Unit groups of maximal orders in totally definite quaternion algebras over real quadratic fields

Li, Qun; Xue, Jiangwei; Yu, Chia-Fu

doi:10.48550/arxiv.1807.04736

Cited by 5 publications

(12 citation statements)

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“…Then O × 24 /O × F is isomorphic to the symmetric group S 4 , and O 24 is the unique maximal order up to conjugation with this property by [28,Proposition 5.1]. We leave it as an exercise to show that O 24 is symmetric, and (4.24)…”

Section: Let Us Identifymentioning

confidence: 99%

“…From [28,Remark 5.8], O 8 ∩ O 24 is an Eichler order of level q, and hence ρ(O 8 , O 24 ) ∈ G sg is represented by [q] + . It follows from (4.12) that O 24 belongs to the principal spinor genus if and only if p ≡ 7 (mod 8).…”

Section: Let Us Identifymentioning

confidence: 99%

“…Suppose that p ≥ 7. According to [28,Theorem 1.6], there is a unique maximal order O 6 in D up to conjugation with O × 6 /O × F isomorphic to the dihedral group of order 6. Different from the cases for O 8 or O 24 , the maximal order O 6 does not admit a unique form [28, §8.2], so we separate the discussion into cases.…”

Section: Let Us Identifymentioning

confidence: 99%

“…Since 2ε ∈ F ×2 , we have ε ≡ −1 (mod 3O F ). From [28,Corollary 6.2.6], O 6 may be taken to be the maximal order in either [28, (6.3.8)] or [28, (6.3.9)] depending on whether p ≡ 7 (mod 24) or not. The proofs of [28, Proposition 6.2.2 and §6.2.5] imply that there is a suitable α ∈ D × such that (αO 6 α −1 ) ∩ O 24 is an Eichler order of level p (or p for another suitable choice of α ∈ D × ).…”

Section: Let Us Identifymentioning

confidence: 99%

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Class number formulas for the norm one group of totally definite quaternion algebras

Xue¹,

Yu²

2019

Preprint

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We present two class number formulas associated to orders in totally definite quaternion algebras in the spirit of the Eichler class number formula. More precisely, let F be a totally real number field, D be a totally definite quaternion F -algebra, and O an O F -order in D. We derive explicit formulas for the following two class numbers associated to O: (1) the class number of the reduced norm one group with respect to O, namely, the cardinality of the double coset space D 1 \ D 1 / O 1 ; (2) the number of locally principal right O-ideal classes within the spinor class of the principal right O-ideals, that is, the cardinality ofBoth class numbers depend only on the spinor genus of O, hence the title of the present paper. The formulas work unconditionally for O F -orders with nonzero Eichler invariants at all finite places of F (e.g. Eichler orders of arbitrary level), and are extendable to other classes of orders as long as certain local condition for optimal embeddings is satisfied. The proofs are made possible by optimal spinor selectivity for quaternion orders.

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Section: Let Us Identifymentioning

confidence: 99%

Section: Let Us Identifymentioning

confidence: 99%

Section: Let Us Identifymentioning

confidence: 99%

Section: Let Us Identifymentioning

confidence: 99%

See 2 more Smart Citations

Class number formulas for the norm one group of totally definite quaternion algebras

Xue¹,

Yu²

2019

Preprint

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“…Here h(D, G) does not depends on the choice of the maximal order O, hence denoted as such. The following result is used in [19]. Proof.…”

Section: Picard Group Action For Totally Definite Quaternion Algebrasmentioning

confidence: 99%

On superspecial abelian surfaces and type numbers of totally definite quaternion algebras

Xue¹,

Yu²

2018

Preprint

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In this paper we determine the number of endomorphism rings of superspecial abelian surfaces over a field Fq of odd degree over Fp in the isogeny class corresponding to the Weil q-number ± √ q . This extends earlier works of T.-C. Yang and the present authors on the isomorphism classes of these abelian surfaces, and also generalizes the classical formula of Deuring for the number of endomorphism rings of supersingular elliptic curves. Our method is to explore the relationship between the type and class numbers of the quaternion orders concerned. We study the Picard group action of the center of an arbitrary Z-order in a totally definite quaternion algebra on the ideal class set of said order, and derive an orbit number formula for this action. This allows us to prove an integrality assertion of Vignéras [Enseign. Math.(2), 1975] as follows. Let F be a totally real field of even degree over Q, and D be the (unique up to isomorphism) totally definite quaternion F -algebra unramified at all finite places of F . Then the quotient h(D)/h(F ) of the class numbers is an integer.

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On superspecial abelian surfaces over finite fields II

Xue¹,

Yang²,

Yu³

2020

J. Math. Soc. Japan

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Extending the results of [24, Asian J. Math.], in [26, Doc. Math. 21, 2016] we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of odd degree over the prime field Fp. A key step was to reduce the calculation to the prime field case, and we calculated the number of isomorphism classes in each isogeny class through a concrete lattice description. In the present paper we treat the even degree case by a different method. We first translate the problem by Galois cohomology into a seemingly unrelated problem of computing conjugacy classes of elements of finite order in arithmetic subgroups, which is of independent interest. We then explain how to calculate the number of these classes for the arithmetic subgroups concerned, and complete the computation in the case of rank two. This complements our earlier results and completes the explicit calculation of superspecial abelian surfaces over finite fields.Date: October 4, 2018. 2010 Mathematics Subject Classification. 11R52, 11G10. 1 This is well known when the ground field k is algebraically closed, and is proved in [29] for arbitrary k 1 Lemma 6.1. Suppose that S × ⊆ N ( R), the normalizer of R in B × . Then the suborder R J := x Rx −1 ∩ B of S J is independent of the choice of x ∈ B × for J, and.Proof. Suppose that J = x ′ S for x ′ ∈ B × as well. Then there exists u ∈ S × such that x ′ = xu. Since S × ⊆ N ( R), we havewhich proves the independence of R J of the choice of x. If I is a locally principal right R-ideal such that IS = J, then R J = O l (I), the associated left order of I.Conjugating by x ∈ B × on the right hand side of (6.3), we obtainThe assumption S × ⊆ N ( R) also implies that R × S × , and hence R × J S × J and R × J S × J . The left action of S × J on the quotient group S × J / R × J factors through S × J /R × J ⊆ S × J / R × J , and its orbits are the right cosets of S × JRemark 6.2. The condition S × ⊆ N ( R) implies that R × S × . However, the converse does not hold in general. It is enough to provide a counterexample locally Corollary 6.3. Keep the notation and assumption of Lemma 6.1. If the natural homomorphism S ×Proof. It is enough to show that π is injective. The surjectivity of S × J → S × J / R × J implies that the monomorphism S × J /R × J ֒→ S × J / R × J is an isomorphism, and hence |π −1 ([J])| = [ S × J / R × J : S × J /R × J ] = 1.

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Unit groups of maximal orders in totally definite quaternion algebras over real quadratic fields

Cited by 5 publications

References 38 publications

Class number formulas for the norm one group of totally definite quaternion algebras

Class number formulas for the norm one group of totally definite quaternion algebras

On superspecial abelian surfaces and type numbers of totally definite quaternion algebras

On superspecial abelian surfaces over finite fields II

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