- Lecturer
- Prof. Dr. Özlem Imamoglu | ozlem.imamoglu@math.ethz.ch
- Coordinator
- Dr. Sebastián Herrero | sebastian.herrero.m@gmail.com

Date | Room | Content | Notes |
---|---|---|---|

21st February 2023 | HG G 5 | Motivation and background | Lecture 1 |

24th February 2023 | HG F 5 | The modular group and its action on \( \mathbb{H} \) | Lecture 2 |

28th February 2023 | HG E 22 | Fundamental domain and generators for the modular group | Lecture 3 |

3rd March 2023 | ML J 37.1 | (Exercise class #1) Review of Lectures 1 to 3 | Exercise class 1 |

7th March 2023 | HG E 22 | Modular forms for \( \mathrm{SL}_2(\mathbb{Z}) \) | Lecture 4 |

10th March 2023 | HG F 5 | Eisenstein series for \( \mathrm{SL}_2(\mathbb{Z}) \) | Lecture 5 |

14th March 2023 | HG E 22 | The valence formula and the structure of \( M_k \) | Lecture 6 |

17th March 2023 | ML J 37.1 | (Exercise class #2) The cotangent identity, the weight 2 Eisenstein series and relations among modular forms | Exercise class 2 |

21st March 2023 | HG E 22 | Klein's \( j \)-invariant and the discriminant modular function | Lecture 7 |

24th March 2023 | HG F 5 | Growth of Fourier coefficients and \( L \)-functions of modular forms | Lecture 8 |

28th March 2023 | HG E 22 | Analytic continuation and functional equation of Riemann's zeta function | Lecture 9 |

31st March 2023 | ML J 37.1 | (Exercise class #3) Algebraic independence of \( E_4 \) and \(E_6 \), infinite product expansion of \( \Delta \) and values of \( j\) | Exercise class 3 |

4th April 2023 | HG E 22 | Modular \(L\)-functions and Poincaré series | Lecture 10 |

Easter break | No classes | ||

18th April 2023 | HG E 22 | Properties of Poincaré series | Lecture 11 |

21st April 2023 | HG F 5 | Hecke operators (part I) | Lecture 12 |

25th April 2023 | HG E 22 | Hecke operators (part II) | Lecture 13 |

28th April 2023 | ML J 37.1 | (Exercise class #4) Duplication formula for \( \Gamma \), modular \( L \)-functions, Hecke eigenforms | Exercise class 4 |

2nd May 2023 | HG E 22 | Hecke operators (part III) | Lecture 14 |

5th May 2023 | HG F 5 | Review of sections 1 to 6 | Lecture 15 |

9th May 2023 | HG E 22 | Class number formula and binary quadratic forms | Lecture 16 |

12th May 2023 | ML J 37.1 | (Exercise class #5) Petersson inner product, Hecke operators are hermitian on cusp forms, Miller basis | Exercise class 5 |

16th May 2023 | HG E 22 | Reduced quadratic forms and class numbers | Lecture 17 |

19th May 2023 | HG F 5 | Number of representations of an integer by a quadratic form | Lecture 18 |

23rd May 2023 | HG E 22 | Number of primitive representations and class number formula | Lecture 19 |

Additional material: Fourier expansion of the non-holomorphic Eisenstein series and Maass forms | Additional notes | ||

26th May 2023 | ML J 37.1 | (Exercise class #6) | |

30th May 2023 | HG E 22 | Applications of modular forms | Lecture 20 |

2nd June 2023 | HG F 5 | Review of section 7 | Lecture 21 |

The new problem sheets will be posted here every other Tuesday. We expect you to look at the problems and to prepare questions for the exercise class. Please hand in to the course coordinator the problems that you want to receive feedback on, either at HG J 16.5 or via e-mail to sebastian.herreromiranda@math.ethz.ch.

Exercise sheet | Hand-in by | Solutions |
---|---|---|

Exercise sheet 1 | 10th March 2023 | Solutions 1 |

Exercise sheet 2 | 24th March 2023 | Solutions 2 (updated!) |

Exercise sheet 3 | 6th April 2023 | Solutions 3 |

Exercise sheet 4 | 5th May 2023 | Solutions 4 |

Exercise sheet 5 | 19th May 2023 | Solutions 5 (updated!) |

Exercise sheet 6 | 2nd June 2023 | Solutions 6 |

- A Course in Arithmetic by J.-P. Serre (Springer 1973)
- Modular Forms by T. Miyake (Springer 1989)
- Introduction to Elliptic Curves and Modular Forms by N. Koblitz (Springer 1993)
- A First Course in Modular Forms by F. Diamond and J. Shurman (Springer 2005)
- Introduction to Modular Forms by S. Lang (Springer 1987)
- The 1-2-3 of Modular Forms by J. Bruinier, G. van der Geer, G. Harder, and D. Zagier (Springer 2008)