Project Details
Projekt
Two applications of the doubling method: The basis problem for Siegel modular forms for the Weil representation and the Kodaira Dimension of A_6
Applicant
Dr. Manuel Karl-Heinz Müller
Subject Area
Mathematics
Term
since 2025
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 566117541
The doubling method is a powerful tool in the theory of modular forms. This project aims to further extend its applications and explore its potential in a new geometric context. In his previous research, the applicant successfully applied the doubling method to the basis problem for elliptic modular forms for the Weil representation. Given this experience, he plans to extend these techniques to Siegel modular forms and solve the basis problem in this setting. Beyond that, he then aims to study the Kodaira dimension of the moduli space A_6 using the doubling method. The Weil representation describes how vector-valued Siegel theta series, special functions associated with even lattices, transform under the action of the metaplectic group. It plays an important role in representation theory, in the theory of modular forms and even in quantum field theory. The basis problem asks whether the space of cusp forms, an important class of modular forms, is spanned by theta series of even lattices. It is a classical question in the theory of modular forms and has applications, for example, to the geometry of orthogonal Shimura varieties. The moduli space A_n parameterizes principally polarized complex abelian varieties of dimension n. The Kodaira dimension of a (quasi-)projective variety measures the size of its canonical model and is a fundamental invariant in algebraic geometry. While the Kodaira dimension of A_n has been known for all n different from 6 for over 40 years, the case n = 6 still remains unresolved. This geometric problem shall be approached by applying the doubling method to suitable Siegel modular forms.
DFG Programme
WBP Fellowship
International Connection
Italy
