Some Conjectures and Theorems about the Shafarevich-Tate Group

Conjecture 2.12 (Shafarevich-Tate)   Let $E$ be an elliptic curve over a number field $K$. Then the group ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/K)$ is finite.

Theorem 2.13 (Rubin)   If $E$ is a CM elliptic curve over $\mathbb{Q}$ with $L(E,1)\neq 0$, then ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$ is finite. (He proved more than just this.)

Thus Rubin's theorem proves that the Shafarevich-Tate group of the CM elliptic curve $y^2 + y = x^3 - 7$ of conductor $27$ is finite.

Theorem 2.14 (Kolyvagin et al.)   If $E$ is an elliptic curve over $\mathbb{Q}$ with $\ord _{s=1} L(E,s) \leq 1$, then ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$ is finite.

Kolyvagin's theorem is proved in a completely different way than Rubin's theorem. It combines the Gross-Zagier theorem, the modularity theorem that there is a map $X_0(N)\to E$, a nonvanishing result about the special values $L(E^D,1)$ of quadratic twists of $E$, and a highly original explicit study of the structure of the images of certain points on $X_0(N)(\overline{\mathbb{Q}})$ in $E(\overline{\mathbb{Q}})$.

Theorem 2.15 (Cassels)   Let $E$ be an elliptic curve over a number field $K$. There is an alternating pairing on ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/K)$, which is nondegenerate on the quotient of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/K)$ by its maximal divisible subgroup. Moreover, if ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/K)$ is finite then $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/K)$ is a perfect square.

For an abelian group $A$ and a prime $p$, let $A(p)$ denote the subgroup of elements of $p$ power order in $A$.

The following problem remains open. It helps illustrate our ignorance about Conjecture 2.12 in any cases beyond those mentioned above.

Problem 2.16   Show that there is an elliptic curve $E$ over $\mathbb{Q}$ with rank $\geq 2$ such that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})(p)$ is finite for infinitely many primes $p$.