This writing is about recently deceased mathematician Sir Michael F. Atiyah’s groundbreaking paper Vector Bundles over Elliptic Curves published in 1957 . He was awarded the Fields Medal the highest honor there is for mathematicians in 1966 and recently passed away on 11 January 2019 . We will introduce one of his well-known works where he classifies the linear and geometric objects called vector bundles on yet another geometric objects called elliptic curves. We will do so through analogies and descriptions rather than precise definitions as that would require a degree in Mathematics. It is the philosophical perspective in a mathematician’s thinking that we would like to convey. Elliptic curves are geometric objects that have the shape of a surface of a doughnut without any fillings. They are one of the most studied mathematical objects but we rather focus on the linear objects on them and see how the geometry of elliptic curves is reflected on vector bundles.
Vector bundles have an important invariant attached to them: rank. The rank of a vector bundle is a natural number, i.e. 1, 2, 3 and so on. Let us start with a description of vector bundles of the lowest rank, the so-called line bundles. Imagine an elliptic curve, i.e. a surface of a doughnut, and then further imagine positioning a line segment on every point of this elliptic curve. You can think of it a sort of thickening. Now imagine these buildings having an infinite height. We will call this geometric object the trivial line bundle. At this point if your imagination fails you, don’t worry. We are so used to thinking in a Euclidean system that any object beyond this system requires taking a leap of faith. “How would a nontrivial line bundle look like on an elliptic curve?” you might ask next. Let us simplify the situation. Rather than an elliptic curve, imagine a circle. What is the trivial line bundle on a circle? Well, it is nothing other than a cylinder. An example of a nontrivial line bundle on a circle is an object you get by taking an infinite band and glueing the opposite sides after a twisting. You might have heard of this object being called an infinite Möbius strip. It might help to search for it online and see some pictures available. After a little practice comes the time to make a correction. When we say a bundle over an elliptic curve what we actually consider is not positioning buildings on every point but rather positioning planes the so-called complex lines at every point.
There are two operations on bundles we can perform: direct sum and tensor product. The former one works in this way: if you take a bundle of rank r and a bundle of rank s, their direct sum is a bundle of rank r+s whereas their tensor product is a bundle of rank rs. You might expect that a tensor product is a bit more technical than the direct sum operation. We will consider the building blocks with respect to the direct sum operation. If a bundle cannot be written as a direct sum of two smaller bundles, we call it indecomposable and that is your first rigorous definition. Atiyah deals with the following question: What are the indecomposable vector bundles on an elliptic curve of fixed rank?
Notice that a line bundle a rank 1 vector bundle is necessarily indecomposable and the above question in this case had already an answer when Atiyah started his work. The line bundles on an elliptic curve are parametrized by a pair: a point on the elliptic curve and an integer. I.e given any point on an elliptic curve and an integer (.., -2, -1, 0, 1, 2), one can create a line bundle associated to it. This association is a bit subtle so we will not really get into that. The integer that appear in this association is another invariant of a bundle called the degree.
There is one more operation we will consider to make the above question more specific: taking the determinant of a bundle. The determinant of a vector bundle is a line bundle, i.e. given a bundle of arbitrary rank r, there is a way to produce a bundle of rank 1. If we already start with a line bundle, this process does nothing at all, i.e. the determinant of a line bundle is the line bundle itself. Let us make the above question more specific and focus on the next simplest case after line bundles by asking “What are the indecomposable bundles of rank 2 and degree 0?” as Atiyah did back then.
Atiyah observes that there is a distinguished rank 2 bundle of degree 0 that I would like to call the rank 2 Atiyah bundle and denote it by F_2. All other such bundles are some twists of F_2, i.e. the tensor product of F_2 with a line bundle of degree 0. Remember that tensoring with a line bundle does in fact preserve the rank so we still get just another rank 2 bundle after twisting. The story is exactly the same in all ranks 2, 3, 4… I.e., we also have the higher rank Atiyah bundles F_3, F_4, F_5. In fact, he defines this bundles inductively so that for example F_3 is obtained from F_2 after a certain process. In retrospect, you might wonder what would F_1 be, i.e. what is the first Atiyah bundle? Well, it is simply the trivial line bundle.
In this work, Atiyah analyzed the ways in which one can extend a bundle of rank r by another bundle of rank s to get a bigger bundle of rank r+s. He gave a definition that shows in a sense these extensions happen over a spectrum. On the one hand, we have the split case corresponding to simply taking the direct sum and on the other hand we have the complete case which one can think as the most non-split case possible. Extensions are subtle for the following reason: One extension might appear as a non-split extension of two smaller vector bundles but it might decompose into some other pair of smaller vector bundles and correspond to a split extension. Atiyah systematically analyzed when a non-split extension really produces an indecomposable bundle. He also gave an existential result on when a bundle admits a trivial bundle as a subbundle. This enabled him to reduce the question to smaller and smaller ranks. With his description, we now have an important class of objects, namely elliptic curves, on which vector bundles have a very explicit description. His work naturally led to many further progress over time making his paper one of the most cited articles in Mathematics.
 Atiyah, M. F.. (14 February 1957) Vector Bundles over an Elliptic Curve. Date of Access: 13 January 2019, https://math.berkeley.edu/~nadler/atiyah.classification.pdf