MAT415: HOMEWORK SET #6
DUE: MONDAY APRIL 20, 2015
Define the finite Adelle ring Af as follows: it is the subring of
(where the product is over all primes p) consisting of those elements
Q
p Qp
~a = (a2 , a3 , a5 , . . . , ap , . . . )
such that for almost all prime p, ap ∈ Zp .
Q
(1) Prove that the diagonal embedding Q ,→ p Qp gives a map Q →
Af . So that Af naturally contains Q as a subring.
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(2) Define a metric on A by |~
α|Af := maxp p− 2 |αp |p . This makes Af
~ := |~
~ A . Prove that Q is dense
into a metric space by d(~
α, β)
α − β|
f
in Af .
(3) In the same way as above, one can consider the diagonal embedding
of Q into Af × R. Prove that the image of Q in Af × R is discrete.
b :=
(4) For an abelian group G, its pontryagin dual is defined to be G
Homcont (G, R/Z), the group of continuous homomorphisms of from
G to R/Z. In what follows, consider Qp , Zp , Af as abelian groups
under addition. Prove that
b ≈ R/Z
(a) Z
\
(b) Q
p /Zp ≈ Zp
Q
d
(c) Q/Z ≈ p Zp
b ≈ (Af × R)/Q.
(d) BONUS: Q
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