Central object of class field theory
In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles[1]) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the global field and is an example of a self-dual topological ring.
An adele derives from a particular kind of idele. "Idele" derives from the French "idèle" and was coined by the French mathematician Claude Chevalley. The word stands for 'ideal element' (abbreviated: id.el.). Adele (French: "adèle") stands for 'additive idele' (that is, additive ideal element).
The ring of adeles allows one to describe the Artin reciprocity law, which is a generalisation of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that
-bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group
. Adeles are also connected with the adelic algebraic groups and adelic curves.
The study of geometry of numbers over the ring of adeles of a number field is called adelic geometry.
Definition
Let
be a global field (a finite extension of
or the function field of a curve
over a finite field). The adele ring of
is the subring
![{\displaystyle \mathbf {A} _{K}\ =\ \prod (K_{\nu },{\mathcal {O}}_{\nu })\ \subseteq \ \prod K_{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2518139360729a4dbd031675a375bbe97ba5c30b)
consisting of the tuples
where
lies in the subring
for all but finitely many places
. Here the index
ranges over all valuations of the global field
,
is the completion at that valuation and
the corresponding valuation ring.[2]
Motivation
The ring of adeles solves the technical problem of "doing analysis on the rational numbers
." The classical solution was to pass to the standard metric completion
and use analytic techniques there.[clarification needed] But, as was learned later on, there are many more absolute values other than the Euclidean distance, one for each prime number
, as was classified by Ostrowski. The Euclidean absolute value, denoted
, is only one among many others,
, but the ring of adeles makes it possible to comprehend and use all of the valuations at once. 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 purpose of the adele ring is to look at all completions of
at once. The adele ring is defined with the restricted product, rather than the Cartesian product. There are two reasons for this:
- For each element of
the valuations are zero for almost all places, i.e., for all places except a finite number. So, the global field can be embedded in the restricted product. - The restricted product is a locally compact space, while the Cartesian product is not. Therefore, there cannot be any application of harmonic analysis to the Cartesian product. This is because local compactness ensures the existence (and uniqueness) of Haar measure, a crucial tool in analysis on groups in general.
Why the restricted product?
The restricted infinite product is a required technical condition for giving the number field
a lattice structure inside of
, making it possible to build a theory of Fourier analysis (cf. Harmonic analysis) in the adelic setting. This is analogous to the situation in algebraic number theory where the ring of integers of an algebraic number field embeds
![{\displaystyle {\mathcal {O}}_{K}\hookrightarrow K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76ea91c1a523586ccb7c121d5e13dca31e56dff0)
as a lattice. 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. Another natural reason for why this technical condition holds can be seen by constructing the ring of adeles as a tensor product of rings. If defining the ring of integral adeles
as the ring
![{\displaystyle \mathbf {A} _{\mathbf {Z} }=\mathbf {R} \times {\hat {\mathbf {Z} }}=\mathbf {R} \times \prod _{p}\mathbf {Z} _{p},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8fac259a3a81e8953445b994e54ad54cfb7451d)
then the ring of adeles can be equivalently defined as
![{\displaystyle {\begin{aligned}\mathbf {A} _{\mathbf {Q} }&=\mathbf {Q} \otimes _{\mathbf {Z} }\mathbf {A} _{\mathbf {Z} }\\&=\mathbf {Q} \otimes _{\mathbf {Z} }\left(\mathbf {R} \times \prod _{p}\mathbf {Z} _{p}\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f6db2a683ce263d7e51742c2a3ea9a8102c3343)
The restricted product structure becomes transparent after looking at explicit elements in this ring. The image of an element
inside of the unrestricted product
is the element
![{\displaystyle \left({\frac {br}{c}},\left({\frac {ba_{p}}{c}}\right)\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef4492345f24f5ddd91f66a637914ed7940a0a0e)
The factor
lies in
whenever
is not a prime factor of
, which is the case for all but finitely many primes
.[3]
Origin of the name
The term "idele" (French: idèle) is an invention of the French mathematician Claude Chevalley (1909–1984) and stands for "ideal element" (abbreviated: id.el.). The term "adele" (French: adèle) stands for additive idele. Thus, an adele is an additive ideal element.
Examples
Ring of adeles for the rational numbers
The rationals
have a valuation for every prime number
, with
, and one infinite valuation ∞ with
. Thus an element of
![{\displaystyle \mathbf {A} _{\mathbf {Q} }\ =\ \mathbf {R} \times \prod _{p}(\mathbf {Q} _{p},\mathbf {Z} _{p})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c344ff5d6969183fb5b9adb2ded421a4eb5ae81)
is a real number along with a p-adic rational for each
of which all but finitely many are p-adic integers.
Ring of adeles for the function field of the projective line
Secondly, take the function field
of the projective line over a finite field. Its valuations correspond to points
of
, i.e. maps over
![{\displaystyle x\ :\ {\text{Spec}}\mathbf {F} _{q^{n}}\ \longrightarrow \ \mathbf {P} ^{1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc9e1fffb6cac62e86597c63e4d3d0b0f5be7c3)
For instance, there are
points of the form
. In this case
is the completed stalk of the structure sheaf at
(i.e. functions on a formal neighbourhood of
) and
is its fraction field. Thus
![{\displaystyle \mathbf {A} _{\mathbf {F} _{q}(\mathbf {P} ^{1})}\ =\ \prod _{x\in X}({\mathcal {K}}_{X,x},{\widehat {\mathcal {O}}}_{X,x}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8233a1a095b38a0a3d3349bef6f0e1ac0fb962d4)
The same holds for any smooth proper curve
over a finite field, the restricted product being over all points of
.
The group of units in the adele ring is called the idele group
.
The quotient of the ideles by the subgroup
is called the idele class group
![{\displaystyle C_{K}\ =\ I_{K}/K^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7d7d04bce09cf5b217237f855edf7574bb11dd)
The integral adeles are the subring
![{\displaystyle \mathbf {O} _{K}\ =\ \prod O_{\nu }\ \subseteq \ \mathbf {A} _{K}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee31a682f3e03aca50fa70cf8303632a31323450)
Applications
Stating Artin reciprocity
The Artin reciprocity law says that for a global field
,
![{\displaystyle {\widehat {C_{K}}}={\widehat {\mathbf {A} _{K}^{\times }/K^{\times }}}\ \simeq \ {\text{Gal}}(K^{\text{ab}}/K)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20b16aa5f7cbc1d6262e6f5882861827edf09775)
where
is the maximal abelian algebraic extension of
and
means the profinite completion of the group.
If
is a smooth proper curve then its Picard group is[4]
![{\displaystyle {\text{Pic}}(X)\ =\ K^{\times }\backslash \mathbf {A} _{X}^{\times }/\mathbf {O} _{X}^{\times }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e195a59357b57d07ed1a6d90d4d6facf6a60f96a)
and its divisor group is
. Similarly, if
is a semisimple algebraic group (e.g.
