Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/120989
Families of orders of quaternion algebras over $\mathbb{Q}$
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Every quaternion algebra contains a set of orders, whose understanding would be helping for the Shimura curves theory development. In this master’s thesis, certain parametric families of orders of quaternion algebras over $\mathbb{Q}$ have been defined, and their relationships with Eichler orders have been studied. In particular, for some given quaternion algebras over $\mathbb{Q}$ , we have defined and studied three families of orders $\mathcal{O}$, $\mathcal{O'}$ and $\mathcal{O''}$ , together with a maximal order $\mathcal{O}^{max}$ belonging to all of the families. As a main result, given a square-free integer $N$ coprime with the discriminant of the quaternion algebra given, it is possible to find an Eichler order of level $N$ belonging to the family $\mathcal{O'}$ and satisfying $\mathcal{O}^{max} \supset \mathcal{O'} \supset \mathcal{O''} \supset \mathcal{O}=\mathbb{Z}+N\mathcal{O}^{max}$, in a way that every quotient is isomorphic to $\mathbb{Z}/N\mathbb{Z}$ as abelian groups, this is, $\mathcla{O}/\mathcal{O'}\cong \mathcla{O'}/\mathcal{O''}\cong \mathcla{O''}/\mathcal{O}\cong\mathbb{Z}/N\mathbb{Z}$.
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Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: Alejandro de Miquel Bleier, Director: Artur Travesa i Grau
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MIQUEL BLEIER, Alejandro de. Families of orders of quaternion algebras over $\mathbb{Q}$. [consulted: 10 of August of 2026]. Available at: https://hdl.handle.net/2445/120989