Quaternion constituents of group algebras

Hyperbolicity of orders of quaternion algebras and group rings

For a given division algebra of the quaternions, we construct two types of units of its Z-orders: Pell units and Gauss units. Also, if K = Q √ −d, d ∈ Z \ {0, 1} is square free and R = IK , we classify R and G such that U1(RG) is hyperbolic. In particular, we prove that U1(RK8) is hyperbolic iff d > 0 and d ≡ 7 (mod 8). In this case, the hyperbolic boundary ∂(U1(RG)) ∼= S, the two dimensional s...

Quaternion Algebras

The additive identity is (0, 0), the multiplicative identity is (1, 0), and from addition and scalar multiplication of real vectors we have (a, b) = (a, 0) + (0, b) = a(1, 0) + b(0, 1), which looks like a+ bi if we define i to be (0, 1). Real numbers occur as the pairs (a, 0). Hamilton asked himself if it was possible to multiply triples (a, b, c) in a nice way that extends multiplication of co...

Levels of Quaternion Algebras

The level of a ring R with 1 6= 0 is the smallest positive integer s such that −1 can be written as a sum of s squares in R, provided −1 is a sum of squares at all. D.W. Lewis showed that any value of type 2n or 2n + 1 can be realized as level of a quaternion algebra, and he asked whether there exist quaternion algebras whose levels are not of that form. Using function fields of quadratic forms...

Selectivity in Quaternion Algebras

Let A be a quaternion algebra over a number field K and assume that A satisfies the Eichler condition; that is, there exists an archimedean prime of K which does not ramify in A. Let Ω be a commutative, quadratic OK-order and let R ⊂ A be an order. We determine the isomorphism classes in the genus of R which admit an embedding of Ω. In particular, we show that the proportion of the genus of R a...

A Description of Quaternion Algebras