À propos de ce cours
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Approx. 54 heures pour terminer

Recommandé : 12 weeks of study, 3-5 hours per week...

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Sous-titres : Anglais

100 % en ligne

Commencez dès maintenant et apprenez aux horaires qui vous conviennent.

Dates limites flexibles

Réinitialisez les dates limites selon votre disponibilité.

Niveau avancé

Approx. 54 heures pour terminer

Recommandé : 12 weeks of study, 3-5 hours per week...

Anglais

Sous-titres : Anglais

Programme du cours : ce que vous apprendrez dans ce cours

Semaine
1
1 heure pour terminer

Introduction to the Course

Welcome to the course! I hope you have an opportunity to reserve some time to explore the course content, course logic and our grading policy. The course consists of 12 lectures. This course will help you to start your progress in the field of the theory of Jacobi modular forms. Best regards, Valery Gritsenko

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1 vidéo (Total 14 min), 4 lectures
1 vidéo
4 lectures
Course Overview10 min
Grading and Logistics10 min
Suggested Readings10 min
About the Instructor10 min
3 heures pour terminer

Jacobi modular forms: motivations

This module is devoted to motivations to study Jacobi forms. We provide some first examples including theta-functions. Also there is a peer review in the end of this module.

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5 vidéos (Total 62 min), 1 quiz
5 vidéos
Theta-function12 min
Modular and abelian transformations12 min
Pullbacks of theta-function12 min
Modular forms12 min
Semaine
2
3 heures pour terminer

Jacobi modular forms: the first definition

This module is devoted to the first definition of Jacobi forms. In this module we also define Jacobi modular group. Also there is a peer review in the end of this module.

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6 vidéos (Total 75 min), 1 quiz
6 vidéos
Definition of Jacobi forms (part 2)13 min
Basic properties of Jacobi forms12 min
Jacobi modular group11 min
Symplectic group11 min
Jacobi modular group (part 2)13 min
Semaine
3
3 heures pour terminer

Jacobi modular group and the second definition of Jacobi forms. Special values of Jacobi modular forms

This module is devoted to the second definition of Jacobi forms. In this module we also consider special values of Jacobi forms. Also there is a peer review in the end of this module.

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6 vidéos (Total 86 min), 1 quiz
6 vidéos
The action of Jacobi modular group14 min
The action of Jacobi modular group (part 2)10 min
The second definition of Jacobi forms15 min
Special values of Jacobi forms11 min
The first theorem20 min
Semaine
4
3 heures pour terminer

Zeros of Jacobi forms. The Jacobi theta-series, the Dedekind eta-function and the first examples of Jacobi modular forms

This module is devoted to zeros of Jacobi modular forms, their Taylor extensions and the first examples of Jacobi forms. Using classical Jacobi theta-series and Dedekind eta-function we construct a series of Jacobi forms. Also there is a peer review in the end of this module.

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6 vidéos (Total 76 min), 1 quiz
6 vidéos
The zeros of Jacobi forms (part 2)11 min
Taylor expansion of Jacobi forms12 min
Taylor expansion of Jacobi forms (part 2)13 min
Dimensions of some spaces of Jacobi forms10 min
Examples of Jacobi modular forms14 min
Semaine
5
3 heures pour terminer

The Jacobi theta-series as Jacobi modular form. The basic Jacobi modular forms

This module is devoted to detailed study of Jacobi theta-series. We will discuss abelian, modular and some other properties of this function. Also there is a peer review in the end of this module.

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6 vidéos (Total 84 min), 1 quiz
6 vidéos
Properties of Jacobi theta-function9 min
Properties of Jacobi theta-function (part 2)13 min
Properties of Jacobi theta-function (part 3)15 min
Construction of Jacobi forms using theta-series15 min
Construction of Jacobi forms using theta-series (part 2)13 min
Semaine
6
4 heures pour terminer

Theta-blocks, theta-quarks and the first Jacobi cusp form of weight 2

This module is devoted to very important notion of theta-blocks and theta-quarks. In this module we also construct the first Jacobi form of weight 2. Also there is a peer review in the end of this module.

