It is the restricted product of all the completions of the global field and is an example of a self-dual topological ring.
Adele (French: "adèle") stands for 'additive idele' (that is, additive ideal element).
-bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group
[2] The ring of adeles solves the technical problem of "doing analysis on the rational numbers
[clarification needed] But, as was learned later on, there are many more absolute values other than the Euclidean distance, one for each prime number
This has the advantage of enabling analytic techniques while also retaining information about the primes, since their structure is embedded by the restricted infinite product.
The restricted infinite product is a required technical condition for giving the number field
With the power of a new theory of Fourier analysis, Tate was able to prove a special class of L-functions and the Dedekind zeta functions were meromorphic on the complex plane.
is called the idele class group The integral adeles are the subring The Artin reciprocity law says that for a global field
John Tate in his thesis "Fourier analysis in number fields and Hecke Zeta functions"[6] proved results about Dirichlet L-functions using Fourier analysis on the adele ring and the idele group.
is a smooth proper curve over the complex numbers, one can define the adeles of its function field
Infinite places of a global field form a finite set, which is denoted by
Thus: Or for short the difference between restricted and unrestricted product topology can be illustrated using a sequence in
is defined as This definition is based on the alternative description of the adele ring as a tensor product equipped with the same topology that was defined when giving an alternate definition of adele ring for number fields.
By composing both isomorphisms, a linear homeomorphism which transfers the two topologies into each other is obtained.
the definition above is consistent with the results about the adele ring of a finite extension
The implication is true, because the two terms of the strong triangle inequality are equal if the absolute values of both integers are different.
The fourth statement is a special case of the strong approximation theorem.
is a topological ring, it is sufficient to show that the inverse map is continuous.
Since the idele group is a locally compact, there exists a Haar measure
The trace and the norm should be transfer from the adele ring to the idele group.
This is possible, because Please, note the abuse of notation: On the left hand side in line 1 of this chain of equations,
As a consequence, the map above induces a surjective homomorphism To prove the second isomorphism, it has to be shown that
The strong approximation theorem tells us that, if one place (or more) is omitted, the property of discreteness of
For polynomials of degree larger than 2 the Hasse principle isn't valid in general.
Define: Then the following map is an isomorphism which respects topologies: With the help of the characters of
[31] John Tate in his thesis "Fourier analysis in Number Fields and Hecke Zeta Functions"[6] proved results about Dirichlet L-functions using Fourier analysis on the adele ring and the idele group.
As a result, the Riemann zeta function can be written as an integral over (a subset of) the adele ring.
To see this note: Based on these identification a natural generalisation would be to replace the idele group and the 1-idele with: And finally where
The product of the local reciprocity maps in local class field theory gives a homomorphism of the idele group to the Galois group of the maximal abelian extension of the global field.