $p$-adic $L$-function at $s=0$, and the $\mathbb Z_p$-corank of the Tate-Shafarevich group for all sufficiently large good ordinary primes $p$.

Keywords: good ordinary reduction, Tate-Shafarevich group, elliptic curves, Iwasawa theory

MSC numbers: 11G05