Everything about Adele Ring totally explained
In
number theory, the
adele ring is a
topological ring containing the
field of
rational numbers (or, more generally, to an
algebraic number field). It involves all the completions of the field.
The word "adele" is short for "additive idele". Adeles were called
valuation vectors or
repartitions before about 1950.
Definitions
The
profinite completion of the integers
and the
real numbers (or in other words as the restricted product of all completions of the rationals). In this case the restricted product means that for an adele
all but a finite number of the
are
p-adic integers.
The adeles of a
function field over a finite field can be defined in a similar way, as the restricted product of all completions.
Properties
The rational adeles
A are a
locally compact group with the rational numbers
Q contained as a discrete co-compact subgroup. The use of adele rings in connection with
Fourier transforms was exploited in
Tate's thesis. One key property of the
additive group of adeles is that it's isomorphic to its
Pontryagin dual.
Applications
The ring
A is much used in advanced parts of
number theory, often as the coefficients in
matrix groups: that is, combined with the theory of
algebraic groups to construct
adelic algebraic groups. The
idele group of
class field theory appears as the group of 1×1 invertible matrices over the adeles. (It is
not given the subset topology, as the inverse isn't continuous in this topology. Instead the ideles are identified with the closed subset of all pairs (
x,
y) of
A×
A
with
xy=1, with the subset topology.)
An important stage in the development of the theory was the definition of the
Tamagawa number for an adelic linear algebraic group. This is a volume measure relating
G(
Q) with
G(
A), saying how
G(
Q), which is a
discrete group in
G(
A), lies in the latter. A
conjecture of André Weil was that the Tamagawa number was always 1 for a
simply connected G. This arose out of Weil's modern treatment of results in the theory of
quadratic forms; the proof was finally completed by Kottwitz.
Meanwhile the influence of the Tamagawa number idea was felt in the theory of
abelian varieties. There the application by no means works, in any straightforward way. But during the formulation of the
Birch and Swinnerton-Dyer conjecture, the consideration that for an
elliptic curve E the group of rational points
E(
Q) might be brought into relation with the
E(
Qp) was one motivation and signpost, on the way from numerical evidence to the conjecture.
Further Information
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