, it also holds for
) then Weil uniformisation says that[5]
![{\displaystyle {\text{Bun}}_{G}(X)\ =\ G(K)\backslash G(\mathbf {A} _{X})/G(\mathbf {O} _{X}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bcc5e16c610ec352d7d37d7642ee11ed14590a5)
Applying this to
gives the result on the Picard group.
Tate's thesis
There is a topology on
for which the quotient
is compact, allowing one to do harmonic analysis on it. John Tate in his thesis "Fourier analysis in number fields and Hecke Zeta functions" proved results about Dirichlet L-functions using Fourier analysis on the adele ring and the idele group. Therefore, the adele ring and the idele group have been applied to study the Riemann zeta function and more general zeta functions and the L-functions.
Proving Serre duality on a smooth curve
If
is a smooth proper curve over the complex numbers, one can define the adeles of its function field
exactly as the finite fields case. John Tate proved[7] that Serre duality on
![{\displaystyle H^{1}(X,{\mathcal {L}})\ \simeq \ H^{0}(X,\Omega _{X}\otimes {\mathcal {L}}^{-1})^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d4b46f7e66362006126b9eb8039154da36442ee)
can be deduced by working with this adele ring
. Here L is a line bundle on
.
Notation and basic definitions
Global fields
Throughout this article,
is a global field, meaning it is either a number field (a finite extension of
) or a global function field (a finite extension of
for
prime and
). By definition a finite extension of a global field is itself a global field.
Valuations
For a valuation
of
it can be written
for the completion of
with respect to
If
is discrete it can be written
for the valuation ring of
and
for the maximal ideal of
If this is a principal ideal denoting the uniformising element by
A non-Archimedean valuation is written as
or
and an Archimedean valuation as
Then assume all valuations to be non-trivial.
There is a one-to-one identification of valuations and absolute values. Fix a constant
the valuation
is assigned the absolute value
defined as:
![{\displaystyle \forall x\in K:\quad |x|_{v}:={\begin{cases}C^{-v(x)}&x\neq 0\\0&x=0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/716ae66ccb980acb337317715bb89c4cf52dd543)
Conversely, the absolute value
is assigned the valuation
defined as:
![{\displaystyle \forall x\in K^{\times }:\quad v_{|\cdot |}(x):=-\log _{C}(|x|).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e80f3cf56dcfda62f03136fa191166e2f7dda6c)
A place of
is a representative of an equivalence class of valuations (or absolute values) of
Places corresponding to non-Archimedean valuations are called finite, whereas places corresponding to Archimedean valuations are called infinite. Infinite places of a global field form a finite set, which is denoted by
Define
and let
be its group of units. Then
Finite extensions
Let
be a finite extension of the global field
Let
be a place of
and
a place of
If the absolute value
restricted to
is in the equivalence class of
, then
lies above
which is denoted by
and defined as:
![{\displaystyle {\begin{aligned}L_{v}&:=\prod _{w|v}L_{w},\\{\widetilde {O_{v}}}&:=\prod _{w|v}O_{w}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a64f4d3d64003cea5b5ebf8e176094769d8291b5)
(Note that both products are finite.)
If
,
can be embedded in
Therefore,
is embedded diagonally in
With this embedding
is a commutative algebra over
with degree
![{\displaystyle \sum _{w|v}[L_{w}:K_{v}]=[L:K].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa310bdb2e63a00ef6c4eaad987d49881ed5192d)
The adele ring
The set of finite adeles of a global field
denoted
is defined as the restricted product of
with respect to the
![{\displaystyle \mathbb {A} _{K,{\text{fin}}}:={\prod _{v<\infty }}^{'}K_{v}=\left\{\left.(x_{v})_{v}\in \prod _{v<\infty }K_{v}\right|x_{v}\in O_{v}{\text{ for almost all }}v\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74c91cb82edc7227c515242898778847af3de763)
It is equipped with the restricted product topology, the topology generated by restricted open rectangles, which have the following form:
![{\displaystyle U=\prod _{v\in E}U_{v}\times \prod _{v\notin E}O_{v}\subset {\prod _{v<\infty }}^{'}K_{v},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4edf3ff1d31b859a87c149154aeabe24eb0601)
where
is a finite set of (finite) places and
are open. With component-wise addition and multiplication
is also a ring.
The adele ring of a global field
is defined as the product of
with the product of the completions of
at its infinite places. The number of infinite places is finite and the completions are either
or
In short:
![{\displaystyle \mathbb {A} _{K}:=\mathbb {A} _{K,{\text{fin}}}\times \prod _{v|\infty }K_{v}={\prod _{v<\infty }}^{'}K_{v}\times \prod _{v|\infty }K_{v}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b33f060661f5bce7e018e798d4b3cd932b36736)
With addition and multiplication defined as component-wise the adele ring is a ring. The elements of the adele ring are called adeles of
In the following, it is written as
![{\displaystyle \mathbb {A} _{K}={\prod _{v}}^{'}K_{v},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819b4d7c7b8ac41c7a6c017e9d5d7aaf82550121)
although this is generally not a restricted product.
Remark. Global function fields do not have any infinite places and therefore the finite adele ring equals the adele ring.
- Lemma. There is a natural embedding of
into
given by the diagonal map: ![{\displaystyle a\mapsto (a,a,\ldots ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2735b11cedf42785607232182f87f7b013eb8a2)
Proof. If
then
for almost all
This shows the map is well-defined. It is also injective because the embedding of
in
is injective for all
Remark. By identifying
with its image under the diagonal map it is regarded as a subring of
The elements of
are called the principal adeles of
Definition. Let
be a set of places of
Define the set of the
-adeles of
as
![{\displaystyle \mathbb {A} _{K,S}:={\prod _{v\in S}}^{'}K_{v}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f265e41dbe0411633ad0cb41fedbca8710124281)
Furthermore, if
![{\displaystyle \mathbb {A} _{K}^{S}:={\prod _{v\notin S}}^{'}K_{v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d599dd6a4239e8e11d0168f48e736143fcf9d28c)
the result is:
The adele ring of rationals
By Ostrowski's theorem the places of
are
it is possible to identify a prime
with the equivalence class of the
-adic absolute value and
with the equivalence class of the absolute value
defined as:
![{\displaystyle \forall x\in \mathbb {Q} :\quad |x|_{\infty }:={\begin{cases}x&x\geq 0\\-x&x<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf029834c02f544eee21deb196ca106de79fa766)
The completion of
with respect to the place
is
with valuation ring
For the place
the completion is
Thus:
![{\displaystyle {\begin{aligned}\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}&={\prod _{p<\infty }}^{'}\mathbb {Q} _{p}\\\mathbb {A} _{\mathbb {Q} }&=\left({\prod _{p<\infty }}^{'}\mathbb {Q} _{p}\right)\times \mathbb {R} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab5d6b07ec636200ad95304660e6e1be01121174)
Or for short
![{\displaystyle \mathbb {A} _{\mathbb {Q} }={\prod _{p\leq \infty }}^{'}\mathbb {Q} _{p},\qquad \mathbb {Q} _{\infty }:=\mathbb {R} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb477da0257fe05f74adf94d6df74e63bf406c45)
the difference between restricted and unrestricted product topology can be illustrated using a sequence in
:
- Lemma. Consider the following sequence in
: ![{\displaystyle {\begin{aligned}x_{1}&=\left({\frac {1}{2}},1,1,\ldots \right)\\x_{2}&=\left(1,{\frac {1}{3}},1,\ldots \right)\\x_{3}&=\left(1,1,{\frac {1}{5}},1,\ldots \right)\\x_{4}&=\left(1,1,1,{\frac {1}{7}},1,\ldots \right)\\&\vdots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5ee32d02bab59b66411d31053717c8b1f33b6b)
- In the product topology this converges to
, but it does not converge at all in the restricted product topology.
