The mysterious

To every elliptic curve one can attach a certain series that we call . To define recall that we have previously discussed points on an elliptic curve over such fields as or . However, the notion of points on an elliptic curve is not limited to these fields. One can consider the number of points on an elliptic curve over for any prime (that does not divide the discriminant of the curve). We denote the number of points on over as . We can now define a sequence of numbers such that . There is also a slightly more complicated way to define for any number. These can be found in PARI by using the ellan command.

Once we have these we can now define :

It is a theorem of Breuil, Conrad, Diamond, Taylor, and Wiles that can be extended to an analytic function on all of . As with any other analytic function we can ask what the order of vanishing of is at any point. It turns out that the order of vanishing of at is a rather interesting story. In fact the Birch and Swinnerton-Dyer conjecture is that the order of vanishing at is exactly equal to the rank of the elliptic curve.

In other words, for any elliptic curve, ,

at . Where here and is such that .

The BSD conjecture is fairly amazing in that it asserts the equality of two seemingly very different quantities.

So far, the BSD conjecture has been proved when ord by Gross, Kolyvagin, Zagier, et al. However, for ord it is still an open problem, and as was mentioned above, it has yet to be proven that any elliptic curve has rank 4.