Working with Dinakar Ramakrishnan means getting more and more involved with the representation-theoretic aspects of number theory. I have often heard that:
Langlands' programme generalises reciprocity
How is that supposed to work? Since the IMSc Institute Seminar Week is next week, I have spent a bit of time figuring out how to frame the joint work with Dinakar in this context. A similar question also came up when I gave a course on "Complex multiplication on Elliptic curves" at CalTech last year.
There are a number of survey articles such as the one by Gelbart and another by Borel. However, I believe that the point of view expressed in my talk may be different. This point of view is derived partly from computation and partly from algebraic geometry --- thereby combining my two interests!
The key idea is that Gauss' reciprocity is a computational jump ahead of Legendre's method. It is this computational jump that we want to achieve in a more general context.
The loosely stated problem is:
Given a system of equations, find out the number of solutions modulo a large prime N.
(More generally, find the behaviour of this system of equations modulo the prime). The Grothendieck-Weil-Lefschetz approach says that this can be done by constructing certain square matrices TN 1 of fixed size for each large prime N. (Such a system of matrices can be loosely be identified with a motive.)
For example, when the equation is xk-1=0, the matrix TN is the N-th power of the "standard" k × k matrix of order k.
What Langlands says is that we expect the matrices TN to be exactly those that arise out of the theory of automorphic representations.
Unfortunately, from the point of view of computations, the latter theory is inadequate since it is analytic. So what one would really like is analogues of the cyclotomic case (xk-1=0) which would cover all motivic systems arising as above. Elliptic curves provide such examples in some of the cases when we are dealing with 2x2 matrices. In the work with Dinakar, we are suggesting some generalisations.
This is still a couple of steps removed from doing something computationally as efficient as Gauss' quadratic reciprocity. That approach reduces the problem to a computation that is "of size k" in time and space after reducing N modulo k.
Schoof's algorithm provides something computationally good for elliptic curves. It takes polynomial in \log(N) steps; unfortunately, it still involves computations with numbers of of size N. Apparently, Bas Edixhoven has some similar results in the more general rank 2 case.
There is also the possibility that one can find a computationally efficient method without all the theory. From my point of view that will be unfortunate!
Actually these are conjugacy classes rather than actual matrices. ↩