23
votes
Accepted
Quaternionic and octonionic analogues of the Basel problem
This isn't really a full answer, but it's too long for a comment, and perhaps it's informative all the same.
Your sum $S_k[\mathcal{O}]$ can be written as the value at $s = k$ of the sum
$$\sum_{0 \ne ...
- 37k
13
votes
A note on orders in quaternion algebras
Two orders need not be isomorphic.
First of all, in number fields $K$ other than $\mathbf Q$ not all orders are isomorphic rings (even if they are isomorphic abelian groups): the full ring of integers ...
- 50.6k
10
votes
Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$
The theta function associated to your quadratic form (here $\theta(\tau)\theta(3\tau)^2$) belongs to the modular form space $M_{3/2}(\Gamma_0(12))$,
and you are in luck because there are no cusp forms,...
- 13.1k
9
votes
Accepted
What is the effect of imaginary quadratic extension on a quaternion algebra's ramified primes
The base change of your quaternion algebra will be ramified at $\mathfrak{P}$ if and only the degree of the extension of completions $K_{\mathfrak{P}}/F_{\mathfrak{p}}$ has odd degree, i.e. case 3. A ...
- 5,382
8
votes
Ramification of quaternion algebras over $\mathbb Q$
I'm not sure how to define "classically" here (Hensel's $p$-adics hail from around 1890s), but the Hilbert symbol gives a way to do this without $p$-adics. (See Chapter 12 of my book, http:/...
- 3,009
8
votes
How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?
A quaternion algebra with center $F$ that is not a division ring is isomorphic to ${\rm M}_2(F)$. See Theorem 4.21 and Corollary 4.24 here. Let's show when $F = \mathbf F_p$ (field of order $p$) for $...
- 50.6k
8
votes
Accepted
The Hilbert symbols of quaternion algebras over a totally real field
Yes.
First choose $a$. You can take any $a$ such that $K = k(\sqrt{a})$ is a splitting field of $B$, so by Grunwald--Wang (or elementary congruences and sign conditions) you can enforce $\sigma(a)>...
- 5,382
7
votes
Accepted
Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$
One of the beautiful (and sometimes flummoxing) aspects of the theory of ternary quadratic forms is that you can find answers and questions coming from many different points of view! Using modular ...
- 3,009
7
votes
How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?
At the risk of being redundant and repeating earlier answers, let me mention that this is explicitly contained in the book "Elementary number theory, group theory and Ramanujan graphs" by ...
- 6,614
6
votes
Accepted
About some property of automorphism of octonions
Seen as a map of $8$-dimensional Euclidean vector spaces, $f$ is obviously (special) orthogonal, so we can find an orthonormal basis on which is has a block diagonal form of $2\times 2$ rotation ...
- 32.4k
6
votes
A note on orders in quaternion algebras
The classification of orders is the topic of chapters 22-24 in my book. Yes, there are plenty more orders than the ones you listed.
One cheeky answer is that isomorphism classes of orders in $\mathrm{...
- 3,009
5
votes
How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?
Yes, you can do this without any advanced theorems.
Does $\mathbb{Z}\mathbb{p}$ mean the ring $\mathbb{Z}_p$ of $p$-adic integers or the finite field $\mathbb{Z}/p\mathbb{Z}=\mathbb{F}_p$ with $p$ ...
- 3,009
5
votes
Units in indefinite quaternionic algebra
If there is only one split real place, you can use the algorithm described by John Voight in Computing fundamental domains for Fuchsian groups. This algorithm is available in Magma.
- 5,382
4
votes
Accepted
Fuchsian group which is derived from a division quaternion algebra, Mixing flows on the quotient space
1 - There is an explicit reference given in the book: Borel, Harish-chandra, arithmetic subgroups of algebraic groups, 1962. This is the general result for matrix groups. A simpler proof has been ...
- 18.7k
4
votes
Are there algorithms for deciding or solving conjugacy in integer quaternion rings?
