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Difference between revisions of "Genus of an element of an arithmetic group"

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The set of elements of the group $G_\mathbf Z$ of units of a connected algebraic $\mathbf Q$-group $G$ that are conjugate to a given element $a$ in the groups $G_\mathbf Q$ and $G_{\mathbf Z_p}$ for all primes $p$, where $\mathbf Q$ and $\mathbf Z_p$ are the field of rational numbers and the ring of $p$-adic integers, respectively. The class of the element $a$ is defined as its conjugacy class in $G_\mathbf Z$. The number of disjoint classes into which the genus of an element $a$ decomposes is finite [[#References|[1]]], is denoted by $f_G(a)$, and is called the number of classes in the genus of the element $a$. The function $f_G$ on $G_\mathbf Z$ arising here is an important arithmetic characteristic expressing the deviation from the local-global principle in questions of conjugacy. In general, $f_G(a)>1$. Moreover, if the group $G$ is semi-simple and the group $G_\mathbf Z$ is infinite, then $\sup_{a\in H}f_G(a)=\infty$ for any arithmetic subgroup $H\subset G_\mathbf Z$ (see [[#References|[1]]], [[#References|[3]]]). The notions considered are a natural modification of the corresponding classical concepts in the theory of quadratic forms and are used in studies of residual approximability of arithmetic groups relative to conjugacy (see [[#References|[2]]]).
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The set of elements of the group $G_\mathbf Z$ of units of a connected algebraic $\mathbf Q$-group $G$ that are conjugate to a given element $a$ in the groups $G_\mathbf Q$ and $G_{\mathbf Z_p}$ for all primes $p$, where $\mathbf Q$ and $\mathbf Z_p$ are the field of rational numbers and the ring of $p$-adic integers, respectively. The class of the element $a$ is defined as its [[conjugacy class]] in $G_\mathbf Z$. The number of disjoint classes into which the genus of an element $a$ decomposes is finite [[#References|[1]]], is denoted by $f_G(a)$, and is called the number of classes in the genus of the element $a$. The function $f_G$ on $G_\mathbf Z$ arising here is an important arithmetic characteristic expressing the deviation from the local-global principle in questions of conjugacy. In general, $f_G(a)>1$. Moreover, if the group $G$ is semi-simple and the group $G_\mathbf Z$ is infinite, then $\sup_{a\in H}f_G(a)=\infty$ for any arithmetic subgroup $H\subset G_\mathbf Z$ (see [[#References|[1]]], [[#References|[3]]]). The notions considered are a natural modification of the corresponding classical concepts in the theory of quadratic forms and are used in studies of residual approximability of arithmetic groups relative to conjugacy (see [[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.P. Platonov, "On the genus problem in arithmetic subgroups" ''Soviet Math. Dokl.'' , '''12''' : 5 (1971) pp. 1503–1507 ''Dokl. Akad. Nauk SSSR'' , '''200''' : 4 (1971) pp. 793–796</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.P. Platonov, G.V. Matveev, "Abelian groups and the finite approximability of linear groups with respect to conjugacy" ''Dokl. Akad. Nauk Bel.SSR'' , '''14''' : 9 (1970) pp. 777–779 (In Russian) {{MR|0274603}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.S. Rapinchuk, "Platonov's conjecture on the genus in arithmetic groups" ''Dokl. Akad. Nauk Bel.SSR'' , '''25''' : 2 (1981) pp. 101–104; 187 (In Russian) (English summary) {{MR|613414}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.P. Platonov, "On the genus problem in arithmetic subgroups" ''Soviet Math. Dokl.'' , '''12''' : 5 (1971) pp. 1503–1507 ''Dokl. Akad. Nauk SSSR'' , '''200''' : 4 (1971) pp. 793–796</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.P. Platonov, G.V. Matveev, "Abelian groups and the finite approximability of linear groups with respect to conjugacy" ''Dokl. Akad. Nauk Bel.SSR'' , '''14''' : 9 (1970) pp. 777–779 (In Russian) {{MR|0274603}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.S. Rapinchuk, "Platonov's conjecture on the genus in arithmetic groups" ''Dokl. Akad. Nauk Bel.SSR'' , '''25''' : 2 (1981) pp. 101–104; 187 (In Russian) (English summary) {{MR|613414}} {{ZBL|}} </TD></TR></table>

Latest revision as of 20:58, 29 November 2014

The set of elements of the group $G_\mathbf Z$ of units of a connected algebraic $\mathbf Q$-group $G$ that are conjugate to a given element $a$ in the groups $G_\mathbf Q$ and $G_{\mathbf Z_p}$ for all primes $p$, where $\mathbf Q$ and $\mathbf Z_p$ are the field of rational numbers and the ring of $p$-adic integers, respectively. The class of the element $a$ is defined as its conjugacy class in $G_\mathbf Z$. The number of disjoint classes into which the genus of an element $a$ decomposes is finite [1], is denoted by $f_G(a)$, and is called the number of classes in the genus of the element $a$. The function $f_G$ on $G_\mathbf Z$ arising here is an important arithmetic characteristic expressing the deviation from the local-global principle in questions of conjugacy. In general, $f_G(a)>1$. Moreover, if the group $G$ is semi-simple and the group $G_\mathbf Z$ is infinite, then $\sup_{a\in H}f_G(a)=\infty$ for any arithmetic subgroup $H\subset G_\mathbf Z$ (see [1], [3]). The notions considered are a natural modification of the corresponding classical concepts in the theory of quadratic forms and are used in studies of residual approximability of arithmetic groups relative to conjugacy (see [2]).

References

[1] V.P. Platonov, "On the genus problem in arithmetic subgroups" Soviet Math. Dokl. , 12 : 5 (1971) pp. 1503–1507 Dokl. Akad. Nauk SSSR , 200 : 4 (1971) pp. 793–796
[2] V.P. Platonov, G.V. Matveev, "Abelian groups and the finite approximability of linear groups with respect to conjugacy" Dokl. Akad. Nauk Bel.SSR , 14 : 9 (1970) pp. 777–779 (In Russian) MR0274603
[3] A.S. Rapinchuk, "Platonov's conjecture on the genus in arithmetic groups" Dokl. Akad. Nauk Bel.SSR , 25 : 2 (1981) pp. 101–104; 187 (In Russian) (English summary) MR613414
How to Cite This Entry:
Genus of an element of an arithmetic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genus_of_an_element_of_an_arithmetic_group&oldid=32457
This article was adapted from an original article by A.S. Rapinchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article