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Abstracts for Vector Valued Modular Forms Day

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Christopher Marks (U of California, Santa Cruz/MPI)

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Speaker: 
Christopher Marks
Affiliation: 
U of California, Santa Cruz/MPI
Date: 
Wed, 2010-03-10 11:15 - 12:15
Location: 
MPIM Lecture Hall
Parent event: 
Vector Valued Modular Forms Day

The main purpose of this talk is to explain the connection between vector-valued modular forms for SL(2,Z) and certain ordinary differential equations in the punctured unit disk, whose coefficient functions are holomorphic modular forms. An important tool arising in this context is a modular version of the familiar Wronskian from the classical ODE theory; I will discuss, for example, how this modular Wronskian provides a lower bound for the weight of a nonzero holomorphic form associated to a given representation of SL(2,Z).

Vector-valued modular forms and matrix-valued hypergeometric functions

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Speaker: 
Terry Gannon
Affiliation: 
U Würzburg/U of Alberta, Edmonton
Date: 
Wed, 2010-03-10 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Vector Valued Modular Forms Day

A general theory of vector-valued modular forms is presented. In my talk I'll focus on weakly holomorphic vector-valued modular functions and their relation to a generalised hypergeometric equation, but time permitting I'll also describe the resulting dimension formulas for holomorphic vector-valued modular forms. I'll also demonstrate with examples how this theory is conducive to explicit calculations of Fourier coefficients.

Strongly reflective modular forms and applications

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Speaker: 
Valery Gritsenko
Affiliation: 
Univ. Lille 1/MPI
Date: 
Wed, 2010-03-10 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
Vector Valued Modular Forms Day

We discuss two new classes of strongly reflective modular forms with respect to orthogonal groups $O(2,n)$. The first class contains 36 functions including the Borcherds form $\Phi_{12}$. The second one has at least 12 functions (this work is in progress) including the Igusa modular form of weight 5. We give some applications of these remarkable modular forms to the algebraic geometry of modular varieties and to the theory of Kac-Moody Lie algebras.

 

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