Gauss's form class groups and Shimura's canonical models

2023/11/14 by Ja Kyung Koo, Koo, Ja Kyung, Dong Hwa Shin +3
Mathematics · #Algebraic Geometry and Number Theory #Advanced Mathematical Identities #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2311.07837

Abstract

Let N be a positive integer and Γ be a subgroup of SL₂(ℤ) containing Γ₁(N). Let K be an imaginary quadratic field and O be an order of discriminant D_O in K. Under some assumptions, we show that Γ induces a form class group of discriminant D_O (or of order O) and level N if and only if there is a certain canonical model of the modular curve for Γ defined over a suitably small number field. In this way we can find an interesting link between two different subjects, which will be useful in the study of certain quadratic Diophantine equations in terms of primes p.

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