Iwasawa theory of elliptic curves and the Birch–Swinnerton-Dyer conjecture

The Birch–Swinnerton-Dyer conjecture is a Millenium Prize Problem and is unquestionably one of the most important open problems in mathematics.

It concerns elliptic curves E, which are indispensable in today’s society as they are extensively used in the form of cryptography.

Iwasawa theory proved to be an extremely powerful tool in modern number theory, and it serves as a bridge between the two mysterious mathematical objects appearing in the conjecture (the Tate–Shafarevich group and the complex L-series) which are completely different in nature, by allowing us to study it at one prime number at a time.In this proposal, E will have complex multiplication, since their L-series are known to be defined at the critical point.

The classical theory only deals with odd primes p.

Even though p = 2 is the most interesting prime for the conjecture, it is always omitted because of serious technical difficulties, preventing classical methods from obtaining the full conjecture for any curve. Furthermore, E is always defined over the rational numbers or the field K of complex multiplication.

A main reason for this is that the Iwasawa modules are not semisimple at p = 2 and p dividing the class number h of K. I will explore and apply the full force of Iwasawa theory and aim to include the eschewed prime.

This is made possible by my recent results on an Iwasawa main conjecture at p = 2 and the vanishing of Iwasawa μ-invariant. The first research project deals with a family of elliptic curves which are defined over extensions of K.

For these curves, I propose to show the p-part of the Birch–Swinnerton-Dyer conjecture for ordinary primes p, which include p = 2 and p dividing h.

The second project explores noncommutative Iwasawa main conjecture at p = 2 by combining the early works of Colmez–Schneps and the recent work of Kings–Sprang.

Finally in the third project, I will use an elliptic curve from the second project to propose and study an analogue of Weber’s class number problem.