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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.
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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 $mathbb{Q}(sqrt{p})$ in all positive characteristic $p otequiv 1pmod{24}$.
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.
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.
Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for CM(K).G_1 under strong assumptions on the ramification in K. Yang later proved this conjecture under slightly stronger assumptions on the ramification. In recent work, Lauter and Viray proved a different formula for CM(K).G_1 for primitive quartic CM fields with a mild assumption, using a method of proof independent from that of Yang. In this paper we show that these two formulas agree, for a class of primitive quartic CM fields which is slightly larger than the intersection of the fields considered by Yang and Lauter and Viray. Furthermore, the proof that these formulas agree does not rely on the results of Yang or Lauter and Viray. As a consequence of our proof, we conclude that the Bruinier-Yang formula holds for a slightly largely class of quartic CM fields K than what was proved by Yang, since it agrees with the Lauter-Viray formula, which is proved in those cases. The factorization of these intersection numbers has applications to cryptography: precise formulas for them allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography.
Let $K$ be a totally real number field of degree $n geq 2$. The inverse different of $K$ gives rise to a lattice in $mathbb{R}^n$. We prove that the space of Schwartz Fourier eigenfunctions on $mathbb{R}^n$ which vanish on the component-wise square root of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres $sqrt{m}S^{n-1}$ for integers $m geq 0$ and, as $m rightarrow infty$, there are $sim c_{K} m^{n-1}$ many points on the $m$-th sphere for some explicit constant $c_{K}$, proportional to the square root of the discriminant of $K$. This contrasts a recent Fourier uniqueness result by Stoller. Using a different construction involving the codifferent of $K$, we prove an analogue of our results for discrete subsets of ellipsoids. In special cases, these sets also lie on spheres with more densely spaced radii, but with fewer points on each. We also study a related question about existence of Fourier interpolation formulas with nodes $sqrt{Lambda}$ for general lattices $Lambda subset mathbb{R}^n$. Using results about lattices in Lie groups of higher rank, we prove that, if $n geq 2$ and if a certain group $Gamma_{Lambda} leq operatorname{PSL}_2(mathbb{R})^n$ is discrete, then such interpolation formulas cannot exist. Motivated by these more general considerations, we revisit the case of one radial variable and prove, for all $n geq 5$ and all real $lambda > 2$, Fourier interpolation results for sequences of spheres $sqrt{2 m/ lambda}S^{n-1}$, where $m$ ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincare type for Hecke groups of infinite covolume, similarly to the construction previously used by Stoller.
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