# Deducing BSD from Gross-Zagier and Kolyvagin

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!

## 3 Answers

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

• It's also worth noting that additional work by Kolyvagin proves that under these conditions, the Tate-Shafarevich group of $E$ over $\mathbf{Q}$ is also finite. – stankewicz Mar 11 '13 at 6:52
• Dear stankewicz, Since GZ use a modular parameterization of their elliptic curve, they probably assumed that the elliptic curves is defined over $\mathbb Q$ (not just over an imag. quad. field), but you are right that they then study its Heegner point over an imag. quad. field. Regards, Matt – Emerton Mar 11 '13 at 7:02

Let $E/\mathbb{Q}$ be an elliptic curve of analytic rank $0$ or $1$. Then indeed the rank part of the BSD conjecture is known, but the exact formula for the leading term is not yet fully proven. The $p$-part of the formula is known for almost all $p$, but a few stubborn ones might need an extra argument. The primes dividing the Tamagawa numbers for instance may cause problems.

I believe the current state of affairs is well written up in Robert Miller's (http://www.rlmiller.org/) paper in the LMS journal of computational mathematics. It will refer to a paper in Compositio by Dimitar Jetchev in 2008. But there might be better results by now that I am not aware of.

For the analytic rank $0$ case, one can also obtain information about the leading term through other means than Gross-Zagier-Kolyvagin.

Some results can be found in this survey: A survey on development of the Gross- Zagier formulas and their applications: Elliptic curves, L-functions, and CM-points by Shou-Wu Zhang.