Proof. In product topology convergence corresponds to the convergence in each coordinate, which is trivial because the sequences become stationary. The sequence doesn't converge in restricted product topology. For each adele
and for each restricted open rectangle
it has:
for
and therefore
for all
As a result
for almost all
In this consideration,
and
are finite subsets of the set of all places.
Alternative definition for number fields
Definition (profinite integers). The profinite integers are defined as the profinite completion of the rings
with the partial order
i.e.,
![{\displaystyle {\widehat {\mathbb {Z} }}:=\varprojlim _{n}\mathbb {Z} /n\mathbb {Z} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8043c70adb1de92d6b4b95d719235e1f7b9ad6)
- Lemma.
![{\displaystyle \textstyle {\widehat {\mathbb {Z} }}\cong \prod _{p}\mathbb {Z} _{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f6406b950c8adf3bbc769fd26e8e49d66e31a7e)
Proof. This follows from the Chinese Remainder Theorem.
- Lemma.
![{\displaystyle \mathbb {A} _{\mathbb {Q} ,{\text{fin}}}={\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01ea48368de5d74cfde7ae585a8046cdc908ea33)
Proof. Use the universal property of the tensor product. Define a
-bilinear function
![{\displaystyle {\begin{cases}\Psi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \to \mathbb {A} _{\mathbb {Q} ,{\text{fin}}}\\\left((a_{p})_{p},q\right)\mapsto (a_{p}q)_{p}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e659948f962d675b560b785d9b9d22fb4bc1d40c)
This is well-defined because for a given
with
co-prime there are only finitely many primes dividing
Let
be another
-module with a
-bilinear map
It must be the case that
factors through
uniquely, i.e., there exists a unique
-linear map
such that
can be defined as follows: for a given
there exist
and
such that
for all
Define
One can show
is well-defined,
-linear, satisfies
and is unique with these properties.
- Corollary. Define
This results in an algebraic isomorphism ![{\displaystyle \mathbb {A} _{\mathbb {Q} }\cong \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/647f505973355ae79e9cbf2e442ef54a0c553d4e)
Proof.
- Lemma. For a number field
![{\displaystyle K,\mathbb {A} _{K}=\mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a82b2ff0d2ff119d8c66a13004d762edebcf520)
Remark. Using
where there are
summands, give the right side receives the product topology and transport this topology via the isomorphism onto
The adele ring of a finite extension
If
be a finite extension, then
is a global field. Thus
is defined, and
can be identified with a subgroup of
Map
to
where
for
Then
is in the subgroup
if
for
and
for all
lying above the same place
of
- Lemma. If
is a finite extension, then
both algebraically and topologically.
With the help of this isomorphism, the inclusion
is given by
![{\displaystyle {\begin{cases}\mathbb {A} _{K}\to \mathbb {A} _{L}\\\alpha \mapsto \alpha \otimes _{K}1\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08b8f70a6b9e55672e4a052efab9d9de8f27c839)
Furthermore, the principal adeles in
can be identified with a subgroup of principal adeles in
via the map
![{\displaystyle {\begin{cases}K\to (K\otimes _{K}L)\cong L\\\alpha \mapsto 1\otimes _{K}\alpha \end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34259e63005b25fb77e95aa9f8538aff613a744a)
Proof.[8] Let
be a basis of
over
Then for almost all
![{\displaystyle {\widetilde {O_{v}}}\cong O_{v}\omega _{1}\oplus \cdots \oplus O_{v}\omega _{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc07854736b23b318a23ebcb5c65a21d7ee4182)
Furthermore, there are the following isomorphisms:
![{\displaystyle K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n}\cong K_{v}\otimes _{K}L\cong L_{v}=\prod \nolimits _{w|v}L_{w}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/364870f9f795fdb378d3a67e28be1f6c81f9f4d3)
For the second use the map:
![{\displaystyle {\begin{cases}K_{v}\otimes _{K}L\to L_{v}\\\alpha _{v}\otimes a\mapsto (\alpha _{v}\cdot (\tau _{w}(a)))_{w}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d850833863b1f0e548b03759fcb6a01463464e2)
in which
is the canonical embedding and
The restricted product is taken on both sides with respect to
![{\displaystyle {\begin{aligned}\mathbb {A} _{K}\otimes _{K}L&=\left({\prod _{v}}^{'}K_{v}\right)\otimes _{K}L\\&\cong {\prod _{v}}^{'}(K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n})\\&\cong {\prod _{v}}^{'}(K_{v}\otimes _{K}L)\\&\cong {\prod _{v}}^{'}L_{v}\\&=\mathbb {A} _{L}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/634967ce164a96f3ddad9a015b2131413b03ab03)
- Corollary. As additive groups
where the right side has
summands.
The set of principal adeles in
is identified with the set
where the left side has
summands and
is considered as a subset of
The adele ring of vector-spaces and algebras
- Lemma. Suppose
is a finite set of places of
and define ![{\displaystyle \mathbb {A} _{K}(P):=\prod _{v\in P}K_{v}\times \prod _{v\notin P}O_{v}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49691af7ca71617dcb45c409386105fe49e75f5e)
- Equip
with the product topology and define addition and multiplication component-wise. Then
is a locally compact topological ring.
Remark. If
is another finite set of places of
containing
then
is an open subring of
Now, an alternative characterisation of the adele ring can be presented. The adele ring is the union of all sets
:
![{\displaystyle \mathbb {A} _{K}=\bigcup _{P\supset P_{\infty },|P|<\infty }\mathbb {A} _{K}(P).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28ada53165f1c55bf917e486c19e71c3a38a5942)
Equivalently
is the set of all
so that
for almost all
The topology of
is induced by the requirement that all
be open subrings of
Thus,
is a locally compact topological ring.
