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Master 2014/2015

Elliptic Modular Forms and Their Applications

Type: Elective course (Mathematics)
Area of studies: Mathematics
When: 1 year, 3, 4 module
Instructors: Sergey Galkin (author, delivers lectures, checks works and administers exams)
Master’s programme: Mathematics
Prerequisites:
Курс мог бы быть полезен всем, кто интересуется арифметикой и геометрией. Не повредят знания основ алгебры, комплексного анализа (формулы Коши) и умение дифференцировать. Course might be useful for those interested in arithmetics and geometry. Prerequisites: knowledge of basic algebra, complex analysis (Cauchy formula) and differentiation.
Language: Russian
ECTS credits: 3
Elliptic Modular Forms and Their Applications

I will loosely follow Don Zagier’s book with same name.
Zagier, Don. "Elliptic modular forms and their applications." *The 1-2-3 of
modular forms*. Springer Berlin Heidelberg, 2008. 1-103.

1. Modular groups, modular functions, modular forms. Fundamental domains.
Finite dimensionality of spaces of modular forms.
2. Eisenstein series, the discriminant.
3. Theta-series.
4. Hecke eigenseries and L-series.
5. Differential operators: derivatives of modular forms, Rankin-Cohen
brackets, quasi-modular forms.
6. Complex multiplication.