# Hilbert symbols, class groups and quaternion algebras

Ted Chinburg; Eduardo Friedman

Journal de théorie des nombres de Bordeaux (2000)

- Volume: 12, Issue: 2, page 367-377
- ISSN: 1246-7405

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topChinburg, 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 -

## References

top- [B] A. Borel, Commensumbility classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa8 (1981), 1-33. Also in Borel's Oeuvres, BerlinSpringer (1983). Zbl0473.57003MR616899
- [CF1] T. Chinburg, E. Friedman, An embedding theorem for quaternion algebras. J. London Math. Soc. (2) 60 (1999), 33-44. Zbl0940.11053MR1721813
- [CF2] T. Chinburg, E. Friedman, The finite subgroups of maximal arithmetic Kleinian groups. Ann. Institut Fourier50 (2000), 1765-1798. Zbl0973.20040MR1817383
- [CFJR] T. Chinburg, E. Friedman, K.N. Jones, A.W. Reid, The arithmetic hyperbolic 3-manifold of smallest volume. Ann. Scuola Norm. Sup. Pisa (to appear). Zbl1008.11015MR1882023
- [R] I. Reiner, Maximal Orders. Acad. PressLondon (1975). Zbl0305.16001MR1972204
- [V] M.-F. Vigngras, Arithmétique des algèbres de Quaternions. Lecture Notes in Math. 800, Springer-VerlagBerlin (1980). Zbl0422.12008MR580949

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