Fix a place
of
Let
be a finite set of places of
containing
and
Define
![{\displaystyle \mathbb {A} _{K}'(P,v):=\prod _{w\in P\setminus \{v\}}K_{w}\times \prod _{w\notin P}O_{w}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea64fd9c37aab2a4c00a7c1eaec23d6f21a198a4)
Then:
![{\displaystyle \mathbb {A} _{K}(P)\cong K_{v}\times \mathbb {A} _{K}'(P,v).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/357dabd62bcd65385403aeb1b75f06a375851164)
Furthermore, define
![{\displaystyle \mathbb {A} _{K}'(v):=\bigcup _{P\supset P_{\infty }\cup \{v\}}\mathbb {A} _{K}'(P,v),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fcfada0ee222ba5a38d29bbdfe20768f5263108)
where
runs through all finite sets containing
Then:
![{\displaystyle \mathbb {A} _{K}\cong K_{v}\times \mathbb {A} _{K}'(v),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57205af77429c05de343ee7facdcaae75341488e)
via the map
The entire procedure above holds with a finite subset
instead of
By construction of
there is a natural embedding:
Furthermore, there exists a natural projection
The adele ring of a vector-space
Let
be a finite dimensional vector-space over
and
a basis for
over
For each place
of
:
![{\displaystyle {\begin{aligned}E_{v}&:=E\otimes _{K}K_{v}\cong K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n}\\{\widetilde {O_{v}}}&:=O_{v}\omega _{1}\oplus \cdots \oplus O_{v}\omega _{n}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/066fe15fb5ff66b7beddb894bf845f29caba38d9)
The adele ring of
is defined as
![{\displaystyle \mathbb {A} _{E}:={\prod _{v}}^{'}E_{v}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdfe6a37957c59ea4bde5519d5e986b79c0e8fe4)
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. Next,
is equipped with the restricted product topology. Then
and
is embedded in
naturally via the map
An alternative definition of the topology on
can be provided. Consider all linear maps:
Using the natural embeddings
and
extend these linear maps to:
The topology on
is the coarsest topology for which all these extensions are continuous.
The topology can be defined in a different way. Fixing a basis for
over
results in an isomorphism
Therefore fixing a basis induces an isomorphism
The left-hand side is supplied with the product topology and transport this topology with the isomorphism onto the right-hand side. The topology doesn't depend on the choice of the basis, because another basis defines a second isomorphism. By composing both isomorphisms, a linear homeomorphism which transfers the two topologies into each other is obtained. More formally
![{\displaystyle {\begin{aligned}\mathbb {A} _{E}&=E\otimes _{K}\mathbb {A} _{K}\\&\cong (K\otimes _{K}\mathbb {A} _{K})\oplus \cdots \oplus (K\otimes _{K}\mathbb {A} _{K})\\&\cong \mathbb {A} _{K}\oplus \cdots \oplus \mathbb {A} _{K}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77d39800819a0fa64a377ff1f13b030e12ce272a)
where the sums have
summands. In case of
the definition above is consistent with the results about the adele ring of a finite extension
[9]
The adele ring of an algebra
Let
be a finite-dimensional algebra over
In particular,
is a finite-dimensional vector-space over
As a consequence,
is defined and
Since there is multiplication on
and
a multiplication on
can be defined via:
![{\displaystyle \forall \alpha ,\beta \in \mathbb {A} _{K}{\text{ and }}\forall a,b\in A:\qquad (\alpha \otimes _{K}a)\cdot (\beta \otimes _{K}b):=(\alpha \beta )\otimes _{K}(ab).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14af1aef8fb6fbafa4734637806af833d656d2d9)
As a consequence,
is an algebra with a unit over
Let
be a finite subset of
containing a basis for
over
For any finite place
,
is defined as the
-module generated by
in
For each finite set of places,
define
![{\displaystyle \mathbb {A} _{A}(P,\alpha )=\prod _{v\in P}A_{v}\times \prod _{v\notin P}M_{v}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57e3d6bf4bf8bce64a2016064794d36fd385248f)
One can show there is a finite set
so that
is an open subring of
if
Furthermore
is the union of all these subrings and for
the definition above is consistent with the definition of the adele ring.
Trace and norm on the adele ring
Let
be a finite extension. Since
and
from the Lemma above,
can be interpreted as a closed subring of
For this embedding, write
. Explicitly for all places
of
above
and for any
Let
be a tower of global fields. Then:
![{\displaystyle \operatorname {con} _{M/K}(\alpha )=\operatorname {con} _{M/L}(\operatorname {con} _{L/K}(\alpha ))\qquad \forall \alpha \in \mathbb {A} _{K}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ea0ebf87767e88762c288897a5bc12511ff194)
Furthermore, restricted to the principal adeles
is the natural injection
Let
be a basis of the field extension
Then each
can be written as
where
are unique. The map
is continuous. Define
depending on
via the equations:
![{\displaystyle {\begin{aligned}\alpha \omega _{1}&=\sum _{j=1}^{n}\alpha _{1j}\omega _{j}\\&\vdots \\\alpha \omega _{n}&=\sum _{j=1}^{n}\alpha _{nj}\omega _{j}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5fe3037577807328fe061622d9fb000bcf80b5c)
Now, define the trace and norm of
as:
![{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&:=\operatorname {Tr} ((\alpha _{ij})_{i,j})=\sum _{i=1}^{n}\alpha _{ii}\\N_{L/K}(\alpha )&:=N((\alpha _{ij})_{i,j})=\det((\alpha _{ij})_{i,j})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db5ab60cdff212a644b2b01ce33070e1462b9fed)
These are the trace and the determinant of the linear map
![{\displaystyle {\begin{cases}\mathbb {A} _{L}\to \mathbb {A} _{L}\\x\mapsto \alpha x\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2337dbbdd069a289c5a3ff09b9a7e7a282b3e1d8)
They are continuous maps on the adele ring, and they fulfil the usual equations:
![{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha +\beta )&=\operatorname {Tr} _{L/K}(\alpha )+\operatorname {Tr} _{L/K}(\beta )&&\forall \alpha ,\beta \in \mathbb {A} _{L}\\\operatorname {Tr} _{L/K}(\operatorname {con} (\alpha ))&=n\alpha &&\forall \alpha \in \mathbb {A} _{K}\\N_{L/K}(\alpha \beta )&=N_{L/K}(\alpha )N_{L/K}(\beta )&&\forall \alpha ,\beta \in \mathbb {A} _{L}\\N_{L/K}(\operatorname {con} (\alpha ))&=\alpha ^{n}&&\forall \alpha \in \mathbb {A} _{K}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ba018845ce5f0e03c00271f4b9917baca826f9)
Furthermore, for ![{\displaystyle \alpha \in L,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6cbdc869fdd0b1030e559b051d0d4240f23a0e0)
and
are identical to the trace and norm of the field extension
For a tower of fields
the result is:
![{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\operatorname {Tr} _{M/L}(\alpha ))&=\operatorname {Tr} _{M/K}(\alpha )&&\forall \alpha \in \mathbb {A} _{M}\\N_{L/K}(N_{M/L}(\alpha ))&=N_{M/K}(\alpha )&&\forall \alpha \in \mathbb {A} _{M}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1a7f24cde7e61e3c86556118ca6b00aaf3f024)
Moreover, it can be proven that:[10]
![{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&=\left(\sum _{w|v}\operatorname {Tr} _{L_{w}/K_{v}}(\alpha _{w})\right)_{v}&&\forall \alpha \in \mathbb {A} _{L}\\N_{L/K}(\alpha )&=\left(\prod _{w|v}N_{L_{w}/K_{v}}(\alpha _{w})\right)_{v}&&\forall \alpha \in \mathbb {A} _{L}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e78336215ffa61fb33d23dd738d417bc87d4d943)
Properties of the adele ring
- Theorem.[11] For every set of places
is a locally compact topological ring.
