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Durham University

Department of Mathematical Sciences

MATH4151 Topics in Algebra and Geometry IV

The aim of this course is to introduce a contemporary topic in pure mathematics and to develop and apply it. Algebra II is a prerequisite. Topics will be one of the following:

  1. Elliptic functions and modular forms
  2. Algebraic curves
  3. Analytic number theory
  4. Riemann surfaces

For 2019/20, the topic that will run is "Algebraic Curves".

Outline of Course

Elliptic functions and modular forms

Aim: To introduce the theory of multiply-periodic functions of one complex variable and the closely related theory of modular forms and to develop and apply it. Modular forms and elliptic functions are closely related topics in complex analysis and are ingredients of much current research, from number theory and geometry to representation theory and theoretical physics. In particular, modular forms play an increasingly central role in modern number theory, for example in the celebrated proof of Fermat's last theorem. The name elliptic functions arises from the problem of finding the arc length of an ellipse, which leads to an integral that cannot be evaluated by elementary functions. The idea of inverting such integrals, rather like finding arcsin(x) as an integral, led to the theory of doubly periodic functions. The development of that theory, by Jacobi and Weierstrass and others, was one of the high points of nineteenth century mathematical achievements.

Texts:

  • J. Serre, A Course in Arithmetic, Springer 1996
  • N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer 1993
  • G.A. Jones, D. Singerman, Complex Functions, Cambridge University Press 1987
  • J.V. Armitage, W.F. Eberlein, Elliptic Functions, Cambridge University Press 2006
  • E. Freitag, R. Busam, Complex Analysis, Springer 2009

Algebraic curves

Aim: To introduce the basic theory of plane curves, with a particular emphasis on elliptic curves and their arithmetic.

The main aim in the Michaelmas Term is to introduce some basics on algebraic curves aiming to state and prove the famous Bezout theorem for plane algebraic curves. This theorem is a vast generalization of two facts that one learns in Linear Algebra I and Complex Analysis II. In a rather informal and vague form, it states that the system of two polynomial equations in two variables over the complex numbers can have at most d1d2 many solutions, where d1 and d2 are the degrees of the two polynomials. This generalizes the well known facts about solutions of linear systems as well as the fundamental theorem of algebra regarding the solutions of a polynomial in one variable over the complex numbers.

In the second term we turn our attention to elliptic curves. This is a special case of algebraic curves (non-singular cubics) which have very rich structure. We will study elliptic curves over different fields (complex, rational and finite fields) and at the end of the term we will be able to state one of the most famous conjectures of modern number theory, the Birch and Swinnerton-Dyer conjecture.

Texts:

  • F. Kirwan, Complex algebraic curves, LMS student Texts 23, Cambridge 1992
  • Reid, Undergraduate algebraic geometry, LMS student text 12, Cambridge 1988
  • J. Cassels, Lectures on elliptic curves, LMS student texts 24, Cambridge 1991
  • Joseph H. Silverman and John Tate, Rational points on elliptic curves.
  • Lawrence C. Washington, Elliptic curves: number theory and cryptography.

Analytic number theory

Aim: To understand important results in analytic number theory related to the distribution of primes. A significant attention will be provided on the theory or Riemann zeta function and Dirichlet series, gearing towards the proof of the prime number theorem. The course will demonstrate how to use tools from complex analysis to derive results about primes.

Texts:

  • T. M. Apostol, Introduction to analytic number theory. Springer, 1976
  • Edwards, Riemann's Zeta Function, Dover
  • H. Davenport, Multiplicative Number Theory, third edition, Springer 2000

Riemann surfaces

Aim: To introduce the theory of multi-valued complex functions and Riemann surfaces.

Texts:

  • R. Miranda, Algebraic curves and Riemann surfaces, AMS, 1995
  • H.Farkas, I.Kra, Riemann surfaces, Soringer, 1980

Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.

Reading List

Please see the Library Catalogue for the MATH4151 reading list.

Examination Information

For information about use of calculators and dictionaries in exams please see the Examination Information page in the Degree Programme Handbook.