文档介绍：APPENDIX A: WEIL’S PROOF OF THEOREM
Adeles
Let P denote the set of all rational primes. For each p ∈ P, the field of p–adic
numbers, Qp, is defined to be pletion of the rationals in the p–adic valuation,
−ordp(r)
|r|p = p ,
where ordp(r) is the highest power of p dividing r. Extend the definition of | · |p
by setting |0|p = 0. Write “p = ∞” for the usual valuation on Q, |r|∞= |r|. Then
Q∞= R. For each p ∈{∞} ∪ P, the field Qp is a non–discrete pact
field, with topology given by the metric dp(x, y) = |x − y|p. For finite p, the metric
is an ultrametric:
dp(x, y) ≤ max{dp(x, z), dp(z, y)}.
It follows that for finite p, Zp = {x ∈ Qp | |x|p ≤ 1} is a pact subring
of Qp, so Qp is a local field.
For finite p, the elements of the field Qp may be represented as formal power
series in p, of the form
x = a pk,
X k
k≥n
−n
where each ak ∈ Fp = {0, 1, . . . , p − 1}, n ∈ Z, and an =6 0. Then |x|p = p , and
x ∈ Zp if and only if n ≥ 0. This representation of the p–es straight from
N k
the definition: the sequence of rational numbers (sn)N≥n, where sN = akp ,
Pk=n
is a Cauchy sequence with respect to | · |p.
Define the adele ring QA of Q by
Q = x ∈ Q | x ∈ Z for all but finitely many p.
A Y p p p
p≤∞
Topologize the ring as follows: for each finite set P , {∞} ⊂ P ⊂{∞} ∪ P, let
Q (P ) = {x ∈ Q | x ∈ Z if p∈/ P } = Q × Z .
A A p p Y p Y p
p∈P