2016 Advanced topics in Algebra E

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Academic unit or major
Graduate major in Mathematics
Mizumoto Shin-Ichiro 
Class Format
Media-enhanced courses
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Course description and aims

This course covers basic topics of single-variable regular automorphic forms. Building on basic undergraduate level knowledge, basic properties of the Riemann zeta function are proven, and students are introduced to the theory of automorphic L-functions. Single-variable regular automorphic forms are then defined, and students become familiar with specific treatments of the materials through several examples. This course is followed by Advanced Topics in Algebra F.
Automorphic forms are the foundation of modern number theory, and are an important mathematical subject related to a variety of fields such as group representation theory, the geometry of numbers, and theoretical physics.

Student learning outcomes

The following concepts are especially important:
Riemann Zeta function (Euler product, analytic continuation, special values), elliptic automorphic form, Fourier coefficient, Eisenstein series.
Students will become familiar with these concepts, and learn the skills for calculating examples on their own.


Modular forms, modular groups, zeta functions

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Standard lecture course

Course schedule/Required learning

  Course schedule Required learning
Class 1 multiplivative functions Details will be provided during each class session
Class 2 Riemann zeta function
Class 3 analytic continuation and special values of the Riemann zeta function
Class 4 modular groups
Class 5 elliptic modular forms
Class 6 examples of modular forms (1): Eisenstein series
Class 7 examples of modular forms (2): Ramanujan's delta function
Class 8 Fourier expansion of Eisenstein series


None required

Reference books, course materials, etc.

T. M. Apostol: Modular Functions and Dirichlet Series in Number Theory (Springer)

Assessment criteria and methods

Course scores are evaluated by homework assignments. Details will be announced during the course.

Related courses

  • MTH.A502 : Advanced topics in Algebra F
  • MTH.C301 : Complex Analysis I
  • MTH.C302 : Complex Analysis II
  • MTH.A201 : Introduction to Algebra I
  • MTH.A202 : Introduction to Algebra II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

basic undergraduate algebra and complex analysis

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