Questions tagged [quaternion-algebras]

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3
votes
3answers
286 views

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
votes
0answers
59 views

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
votes
1answer
120 views

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 $...
3
votes
0answers
44 views

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 ...
2
votes
1answer
105 views

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 ...
27
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1answer
669 views

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\...
1
vote
1answer
244 views

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 ...
3
votes
1answer
130 views

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 ...
1
vote
0answers
103 views

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
vote
1answer
145 views

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 ...
4
votes
0answers
119 views

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 ...
2
votes
2answers
361 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 ...
4
votes
1answer
325 views

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 ...
6
votes
1answer
484 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 ...
5
votes
4answers
545 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 ...
2
votes
2answers
186 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 ...
5
votes
1answer
138 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 ...
4
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0answers
275 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 $x,y$ ...
1
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1answer
103 views

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 $...
4
votes
1answer
112 views

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 ...
3
votes
0answers
84 views

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 ...
2
votes
0answers
126 views

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 $$ i^2 = a, \quad j^2 = b, \quad k = ij = -ji. $$ ...
4
votes
1answer
213 views

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 ...
5
votes
2answers
466 views

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?
2
votes
1answer
239 views

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 ...
0
votes
1answer
164 views

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 ...
2
votes
1answer
332 views

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
votes
0answers
211 views

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 $\...
14
votes
2answers
1k views

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 ...
6
votes
1answer
288 views

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 ...