Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/ium/postscript/f10/conrad-set5eng.pdf Дата изменения: Wed Oct 27 17:20:45 2010 Дата индексирования: Sun Feb 13 18:39:16 2011 Кодировка: Поисковые слова: arp 220

The Classical and Adelic Point of View in Number Theory K. Conrad

10/27/2010 Problem Set 5

1. Let G be a locally compact group and µ be a left-invariant Borel measure on G. (There are no regularity assumptions about the measure.) Show the following conditions are equivalent: a) µ(K ) < for all compact subsets K of G, b) µ(U0 ) < for some nonempty open subset U0 of G. Therefore in the definition of a left Haar measure, instead of requiring all compact subsets have finite measure we can require there is some nonempty open subset with finite measure. 2. For each number field K , show AK and JK are both -compact: each can be written as the union of countably many compact subsets. 3. Let µp = {z C : z p = 1 for some n}, but we view it as a discrete group, which is not its usual topology as a subset of C. Prove µp Zp as topological = groups using an explicit isomorphism. (Do not use Pontryagin duality!!). 4. a) In the group Q/Z, show the subgroup of elements of p-power order is isomorphic to Qp /Zp by an explicit isomorphism. (There is no topology here.) b) The function r e2ir induces an isomorphism of Q/Z with the group µ of all roots of unity in C as abstract groups. This isomorphism and part a gives us an isomorphism f : Qp /Zp µp as abstract groups.1 Show the composite map f Qp - - Qp /Zp - - µp S 1 - -- is precisely the standard character x e2 5. Let F be a finite extension of Qp . a) Use a Qp -basis of F to show a topological group isomorphism Qp Qp = F . This shows F is self-dual implies a topological group isomorphism F = without using a "natural" isomorphism.
Since the quotient topology on Qp /Zp is discrete, Qp /Zp is the correct model for µp topological group with the discrete topology.
1


n

i{x}

p

on Qp .

as a


b) For each y F , define y : F S 1 by y (x) = e2i{TrF /Qp (xy)}p . Show y y is a topological group isomorphism F F , using part a. Here we are giving a = "natural" self-duality of F . (Hint: On any separable field extension L/K , every K -linear map L K has the form x TrL/K (xy ) for a unique y L.) c) Set F = Q3 ( 6) and (a + b 6) = e2i{a}3 e-2i{5b}3 . For which explicit number y F do we have (x) = e2i{TrF /Q3 (xy)}3 for all x F ?