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Jacobi modular forms: 30 ans après
This is a master course given in Moscow at the Laboratory of Algebraic Geometry of the National Research University Higher School of Economics by Valery Gritsenko, a professor of University Lille 1, France.
Jacobi modular group and the second definition of Jacobi forms. Special values of Jacobi modular forms
Zeros of Jacobi forms. The Jacobi theta-series, the Dedekind eta-function and the first examples of Jacobi modular forms
The Jacobi theta-series as Jacobi modular form. The basic Jacobi modular forms
Theta-blocks, theta-quarks and the first Jacobi cusp form of weight 2
Jacobi forms in many variables and the Eichler-Zagier Jacobi forms
Jacobi forms in many variables and the splitting principle. Theta-quarks as a pull-back. Weak Jacobi forms in many variables
The Weil representation and vector valued modular forms. Jacobi forms of singular weight
Quasi-modular Eisenstein series. The automorphic correction of Jacobi forms and Taylor expansions
Modular differential operators. The graded ring of the weak Jacobi modular forms
Jacobi type forms and the generalisation of the Cohen-Kuznetsov-Zagier operator
This is a master course given in Moscow at the Laboratory of Algebraic Geometry of the National Research University Higher School of Economics by Valery Gritsenko, a professor of University Lille 1, France.
Jacobi forms are holomorphic functions in two complex variables. They are modular in one variable and abelian (or double periodic) in another variable. The theory of Jacobi modular forms became an independent research subject after the famous book of Martin Eichler and Don Zagier “Jacobi modular forms” (Progress in Mathematics, vol. 55, 1985) which was cited more than a thousand times in research papers. This is due to many applications of Jacobi forms in arithmetic, topology, algebraic and differential geometry, mathematical and theoretical physics, in the theory of Lie algebras, etc. The list of mentioned subjects shows that my course might be useful for master and Ph.D. students working in different directions.
Motivated undergraduate students can also study this subject. To follow the course one has to know only elementary basic facts from the theory of modular forms (for example, the paragraphs 1-4 of the chapter VII of Serre’s “A Course in Arithmetic” are enough).
The main hero of the course is the Jacobi theta-series. Using it we will construct a lot of concrete examples of Jacobi forms in one or many abelian variables, in particular, Jacobi forms for root systems.
For some of you, who will be successful with the theoretical exercises of the course, I am ready to formulate research problems for Master or Ph.D. thesis. (Ph.D. support might be available at CEMPI in Lille or at the Faculty of Mathematics of National Research University Higher School of Economics in Moscow)
Do you have technical problems? Write to us: [email protected]
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Advanced Level
Approx. 62 hours to complete
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