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General


Description

Modular forms (and automorphic forms/representations) play an increasingly central role in modern number theory, but also in other branches of mathematics and even in physics. This course gives an introduction to the subject. Here is a sample of topics we plan to cover:
  • Modular curves, also as Riemann surfaces and as moduli space of elliptic curves (over C);
  • Modular functions and forms, basic properties, Eisenstein series, eta-function;
  • Hecke operators, Petersson scalar product;
  • Modular forms and Dirichlet series, functional equation;
  • Theta series, arithmetic applications;
There are now several good introductory texts on modular forms (each with somewhat different focus) such as A First Course in Modular Forms by Diamond and Shurman, Topics in Classical Automorphic Forms by Iwaniec, Introduction to Elliptic Curves and Modular Forms by Koblitz, and Modular Forms by Miyake. Of course there is also the classical text by Serre and the 1971 book by Shimura.

Prerequisites: Good command of complex analysis and algebra. Occasionally, some knowledge of algebraic number theory and Riemann surface theory would be helpful.
Semester

Spring 2012 (Monday, January 16 to Friday, March 23)

Timetable
  • Tue 15:05 - 15:55

Lecturer


Jens Funke
Email jens.funke@durham.ac.uk
Phone (0191) 3343063
vcard
Photo of Jens Funke


Students


Photo of Steven Charlton
Steven Charlton
(Durham)
Photo of Neslihan Delice
Neslihan Delice
(Leeds)
Photo of Maxwell Fennelly
Maxwell Fennelly
(Southampton)
Photo of Adam Firkin
Adam Firkin
(Birmingham)
Photo of Daniel Fretwell
Daniel Fretwell
(Sheffield)
Photo of Kostas Georgiadis
Kostas Georgiadis
(Loughborough)
Photo of Alena Jassova
Alena Jassova
(Liverpool)
Photo of Andrew Jones
Andrew Jones
(Sheffield)
Photo of Amin Saied
Amin Saied
(Southampton)
Photo of Joe Tait
Joe Tait
(Southampton)
Photo of Konstantinos Tsaltas
Konstantinos Tsaltas
(Sheffield)
Photo of Peng Xu
Peng Xu
(East Anglia)


Prerequisites


Good command of complex analysis and algebra. Occasionally, some knowledge of algebraic number theory and Riemann surface theory would be helpful.

Syllabus


No syllabus information is available yet.

Bibliography


A first course in modular forms Diamond and Shurman
Topics in Classical Automorphic Forms Iwaniec
Introduction to elliptic curves and modular forms Koblitz
Modular Forms Miyake
Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

Assessment


There will be two homework assignments which both need to be passed satisfactorily.
HW1: Due March 9
HW2: Due April 13

Assignments


Assignment 1

Deadline: Friday 9 March 2012 (924.1 days ago)
Instructions:Do the following problems from the problem sheets:
HW-week4:(3)
HW-week5:(3) and (5)
HW-week6:(2) and (3)


Homework 2

Deadline: Friday 13 April 2012 (889.1 days ago)
Instructions:Do the following problems from the HW-sheets:
1) Week 6, (3), (iii)-(viii). Yes, again. You are allowed to use (i) and (ii) but NOT the general dimension formula which wasn't proved anyway in the lectures. Only the consequence from the "k/2-formula". Also do not use the Jacobi function (just yet, note that ^2 is the form of weight 1 for (iv)).
2) Week 10, (6). (in (iv) there is a small gap, namely, one would need to assume week 10, (5), which is ok).