Overview
This study focuses on the construction of holomorphic differential forms on specific orthogonal modular varieties. The investigation centers on modular varieties linked to even lattices possessing a signature of $(2, n)$, with $n$ satisfying the congruences $n \equiv 1 \pmod 8$ or $n \equiv 3 \pmod 8$, and a discriminant of $-2$. The range of $n$ considered in this context is $n \geq 25$. The constructed holomorphic differential forms encompass a variety of degrees, including what is identified as the minimum possible degree.
Research Context
The domain of orthogonal modular varieties is a key area within mathematics, particularly related to the study of automorphic forms and algebraic geometry. Within this context, the presence and characteristics of holomorphic differential forms are of considerable interest. Prior to this research, observations of holomorphic differential forms on orthogonal modular varieties had not specifically identified instances of such forms existing in non-top degrees.
Approach
The methodology employed for the construction of these holomorphic differential forms relies on the Arthur multiplicity formula. This formula is a component of the broader theory of automorphic representations. The application of this formula facilitated the identification and construction of the specified differential forms on the target modular varieties.
Findings
- Holomorphic differential forms were constructed on modular varieties associated with even lattices of signature $(2, n)$.
- These lattices had a discriminant of $-2$.
- The parameter $n$ in the signature $(2, n)$ satisfied the conditions $n \equiv 1 \pmod 8$ or $n \equiv 3 \pmod 8$.
- The construction was achieved for $n \geq 25$.
- The constructed forms include those of the minimum possible degree, alongside other degrees.
- This research marks the first instance of observing holomorphic differential forms of non-top degree on orthogonal modular varieties.