3-Selmer groups and the rank of elliptic curves of $j$-invariant zero

In this talk I will discuss recent results concerning the rank of an infinite family of elliptic
curves of $j$-invariant zero. I will describe an explicit relation between the 3-Selmer group
of these elliptic curves, and the 3-Selmer group of quadratic number fields with non-trivial
3-class group. By combining with results of T. Honda and Ph. Satgé, I will show how I
derived lower and upper bounds for the rank of these elliptic curves through the method of
descent, and by assuming finiteness of the Tate-Shafarevich group. Finally, I will describe
the torsors for these curves, and give a new example of genus-1 curves that violate the Hasse
Principle.