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7 vidéos (Total 96 min), 1 quiz
7 vidéos
Theta-blocks (part 2)13 min
Theta-quarks14 min
Dimension of the space of Jacobi forms of odd weight14 min
Theta-quarks (part 2)11 min
The proof of theorem about theta-quarks16 min
The first Jacobi cusp form of weight 210 min
Semaine
7
4 heures pour terminer

Jacobi forms in many variables and the Eichler-Zagier Jacobi forms

This module is devoted to Jacobi forms in many variables. In this module we also define classical Eichler-Zagier Jacobi forms in terms of Jacobi forms in many variables. Also there is a peer review in the end of this module.

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6 vidéos (Total 93 min), 1 quiz
6 vidéos
Jacobi theta-series and Jacobi forms in many variables (part 2)15 min
Hyperbolic reformulation14 min
Examples of Jacobi forms in many variables14 min
Examples of Jacobi forms in many variables (part 2)16 min
Examples of Jacobi forms in many variables (part 3)16 min
Semaine
8
3 heures pour terminer

Jacobi forms in many variables and the splitting principle. Theta-quarks as a pull-back. Weak Jacobi forms in many variables

In this module we continue studying Jacobi forms in many variables. Among other things we discuss splitting principle and realize theta-quarks as a pull-back. Also there is a peer review in the end of this module.

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6 vidéos (Total 90 min), 1 quiz
6 vidéos
Theta-quarks as pull-backs of Jacobi forms (part 2)16 min
Theta-quarks as pull-backs of Jacobi forms (part 3)13 min
The splitting principle15 min
The splitting principle (part 2)16 min
Weak Jacobi forms in many variables13 min
Semaine
9
3 heures pour terminer

The Weil representation and vector valued modular forms. Jacobi forms of singular weight

This module is devoted to very useful notion of the Weil representation and vector-valued modular forms. In this module we also define Jacobi forms of singular weight. Also there is a peer review in the end of this module.

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6 vidéos (Total 85 min), 1 quiz
6 vidéos
The Weil representation (part 2)15 min
Jacobi forms of singular weight14 min
Jacobi forms of singular weight (part 2)15 min
Jacobi forms of critical weight15 min
Jacobi forms of critical weight (part 2)12 min
Semaine
10
3 heures pour terminer

Quasi-modular Eisenstein series. The automorphic correction of Jacobi forms and Taylor expansions

This module is devoted to Quasi-modular Eisenstein series. In this module we also define the automorpic correction of Jacobi forms and its Taylor expansion that gives us the way to construct the series of Jacobi forms. Also there is a peer review in the end of this module.

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6 vidéos (Total 84 min), 1 quiz
6 vidéos
Differential operator D15 min
Quasi-modular Eisenstein series15 min
Automorphic correction and Taylor expansions13 min
Automorphic correction of Jacobi forms in many variables12 min
Automorphic correction of Jacobi forms in many variables (part 2)12 min
Semaine
11
3 heures pour terminer

Modular differential operators. The graded ring of the weak Jacobi modular forms

This module is devoted to Modular differential operators. In this module we also consider the Jacobi forms as the space with the structure of the bigraded ring. Also there is a peer review in the end of this module.

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6 vidéos (Total 90 min), 1 quiz
6 vidéos
Heat operator14 min
Heat operator (part 2)15 min
Applications of differential operators for Jacobi forms11 min
The graded ring of the weak Jacobi modular forms17 min
Jacobi type forms in many variables14 min
Semaine
12
3 heures pour terminer

Jacobi type forms and the generalisation of the Cohen-Kuznetsov-Zagier operator

The last module is devoted to Jacobi type forms. In this module we also consider the generalisation of the Cohen-Kuznetsov-Zagier operator. Also there is a peer review in the end of this module.

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4 vidéos (Total 52 min), 1 quiz
4 vidéos
A new proof of Cohen-Kuznetsov-Zagier theorem13 min
A new proof of Cohen-Kuznetsov-Zagier theorem (part 2)9 min
Cohen-Kuznetsov-Zagier theorem in case of many variables14 min

Enseignant

Avatar

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

À propos de Université nationale de recherche, École des hautes études en sciences économiques

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communicamathematics, engineering, and more. Learn more on www.hse.ru...

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