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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

Autumn 2016 (Monday, October 3 to Friday, December 9)

Timetable

  • Tue 10:05 - 10:55

Prerequisites

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

Syllabus


    (1) Modular curves, also as Riemann surfaces and as moduli space of elliptic curves (over C)
    (2) Modular functions and forms, basic properties, Eisenstein series, eta-function
    (3) Theta series, arithmetic applications
    (4) Modular forms and Dirichlet series, functional equation
    (5) Hecke operators, Petersson scalar product

Lecturer


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


Students


Photo of Daniel Evans
Daniel Evans
(Liverpool)
Photo of Maxime Fairon
Maxime Fairon
(Leeds)
Photo of Robert Little
Robert Little
(Durham)
Photo of David Pescod
David Pescod
(Newcastle)
Photo of Matthew Poulter
Matthew Poulter
(Lancaster)
Photo of Stefano Sannella
Stefano Sannella
(Birmingham)
Photo of Roberto Sisca
Roberto Sisca
(Surrey)
Photo of Kathryn Spalding
Kathryn Spalding
(Loughborough)
Photo of Raffael Stenzel
Raffael Stenzel
(Leeds)
Photo of Matty Van Son
Matty Van Son
(Liverpool)
Photo of James Whitley
James Whitley
(Birmingham)
Photo of Di Zhang
Di Zhang
(Sheffield)


Bibliography


The 1-2-3 of Modular FormsBruinier, van der Geer, Harder, Zagier
A Course in ArithmeticJean-Pierre Serre
Introduction to elliptic curves and modular formsKoblitz
Modular FormsMiyake
Topics in Classical Automorphic FormsIwaniec
Introduction to the Arithmetic Theory of Automorphic FunctionsGoro Shimura


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



The assessment for this course will be via a single take-home paper in January with 2 weeks to complete and submit online. There will be questions worth 100 marks and you will need 50 marks to pass.



Files:Exam paper
Released: Monday 9 January 2017 (44.6 days ago)
Deadline: Sunday 22 January 2017 (30.6 days ago)


Recorded Lectures


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