Introduction to Drinfeld Module
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.
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.
Lecturer
Date
30th September, 2024 ~ 8th January, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 15:20 - 16:55 | A3-3-301 | ZOOM 11 | 435 529 7909 | BIMSA |
Prerequisite
Number Field Theory, Function Field Theory, Class Field Theory, Basic Algebraic Geometry.
Reference
Mihran Papikian, Drinfeld Moudle;
David Goss, Basic Structures Of Function Field;
David Goss, Basic Structures Of Function Field;
Audience
Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
Chinese
Lecturer Intro
Hu chuangqiang joined Bimsa in the autumn of 2021. The main research fields include: coding theory, function field and number theory, singularity theory. In recent years, he has made a series of academic achievements in the research of quantum codes, algebraic geometric codes, Drinfeld modules, elliptic singular points, Yau Lie algebras and other studies. He has published 13 papers in famous academic journals such as IEEE Trans. on IT., Final Fields and their Applications, Designs, Codes and Cryptography. He has been invited to attend domestic and international academic conferences for many times and made conference reports.