Supersingular abelian surfaces and Eichler class number formula

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

doi:10.48550/arxiv.1404.2978

Cited by 9 publications

(28 citation statements)

References 9 publications

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“…Therefore, up to isomorphism, B = B 3,2 is the unique CM proper A-order with ker(i B/A ) nontrivial. The same proof as that in [37,Subsection 6.2.5] shows that m 2 (B 3,2 ) = 1 for O = O 8 . It follows from Theorem 3.7 that (4.17)…”

Section: Calculation Of Type Numberssupporting

confidence: 54%

“…Now let q be an odd power of p, and let Sp( √ q ) be the set of isomorphism classes of superspecial abelian surfaces over F q in the isogeny class corresponding to the Weil number ± √ q . As a generalization of (1.1), T.-C. Yang and the present authors [37,Theorem 1.2] (also see [39,Theorem 1.3]) show the following explicit formula for | Sp( √ q )|.…”

Section: Introductionmentioning

confidence: 56%

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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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Section: Calculation Of Type Numberssupporting

confidence: 54%

Section: Introductionmentioning

confidence: 56%

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

Xue¹,

Yu²

2018

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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: 73%

“…Lastly, if O ⋆ ⋆ ⋆ ≃ D 5 or A 5 , then O contains a minimal D 5 -order, which implies (by the proof of Proposition 4.3.5) that H ′ = {Q(ζ 5 ), −1}. A direct calculation shows that {Q(ζ 5 ), −1} ≃ H ∞1,∞2 .Thanks to[38] (see also[44, Theorem 1.3]), we have…”

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confidence: 72%

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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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On Counting Certain Abelian Varieties Over Finite Fields

Xue

2020

Acta. Math. Sin.-English Ser.

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This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers [32,31,33,34], the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number √ q. This establishes a key step that one may extend our previous explicit calculations of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the results of [40] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigation.

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Supersingular abelian surfaces and Eichler class number formula

Cited by 9 publications

References 9 publications

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

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

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

On Counting Certain Abelian Varieties Over Finite Fields

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