arXiv subjects

Jiangwei Xue

Publications and source records attributed to Jiangwei Xue.

At least 19 recordsLinked to original sources

arXiv2026

Ribet bimodules and principally polarized superspecial abelian varieties with quaternion action

In an influential paper [K. Ribet, Bimodules and abelian surfaces, in Algebraic number theory, 359-407, Adv. Stud. Pure Math., Vol. 17, 1989] on the bad reduction of Shimura curves, Ribet studies certain superspecial abelian surfaces over $\overline{\mathbb{F}}_p$ with quaternion multiplication by a maximal order $\mathcal{O}$ in an indefinite quaternion $\mathbb{Q}$-algebra ramified at $p$. In particular, he classifies the $p$-divisible groups of such $\mathcal{O}$-abelian surfaces by classifying $(\mathcal{O}_p, \mathcal{O}_p)$-bimodules $L_p$ that are free over $\mathbb{Z}_p$ (i.e.bilattices) under an additional admissible assumption. In this paper, we generalize Ribet's result by removing the admissible assumption and producing a complete classification of $(\mathcal{O}_p, \mathcal{O}_p)$-bilattices $L_p$. Equip the right order $\mathcal{O}_p$ with the canonical involution, and suppose additionally that the left order $\mathcal{O}_p$ is equipped with an orthogonal involution $*$. We derive the necessary and sufficient condition for the existence of a perfect quaternion hermitian form $\langle~,~\rangle_p:L_p\times L_p\to \mathcal{O}_p$ on the right $\mathcal{O}_p$-lattice $L_p$ inducing the given involution $*$ on the left order $\mathcal{O}_p$, and give a complete classification of such self-dual quaternion hermitian $(\mathcal{O}_p, *, \mathcal{O}_p)$-bilattices $(L_p, \langle~,~\rangle_p)$. Globally, we apply these classification results to the study of the existence of principal polarizations on superspecial abelian varieties over $\overline{\mathbb{F}}_p$ equipped with $\mathcal{O}$-action.

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arXiv2026

Superspecial Points on Shimura Curves

Let $X$ be the Shimura curve attached to an indefinite quaternion $\mathbb{Q}$-algebra $B$ with a maximal order $O_B$. This paper investigates the reduction $X\otimes \mathbb{F}_p$ of $X$ modulo an arbitrary prime $p$, focusing particularly on its superspecial locus. We give an explicit criterion for the existence of superspecial $\mathbb{F}_q$-rational points on $X$. Furthermore, we compute both the number of geometric superspecial points and the number of $\mathbb{F}_p$-rational superspecial points, through the Eichler class number formula and the Selberg trace formula. As a key ingredient, we classify the Dieudonn\'e modules attached to superspecial $O_B$-abelian surfaces, which generalizes Ribet's classification of admissible quaternion bimodules of rank $2$ by dropping the admissible hypothesis. These results generalize Deuring's explicit formula for supersingular elliptic curves over $\mathbb{F}_p$ and give the Shimura-curve analogue of the Ibukiyama-Katsura formulas for principally polarized superspecial abelian surfaces over $\mathbb{F}_p$.

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arXiv2026

Polarized superspecial abelian varieties over $\mathbb{F}_p$ via hermitian lattices

We study the set of isomorphism classes of polarized superspecial abelian varieties $(A,\lambda)$ of a fixed dimension over $\mathbb{F}_p$ with Frobenius endomorphism $\pi_A=\sqrt{-p}$ and $\ker \lambda =\ker \pi_A$. This set plays an important role in the geometry of the supersingular locus, and the generalizations of Deuring's $2T-H$ Theorem by Ibukiyama and Katsura. We determine when this set is nonempty and classify its genera. Our method reduces the problems of superspecial abelian varieties to those of certain hermitian lattices by the lattice description established by Jordan et. al and Ibukiyama--Karemaker--Yu, and we treat these problems on the lattices concerned by arithmetic methods.

