NONVANISHING TWISTS AND VISIBLE SHAFAREVICH-TATE GROUPS¹

William A. Stein

Date: October 1, 2001

Let $E$ be an elliptic curve over  $\mathbb{Q}$. Our long-term goal is to find, for many primes $p$, rank 0 abelian varieties $A$ such that

$$0 \rightarrow A \rightarrow J \rightarrow E \rightarrow 0$$

induces an isomorphism

$$E(\mathbb{Q})/p E(\mathbb{Q}) \cong \Vis_J({\mbox{{\fontencoding... ...coding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(J)).$$

Such results are useful in connecting the rank 0 BSD formula to the conjecture that $\ord_{s=1}L(E,s)=$   rk$(E)$.