Remark. The result above also holds for the adele ring of vector-spaces and algebras over
- Theorem.[12]
is discrete and cocompact in
In particular,
is closed in ![{\displaystyle \mathbb {A} _{K}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b21c53be4f5448918d406551bf1500d3836e5c45)
Proof. Prove the case
To show
is discrete it is sufficient to show the existence of a neighbourhood of
which contains no other rational number. The general case follows via translation. Define
![{\displaystyle U:=\left\{(\alpha _{p})_{p}\left|\forall p<\infty :|\alpha _{p}|_{p}\leq 1\quad {\text{and}}\quad |\alpha _{\infty }|_{\infty }<1\right.\right\}={\widehat {\mathbb {Z} }}\times (-1,1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd0b9c50dd3e04d76cc0451db7d5d7713fd377c6)
is an open neighbourhood of
It is claimed that
Let
then
and
for all
and therefore
Additionally,
and therefore
Next, to show compactness, define:
![{\displaystyle W:=\left\{(\alpha _{p})_{p}\left|\forall p<\infty :|\alpha _{p}|_{p}\leq 1\quad {\text{and}}\quad |\alpha _{\infty }|_{\infty }\leq {\frac {1}{2}}\right.\right\}={\widehat {\mathbb {Z} }}\times \left[-{\frac {1}{2}},{\frac {1}{2}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dfc0a75eff4cf80cbda2b1400817c3495e16c80)
Each element in
has a representative in
that is for each
there exists
such that
Let
be arbitrary and
be a prime for which
Then there exists
with
and
Replace
with
and let
be another prime. Then:
![{\displaystyle \left|\alpha _{q}-r_{p}\right|_{q}\leq \max \left\{|a_{q}|_{q},|r_{p}|_{q}\right\}\leq \max \left\{|a_{q}|_{q},1\right\}\leq 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6841002114fa74d22b23185447b33f11ade5e7f)
Next, it can be claimed that:
![{\displaystyle |\alpha _{q}-r_{p}|_{q}\leq 1\Longleftrightarrow |\alpha _{q}|_{q}\leq 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63d8779f6ad1ab941ee3e04a865c9c912cea24c0)
The reverse implication is trivially true. The implication is true, because the two terms of the strong triangle inequality are equal if the absolute values of both integers are different. As a consequence, the (finite) set of primes for which the components of
are not in
is reduced by 1. With iteration, it can be deduced that there exists
such that
Now select
such that
Then
The continuous projection
is surjective, therefore
as the continuous image of a compact set, is compact.
- Corollary. Let
be a finite-dimensional vector-space over
Then
is discrete and cocompact in ![{\displaystyle \mathbb {A} _{E}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ccdb1ba27215e0ae8d8fc61df305b7d4b90073f)
- Theorem. The following are assumed:
![{\displaystyle \mathbb {A} _{\mathbb {Q} }=\mathbb {Q} +\mathbb {A} _{\mathbb {Z} }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70ce8b421c9ea5c3e6afc2308d138ab5122b75a6)
![{\displaystyle \mathbb {Z} =\mathbb {Q} \cap \mathbb {A} _{\mathbb {Z} }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb27179f557b1acd76415279380c36ffd579466)
is a divisible group.[13]
is dense.
Proof. The first two equations can be proved in an elementary way.
By definition
is divisible if for any
and
the equation
has a solution
It is sufficient to show
is divisible but this is true since
is a field with positive characteristic in each coordinate.
For the last statement note that
because the finite number of denominators in the coordinates of the elements of
can be reached through an element
As a consequence, it is sufficient to show
is dense, that is each open subset
contains an element of
Without loss of generality, it can be assumed that
![{\displaystyle V=\prod _{p\in E}\left(a_{p}+p^{l_{p}}\mathbb {Z} _{p}\right)\times \prod _{p\notin E}\mathbb {Z} _{p},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/183ead5d2668e6fe9d176afb1cacbf344f20b37d)
because
is a neighbourhood system of
in
By Chinese Remainder Theorem there exists
such that
Since powers of distinct primes are coprime,
follows.
Remark.
is not uniquely divisible. Let
and
be given. Then
![{\displaystyle {\begin{aligned}x_{1}&=(0,0,\ldots )+\mathbb {Z} \\x_{2}&=\left({\tfrac {1}{n}},{\tfrac {1}{n}},\ldots \right)+\mathbb {Z} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f63ff342264dfa52f9edecb0f4db2599a0b1b033)
both satisfy the equation
and clearly
(
is well-defined, because only finitely many primes divide
). In this case, being uniquely divisible is equivalent to being torsion-free, which is not true for
since
but
and
Remark. The fourth statement is a special case of the strong approximation theorem.
Haar measure on the adele ring
Definition. A function
is called simple if
where
are measurable and
for almost all
- Theorem.[14] Since
is a locally compact group with addition, there is an additive Haar measure
on
This measure can be normalised such that every integrable simple function
satisfies: ![{\displaystyle \int _{\mathbb {A} _{K}}f\,dx=\prod _{v}\int _{K_{v}}f_{v}\,dx_{v},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0065bac138c0a162f2fa83b5da375de3335660f)
- where for
is the measure on
such that
has unit measure and
is the Lebesgue measure. The product is finite, i.e., almost all factors are equal to one.