If you are talking about quaternions, then you can rewrite the equation $y = z^{-1}xz$ as $zy - xz = 0$ which is $\mathbb{R}$-linear in $z$. Therefore both problems can be solved quite easily using ...
- 8,597
4
votes
How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?
As Keith Conrad says, use the pigeonhole principle to find $a$ and $b$ satisfying $a^2+b^2+1 \equiv 0 \bmod p$. Set
$$I = \begin{bmatrix} 0&-1 \\ 1&0 \end{bmatrix} \qquad J = \begin{bmatrix} a&...
- 156k
4
votes
Accepted
MAGMA (pseudo)basis for maximal order in quaternion algebra
(1) Yes, your interpretation is correct.
(2) Computing a single generator (when it even exists) of a fractional ideal in a number field can be very costly (not polynomial time) and yield a very large ...
- 5,382
3
votes
Accepted
Are there any central simple algebras admitting a standard basis?
As suggested by @Kimball I develop my comment. An important class of central simple algebras consists of the cyclic algebras: assume that the field $k$ contains a primitive $n$-th root of unity $\zeta ...
- 38k
3
votes
Ramification of quaternion algebras over $\mathbb Q$
Another answer to your second question is as follows:
if $Q$ splits at $p$, then $\mathcal{O}/p\mathcal{O} \cong M_2(\mathbb{F}_p)$, and
if $Q$ ramifies at $p$, then $\mathcal{O}/p\mathcal{O} \cong A$...
- 5,382
3
votes
Ramification of quaternion algebras over $\mathbb Q$
The concept of a quaternion algebra over a field, going beyond the case of Hamilton's quaternions, is due to Dickson in 1906 and he introduced the concept of maximal orders in finite-dimensional ...
- 50.6k
3
votes
Accepted
Subring of quaternion algebra
That's because $w$ is a valuation (the main point being the nonarchimedean triangle inequality), which can be proved by reducing to the commutative case; see for instance Lemma 13.3.2 in https://math....
- 5,382
3
votes
Accepted
Does a quaternion algebra exist over a number field that is split over some infinite real places, but not others?
Let $a$ be any totally negative element and pick $b$ to be an element such that $\sigma_1(b),\sigma_2(b)$ are positive, while $\sigma(b)$ is negative for all other $\sigma:F\to\mathbb R$. Such ...
- 28.2k
3
votes
How can I find explicit examples of maximal orders of quaternion algebras that are not isomorphic?
Ben explained what happens for indefinite quaternion algebras, and in particular that you don't get such examples if your base field has narrow class number one. Let me discuss the (totally) definite ...
- 6,039
3
votes
Finite group of units in quaternion orders
By the ramification hypothesis, the reduced norm $\mathrm{nrd}$ gives a positive definite quadratic form $\mathrm{Tr}\circ \mathrm{nrd} : \mathcal{O}\to \mathbb{Q}$. On the other hand, the reduced ...
- 5,382
3
votes
Computing endomorphism rings of supersingular elliptic curves
For the computation of a basis of the endomorphism algebra, you should read David Kohel's thesis, see for instance Theorem 2.
To find the endomorphism ring, you can repeat Kohel's methods to find ...
- 5,382
3
votes
Accepted
A generalization of Witt's theorem for quaternion algebra isomorphism
Norms of quaternion algebras are particular cases of n-fold Pfister forms for n=2 (in any characteristic), and the result holds true for any Pfister forms. See, for example, Elman, Karpenko, Merkurjev ...
- 1,602
2
votes
Accepted
Integrality of the support of matrix coefficients?
According to the comments, I understand you to mean the following local question: Say $D$ is the quaternion division algebra over an $p$-adic field $F$,
and $\pi$ is a smooth representation of $D^\...
- 6,039
2
votes
Involution on false elliptic curve
I think that you can construct an idempotente $e=e^*$ in some quadratic base change $B\otimes F$ where $F$ is a subfied of $B$ fixed by the involution.
Now, if you choose $F$ so the prime $p$ splits ...
- 221
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