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arXiv2026

The spinor type number formula for totally definite quaternion orders

Let $D$ be a totally definite quaternion algebra over a totally real number field $F$, and $\mathcal{O}$ be an $O_F$-order (of full rank) in $D$. The type number $t(\mathcal{O})$ is an important arithmetic invariant of $\mathcal{O}$ that counts the number of isomorphism classes of orders belonging to the same genus as $\mathcal{O}$ (i.e. locally isomorphic to $\mathcal{O}$ at every finite place $\mathfrak{p}$ of $F$). The type number formula has been studied by Eichler, Peters, Pizer, Vigneras, K\"orner and many others. As the genus of $\mathcal{O}$ further divides into spinor genera, one naturally seeks a finer type number formula for the number of isomorphism classes of orders belonging to the same spinor genus of $\mathcal{O}$. The main goal of this paper is to provide such a refinement for a large class of quaternion $O_F$-orders $\mathcal{O}$ that includes all Eichler orders. This enables us to prove that $t(\mathcal{O})$ is divisible by the order of a quotient group $\mathrm{WSG}(\mathcal{O})$ of the Gauss genus group $\mathrm{Cl}^+(O_F)/\mathrm{Cl}^+(O_F)^2$ naturally attached to $\mathcal{O}$. Similarly, we show that the trace of the $\mathfrak{n}$-Brandt matrix $\mathfrak{B}(\mathcal{O}, \mathfrak{n})$ is divisible by the class number $h(F)$ for any nonzero integral $O_F$-ideal $\mathfrak{n}$. In particular, the class number $h(\mathcal{O})=\mathrm{Tr}(\mathfrak{B}(\mathcal{O}, O_F))$ is always divisible by $h(F)$ for such quaternion orders. This generalizes the divisibility result of $h(\mathcal{O})$ proved in a different way by Chia-Fu Yu and the second named author [Indiana Univ. Math. J., Vol. 70, No. 2 (2021)] in the case when $\mathcal{O}$ is a maximal $O_F$-order in a totally definite quaternion algebra unramified at all the finite places.

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arXiv2024

Polarized superspecial simple abelian surfaces with real Weil numbers

Let $q$ be an odd power of a prime $p\in \mathbb{N}$, and $\mathrm{PPSP}(\sqrt{q})$ be the finite set of isomorphism classes of principally polarized superspecial abelian surfaces in the simple isogeny class over $\mathbb{F}_q$ corresponding to the real Weil $q$-numbers $\pm \sqrt{q}$. We produce explicit formulas for $\mathrm{PPSP}(\sqrt{q})$ of the following kinds: (i) the class number formula, i.e.~the cardinality of $\mathrm{PPSP}(\sqrt{q})$; (ii) the type number formula, i.e. the number of endomorphism rings up to isomorphism of the underlying abelian surfaces of $\mathrm{PPSP}(\sqrt{q})$. Similar formulas are obtained for other collections of polarized superspecial members of this isogeny class grouped together according to their polarization modules. We observe several surprising identities involving the arithmetic genus of certain Hilbert modular surface on one side and the class number or type number of $(P, P_+)$-polarized superspecial abelian surfaces in this isogeny class on the other side.

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arXiv2023

On the supersingular locus of Shimura varieties for quaternionic unitary groups

We study a Shimura variety attached to a unitary similitude group of a skew-Hermitian form over a totally indefinite quaternion algebra over a totally real number field. We give a necessary and sufficient condition for the existence of skew-Hermitian self-dual lattices. Under this condition we show that the superspecial locus in the fiber at $p$ of the associated Shimura variety is non-empty. We also give an explicit formula for the number of irreducible components of the supersingular locus when $p$ is odd and unramified in the quaternion algebra.

