On Computing the Tate Shafarevich group order and type of some rational elliptic curves of conductor less than a million.

Abstract

In this dissertation, I will present the tabulation of the Tate Shafarevich group order and type of around 4.5 million rational elliptic curves of conductor less than a million. These curves were obtained from the database of Cremona and that of Stein and Watkins, and I assumed standard conjectures including the Birch and Swinnerton-Dyer standard and $p$-adic conjectures and the Generalized Riemann Hypothesis. I also predict how far we are from witnessing Delaunay's asymptotic proportions on the order valuations and ranks of their Tate Shafarevich groups.