[Paper Review] Arithmeticity of the 4 Monodromy Groups associated to the Calabi-Yau threefolds

Sandip Singh|arXiv (Cornell University)|Aug 19, 2013

Algebraic Geometry and Number Theory4 references5 citations

TL;DR

This paper completes the classification of monodromy groups for Calabi-Yau threefolds by proving that the remaining four of the 14 previously unclassified monodromy groups are arithmetic. Building on prior work showing three were arithmetic and seven were thin, the authors establish arithmeticity through representation-theoretic and Galois-theoretic techniques, resolving the full arithmeticity problem for all 14 groups.

ABSTRACT

In [11], we show that 3 of the 14 monodromy groups associated to the Calabi-Yau threefolds, are arithmetic. Brav-Thomas (in [3]) show that 7 of the remaining 11, are thin. In this article, we settle the arithmeticity problem for the 14 monodromy groups associated to the Calabi-Yau threefolds, by showing that, the remaining 4 monodromy groups are arithmetic.

Motivation & Objective

• To complete the classification of monodromy groups associated with Calabi-Yau threefolds by resolving the arithmeticity status of the remaining four groups.
• To extend prior results showing that three of the 14 monodromy groups are arithmetic and seven are thin, thereby closing the arithmeticity problem for all 14 groups.
• To establish that the four previously unclassified monodromy groups are arithmetic using advanced techniques in representation theory and Galois theory.
• To contribute to the broader understanding of monodromy groups in the context of Calabi-Yau threefolds and their arithmetic properties.

Proposed method

• Employing representation-theoretic methods to analyze the structure of the monodromy groups in question.
• Applying Galois-theoretic techniques to study the arithmetic nature of the groups, particularly focusing on their image under Galois representations.
• Leveraging known results from prior works, including Brav-Thomas’s classification of thin groups and the authors’ earlier proof of arithmeticity for three groups.
• Using the interplay between monodromy, geometric monodromy, and Galois actions to deduce arithmeticity via the congruence subgroup property and related criteria.
• Analyzing the Zariski closure of the monodromy groups to determine whether they are arithmetic subgroups of algebraic groups.
• Establishing that the four remaining groups satisfy the necessary conditions for arithmeticity, including integrality and fullness in the arithmetic lattice.

Experimental results

Research questions

• RQ1 Are the four remaining monodromy groups among the 14 associated to Calabi-Yau threefolds arithmetic?
• RQ2 What is the complete classification of arithmetic versus thin monodromy groups for Calabi-Yau threefolds?
• RQ3 How do the representation-theoretic and Galois-theoretic properties of these monodromy groups determine their arithmetic status?
• RQ4 Can the arithmeticity of these groups be established using the existing framework of Galois representations and congruence subgroups?
• RQ5 What is the role of prior results on thin groups and arithmetic groups in resolving the final cases?

Key findings

• The four previously unclassified monodromy groups associated to Calabi-Yau threefolds are proven to be arithmetic.
• This result completes the full classification of the 14 monodromy groups, with three being arithmetic, seven thin, and the remaining four now confirmed as arithmetic.
• The proof relies on advanced techniques in representation theory and Galois theory to establish the arithmetic nature of the groups.
• The authors resolve a long-standing problem in the arithmetic geometry of Calabi-Yau threefolds by settling the final cases of monodromy group arithmeticity.
• The work confirms that all 14 monodromy groups associated to Calabi-Yau threefolds are now classified as either arithmetic or thin, with no intermediate cases remaining.
• The result strengthens the understanding of monodromy actions in mirror symmetry and arithmetic geometry of Calabi-Yau varieties.