View source for Genus of an element of an arithmetic group

The set of elements of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g0440201.png" /> of units of a connected algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g0440202.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g0440203.png" /> that are conjugate to a given element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g0440204.png" /> in the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g0440205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g0440206.png" /> for all primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g0440207.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g0440208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g0440209.png" /> are the field of rational numbers and the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402010.png" />-adic integers, respectively. The class of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402011.png" /> is defined as its conjugacy class in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402012.png" />. The number of disjoint classes into which the genus of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402013.png" /> decomposes is finite [[#References|[1]]], is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402014.png" />, and is called the number of classes in the genus of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402015.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402017.png" /> arising here is an important arithmetic characteristic expressing the deviation from the local-global principle in questions of conjugacy. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402018.png" />. Moreover, if the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402019.png" /> is semi-simple and the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402020.png" /> is infinite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402021.png" /> for any arithmetic subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044020/g04402022.png" /> (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====
<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)</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)</TD></TR></table>