During the Summer of 2009, I read some papers of Cornut, Vatsal, and Jetchev-Kane that were motivated by Mazur's conjecture about Heegner points. Meanwhile, I was thinking about Kolyvagin's conjecture on nontriviality of his Euler system of Heegner points. This is a conjecture about any elliptic curve E over Q along with any prime p and a fixed choice of quadratic imaginary field K that satisfies the Heegner hypothesis (e.g., each prime dividing the conductor of E splits in K). The conjecture is simply that one of Kolyvagin's cohomology classes in H^1(K, E[p^n]) is nonzero, for some n. When E has analytic rank 1 over K this conjecture holds for the class tau_1 for n sufficiently large, by the Gross-Zagier formula.
Three years ago, I wrote a paper with Dimitar Jetchev and Kristen Lauter where we gave *numerical evidence* that Kolyvagin's conjecture holds for the curve 389a with p=3, K=Q(sqrt(-7)), n=1, and tau_5!=0. But we didn't *prove* that tau_5 is nonzero. Moreover, our numerical approach was so inefficient that we couldn't have done many more examples. Nonetheless, this was good news - at least there was evidence thst Kolyvagin's construction doesn't completely degenerate for curves of rank > 1.
Last summer, when reading Cornut, Jetchev-Kane, etc., I figured out how to make some of what they do explicit and machine computable, and I implemented code. This was several weeks of focused programming work in Sage, starting from scratch, partly because I refuse to used the closed Magma math software system (which had maybe half of what I would have needed).
Anyway, running all this code for a week on my (big NSF funded!) cluster, and also Jon Hanke's at Univ of Georgia resulted in provable verification of Kolyvagin's conjecture for around 100 triples (E,K,p) with E of rank 2, and K of class number 1. I am right now working on polishing the write up of this for publication (I plan to submit the paper sometime in the next month).
At a Sage Days we had out on Lopez in late summer 2009, Jared Weinstein got interested in this project, and with some input from Karl Rubin about how to apply Kolyvagin systems, we figured out how to compute the distribution of Kolyvagin classes which massively clarified the numerical data I had obtained. He also found a completely different *conditional* algorithm to compute Kolyvagin classes: when I implemented it the results exactly matched what I got with my algorithm the previous month: nice.
In December 2009, at another Sage Days at the Clay Math Institute in Cambridge, MA, Jennifer Balakrishnan became interested in the project and ran my code (with some modifications) to verify Kolyvagin's conjecture for the rank 3 curve 5077a and p=3. The relevant quadratic imaginary field K had larger class number so we had to improve the theory a bit. The computer calculations also took many hours. Meanwhile, Jared Weinstein came up with some preliminary rank 3 analogues of his results from the previous summer, which gave Jen and me confidence that our rank 3 verification of Kolyvagin's conjecture was correct. We also plan to write a paper about this soon, but still haven't really done much (Jen has though, actually).
During the next few months (early 2010) Jared worked on refining and generalizing his density arguments, with some input from me. I also finally read the Mazur-Rubin book on Kolyvagin systems.
In June at Sage Days 22 at MSRI Jared ran a very coherent 2 week minicourse on Kolyvagin and had several students do projects making explicit subtle cases of explicit computation with Kolyvagin's Euler system when p=3. They were forced to use Magma, since they had to adapt some general 3-descent code, and there is only one implementation of 3-descent in the world: Michael Stoll's in Magma. (My student Robert Miller started along the path to a free open implementation last year, but we decided he should finish his thesis first. That said, it will be a great contribution to math when somebody finally implements a free 3-descent.) Anyway, I think their Magma computation with a modified 3-descent led to some great new data related to Kolyvagin's Euler system, I think when Sha(E/K) is nontrivial.
There are a few other related directions that I haven't mentioned above. First it is possible to use similar techniques to verify the natural analogue of Kolyvagin's conjecture for modular abelian varieties of larger rank and I have done this for a few examples. It would also be interested to extend the Stein-Weinstein density business to this case. There is probably an analogue of everything for modular motives (in the sense of Scholl) and maybe Kimberly Hopkins would have some interest in this.
Another interesting direction to pursue is totally real fields. Around 2001, Zhang generalized some of the Gross-Zagier formula and also some of Kolyvagin's Euler system to this setting. A natural Ph.D. (?) project is to generalize enough to state an analogue of Kolyvagin's conjecture, then verify the conjecture in at least one case of a curve of rank 2. This will inevitably involve Hilbert modular forms, and certain explicit computation with quaternion algebras over a totally real field. Currently all existing code in this direction is in Magma, so step 1 is to understand enough of the recent work of Voight and Dembelle to implement open source code in Sage (it appears neither Dembelle or Voight have any plans to do this, unfortunately). But for this project doing only one single verification of Kolyvagin means that one doesn't have to write very general code, so this should be much easier than a general implementation (which would likely take at least a year fulltime work). Probably my Ph.D. student Aly Deines will take on this project.
I will close this blog post with some remarks about the whole point of this line of investigation. In first learned about Kolyvagin's conjecture from Christophe Cornut back in 2001 when we were both B.P. Assistant Professors at Harvard together. At the time, I think Christophe was perhaps trying to prove the conjecture, maybe using similar tools (e.g. Ratner's theorem) to what he used to prove Mazur's conjecture on nontriviality of Heegner points. I only wondered about verifying the conjecture in a few cases later in 2007, since I started wondering about BSD for rank 2 curves, and it is natural to wonder whether or not Kolyvagin's approach just falls apart for rank 2 or not. (Answer: it doesn't!) Anyway, I stumbled on a way of checking the conjecture computationally in specific cases, and this has led to some potentially interesting questions and results that explain the computations. Also, one gets a new algorithm to compute something about Selmer groups, which applies in some cases where p-adic techniques don't, e.g. additive primes.
I personally do not think anything mentioned above will prove Kolyvagin's conjecture in general. I.e. the above techniques are very unlikely to say anything much about the question that Cornut was attacking back in 2001. I published in IMRN a (fairly naive and formal!) conjectural analogue of the Gross-Zagier formula for arbitrary rank. This is in a totally diffent direction to the work of Zhang, Yang, etc. generalizing Gross-Zagier. A "suitably form" of this conjectural formula does imply Kolyvagin's conjecture (and the full Birch and Swinnerton-Dyer rank conjecture). I personally think the most likely way in which Kolyvagin's conjecture will be proved is that somebody will prove something nontrivial toward the above mentioned generalization of Gross-Zagier.