Hilbert symbols, class groups and quaternion algebras

Chinburg, Ted, and Friedman, Eduardo. "Hilbert symbols, class groups and quaternion algebras." Journal de théorie des nombres de Bordeaux 12.2 (2000): 367-377. <http://eudml.org/doc/248490>.

@article{Chinburg2000,
abstract = {Let $B$ be a quaternion algebra over a number field $k$. To a pair of Hilbert symbols $\lbrace a, b\rbrace $ and $\lbrace c, d\rbrace $ for $B$ we associate an invariant $\rho = \rho _R \left([\mathcal \{D\}(a, b)], [\mathcal \{D\}(c, d)]\right)$ in a quotient of the narrow ideal class group of $k$. This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders $\mathcal \{D\}(a, b)$ and $\mathcal \{D\}(c, d)$ in $B$ associated to $\lbrace a, b\rbrace $ and $\lbrace c,d\rbrace .$ If $a = c$, we compute $\rho _R ([\mathcal \{D\}(a, b)], [\mathcal \{D\}(c, d)])$ by means of arithmetic in the field $k (\sqrt\{a\}).$ The problem of extending this algorithm to the general case leads to studying a finite graph associated to different Hilbert symbols for $B$. An example arising from the determination of the smallest arithmetic hyperbolic $3$-manifold is discussed.},
author = {Chinburg, Ted, Friedman, Eduardo},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {quaternion algebras; ideal class groups; maximal orders},
language = {eng},
number = {2},
pages = {367-377},
publisher = {Université Bordeaux I},
title = {Hilbert symbols, class groups and quaternion algebras},
url = {http://eudml.org/doc/248490},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Chinburg, Ted
AU - Friedman, Eduardo
TI - Hilbert symbols, class groups and quaternion algebras
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 2
SP - 367
EP - 377
AB - Let $B$ be a quaternion algebra over a number field $k$. To a pair of Hilbert symbols $\lbrace a, b\rbrace $ and $\lbrace c, d\rbrace $ for $B$ we associate an invariant $\rho = \rho _R \left([\mathcal {D}(a, b)], [\mathcal {D}(c, d)]\right)$ in a quotient of the narrow ideal class group of $k$. This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders $\mathcal {D}(a, b)$ and $\mathcal {D}(c, d)$ in $B$ associated to $\lbrace a, b\rbrace $ and $\lbrace c,d\rbrace .$ If $a = c$, we compute $\rho _R ([\mathcal {D}(a, b)], [\mathcal {D}(c, d)])$ by means of arithmetic in the field $k (\sqrt{a}).$ The problem of extending this algorithm to the general case leads to studying a finite graph associated to different Hilbert symbols for $B$. An example arising from the determination of the smallest arithmetic hyperbolic $3$-manifold is discussed.
LA - eng
KW - quaternion algebras; ideal class groups; maximal orders
UR - http://eudml.org/doc/248490
ER -