Numerical Invariants of Totally Imaginary Quadratic
            $\mathbb{Z}[\sqrt{p}]$-orders

Xue, Jiangwei; Yang, Tse-Chung; Yu, Chia-Fu

doi:10.11650/tjm.20.2016.6464

Cited by 16 publications

(20 citation statements)

References 18 publications

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“…In [26] we calculated explicitly the number of superspecial abelian surfaces over an arbitrary finite field F q of odd degree over F p . This extended our earlier works [24,25] and [31], which contributed to the study of superspecial abelian varieties over finite fields. In this paper we treat the even degree case.…”

Section: Introductionsupporting

confidence: 78%

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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Section: Introductionsupporting

confidence: 78%

On superspecial abelian surfaces over finite fields II

Xue¹,

Yang²,

Yu³

2020

J. Math. Soc. Japan

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“…Lastly, we apply the above results to the study of superspecial abelian surfaces [24, Definition 1.7, Ch.1]. Indeed, one of our motivations is to count the number of certain superspecial abelian surfaces with a fixed reduced automorphism group G. This extends results of our earlier works [44,45,46,47] where we compute explicitly the number of these abelian surfaces over finite fields. We also construct superspecial abelian surfaces X over some field K of characteristic p with endomorphism algebra End 0 (X) = Q( √ p ), provided that p ≡ 1 (mod 24).…”

Section: Introductionmentioning

confidence: 71%

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

Xue

2018

Preprint

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We study a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes (resp. unit groups of maximal orders). For each finite noncyclic group G, we give an explicit formula for the number of conjugacy classes of maximal orders whose unit groups modulo center are isomorphic to G, and write down a representative for each conjugacy class. This leads to a complete recipe (even explicit formulas in special cases) for the refined class number formula for all finite groups. As an application, we prove the existence of superspecial abelian surfaces whose endomorphism algebras coincide with Q( √ p ) in all positive characteristic p ≡ 1 (mod 24).

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“…In a series of papers [25,26,27,28], Tse-Chung Yang and the first two current authors attempt to calculate the cardinality |SSp d (F q )| explicitly in the case d = 2. More precisely, it is shown in [26] that for every fixed d > 1, |SSp d (F q )| depends only on the parity of the degree a = [F q : F p ], and an explicit formula of |SSp 2 (F q )| is provided for the odd degree case.…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, it is shown in [26] that for every fixed d > 1, |SSp d (F q )| depends only on the parity of the degree a = [F q : F p ], and an explicit formula of |SSp 2 (F q )| is provided for the odd degree case. The most involving part of this explicit calculation is carried out prior in [25,27], which counts the number of isomorphism classes of abelian surfaces over F p within the simple isogeny class corresponding to the Weil p-numbers ± √ p . For the even degree case, an explicit formula of |SSp 2 (F q )| is obtained in [28] under a mild condition on p (see Remark 3.7 of loc.…”

Section: Introductionmentioning

confidence: 99%