Rank Jumps and Growth of Shafarevich--Tate Groups for Elliptic Curves in $\mathbb{Z}/p\mathbb{Z}$-Extensions

Abstract: In this paper, we use techniques from Iwasawa theory to study questions about rank jump of elliptic curves in cyclic extensions of prime degree. We also study growth of the $p$-primary Selmer group and the Shafarevich--Tate group in cyclic degree-$p$ extensions and improve upon previously known results in this direction.