Mathematics Department Topics in Algebraic Number Theory (EPSRC/LMS Short Course)

TOPICS IN ALGEBRAIC NUMBER THEORY

LMS/EPSRC Short Course

King's College London, 2-6 September 2002

Organiser: David Burns

Algebraic number theory has a long and distinguished history and remains one of the most significant areas of research in mathematics. The subject has in particular enjoyed spectacular advances in recent years, with Wiles' proof of Fermat's last theorem standing as one of the undisputed milestones of twentieth century mathematics. The analysis of problems in number theory, even those of a seemingly concrete and explicit nature, may well however involve the interplay of results and techniques from may different branches of pure mathematics. In conjunction with the increasing pace of current developments this means that it is all too easy to feel relatively isolated from the fundamental advances which are being achieved today.

With this problem in mind, the lectures at this short course aim to provide students with a grounding in some of the areas which are of central importance in both algebraic number theory and arithmetic algebraic geometry. The topics to be discussed have been chosen both because they have been of pivotal significance to recent developments and also because they illustrate well the wide variety of techniques and the nature of the problems which arise in much of the fundamental research which is being conducted today. The lecturers and course titles are:

• Local Fields : Ivan Fesenko (University of Nottingham)
• Iwasawa Theory : David Burns (King's College, London)
• Modular Forms : Kevin Buzzard (Imperial College, London)

The course on Local Fields will discuss the basic properties of rings which arise naturally in all areas of arithmetic algebraic geometry and will also describe the foundational aspects of class field theory; the course on Iwasawa Theory will introduce students to one of the most useful techniques currently available to number theorists; the course on Modular Forms will take students to the point at which they can understand the statement of the Shimura-Taniyama-Weil Conjecture, its connections to Fermat?s Last Theorem and the statement of results of Wiles and others related to this conjecture. Each course comprises six lectures and will start from a discussion of basic definitions and concepts. Prerequisites will be kept to a minimum so that the courses are for the most part accessible to beginning graduate students, presupposing only a thorough knowledge of undergraduate material. The lectures will be illustrated by the careful treatment of concrete examples. In addition, worksheets and exercises will be supplied, to be discussed with post-doctoral tutors in afternoon sessions.

The registration fee is £60, which for all UK-based research students includes the cost of course accommodation and meals. Participants must pay their own travel costs. EPSRC-supported students can expect that their registration fees and travel costs will be met by their departments from the EPSRC Research Training and Support Grant that is paid to universities with each studentship award (or from the Doctoral Training Account in the case of first-year students).

Application forms are available as an on-line form to be printed off and posted to:

or as an RTF file from the electronic archive which can be conveniently emailed to spoor@lms.ac.uk

Numbers will be limited and those interested are advise to make an early application. The closing date for applications is 21 June 2002

Recommended literature

This book will be available from the AMS bookshop page later in the year. Participants can view or print parts of the book via Ivan Fesenko's webpage.

Modular Forms

This book is very good for the basic definitions.

The next two books are for the (much) more ambitious:

Iwasawa Theory

There are by now quite a few books containing a good account of basic Iwasawa theory. See for example Chapter 13 and the Appendix of

or the relevant sections of