Elliptic curves, and the machinery involved in them have been a hot topic in modern mathematics for quite some time. They came to the fore in the public consciousness most prevalently because of their deep involvement in Andrew Wiles's 1994 proof of Fermat's Last Theorem. This is in fact when I first became aware of them. Elliptic curves have become popular perhaps not only because of their deep and interesting properties, but also because of their fairly simple definition. The notion of an -series attached to an elliptic curve is also a fairly simple notion (as will be explained below), but the study of the correlation between these two objects has led to many of the biggest unsolved problems in mathematics today.
The Birch and Swinnerton-Dyer (BSD) conjecture is the statement that the rank (a simple algebraic invariant) of an elliptic curve is equal to the order of vanishing at zero (a simple analytic property) of the -series attached to that curve. The conjecture was first formulated in the early 1960s, and today we still don't have a good way of approaching the problem. With Fermat's Last Theorem, even before it was proved, the conjecture was known to hold for many specific cases. With BSD we don't even know how to show it is true for some very seemingly simple cases (for example, curves of rank 4).
Until 1986 it was not even known if the BSD conjecture held for a curve as simple as . In fact, the proof that it did hold was so deep that it provided an effective solution to the Gauss class number problem. A seemingly simple problem such as showing that there existed an elliptic curve whose -series had order 3 was a very deep theorem of Gross and Zagier, and as of this writing, it is an open problem to prove that there is an elliptic curve whose -series has order 4 or higher.
It is with these questions in mind that we approach the idea of graphing the -series attached to elliptic curves. While we doubt that anything can be proven through pictures, having pictures as a reference is a very helpful tool when dealing with these series, and perhaps these pictures will give us more insight into what is happening with .