The idele group
Definition. Define the idele group of
as the group of units of the adele ring of
that is
The elements of the idele group are called the ideles of
Remark.
is equipped with a topology so that it becomes a topological group. The subset topology inherited from
is not a suitable candidate since the group of units of a topological ring equipped with subset topology may not be a topological group. For example, the inverse map in
is not continuous. The sequence
![{\displaystyle {\begin{aligned}x_{1}&=(2,1,\ldots )\\x_{2}&=(1,3,1,\ldots )\\x_{3}&=(1,1,5,1,\ldots )\\&\vdots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49e1b265439f6e335df7d7370c7a236e7883d2fe)
converges to
To see this let
be neighbourhood of
without loss of generality it can be assumed:
![{\displaystyle U=\prod _{p\leq N}U_{p}\times \prod _{p>N}\mathbb {Z} _{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1446a3620ec9583664fd7964f53f659e46e3d181)
Since
for all
for
large enough. However, as was seen above the inverse of this sequence does not converge in
- Lemma. Let
be a topological ring. Define: ![{\displaystyle {\begin{cases}\iota :R^{\times }\to R\times R\\x\mapsto (x,x^{-1})\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6417371f6da5b64a5a57362029ba42aa892d7cf8)
- Equipped with the topology induced from the product on topology on
and
is a topological group and the inclusion map
is continuous. It is the coarsest topology, emerging from the topology on
that makes
a topological group.
Proof. Since
is a topological ring, it is sufficient to show that the inverse map is continuous. Let
be open, then
is open. It is necessary to show
is open or equivalently, that
is open. But this is the condition above.
The idele group is equipped with the topology defined in the Lemma making it a topological group.
Definition. For
a subset of places of
set:
- Lemma. The following identities of topological groups hold:
![{\displaystyle {\begin{aligned}I_{K,S}&={\prod _{v\in S}}^{'}K_{v}^{\times }\\I_{K}^{S}&={\prod _{v\notin S}}^{'}K_{v}^{\times }\\I_{K}&={\prod _{v}}^{'}K_{v}^{\times }\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49313e17729c05f58ff51dbc618982ec290f4935)
- where the restricted product has the restricted product topology, which is generated by restricted open rectangles of the form
![{\displaystyle \prod _{v\in E}U_{v}\times \prod _{v\notin E}O_{v}^{\times },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da18027578ee59141a047a4a5bf7655594c57280)
- where
is a finite subset of the set of all places and
are open sets.
Proof. Prove the identity for
; the other two follow similarly. First show the two sets are equal:
![{\displaystyle {\begin{aligned}I_{K}&=\{x=(x_{v})_{v}\in \mathbb {A} _{K}:\exists y=(y_{v})_{v}\in \mathbb {A} _{K}:xy=1\}\\&=\{x=(x_{v})_{v}\in \mathbb {A} _{K}:\exists y=(y_{v})_{v}\in \mathbb {A} _{K}:x_{v}\cdot y_{v}=1\quad \forall v\}\\&=\{x=(x_{v})_{v}:x_{v}\in K_{v}^{\times }\forall v{\text{ and }}x_{v}\in O_{v}^{\times }{\text{ for almost all }}v\}\\&={\prod _{v}}^{'}K_{v}^{\times }\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7978834a0fa8c80c7b3229796cd9eb751f00dcdc)
In going from line 2 to 3,
as well as
have to be in
meaning
for almost all
and
for almost all
Therefore,
for almost all
Now, it is possible to show the topology on the left-hand side equals the topology on the right-hand side. Obviously, every open restricted rectangle is open in the topology of the idele group. On the other hand, for a given
which is open in the topology of the idele group, meaning
is open, so for each
there exists an open restricted rectangle, which is a subset of
and contains
Therefore,
is the union of all these restricted open rectangles and therefore is open in the restricted product topology.
- Lemma. For each set of places,
is a locally compact topological group.
Proof. The local compactness follows from the description of
as a restricted product. It being a topological group follows from the above discussion on the group of units of a topological ring.
A neighbourhood system of
is a neighbourhood system of
Alternatively, take all sets of the form:
![{\displaystyle \prod _{v}U_{v},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5134bb4968b9a89c07327e2925347028460ae246)
where
is a neighbourhood of
and
for almost all
Since the idele group is a locally compact, there exists a Haar measure
on it. This can be normalised, so that
![{\displaystyle \int _{I_{K,{\text{fin}}}}\mathbf {1} _{\widehat {O}}\,d^{\times }x=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/369efcad1e906d690944a632b961d75406026c5f)
This is the normalisation used for the finite places. In this equation,
is the finite idele group, meaning the group of units of the finite adele ring. For the infinite places, use the multiplicative lebesgue measure
The idele group of a finite extension
- Lemma. Let
be a finite extension. Then: ![{\displaystyle I_{L}={\prod _{v}}^{'}L_{v}^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16935daed736e5f81439039aa2f758a51bcdb756)
- where the restricted product is with respect to
![{\displaystyle {\widetilde {O_{v}}}^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14ff721c9541e3ecb225142396ee04b43d0545aa)
- Lemma. There is a canonical embedding of
in ![{\displaystyle I_{L}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b035450ba6af5f0fa5678fb8e48f9a3cb986d41b)
Proof. Map
to
with the property
for
Therefore,
can be seen as a subgroup of
An element
is in this subgroup if and only if his components satisfy the following properties:
for
and
for
and
for the same place
of
The case of vector spaces and algebras
[15]
The idele group of an algebra
Let
be a finite-dimensional algebra over
Since
is not a topological group with the subset-topology in general, equip
with the topology similar to
above and call
the idele group. The elements of the idele group are called idele of
- Proposition. Let
be a finite subset of
containing a basis of
over
For each finite place
of
let
be the
-module generated by
in
There exists a finite set of places
containing
such that for all ![{\displaystyle v\notin P_{0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a54802be7ba420ceb16c8d647c07f9d04113e59)
is a compact subring of
Furthermore,
contains
For each
is an open subset of
and the map
is continuous on
As a consequence
maps
homeomorphically on its image in
For each
the
are the elements of
mapping in
with the function above. Therefore,
is an open and compact subgroup of
[16]
Alternative characterisation of the idele group
- Proposition. Let
be a finite set of places. Then ![{\displaystyle \mathbb {A} _{A}(P,\alpha )^{\times }:=\prod _{v\in P}A_{v}^{\times }\times \prod _{v\notin P}\alpha _{v}^{\times }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5453c7012c0dda545b5dfe12e8efa6246ddc48f)
- is an open subgroup of
where
is the union of all
[17]
- Corollary. In the special case of
for each finite set of places
![{\displaystyle \mathbb {A} _{K}(P)^{\times }=\prod _{v\in P}K_{v}^{\times }\times \prod _{v\notin P}O_{v}^{\times }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b382ec51a14fe73265362a51f97854072e4869d)
- is an open subgroup of
Furthermore,
is the union of all ![{\displaystyle \mathbb {A} _{K}(P)^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1dce05028522164e491b2333f169105211e2c7)
Norm on the idele group
The trace and the norm should be transfer from the adele ring to the idele group. It turns out the trace can't be transferred so easily. However, it is possible to transfer the norm from the adele ring to the idele group. Let
Then
and therefore, it can be said that in injective group homomorphism
![{\displaystyle \operatorname {con} _{L/K}:I_{K}\hookrightarrow I_{L}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/098f4ce838ac0b772f024fa60c34606ae455aabf)
Since
it is invertible,
is invertible too, because
Therefore
As a consequence, the restriction of the norm-function introduces a continuous function:
![{\displaystyle N_{L/K}:I_{L}\to I_{K}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc2780eaf9f61ee1265657f8b47e07d5f7727b7)
The Idele class group
- Lemma. There is natural embedding of
into
given by the diagonal map: ![{\displaystyle a\mapsto (a,a,a,\ldots ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11b163448f8115fea9a3bab9e07a76fefbb85405)
Proof. Since
is a subset of
for all
the embedding is well-defined and injective.
