Does anyone know which papers deduce BSD for elliptic curves $E/\mathbb{Q}$ of rank 0 or 1 from the papers by Gross and Zagier and Kolyvagin? If I understand these theories right, there is still a non-trivial amount of computation left to obtain the precise formula of the BSD conjecture. A paper by Grigorov, et al., describes an explicit computational verification of the BSD conjecture for curves of small conductor, but there are still many more cases. Relatedly, has anyone collected all the results for the known part of BSD in a single expository work?
Thanks!
Best Answer
No papers because it's not proven for elliptic curves of rank zero or one.
The work of Kolyvagin, together with the work of Gross and Zagier almost proves that if $E_/{\mathbf{Q}}$ is an elliptic curve of analytic rank zero or one then the rank is also zero or one. Depending on the form of BSD you use, this may or may not prove it.
To take care of the almost, you need to keep in mind the following. Kolyvagin's work proves that if an elliptic curve over $\mathbf{Q}$ has analytic rank zero, then its rank is zero. Gross and Zagier's work proves that if an elliptic curve over an imaginary quadratic field has rank one, then its rank over that field is one. To bridge the gap, you need to say that if $E$ is an elliptic curve over $\mathbf{Q}$ whose analytic rank is one, then there is an imaginary quadratic field $K$ such that the twist of $E$ by $K$ has analytic rank zero, and thus the base change of $E$ to $K$ has analytic rank one.
This last bit was proven independently by either
the brothers Murty : http://www.mast.queensu.ca/~murty/murty-murty-annals.pdf
or
Bump, Friedberg, and Hoffstein: http://wintrac.sagemath.org/sage_summer/bsd_comp/Bump-Friedberg-Hoffstein-Nonvanishing_theorems_of_L-functions_of_modular_forms_and_their_derivates.pdf