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Fall Tutorial 2014

Complex multiplication

Description: The theory of complex multiplication is not only the most beautiful part of mathematics but of all sciences, according to David Hilbert apparently. It is an elegant and deep interaction between complex analysis, algebraic geometry and number theory. In its classical form, the theory concerns the study of the arithmetic properties of some special values of the modular function j, which is a transcendental holomorphic functions defined on the upper half plane. The j function is closely related to holomorphic or meromorphic function defined on C with double periods, and the special values of j are studied in relation with these functions. From a somewhat modern perspective, a more convenient starting point of the whole theory is the concept of elliptic curves. These objects have surprisingly rich properties and stand in the confluence of various fields in mathematics. They have emerged naturally in mathematics from various different points of view, explicitly or not, in the past two or three centuries. Notably elliptic curves also played a central role in the famous final proof of Fermat's last theorem by Andrew Wiles. We will talk about elliptic curves in length from different perspectives, and in particular emphasize the algebraic nature of them. After that we will realize that the function j is none other than a parameter that classifies the elliptic curves over C, and the special values of j mentioned above correspond to those elliptic curves that enjoy unusual symmetries. We will study the arithmetic significance of these special elliptic curves and prove the so called Main Theorem of Complex Multiplication. After this we will study the intimate relation between the theory of complex multiplication and the theory of modular forms, which can be viewed as a baby case of the profound grand program of Langlands. After all this we will take a tour to the more advanced and more exciting topics which have developed from the theory of complex multiplication and are central in current research in number theory.

In this tutorial we will see how seemingly transcendental objects such as the j-function are intimately related to arithmetic. On the way we will touch upon the foundations of many topics, which have now become ubiquitous in modern number theory.

Prerequisites: Knowledge of abstract algebra and some Galois theory is required. Some experience with algebraic number theory is very desirable, although we will review some fundamental concepts in the first couple of lectures. We do not assume knowledge about Riemann surfaces, algebraic curves, elliptic curves, modular forms, or class field theory.

Contact: Rong Zhou and/or Yihang Zhu (rzhou@math.harvard.edu, yihang@math.harvard.edu)

Spring Tutorial 2015

Cyclotomic fields and Fermat's Last Theorem

Suppose one extends the field of rational numbers to include an extra number whose nth power is 1. Then one obtains a canonical (Galois) field extension whose arithmetic is very important in number theory but which can also be studied very explicitly. In this tutorial we will see how a very natural historical attempt at proving Fermat's Last Theorem leads one to attempt to establish a suitable "unique factorization" property for such fields. To study this question we develop the basic theory of cyclotomic fields, along the way obtaining a very natural proof of Gauss' theorem of quadratic reciprocity. We will see that the required unique factorization property does not hold in all cases, but we will establish Kummer's theorem giving us a recipe relating such questions to easily resolved questions about a very explicit sequence of rational numbers called the Bernoulli numbers (closely related to the Riemann zeta function). We will then discuss the Herbrand-Ribet theorem, which gives a more refined version of Kummer's result.

This subject is the start of two important stories in number theory. On the one hand the basic theory of cyclotomic fields is precisely "class field theory" for the rational numbers, and class field theory is to abelian Galois extensions of number fields what the "Langlands program me" is to arbitrary extensions. On the other hand, our study of the relation of Bernoulli numbers to class groups leads historically into the subject of "Iwasawa theory" which provided much of the inspiration for Wiles' eventual successful proof of Fermat's last theorem and is one of the key techniques being currently used to work on the conjecture of Birch and Swinnerton-Dyer.

Prerequisites: Mathematics 123 (and will complement Mathematics 129). Some complex analysis would be helpful but not essential.

Contact: Tom Lovering (lovering@math.harvard.edu)