# Teaching

**2021 Winter Semester: Modular Forms 1**

The lectures of this course will take place at the indicated times in video-conference via Zoom. You can register on the Moodle page with the enrolling key: MF1WS20/21.

Lectures: Tuesday 12h-14h, Friday 10h-12h; Exercise sessions: Friday 16h-18h.

The aim of the course is to introduce the basic theory of modular forms and related topics. We will talk about the definition of modular forms, their q-expansions, Hecke operators, L-functions, and modular curves. If time permits, we shall talk about Maass forms and automorphic forms for GL_2. Basic knowledge of complex analysis is requested.

References:

[1] F. Diamond and J. Shurman, A first course in modular forms, GTM 228, Springer, 2005.

[2] D. Bump, Automorphic forms and representations, CSAM 55, Cambridge University Press, 1997.

[3] N. Koblitz, Introduction to elliptic curves and modular forms, GTM 97, Springer, 1984.

[4] D. Goldfeld and J. Hundley, Automorphic Representations and L-functions for the General Linear Group, CSAM 129, Cambridge University Press, 2011.

**2020 Summer Semester: Seminar on Beilinson's conjecture**

In this seminar, we are going to read J. Nekovář's survey article on Beilinson's conjectures. The seminar takes place each Wednesday at 14h15-15h30 online via zoom. You can find the detailed program and the notes here: http://www.ihes.fr/~linjie/IHES/Seminar_SS2020.html.

Talk 1 (29 April, by Xiaoyu Zhang). Introduction and overview.

Talk 2. (6 May, by Jie Lin) Motives and realizations.

Talk 3. (13 May, by Jie Lin) Deligne's period map.

Talk 4. (20 May, by Jie Lin) Motivic cohomology.

Talk 5. (3 June, by Chirantan Chowdhury) Mixed motives and examples.

Talk 6. (10 June, by Chirantan Chowdhury) K-theoretic regulator map.

Talk 7. (17 June, by Luca Dall'Ava) Beilinson's conjectures.

Talk 8. (1 July, by Xiaoyu Zhang) Scholl's version of Beilinson's conjectures.

Talk 9. (8 July, by Quentin Gazda) Some recent progress towards these conjectures.

**2020 Summer Semester: Modular Forms 2**

The course will start from the week of April 20 and will probably be online. Please write to jie.lin@uni-due.de or/and xiaoyu.zhang@uni-due.de if you are interested in the course.

Lectures: Tuesday 12h-14h, Friday 10h-12h; Exercise sessions: Friday 12h-14h.

The first half of this course (by Dr. Xiaoyu Zhang) will be a continuation of the course Modular Form 1 of the last Winter semester. We will continue to introduce some basic notions and results in the theory of modular forms on the upper half plane, in particular Hecke operators, Petersson products, oldforms and newforms (the theory of Atkin-Lehner), the Eichler-Shimura isomorphism. After that, we will study the algebraicity of certain Dirichlet L-functions via Eisenstein series, certain standard L-functions of modular forms and then pass to the construction of p-adic interpolations of these L-functions using modular symbols. If times permits, we will study the p-adic interpolation of certain families of modular forms (the Hida theory) and briefly discuss some of its applications.

In the second half (by Dr. Jie Lin), we will talk about automorphic forms for GL_2 over the real field and over the ring of adeles. We will start from Maass forms, their Fourier-Whittaker expansions, and the spectrum. We will then talk about the ring of adeles and the automorphic forms over it. In particular, we will discuss the adelic lifting of classical forms. In the end, we will introduce the theory of automorphic representations.

References:

[1] F. Diamond and J. Shurman, A first course in modular forms, GTM 228, Springer, 2005.

[2] H. Hida, Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts 26, 1993.

[3] D. Bump, Automorphic forms and representations, CSAM 55, Cambridge University Press, 1997.

[4] D. Goldfeld and J. Hundley, Automorphic Representations and L-functions for the General Linear Group, CSAM 129, Cambridge University Press, 2011.

Prerequisites: Complex analysis, Modular form 1 (basic definitions, Hecke operators, basic of L-functions), Complex elliptic curves

**2019 Summer Semester: Modular Forms 2**

** **I will teach until June 7 and then the course will be continued with Prof. Dr. Jasmin Matz.

Lectures: Tuesdays 12h00-13h30, WSC-S-U-3.03 and Fridays 14h00-15h30, WSC-S-U-3.01

Contents: The aim of the course is to introduce the basic theory of modular forms and related topics. We shall discuss how they can be used in classical number theory problems as well as their generalizations from representation theoretic point of view. We will talk about Rankin-Selberg L-function and the converse theorem. If time permits, we will introduce briefly automorphic representations and Langlands functoriality.

Prerequisites: Complex analysis, Riemann surface, Algebraic Number Theory 1 (NO need of Modular form 1)

References:

[1] D. Bump, Automorphic forms and representations, CSAM 55, Cambridge University Press, 1997.

[2] F. Diamond and J. Shurman, A first course in modular forms, GTM 228, Springer, 2005.

[3] N. Koblitz, Introduction to elliptic curves and modular forms, GTM 97, Springer, 1984.

**2018/19 Winter Semester: **

Algebraic Number Theory 3: Tuesday 14h-16h, WSC-S-3.14 and Friday 14h-16h WSC-O-3.46.

Topic of the course: class field theory, Tate thesis