2000 character limit reached
An Infinite Family of Real Quadratic Fields with Three Classes of Perfect Unary Forms
Published 2 Apr 2024 in math.NT | (2404.01538v1)
Abstract: In this paper, we revisit the theory of perfect unary forms over real quadratic fields. Specifically, we deduce an infinite family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ when $d=2$ or $3$ mod $4$, such that there are three classes of perfect unary forms up to homothety and equivalence. This work, along with the work in \cite{unitred}, seems to suggest that the number of classes of perfect unary forms is related to the fundamental unit of $K$.
- Bae, S.: “Real Quadratic Function Fields of Richaud-Degert Type with Ideal Class Number One”. Proc. Am. Math. Soc., vol. 140, no. 2, pp. 403–414 (2012).
- Koecher, M.: “Beiträge zu einer Reduktionstheorie in Positivitätsbereichen. I”. Math. Ann., vol. 141, pp. 384–432 (1960).
- Leibak, A., Porter, C., Ling, C.: “Unit Reducible Fields and Perfect Unary Forms”. Journal de Théorie des Nombres de Bordeaux, vol. 35, no. 6, pp. 867–895 (2024).
- Martinet, J.: “Perfect lattices in Euclidean spaces”. Grundlehren der mathematischen Wissenschaften, vol. 327 (2003).
- Yasaki, D.: “Perfect unary forms over real quadratic fields”. Journal de Théorie des Nombres de Bordeaux 25 (2013) pp. 759–775.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.
