Algebraic Groups Constructed from Orders of Quaternion Algebras

与判别式为$2$的定四元数代数相关的代数模形式的计算

Computations of algebraic modular forms associated with the definite quaternion algebra of discriminant $2$

本文提出一种计算判别式为$2$的定四元数代数极大序上代数模形式空间基的算法，构建了相应基的数据库，以支持代数模形式同余关系的研究。

Bicomplex k-Fibonacci quaternions

In this paper, bicomplex k-Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex k-Fibonacci quaternions which are connected with bicomplex numbers and k-Fibonacci numbers are investigated. Furthermore, the Honsberger identity, the d'Ocagne's identity, Binet's formula, Cassini's identity, Catalan's identity for these quaternions are given.

Orders of Quaternion Algebras with Involution

We introduce the notion of maximal orders over quaternion algebras with orthogonal involution and give a classification over local fields, and a partial classification over algebraic number fields.

例外群上模形式的自动收敛与算术性

Automatic convergence and arithmeticity of modular forms on exceptional groups

本研究证明F₄与Eₙ型群上尖点四元数模形式空间具纯代数刻画，核心是证明相关无穷和绝对收敛，进而推出其存在傅里叶系数全为代数数的基，推动了例外群上模形式的算术性质研究。

Realizing algebraic invariants of hyperbolic surfaces

Let $S_g$ ($g\geq 2$) be a closed surface of genus $g$. Let $K$ be any real number field and $A$ be any quaternion algebra over $K$ such that $A\otimes_K\mathbb{R}\cong M_2(\mathbb{R})$. We show that there exists a hyperbolic structure on $S_g$ such that $K$ and $A$ arise as its invariant trace field and invariant quaternion algebra.

On the arithmetic and geometry of binary Hamiltonian forms

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describ…