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Register a new userOptimal spinor selectivity for quaternion Bass orders
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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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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 $mathcal{O}$ be an $O_F$-order in $D$. Assume that $mathcal{O}$ has nonzero Eichler invariants at all finite places of $F$ (e.g. $mathcal{O}$ is an Eichler order of arbitrary level). We derive explicit formulas for the following two class numbers associated to $mathcal{O}$: (1) the class number of the reduced norm one group with respect to $mathcal{O}$, namely, the cardinality of the double coset space $D^1backslashwidehat{D}^1/widehat{mathcal{O}}^1$; (2) the number of locally principal right $mathcal{O}$-ideal classes within the spinor class of the principal right $mathcal{O}$-ideals, that is, the cardinality of $D^timesbackslashbig(D^timeswidehat{D}^1widehat{mathcal{O}}^timesbig)/widehat{mathcal{O}}^times$. Both class numbers depend only on the spinor genus of $mathcal{O}$, hence the title of the present paper. The proofs are made possible by optimal spinor selectivity for quaternion orders.
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 $p otequiv 1pmod{24}$.
Given a newform f, we extend Howards results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner points on towers of Shimura curves. The novelty of our approach, which systematically exploits the theory of optimal embeddings, consists in treating both the case of definite quaternion algebras and the case of indefinite quaternion algebras in a uniform way. We prove results on the size of Nekovav{r}s extended Selmer groups attached to suitable big Galois representations and we formulate two-variable Iwasawa main conjectures both in the definite case and in the indefinite case. Moreover, in the definite case we propose refined conjectures `a la Greenberg on the vanishing at the critical points of (twists of) the L-functions of the modular forms in the Hida family of f living on the same branch as f.
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 cite{Chevalley-book} 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 $[L cap widetilde K : K]^{-1}$ where $widetilde K$ is the Hilbert class field of $K$. Chinburg and Friedman (cite{Chinburg-Friedman}) consider arbitrary quadratic orders in quaternion algebras satisfying the Eichler condition, and Arenas-Carmona cite{Arenas-Carmona} 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 Friedmans argument was the structure of the tree of maximal orders for $SL_2$ over a local field. In this work, we generalize Chinburg and Friedmans results replacing the tree by the Bruhat-Tits building for $SL_p$.
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 $pmsqrt{q}$. 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 Vigneras [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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