[Paper Review] Finiteness of stable orthogonal modular varieties of non-general type

Shouhei Ma|arXiv (Cornell University)|Sep 27, 2013

Advanced Algebra and Geometry10 references3 citations

TL;DR

This paper proves that for even lattices $L$ of signature $(2,n)$ with $n \geq 15$, there are only finitely many such lattices for which the stable orthogonal modular variety $\mathcal{F}_L$ is not of general type. The proof reduces the problem to quasi-cyclic lattices using discriminant group properties and employs modular forms and cusp form theory to establish finiteness, showing that $\mathcal{F}_L$ is of general type for $n \geq 42$. This confirms that non-general type modular varieties are exceptional in high dimensions.

ABSTRACT

We prove that there are only finitely many even lattices L of signature (2,n) with n>14 such that the modular variety defined by the stable orthogonal group of L is not of general type.

Motivation & Objective

• To establish that modular varieties $\mathcal{F}_L$ associated with even lattices $L$ of signature $(2,n)$ are almost always of general type when $n$ is large.
• To prove that only finitely many such lattices $L$ with $n \geq 15$ yield $\mathcal{F}_L$ that are not of general type.
• To reduce the general case to the subclass of quasi-cyclic lattices, which admit controlled discriminant group behavior and allow cusp form lifting.
• To demonstrate that the birational type of $\mathcal{F}_L$ is generically general type, highlighting the rarity of non-general type examples in high-dimensional orthogonal modular varieties.

Proposed method

• Reduction of the main problem to quasi-cyclic lattices via the existence of quasi-cyclic overlattices with controlled discriminant group exponent.
• Use of the stable orthogonal group $\widetilde{\rm O}^+(L)$ to define the modular variety $\mathcal{F}_L = \widetilde{\rm O}^+(L)\backslash \mathcal{D}_L$, a quasi-projective variety of dimension $n$.
• Application of Jacobi lifting to cusp forms, relying on the quasi-cyclic condition to ensure that lifts remain cusp forms.
• Local analysis at each prime $p$, especially $p=2$, to construct isotropic subgroups $G$ in the discriminant group $A_L$ such that $G^\perp/G$ is quasi-cyclic and retains high exponent.
• Use of modular forms and their vanishing order at cusps to detect non-general type behavior, particularly through the non-vanishing of certain modular forms.
• Leveraging results from Gritsenko, Hulek, and Sankaran on Kodaira dimension of orthogonal modular varieties to inform the finiteness argument.

Experimental results

Research questions

• RQ1 Are there only finitely many even lattices $L$ of signature $(2,n)$ with $n \geq 15$ for which the modular variety $\mathcal{F}_L$ is not of general type?
• RQ2 Can the finiteness of non-general type modular varieties be established by reducing to the subclass of quasi-cyclic lattices?
• RQ3 What is the role of the discriminant group's exponent and structure in determining the birational type of $\mathcal{F}_L$?
• RQ4 How does the stable orthogonal group $\widetilde{\rm O}^+(L)$ influence the birational geometry of $\mathcal{F}_L$?
• RQ5 To what extent can the method of modular forms and cusp form lifting be used to detect non-general type behavior in orthogonal modular varieties?

Key findings

• There are only finitely many even lattices $L$ of signature $(2,n)$ with $n \geq 15$ such that the modular variety $\mathcal{F}_L$ is not of general type.
• The modular variety $\mathcal{F}_L$ is of general type for all $n \geq 42$, establishing a sharp effective bound.
• The finiteness result is effective in principle, as one could in theory enumerate all exceptional $L$ with sufficient computation.
• For any even lattice $L$ of signature $(2,n)$ with $n \geq 15$, if the exponent of its discriminant group $A_L$ exceeds $2D$ for a certain $D$, then $\mathcal{F}_L$ is of general type.
• The class of quasi-cyclic lattices is sufficient to reduce the general problem, as any lattice admits a quasi-cyclic overlattice with controlled discriminant group exponent.
• The discriminant group of a quasi-cyclic form on a 2-group has length at most 5, which bounds the complexity of such forms.