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401-2110-24L 4 Credits BSC , MSC D-MATH

Elliptic Functions and Modular Forms

Lecturers & Examiners: Dr. Markus Schwagenscheidt
Number of participants limited to 24.
VVZ CR n/a

Last Updated: 2026-02-05 16:37:24

Abstract

Seminar about the basic theory of Elliptic Functions and Modular Forms.

Objective

We start with the basic theory of elliptic functions, which are doubly periodic meromorphic functions on the complex plane. As a fundamental example we will construct the Weierstrass p-function, which serves as a building block for all elliptic functions, and which parametrizes complex elliptic curves. We will also discuss the Weierstrass sigma and zeta functions and the Jacobi theta function, which can be used to construct elliptic functions with prescribed zeros and poles. From elliptic functions one is naturally led to the study of modular forms. These are holomorphic functions on the complex upper half-plane that have infinitely many symmetries under Moebius transformations. Using the residue theorem one can show the remarkable fact that the vector spaces of modular forms of fixed weight are finite-dimensional. As examples of modular forms we will study Eisenstein series, the Delta-function, and the j-invariant. By computing the Fourier expansions of modular forms and using the finite-dimensionality of the spaces of modular forms, one obtains interesting applications in number theory. For example, we will prove the fact that every natural number can be written as a sum of four squares. At the end of the seminar we will define Hecke operators and use them to prove that the Fourier coefficients of the Delta-function are multiplicative. Finally, we study the L-functions associated with modular forms and prove their analytic continuation and functional equation.

Resources

Lecture Notes

See the list of topics on the website of the seminarhttps://people.math.ethz.ch/~mschwagen/ellipticfunctionsmodularforms

Literature

See the list of topics on the website of the seminar https://people.math.ethz.ch/~mschwagen/ellipticfunctionsmodularforms

Learning Materials (Links)

General Information

Language
English
Levels
BSC , MSC

Examination

Type
ungraded semester performance

Registration & Places

Limited places (Special selection)
Signup Start
01.01.2024
Signup End
10.02.2024
Priority: Registration for the course unit is only possible for the primary target group

Course Components

Type Title Time & Place Hours
seminar Elliptic Functions and Modular Forms
  • Wed 10:15-12:00 (HG G 26.5)
2 h weekly

Offered In

  • Mathematics Bachelor
    • Seminars (NOTICE: The number of seminar places is limited, and the special selection procedure should help to allocate the places not primarily according to the registration time. Everybody is waitlisted first when he/she tries to register for a seminar in myStudies. Moreover: Only one mathematics seminar can be chosen per semester. Notice also the course unit 401-0002-99L Generic Seminar - Second Priority / Third Priority.)
  • Mathematics Master
    • Seminars and Semester Papers
      • Seminars (NOTICE: The number of seminar places is limited, and the special selection procedure should help to allocate the places not primarily according to the registration time. Everybody is waitlisted first when he/she tries to register for a seminar in myStudies. Moreover: Only one mathematics seminar can be chosen per semester. In case you need to attend 2 seminars in this semester, please take contact with the Study Administration (email: ). Notice also the course unit 401-0002-99L Generic Seminar - Second Priority / Third Priority.)