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 On the Tate-Shafarevich group of elliptic curves over $\mathbb Q$ Bull. Korean Math. Soc. 2012 Vol. 49, No. 1, 155-163 https://doi.org/10.4134/BKMS.2012.49.1.155Published online January 1, 2012 Dohyeong Kim Pohang University of Science and Technology Abstract : 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 Downloads: Full-text PDF