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2020/2021

Research Seminar "Elementary Introduction to the Theory of Automorphic Forms"

Type: Optional course (faculty)
When: 3, 4 module
Instructors: Andrey Levin
Language: English
ECTS credits: 3

Course Syllabus

Abstract

The automorphic forms in rather misterious way appear in different branches of mathematics from number theory to mathematical physics. At the other hand, the technique of this theory includes methods from various parts of mathematics like complex analysis, topology, differential geometry, Lie groups theory, number theory. Nevertheless, the main notions and concepts of this deep theory can be illustrated without use of comlicated tools.
Learning Objectives

Learning Objectives

  • Demonstration of the unity of the mathematics as science.Study theory of modularar forms in one variable with application to number theory and mathematical physics. Exploring the Eisenstein construction of trigonometric and elliptic functions. Realization quadratic divisors as the Hilbert surfaces in the Siegel 3-fold.
Expected Learning Outcomes

Expected Learning Outcomes

  • Creating the vision of mathematics as unified science and interrelations between its different bruches.
  • Realization of the technique of the universal object in the description of configuration of points in a linear space.
  • Realization of the concept of compactification for explicit example.
  • Demonstration of possible variability of the basic defenition for adoption to explicit problems.
  • Demonstration of the averaging method in different contexts.
  • Application of the group-theoretic constructions in geometry.
  • Introduction to the basic theory of geometric groups.
Course Contents

Course Contents

  • Elliptic Integrals and Elliptic Functions.
    Elliptic Functions as invertions of multi-valued integrals with geometrical and physical motivation.
  • 2-Lattices in the Complex Plane.
    2-Lattices as periods of the elliptic functions. Description of framed lattices as points of the upper half-plane. Cange of framing and action of the group of the integer unimodular matrices. The modular set and the modular figure.
  • Degeneration of the Lattices and Comactification.
    Degeneration of the lattices and rational points oe the projective line. Cusp points and comactification of the modular set.
  • Modular Forms in One Variable
    Homogeneous functions of latticis and automorphic forms. Development in the neighbourhood of the compactification point.
  • The Eisenstein Series
    Two realization of the Eisenstein series: via latticis and group-theoretical approuch. Deduction of the development near cusp from the Eisenstein construction of the trigonometric functions.
  • Double coset construction
    Transitivity of the action of the real unimodular on the upper half-plane. The orthoganal group as the stabilizer of the point i. Modular set as double coset. Interpretation of automorphic forms.
  • Automorphic Triples of Groups
    Topologic groups and Lie groups. Classification of the Lie groups. Examples: classical groups. Discrete and compact subgroups: integer classical groups and orthoganal groups.
Assessment Elements

Assessment Elements

  • non-blocking Home Work
  • non-blocking Home Test
  • non-blocking Intermedite Home Test
Interim Assessment

Interim Assessment

  • Interim assessment (4 module)
    0.7 * Home Work + 0.3 * Intermedite Home Test
Bibliography

Bibliography

Recommended Core Bibliography

  • Абелевы многообразия, Мамфорд, Д., Манина, Ю. И., 1971
  • Группы и алгебры Ли : алгебры Ли, свободные алгебры Ли и группы Ли, Бурбаки, Н., Бахтурина, Ю. А., 1976

Recommended Additional Bibliography

  • Группы и алгебры Ли : группы Кокстера и системы Титса. Группы, порожденные отражениями системы корней, Бурбаки, Н., Кострикина, А. И., 1972
  • Группы и алгебры Ли : подалгебры Картана, регулярные элементы, расщепляемые полупростые алгебры Ли, Бурбаки, Н., Рудакова, А. Н., 1978