Applications of spinor class fields: embeddings of orders and quaternionic lattices

Arenas-Carmona, Luis

doi:10.5802/aif.1999

Cited by 28 publications

(86 citation statements)

References 5 publications

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“…We consider the case where B is a central simple algebra having dimension p 2 (p be an odd prime) over a number field K; this part of the setup is of course a special case of the one in [2]. On the other hand, there are other substantive differences between [2] and this work.…”

Section: Theorem the Ratio Of The Number Of Isomorphism Classes Of Mmentioning

confidence: 99%

“…The setting was a central simple algebra B over a number field K of dimension n 2 , n ≥ 3 with the proviso that the completions of B (at the non-archimedean primes) have the form (in the notation above) B ν ∼ = M κν (D ν ), n = κ ν m ν , with κ ν = 1 or n. He considered embeddings of the ring of integers O L of an extension L/K of degree n into maximal orders of B and proves a result (Theorem 1 of [2]) analogous to Chevalley's with the Hilbert class field replaced by a spinor class field. Indeed, Arenas-Carmona proves and uses the key observation (Lemma 2.0.1 of [2]) that for n ≥ 3, the set of isomorphism classes of maximal orders in B equals the set of spinor genera in the genus of any maximal order of B, which enables him to leverage results about spinor genera to deduce results about embedding in maximal orders.…”

Section: Theorem the Ratio Of The Number Of Isomorphism Classes Of Mmentioning

confidence: 99%

“…The first work beyond Chevalley's in the non-quaternion setting was by Arenas-Carmona [2]. The setting was a central simple algebra B over a number field K of dimension n 2 , n ≥ 3 with the proviso that the completions of B (at the non-archimedean primes) have the form (in the notation above) B ν ∼ = M κν (D ν ), n = κ ν m ν , with κ ν = 1 or n. He considered embeddings of the ring of integers O L of an extension L/K of degree n into maximal orders of B and proves a result (Theorem 1 of [2]) analogous to Chevalley's with the Hilbert class field replaced by a spinor class field.…”

Section: Theorem the Ratio Of The Number Of Isomorphism Classes Of Mmentioning

confidence: 99%

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Embedding orders into central simple algebras

Linowitz¹,

Shemanske²

2012

J. Théor. Nombres Bordeaux

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The question of embedding fields into central simple algebras B over a number field K was the realm of class field theory. The subject of embedding orders contained in the ring of integers of maximal subfields L of such an algebra into orders in that algebra is more nuanced. The first such result along those lines is an elegant result of Chevalley [6] which says that with B = M n (K) the ratio of the number of isomorphism classes of maximal orders in B into which the ring of integers of L can be embedded (to the total number of classes) is [L∩ K : K] −1 where K is the Hilbert class field of K. Chinburg and Friedman ([7]) consider arbitrary quadratic orders in quaternion algebras satisfying the Eichler condition, and Arenas-Carmona [2] considers embeddings of the ring of integers into maximal orders in a broad class of higher rank central simple algebras. In this paper, we consider central simple algebras of dimension p 2 , p an odd prime, and we show that arbitrary commutative orders in a degree p extension of K, embed into none, all or exactly one out of p isomorphism classes of maximal orders. Those commutative orders which are selective in this sense are explicitly characterized; class fields play a pivotal role. A crucial ingredient of Chinberg and Friedman's argument was the structure of the tree of maximal orders for SL 2 over a local field. In this work, we generalize Chinburg and Friedman's results replacing the tree by the Bruhat-Tits building for SL p .

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Section: Theorem the Ratio Of The Number Of Isomorphism Classes Of Mmentioning

confidence: 99%

Section: Theorem the Ratio Of The Number Of Isomorphism Classes Of Mmentioning

confidence: 99%

Section: Theorem the Ratio Of The Number Of Isomorphism Classes Of Mmentioning

confidence: 99%

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Embedding orders into central simple algebras

Linowitz¹,

Shemanske²

2012

J. Théor. Nombres Bordeaux

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“…If n is even, and if ℘ is a prime ideal whose order is exactly n in the class group, then the generator b of ℘ n defines an order D(1, b) not isomorphic to D (1,1). If x ∈ A is as in (1), the field F is isomorphic to a quotient of the…”

mentioning

confidence: 99%

“…The Hilbert symbol is a map that associates a central simple algebra (a, b) η of dimension n 2 to every element (a, b) in k * × k * [9]. It is defined as the k-algebra with generators x and y satisfying the relations (1) x n = a, y n = b, and xy = ηyx.…”

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confidence: 99%

Arithmetical Hilbert symbols, distance maps and cohomology

Arenas-Carmona

2009

Acta Arith.

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Relative spinor class fields: A counterexample

Arenas-Carmona

2008

Arch. Math.

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It is known that classes of indefinite quadratic forms in a genus are classified by the Galois group of a spinor class field [4]. Hsia has proved the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. Spinor class fields can also be used to classify conjugacy classes of maximal orders in a central simple algebra. In [1] we left open the issue of whether for every fixed given non-maximal order h in a central simple division algebra there exists a representation field L with the property that h embeds into a given maximal order if and only if the corresponding element of the Galois group is trivial on L. In this work we give a negative answer to this question for central simple division algebras of dimension >= 3(2). The case of non-division algebras is also treated by replacing the phrase embeds into by is contained in a conjugate of. As a byproduct of the techniques used in this paper we compute the representation field of an Eichler order in a quaternion algebra.Fondecyt 108501

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Applications of spinor class fields: embeddings of orders and quaternionic lattices

Cited by 28 publications

References 5 publications

Embedding orders into central simple algebras

Embedding orders into central simple algebras

Arithmetical Hilbert symbols, distance maps and cohomology

Relative spinor class fields: A counterexample

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