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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 $p otequiv 1pmod{24}$.
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 $F$-alge
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
We study an analogue of Serres modularity conjecture for projective representations $overline{rho}: operatorname{Gal}(overline{K} / K) rightarrow operatorname{PGL}_2(k)$, where $K$ is a totally real number field. We prove new cases of this conjecture
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 r
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