Construction of Theta Series of Any Vector-Valued Weight and Applications to Lifts and Congruences

ABSTRACT

It is well known that one can construct Siegel modular forms by theta series with pluriharmonic polynomials which realize the representation of that weight. But it seems we do not have enough reference how to choose these pluriharmonic polynomials for a concrete weight in complicated cases. We explain a general simple way to choose pluriharmonic polynomials concretely for arbitrary fixed vector-valued weight. We also give some examples of degree 3 constructed by this method in the case when the weight is more complicated than the determinant power cross the symmetric tensor. This gives explicit Siegel modular forms which appear in conjectures of liftings by Bergström, Faber, and van der Geer. We also prove a new congruence between examples of the above lifts and other forms.

KEYWORDS:

• Vector valued Siegel modular forms
• Lift
• Congruences
• Theta series

2010 AMS SUBJECT CLASSIFICATION:

• Primary 11F27
• 11F33
• 11F46

Additional information

Funding

Tomoyoshi Ibukiyama was supported by JSPS KAKENHI Grant Number 25247001. Sho Takemori was supported by JSPS KAKENHI Grant Number 16H03919.