Mathematical Research Letters
Volume 16 (2009)
Elliptic curves with large Tate-Shafarevich groups over a number field
Pages: 449 – 461
Let $p$ be a prime number and let $K$ be a cyclic Galois extension of $\Q$ of degree $p$. We prove that the $p$-rank of the Tate-Shafarevich group over $K$ of elliptic curves defined over $\Q$ can be arbitrarily large.