ABC Conjecture

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 N^1/2+ε. In our talk, we showed the correct bound to be N^3/4+ε. Some material from our talk is available:

Slides (version: March 1999) from our March 1998 talk:

48K

202K

324K

Some GP/PARI code for computing the Szpiro number:

• szpiro.gp [GP/PARI code, not user friendly]
• szpiro.out [output from a sample run]

Related work:

• Dorien Goldfeld and Lucien Szpiro. Bounds for the order of the Tate--Shafarevich group. Compositio Mathematica 97 (1995), 71–87.