# Questions tagged [quaternion-algebras]

The quaternion-algebras tag has no usage guidance.

40
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### Proof that the left ideal I of prime norm in maximal order can be written as I = ON + Oα

It seems there is a well-known fact that if $O$ is a maximal order in quaternion algebra $B$ and $I$ is a left $O-$ideal such that $nrd(I) = N$ is
prime, then $I = ON + Oα$ with $gcd(N^2, nrd(α)) = N$....

2
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### Group of invertible elements in a degree 4 central simple algebra with symplectic involution with norm in a center

Let $A$ be a central simple algebra of degree 4 (i.e. dimension 16) over a field $F$ with $\mathrm{char}(F) \neq 2$. It is known that any such algebra is a tensor product $D_1 \otimes D_2$ of two ...

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### A note on orders in quaternion algebras

Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$, where $i^2=a,j^2=b;a,b\in F^\times$.
...

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### Maximal orders and surface subgroups of even genus

Let $A$ be a quaternion algebra over a totally real number field $k$. Suppose that $A$ splits at exactly one real place of $A$. Let $\mathcal{O}$ be a maximal order in $A$. Then $\mathcal{O}$ contains ...

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### Motivating unpublished statements of Gauss about congruences and quaternions

Background
Modern claims that Gauss anticipated the quaternions algebra are based primarily on an unpublished fragment of Gauss dated to 1819 and entitled "rotations of space". In this ...

3
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### Abelianizations of arithmetic Fuchsian groups

Let $a,b$ be positive integers with $x^2-ay^2-bz^2+abv^2=0$ having only the zero solution over $\mathbb Z$ and consider the Fuchsian group
\begin{equation*}
\Gamma=\left\{\begin{bmatrix}
k+\sqrt{a}l &...

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### Number of cosets by quaternion order

Let $D$ be a quaternion division algebra over $\mathbb Q$ and let $\Lambda\subset D$ be a maximal order.
For $n\in\mathbb Z$ let $\Lambda^{(n)}$ denote the set of all $\lambda\in\Lambda$ with reduced ...

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### Is the isomorphism class of Eichler order with prime level in $B_{p,\infty}$ with extra units at most unique?

Let $B_{p, \infty}$ be the quaternion algebra over $\mathbb{Q}$ ramified exactly at $p$ and $\infty$. Let $\mathcal{O}$ be an order in $B_{p, \infty}$. As we know, if $\mathcal{O}$ is maximal, then ...

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### How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?

How to prove without using advanced theorems that quaternions algebra $H = \genfrac(){}{}{-1,-1}{\mathbb{Z}_p}$, where $p$ is prime that $H \cong\operatorname{Mat}_2({\mathbb{Z}_p})$?
My ideas: I ...

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### Are there any central simple algebras admitting a standard basis?

Are there any central simple algebras admitting a standard basis?
By a standard basis I mean a normal basis that has a cyclic property generalizing that of the familiar basis $1, i, j, k$ for ...

3
votes

3
answers

609
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### Ramification of quaternion algebras over $\mathbb Q$

I'm reading on the classification of quaternion algebras over $\mathbb{Q}$. In the most common definition, we have a quaternion algebra $Q = \left(\frac{a,b}{\mathbb{Q}} \right)$ splits at a finite ...

2
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### Bézout theorem for complex solutions of systems of homogeneous quaternionic polynomials

Suppose that I have a system of $n$ homogeneous polynomials in $n+1$ variables $p_{1},\dots,p_{n}\in\mathbb{H}[x_{0},\dots,x_{n}]$ with coefficients at both sides in $\mathbb{H}.$ I want to know if ...

2
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1
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### Does a quaternion algebra exist over a number field that is split over some infinite real places, but not others?

Let $F$ be a totally real number field having at least two different real embeddings $\sigma_1 : F \hookrightarrow \mathbb{R}$ and $\sigma_2 : F \hookrightarrow \mathbb{R}$.
Does a quaternion algebra $...

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### Conceptual meaning of the Dickson construction

The Cayley-Dickson construction (a.k.a. Dickson construction or Dickson doubling) constructs a new *-algebra $A'$ out of a given *-algebra $A$. As a vector space $A' = A \oplus A$, and the ...

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### Quaternions algebras whose class groupoid is actually a class group

Let $A$ be a finite-dimensional $\mathbb{Q}$-algebra, and let $\mathcal{O}$ be a maximal order in $A$. Like the case of number fields, we can define an equivalence relation to define something similar ...

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### Quaternionic and octonionic analogues of the Basel problem

I asked this question in MSE around 3 months ago but I have received no answer yet, so following the suggestion in the comments I decided to post it here.
It is a well-known fact that
$$\sum_{0\neq n\...

