北京雁栖湖应用数学研究院

Drinfeld modules are certain function field analogues of elliptic curves. In fact, due to this analogy, in his seminal paper, Vladimir Drinfeld introduced
these objects under the name of “elliptic modules.” Drinfeld’s motivation was to construct function field analogues of classical modular curves classifying elliptic curves with some additional data and use these to relate automorphic forms and Galois representations, in line with the program envisioned by Langlands. The theory of Drinfeld modules and their generalizations, called shtukas, has since led to a successful resolution of the Langlands correspondence over function fields, and it continues to play a central role in number theory because of its applications to many other important problems, such as the
Birch and Swinnerton-Dyer conjecture. We do not discuss any of the applications of Drinfeld modules and their modular varieties to the Langlands program. Our goal is much more modest: to develop the theory of Drinfeld modules in parallel with the theory of elliptic curves assuming as little background material as possible. In this lecture, we only assume that the reader is familiar with graduate-level abstract algebra as presented, for example, in the book by Dummit and Foote; we review some of the algebra results important for our purposes. In addition, in some proofs we use facts from algebraic number theory, which can be found, for example, in the book by Marcus.