2016 Advanced topics in Algebra F

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Academic unit or major
Graduate major in Mathematics
Mizumoto Shin-Ichiro 
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Syllabus updated
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Course description and aims

In this course the instructor explains basics topics of L-functions associated with single-variable regular automorphic forms. Knowledge of the definition and examples of single-variable regular automorphic forms is assumed, and the instructor covers the space structures formed by automorphic forms as a whole, and Hecke operators that act on them. Using Hecke operators, automorphic L-functions are then defined, and the instructor discusses Euler product representations and analytic continuation. This course follows Advanced Topics in Algebra E, which is held immediately before it.
Automorphic L-functions are a mathematical subject at the center of modern number theory research, and are even now the subject of active research.

Student learning outcomes

The following notions are impotant:
elliptic modular forms, graded ring of modular forms, Poincare series, Hecke operators, automorphic L-functions.
The aim of this course is help the students become acquainted with these notions through concrete examples.


elliptic modular forms, Poincare series, Hecke operators, automorphic L-functions

Competencies that will be developed

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

Standard lecture course

Course schedule/Required learning

  Course schedule Required learning
Class 1 fundamental domains Details will be provided during each class session
Class 2 dimension of the space of modular forms
Class 3 structure of the graded ring of modular forms
Class 4 Poincare series
Class 5 Hecke operators
Class 6 automorphic L-functions (1): Euler products
Class 7 automorphic L-functions (2): analytic continuation
Class 8 supplements and prospects


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.A501 : Advanced topics in Algebra E
  • 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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