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arXiv2023

Trace formulas for the norm one group of totally definite quaternion algebras

In his pioneering work [Crelle's Journal, 1955], Eichler established the theory of trace formulas for Brandt matrices of quaternion orders. From it he derived a class number formula for Eichler orders in a totally definite quaternion algebra $D$. Extending Eichler's work, Pizer [Crelle's Journal, 1973] proved a formula for the type number of Eichler orders in $D$. In this paper, we extend their results to the norm one group of $D$. More precisely, we present a class number formula for the norm one group of $D$ with respect to a class of orders $\mathcal{O}$, called \emph{residually unramified orders}, which includes all Eichler orders. Our second result gives a formula for the number of ideal classes in the spinor class of $\mathcal{O}$, which refines Eichler's class number formula. It is worth mentioning that these class number formulas not only depend on the genus of orders as Eichler and Pizer's formulas, but also depend on the orders themselves. We introduce certain auxiliary invariants in order to keep track of the global information on the relationship between certain CM orders and $\mathcal{O}$, and use them to describe our formulas. Both our class number formulas make use of the optimal spinor selectivity theory for quaternion orders.

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arXiv2022

Optimal spinor selectivity for quaternion orders

Let $D$ be a quaternion algebra over a number field $F$, and $\mathscr{G}$ be an arbitrary genus of $O_F$-orders of full rank in $D$. Let $K$ be a quadratic field extension of $F$ that embeds into $D$, and $B$ be an $O_F$-order in $K$ that can be optimally embedded into some member of $\mathscr{G}$. We provide a necessary and sufficient condition for $B$ to be optimally spinor selective for the genus $\mathscr{G}$, which generalizes previous existing optimal selectivity criterions for Eichler orders as given by Arenas, Arenas-Carmona and Contreras, and by Voight independently. This allows us to obtain a refinement of the classical trace formula for optimal embeddings, which will be called the spinor trace formula. When $\mathscr{G}$ is a genus of Eichler orders, we extend Maclachlan's relative conductor formula for optimal selectivity from Eichler orders of square-free levels to all Eichler orders.

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arXiv2022

On $\mathbb{F}_p$-roots of the Hilbert class polynomial modulo $p$

The Hilbert class polynomial $H_{\mathcal{O}}(x)\in \mathbb{Z}[x]$ attached to an order $\mathcal{O}$ in an imaginary quadratic field $K$ is the monic polynomial whose roots are precisely the distinct $j$-invariants of elliptic curves over $\mathbb{C}$ with complex multiplication by $\mathcal{O}$. Let $p$ be a prime inert in $K$ and strictly greater than $|\operatorname{disc}(\mathcal{O})|$. We show that the number of $\mathbb{F}_p$-roots of $H_\mathcal{O}(x)\!\! \pmod{p}$ is either zero or $|\operatorname{Pic}(\mathcal{O})[2]|$ by exhibiting a free and transitive action of $\operatorname{Pic}(\mathcal{O})[2]$ on the set of $\mathbb{F}_p$-roots of $H_\mathcal{O}(x)\!\! \pmod p$ whenever it is nonempty. We also provide a concrete criterion for the nonemptiness of the set of $\mathbb{F}_p$-roots. A similar result was first obtained by Xiao et al.~[Int. J. Number Theory, DOI: 10.1142/S1793042122500555] and generalized much further by Li et al.~[arXiv:2108.00168] (that covers the current result) with a different approach.

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arXiv2021

Optimal spinor selectivity for quaternion Bass orders

Let $A$ be a quaternion algebra over a number field $F$, and $\mathcal{O}$ be an $O_F$-order of full rank in $A$. Let $K$ be a quadratic field extension of $F$ that embeds into $A$, and $B$ be an $O_F$-order in $K$. Suppose that $\mathcal{O}$ is a Bass order that is well-behaved at all the dyadic primes of $F$. We provide a necessary and sufficient condition for $B$ to be optimally spinor selective for the genus of $\mathcal{O}$. This partially generalizes previous results on optimal (spinor) selectivity by C. Maclachlan [Optimal embeddings in quaternion algebras. J. Number Theory, 128(10):2852-2860, 2008] for Eichler orders of square-free levels, and independently by M. Arenas et al. [On optimal embeddings and trees. J. Number Theory, 193:91-117, 2018] and by J. Voight [Chapter 31, Quaternion algebras, volume 288 of Graduate Texts in Mathematics. Springer-Verlag, 2021] for Eichler orders of arbitrary levels.