- Corollary.
is a discrete subgroup of ![{\displaystyle \mathbb {A} _{A}^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f74faecfb759ba1181eb4ad05374be21c181518c)
Defenition. In analogy to the ideal class group, the elements of
in
are called principal ideles of
The quotient group
is called idele class group of
This group is related to the ideal class group and is a central object in class field theory.
Remark.
is closed in
therefore
is a locally compact topological group and a Hausdorff space.
- Lemma.[18] Let
be a finite extension. The embedding
induces an injective map: ![{\displaystyle {\begin{cases}C_{K}\to C_{L}\\\alpha K^{\times }\mapsto \alpha L^{\times }\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6298980ca1bc4ae7afddf7f0eaf746ca430a114)
Properties of the idele group
Absolute value on the idele group of K and 1-idele
Definition. For
define:
Since
is an idele this product is finite and therefore well-defined.
Remark. The definition can be extended to
by allowing infinite products. However, these infinite products vanish and so
vanishes on
will be used to denote both the function on
and
- Theorem.
is a continuous group homomorphism.
Proof. Let
![{\displaystyle {\begin{aligned}|\alpha \cdot \beta |&=\prod _{v}|(\alpha \cdot \beta )_{v}|_{v}\\&=\prod _{v}|\alpha _{v}\cdot \beta _{v}|_{v}\\&=\prod _{v}(|\alpha _{v}|_{v}\cdot |\beta _{v}|_{v})\\&=\left(\prod _{v}|\alpha _{v}|_{v}\right)\cdot \left(\prod _{v}|\beta _{v}|_{v}\right)\\&=|\alpha |\cdot |\beta |\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6de4d8a0dde96271844df5c24e8e0847c37f03)
where it is used that all products are finite. The map is continuous which can be seen using an argument dealing with sequences. This reduces the problem to whether
is continuous on
However, this is clear, because of the reverse triangle inequality.
Definition. The set of
-idele can be defined as:
![{\displaystyle I_{K}^{1}:=\{x\in I_{K}:|x|=1\}=\ker(|\cdot |).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb771158cfed7ae622139d3eee206cede17007e)
is a subgroup of
Since
it is a closed subset of
Finally the
-topology on
equals the subset-topology of
on
[19][20]
- Artin's Product Formula.
for all ![{\displaystyle k\in K^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c444f1170b65ab2b097a978927c4bb40b16a575d)
Proof.[21] Proof of the formula for number fields, the case of global function fields can be proved similarly. Let
be a number field and
It has to be shown that:
![{\displaystyle \prod _{v}|a|_{v}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2841bb75febfc054d6574a27dc5cb64a6094d879)
For finite place
for which the corresponding prime ideal
does not divide
,
and therefore
This is valid for almost all
There is:
![{\displaystyle {\begin{aligned}\prod _{v}|a|_{v}&=\prod _{p\leq \infty }\prod _{v|p}|a|_{v}\\&=\prod _{p\leq \infty }\prod _{v|p}|N_{K_{v}/\mathbb {Q} _{p}}(a)|_{p}\\&=\prod _{p\leq \infty }|N_{K/\mathbb {Q} }(a)|_{p}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/909b6ac85b8ad1f0cff2539fa1a434de0f098f62)
In going from line 1 to line 2, the identity
was used where
is a place of
and
is a place of
lying above
Going from line 2 to line 3, a property of the norm is used. The norm is in
so without loss of generality it can be assumed that
Then
possesses a unique integer factorisation:
![{\displaystyle a=\pm \prod _{p<\infty }p^{v_{p}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de9c8a254500e733424fb1b0ee6d9809100678ac)
where
is
for almost all
By Ostrowski's theorem all absolute values on
are equivalent to the real absolute value
or a
-adic absolute value. Therefore:
![{\displaystyle {\begin{aligned}|a|&=\left(\prod _{p<\infty }|a|_{p}\right)\cdot |a|_{\infty }\\&=\left(\prod _{p<\infty }p^{-v_{p}}\right)\cdot \left(\prod _{p<\infty }p^{v_{p}}\right)\\&=1\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f57496d62bf49dd520a33e1a1802874430897037)
- Lemma.[22] There exists a constant
depending only on
such that for every
satisfying
there exists
such that
for all ![{\displaystyle v.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7b87f7afb1452b9d0e287f8ef746a9912c8333)
- Corollary. Let
be a place of
and let
be given for all
with the property
for almost all
Then there exists
so that
for all ![{\displaystyle v\neq v_{0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab93edc3647d33ce8f175020687e8048fe7fbbc)
Proof. Let
be the constant from the lemma. Let
be a uniformising element of
Define the adele
via
with
minimal, so that
for all
Then
for almost all
Define
with
so that
This works, because
for almost all
By the Lemma there exists
so that
for all
- Theorem.
is discrete and cocompact in ![{\displaystyle I_{K}^{1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09f34a4765cdda3c0348231c759d469398b4d3e7)
Proof.[23] Since
is discrete in
it is also discrete in
To prove the compactness of
let
is the constant of the Lemma and suppose
satisfying
is given. Define:
![{\displaystyle W_{\alpha }:=\left\{\xi =(\xi _{v})_{v}\in \mathbb {A} _{K}||\xi _{v}|_{v}\leq |\alpha _{v}|_{v}{\text{ for all }}v\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4067c084d75667a3e5a0fb5a8abf920abdefec2)
Clearly
is compact. It can be claimed that the natural projection
is surjective. Let
be arbitrary, then:
![{\displaystyle |\beta |=\prod _{v}|\beta _{v}|_{v}=1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd4195216e5c6bebff233fb3eb1009e6f28ae1a)
and therefore
![{\displaystyle \prod _{v}|\beta _{v}^{-1}|_{v}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a564fbd53bc894675aabfa223baf71a1619b87a)
It follows that
![{\displaystyle \prod _{v}|\beta _{v}^{-1}\alpha _{v}|_{v}=\prod _{v}|\alpha _{v}|_{v}>C.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2dab33a452f92a00abf45e013bf44efea90443e)
By the Lemma there exists
such that
for all
and therefore
proving the surjectivity of the natural projection. Since it is also continuous the compactness follows.
