Three Curves

A story of congruences, multiplicity one, and visibility of invisible Shafarevich-Tate groups

by William Stein

The elliptic curves 54B, 431A, and 5389A star in Three Curves, the story of a small group of adventurous elliptic curves who are determined to provide counterexamples to three tempting assertions. Finding a map they believe will take them to the gold, they embark on a journey that leads to unexpected discoveries, enabling them to rise to heroic challenges that drastically change their lives.

Contents: The modular degree and the congruence number
Multiplicity one
Visibility of Shafarevich-Tate groups

Unless otherwise noted, the examples in this talk were discovered within the last three months using the HECKE/MAGMA package.

Last modified: May 16, 2000