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arXiv2021

On superspecial abelian surfaces over finite fields III

In the paper [On superspecial abelian surfaces over finite fields II. J. Math. Soc. Japan, 72(1):303--331, 2020], Tse-Chung Yang and the first two current authors computed explicitly the number $\lvert \mathrm{SSp}_2(\mathbb{F}_q)\rvert$ of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field $\mathbb{F}_q$ of even degree over the prime field $\mathbb{F}_p$. There it was assumed that certain commutative $\mathbb{Z}_p$-orders satisfy an étale condition that excludes the primes $p=2, 3, 5$. We treat these remaining primes in the present paper, where the computations are more involved because of the ramifications. This completes the calculation of $\lvert \mathrm{SSp}_2(\mathbb{F}_q)\rvert$ in the even degree case. The odd degree case was previous treated by Tse-Chung Yang and the first two current authors in [On superspecial abelian surfaces over finite fields. Doc. Math., 21:1607--1643, 2016]. Along the proof of our main theorem, we give the classification of lattices over local quaternion Bass orders, which is a new input to our previous works.

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arXiv2019

On counting certain abelian varieties over finite fields

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, 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 $\sqrt{q}$. 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 ideal complexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the results of Yu 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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arXiv2019

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

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 $\mathbb{Q}(\sqrt{p})$ in all positive characteristic $p\not\equiv 1\pmod{24}$.

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arXiv2018

On superspecial abelian surfaces over finite fields II

Extending the results of [Asian J. Math. 2019], in [Doc. Math. \textbf{21}, 2016] we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of \textit{odd} degree over the prime field $\mathbb{F}_p$. 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 \textit{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 tocalculate 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.

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arXiv2018

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

In this paper we determine the number of endomorphism rings of superspecial abelian surfaces over a field $\mathbb{F}_q$ of odd degree over $\mathbb{F}_p$ in the isogeny class corresponding to the Weil $q$-number $\pm\sqrt{q}$. 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 $\mathbb{Z}$-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 $\mathbb{Q}$, 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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arXiv2016

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

Let $A$ be a real quadratic order of discriminant $p$ or $4p$ with a prime $p$. In this paper we classify all proper totally imaginary quadratic $A$-orders $B$ with index $w(B)=[B^\times: A^\times]>1$. We also calculate numerical invariants of these orders including the class number, the index $w(B)$ and the numbers of local optimal embeddings of these orders into quaternion orders. These numerical invariants are useful for computing the class numbers of totally definite quaternion algebras.

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arXiv2016

On superspecial abelian varieties over finite fields

In this paper we establish a new lattice description for superspecial abelian varieties over a finite field $\mathbb {F}_q$ of $q=p^a$ elements. Our description depends on the parity of the exponent $a$ of $q$. When $q$ is an odd power of the prime $p$, we give an explicit formula for the number of superspecial abelian surfaces over $\mathbb F_q$.

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arXiv2015

Supersingular abelian surfaces and Eichler class number formula

Let $F$ be a totally real field with ring of integers $O_F$, and $D$ be a totally definite quaternion algebra over $F$. A well-known formula established by Eichler and then extended by Körner computes the class number of any $O_F$-order in $D$. In this paper we generalize the Eichler class number formula so that it works for arbitrary $\mathbb{Z}$-orders in $D$. The motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a prime finite field $\mathbb{F}_p$. We give explicit formulas for the number of these isomorphism classes for all primes $p$.

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