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units_class_no

# units_class_no - Fujisakis lemma units theorem class number...

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( January 19, 2010 ) Fujisaki’s lemma, units theorem, class number Paul Garrett [email protected] http: / /www.math.umn.edu/˜garrett/ Fujisaki’s lemma asserts the compactness of J 1 /k × , where J 1 is the ideles of idele-norm 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 non-zero 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 non-zero 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 ideal class group factors through the idele class group
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