Roots of unity in definite quaternion orders

Arenas-Carmona, Luis

doi:10.4064/aa170-4-5

Cited by 7 publications

(2 citation statements)

References 9 publications

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“…Basse-Serre Theory allows us to determine the structure of the group Γ in terms of the quotient graphs in some cases, see [16] for an account of this subject. If the action has inversions we can still define a quotient graph by replacing G by its barycentric subdivision and ignoring the new vertices unless they become endpoints in the quotient, where they are called virtual endpoints, see [3] or [4] for details. The edge joining a vertex and a virtual endpoint is called a half-edge.…”

Section: Eichler Orders and Treesmentioning

confidence: 99%

Simultaneous diagonalization of vector bundles

Arenas-Carmona

2016

Preprint

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Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line splits as the sum of one dimensional vector bundles. In this work we study simultaneous splittings of two dimensional vector bundles over a finite field using the theory of Eichler orders. In the sheaf-theoretical context, an Eichler order in a matrix algebra is the intersection of the sheaves of endomorphisms of two vector bundles, so characterizing split Eichler orders solves the problem of simultaneous splitting. We caracterize both the genera of Eichler orders containing only split Eichler orders and the genera containing only a finite number of non-split clases, in terms of a divisor-valued distance parametrizing the genera.

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Section: Eichler Orders and Treesmentioning

confidence: 99%

Simultaneous diagonalization of vector bundles

Arenas-Carmona

2016

Preprint

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“…The corresponding quotient graphs are closely related to the function field analog of Shimura curves [13]. The extension of this theory to number fields also encodes important arithmetic information, for example in [7] we characterize, in a definite quaternion algebra, the orders of maximal rank containing cubic roots of unity, in terms of the corresponding quotient graphs. Describing the set of orders in a genus containing a copy of a given suborder is known as the selectivity problem, and its understanding is critical for the construction of isospectral but non-isometric Riemannian manifolds.…”

Section: Introductionmentioning

confidence: 99%

Representation fields for orders of small rank

Arenas-Carmona

2017

Proyecciones (Antofagasta)

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A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we have proved the existence of the representation field for several important families of suborders, like commutative orders, while we have also found examples where the representation field fails to exist. To be precise, we have found full-rank orders, in central simple algebras of dimension 9 or larger over a suitable field, whose representation field is undefined. In this article, we prove that the representation field is defined for any order H of rank r ≤ 7. This is done by defining representation fields for arbitrary representations of orders into central simple algebra and showing that the computation of these generalized representation fields can be reduced to the case of irreducible representations. The same technique yields the existence of representation fields for any order in an algebra whose semi-simple reduction is commutative. We also construct a rank-8 order, in a 16-dimensional matrix algebra, whose representation field is not defined.

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Selectivity in definite quaternion algebras

Quiroz

2018

Journal of Number Theory

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Roots of unity in definite quaternion orders

Cited by 7 publications

References 9 publications

Simultaneous diagonalization of vector bundles

Simultaneous diagonalization of vector bundles

Representation fields for orders of small rank

Selectivity in definite quaternion algebras

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