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1
answer

257
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### Are there algorithms for deciding or solving conjugacy in integer quaternion rings?

I am doing some research on the quaternions and their role in Non-commutative cryptography. I have found a number of articles, but it is still unclear to me if there is a known solution to the ...

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### Subring of quaternion algebra

I am following the book Introduction to Quadratic Forms over Fields by T. Y. Lam. In section VI.2, the author proves that, over an arbitrary local field $F$, there is a unique quaternion division ...

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### Octonion algebras over number fields [closed]

Is there any textbook or paper about Arithmetic of Octonion Algebras or Octonion Algebras constructed over number fields? I know J. Voight book and K. Martin notes about quaternion algebras but I was ...

1
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179
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### Lattices from quaternion algebras (MAGMA software)

I am studying the paper "Lattice Packing from Quaternion Algebras" from 2012 about the construction of ideal lattices.
In Section 3.3 the authors construct very interesting examples of lattices using ...

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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 ...

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### 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 ...

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### Possible values of symmetric functions evaluated on quaternions

$\DeclareMathOperator\sym{sym}$Let $i$, $j$, $k$ be 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 ...

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562
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### 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 ...

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answers

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### 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 ...

2
votes

2
answers

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### 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 ...

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### 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 ...

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### 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 $x,y$ ...

3
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1
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### Fractional ideals of maximal orders in quaternion algebras

Let D be a skew field that is central and finite-dimensional over a number field F (in particular: a quaternion algebra over F). Let $\Delta$ $\subseteq$ D be a maximal $\mathcal{O}$$_{F}$-order. Let $...

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### Integrality of the support of matrix coefficients?

Consider a division quaternion algebra $D$ over a number field $F$. For an automorphic representation $\pi$ of $D$, I am interested in the associated matrix coefficients
$$f : \gamma \in G \longmapsto ...

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### Length of $\mathbb{Z}_p$-modules for quaternion algebras

Consider $G$ the group of units of a division quaternion algebra $D$ over a number field $F$, say $\mathbb{Q}$ for simplicity. Consider a non-central element $x \in G(\mathbb{Z}_p)$: hence, it is ...

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### When is $SL_1$ of a quaternion division algebra perfect?

Let $F$ be a field, let $a,b \in F$ and let $\mathcal A:= \left(\frac{a,b}{F}\right)$ be the quaternion algebra over $F$ determined by the relations
$$
i^2 = a, \quad j^2 = b, \quad k = ij = -ji.
$$
...

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### Fuchsian group which is derived from a division quaternion algebra, Mixing flows on the quotient space

Suppose a Fuchsian group $\Gamma$ is derived from a division
quaternion algebra. Then the quotient space $\Gamma\backslash \mathcal{H}$ is compact.
I am reading the book "Fuchsian Groups" of ...

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### About some property of automorphism of octonions

Let $f$ be an automorphism of the algebra of octonions. Is it true that $f$ preserves some quaternionic subalgebra? Has the statement an elementary proof?

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### What is the effect of imaginary quadratic extension on a quaternion algebra's ramified primes

A smart man once explained to me how to solve the following problem, then I forgot.
Let $F\subset\mathbb{R}$
be a number field,
let $d\in F^+$,
and let $K=F(\sqrt{-d})$.
Denote the rings of integers ...

1
vote

1
answer

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### Units in indefinite quaternionic algebra

This is the opposite to my last question case.
Let $F$ be a totally real number field, $R$ is a quaternion algebra over $F$ unramified in at least one infinite place of $F$. Let $\mathcal{O}⊂R$ be an ...

3
votes

1
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### Finite group of units in quaternion orders

Let $F$ be a totally real number field, $R$ is a quaternion algebra over $F$ ramified in all infinite places of $F$. Let $\mathcal{O}\subset R$ be an order. By assumption on $R$ its group of units $\...

2
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0
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### A Quaternions version of the Gauss Lucas theorem

Edit: My serious thanks to Christian, Todd and Yemon for their comments to the previous version.
The derivative is the same as Todd says: $(az^{n})'=naz^{n-1}$.
The polynomial is in the form of $\...

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### Computing endomorphism rings of supersingular elliptic curves

I would like to know what algorithms there are to compute the linearly independent generators $(1,i,j,k)$ for quaternion algebra containing the endomorphism ring of a supersingular curve. The curve in ...

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### Compact hyperbolic 3-manifolds with prescribed quaternion algebra, quaternion parameters as ramification condition

What is an interesting class of examples of hyperbolic 3-manifolds,
each of which satisfies the following conditions?
1. It is compact
2. Its trace field contains a unique imaginary quadratic ...