# 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:

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:

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