# Questions tagged [quaternion-algebras]

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### System of eigenvalues of partial Laplacians and the Jacquet Langlands correspondence

I'm looking for a precise reference. I could not dig it up in the original paper of Jacquet-Langlands. Let $F$ be a totally real number field with $[F:\mathbb{Q}]=r$, let $B_1=M_2(F)$ be the split ...
140 views

### How can I find explicit examples of maximal orders of quaternion algebras that are not isomorphic?

I know that there exist algorithms that will construct maximal orders of a quaternion algebra over, say, $\mathbb{Q}$. However, the implemented algorithms that I know of require that you input an ...
177 views

### Possible values of symmetric functions evaluated on quaternions

Let $i,j,k$ the units of quaternions, in particular $i^2=j^2=k^2=-1, ijk=-1$. We will use non commutative variables $x,y,z$. Define $sym_{a,b,c}$ to be the polynomial made of the sum of monomials ...
430 views

### The Hilbert symbols of quaternion algebras over a totally real field

Let $k$ be a totally real number field. Every quaternion algebra $B$ over $k$ can be written as $$B = \left(\frac{a,b}{k}\right),$$ for some constants $a,b \in k^\times$. My question is, can I always ...
481 views

### Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$

For a fixed positive integer $n$, the Diophantine equation $$x^2 + y^2 + z^2 = n$$ was studied by Gauss in Disquisitiones Arithmeticae. As is known, this equation is intimately connected to the ...
168 views

### Involution on false elliptic curve

Let B be a indefinite quaternion algebra with discriminant $d>1$, maximal order $\mathcal{O}$ and standard involution $'$, then there exists $t\in{B}$ such that $t^2=-d$ and a new involution on B ...
101 views

### A generalization of Witt's theorem for quaternion algebra isomorphism

Let $Q$ be a quaternion $k$-algebra (namely, a dimension 4 $k$-central simple algebra). Then it is possible to (canonically) attach a smooth projective conic $C_Q\subseteq \mathbf{P}_k^2$ to $Q$: if ...
182 views

### Quaternion algebras in characteristic 2

Let $k$ be a field and let $Q$ be a quaternion algebra over $k$. It is well known that, if $\mathrm{char}\,k\neq 2$, one can define $Q$ as the $k$-algebra of dimension $4$ generated by elements $x,y$ ...
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