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(
January 19, 2010
)
Fujisaki’s lemma, units theorem, class number
Paul Garrett
garrett@math.umn.edu http:
/
/www.math.umn.edu/˜garrett/
Fujisaki’s lemma asserts the compactness of
J
1
/k
×
, where
J
1
is the ideles of idelenorm 1 of a number ﬁeld
k
. This is basic in the harmonic analysis of automorphic forms. It is a corollary of existence and uniqueness
of Haar measure. It
implies
ﬁniteness of class numbers and the units theorem. We recall the standard
argument, apparently ﬁrst enunciated by Iwasawa.
[1]
Let
i
be the
ideal map
from ideles to nonzero fractional ideals of the integers
o
of
k
. That is,
i
(
α
) =
Y
v<
∞
p
ord
v
α
v
(for
α
∈
J
)
where
p
v
is the prime ideal in
o
attached to the place
v
. The subgroup
J
1
of
J
still surjects to the group of
nonzero fractional ideals. The kernel in
J
of the ideal map is
H
=
Y
v
∞
k
×
v
×
Y
v<
∞
o
×
v
and the kernel on
J
1
is
H
1
=
H
∩
J
1
. The principal ideals are the image
i
(
k
×
). The map of
J
1
to the
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This note was uploaded on 10/26/2011 for the course MATH 8410 taught by Professor Staff during the Fall '11 term at University of Georgia Athens.
 Fall '11
 Staff
 Math, Number Theory

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