A Short Introduction to the Algebra, Geometry, Number Theory and Physics of Moonshine

Duncan, John F. R.

doi:10.1007/978-3-030-42400-8_1

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2020

2023

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“…We refer to [1, 13, 15] for general reviews of moonshine, and refer to [7, § 6.3], [17, §§ 1.5–1.6], and also [14], for more detailed expository discussions of optimality.…”

Section: Introductionmentioning

confidence: 99%

Class numbers, cyclic simple groups, and arithmetic

Cheng

Duncan

Mertens

2023

Journal of London Math Soc

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Here, we initiate a program to study relationships between finite groups and arithmetic–geometric invariants in a systematic way. To do this, we first introduce a notion of optimal module for a finite group in the setting of holomorphic mock Jacobi forms. Then, we classify optimal modules for the cyclic groups of prime order, in the special case of weight 2 and index 1, where class numbers of imaginary quadratic fields play an important role. Finally, we exhibit a connection between the classification we establish and the arithmetic geometry of imaginary quadratic twists of modular curves of prime level.

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“…We refer to [1, 13, 15] for general reviews of moonshine, and refer to [7, § 6.3], [17, §§ 1.5–1.6], and also [14], for more detailed expository discussions of optimality.…”

Section: Introductionmentioning

confidence: 99%