For the 1998 Arizona Winter School, a group of students at The University of Arizona studied a paper by Dorian Goldfeld and Lucien Szpiro, which claims to asymptotically bound the order of the Tate--Shafarevich group by N1/2+ε. In our talk, we showed the correct bound to be N3/4+ε. Some material from our talk is available:
Slides (version: March 1999) from our March 1998 talk:
Some GP/PARI code for computing the Szpiro number:
- Bounds for the order of the Tate--Shafarevich group. Compositio Mathematica 97 (1995), 71–87. and .