- Theorem.[24] There is a canonical isomorphism
Furthermore,
is a set of representatives for
and
is a set of representatives for ![{\displaystyle I_{\mathbb {Q} }/\mathbb {Q} ^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa4609187dd81fb3dfc4358aa25735d12c82c5dd)
Proof. Consider the map
![{\displaystyle {\begin{cases}\phi :{\widehat {\mathbb {Z} }}^{\times }\to I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }\\(a_{p})_{p}\mapsto ((a_{p})_{p},1)\mathbb {Q} ^{\times }\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2710f829881489beccfb64d5bad06b56f52cd07e)
This map is well-defined, since
for all
and therefore
Obviously
is a continuous group homomorphism. Now suppose
Then there exists
such that
By considering the infinite place it can be seen that
proves injectivity. To show surjectivity let
The absolute value of this element is
and therefore
![{\displaystyle |\beta _{\infty }|_{\infty }={\frac {1}{\prod _{p}|\beta _{p}|_{p}}}\in \mathbb {Q} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/888b8f5a810eb0d65572a777faad808e98183f77)
Hence
and there is:
![{\displaystyle ((\beta _{p})_{p},\beta _{\infty })\mathbb {Q} ^{\times }=\left(\left({\frac {\beta _{p}}{\beta _{\infty }}}\right)_{p},1\right)\mathbb {Q} ^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17204cf444f20fff6a7588b333589abd94fefa1c)
Since
![{\displaystyle \forall p:\qquad \left|{\frac {\beta _{p}}{\beta _{\infty }}}\right|_{p}=1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d56af12278a4a5a5fa16c4a0b9cb929d68d2246e)
It has been concluded that
is surjective.
- Theorem.[24] The absolute value function induces the following isomorphisms of topological groups:
![{\displaystyle {\begin{aligned}I_{\mathbb {Q} }&\cong I_{\mathbb {Q} }^{1}\times (0,\infty )\\I_{\mathbb {Q} }^{1}&\cong I_{\mathbb {Q} ,{\text{fin}}}\times \{\pm 1\}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8be605e30182aa35b8da58097944f97673ea741a)
Proof. The isomorphisms are given by:
![{\displaystyle {\begin{cases}\psi :I_{\mathbb {Q} }\to I_{\mathbb {Q} }^{1}\times (0,\infty )\\a=(a_{\text{fin}},a_{\infty })\mapsto \left(a_{\text{fin}},{\frac {a_{\infty }}{|a|}},|a|\right)\end{cases}}\qquad {\text{and}}\qquad {\begin{cases}{\widetilde {\psi }}:I_{\mathbb {Q} ,{\text{fin}}}\times \{\pm 1\}\to I_{\mathbb {Q} }^{1}\\(a_{\text{fin}},\varepsilon )\mapsto \left(a_{\text{fin}},{\frac {\varepsilon }{|a_{\text{fin}}|}}\right)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d3163eea78aad0f9f3372defefa26912395b2e)
Relation between ideal class group and idele class group
- Theorem. Let
be a number field with ring of integers
group of fractional ideals
and ideal class group
Here's the following isomorphisms ![{\displaystyle {\begin{aligned}J_{K}&\cong I_{K,{\text{fin}}}/{\widehat {O}}^{\times }\\\operatorname {Cl} _{K}&\cong C_{K,{\text{fin}}}/{\widehat {O}}^{\times }K^{\times }\\\operatorname {Cl} _{K}&\cong C_{K}/\left({\widehat {O}}^{\times }\times \prod _{v|\infty }K_{v}^{\times }\right)K^{\times }\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f55be4c742283184290e83eb44b100dcd985ac2d)
- where
has been defined.
Proof. Let
be a finite place of
and let
be a representative of the equivalence class
Define
![{\displaystyle {\mathfrak {p}}_{v}:=\{x\in O:|x|_{v}<1\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a723a8818f633088911bf0a2e902d723ca37bf)
Then
is a prime ideal in
The map
is a bijection between finite places of
and non-zero prime ideals of
The inverse is given as follows: a prime ideal
is mapped to the valuation
given by
![{\displaystyle {\begin{aligned}v_{\mathfrak {p}}(x)&:=\max\{k\in \mathbb {N} _{0}:x\in {\mathfrak {p}}^{k}\}\quad \forall x\in O^{\times }\\v_{\mathfrak {p}}\left({\frac {x}{y}}\right)&:=v_{\mathfrak {p}}(x)-v_{\mathfrak {p}}(y)\quad \forall x,y\in O^{\times }\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0d7d05420fa91b28b2b232436674f0f2a687db)
The following map is well-defined:
![{\displaystyle {\begin{cases}(\cdot ):I_{K,{\text{fin}}}\to J_{K}\\\alpha =(\alpha _{v})_{v}\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})},\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5461bef101531ec906a53f0dc573032eea004e)
The map
is obviously a surjective homomorphism and
The first isomorphism follows from fundamental theorem on homomorphism. Now, both sides are divided by
This is possible, because
![{\displaystyle {\begin{aligned}(\alpha )&=((\alpha ,\alpha ,\dotsc ))\\&=\prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha )}\\&=(\alpha )&&{\text{ for all }}\alpha \in K^{\times }.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72809841d7c3838b74db50f328868bd44aea54dd)
Please, note the abuse of notation: On the left hand side in line 1 of this chain of equations,
stands for the map defined above. Later, the embedding of
into
is used. In line 2, the definition of the map is used. Finally, use that
is a Dedekind domain and therefore each ideal can be written as a product of prime ideals. In other words, the map
is a
-equivariant group homomorphism. As a consequence, the map above induces a surjective homomorphism
![{\displaystyle {\begin{cases}\phi :C_{K,{\text{fin}}}\to \operatorname {Cl} _{K}\\\alpha K^{\times }\mapsto (\alpha )K^{\times }\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/610eebb827ae08ecaef496b8c4a164a34cf54784)
To prove the second isomorphism, it has to be shown that
Consider
Then
because
for all
On the other hand, consider
with
which allows to write
As a consequence, there exists a representative, such that:
Consequently,
and therefore
The second isomorphism of the theorem has been proven.
For the last isomorphism note that
induces a surjective group homomorphism
with
![{\displaystyle \ker({\widetilde {\phi }})=\left({\widehat {O}}^{\times }\times \prod _{v|\infty }K_{v}^{\times }\right)K^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/722a7f0578a495f431f9cc13d938104e9e646c08)
Remark. Consider
with the idele topology and equip
with the discrete topology. Since
is open for each
is continuous. It stands, that
is open, where
so that
Decomposition of the idele group and idele class group of K
- Theorem.
![{\displaystyle {\begin{aligned}I_{K}&\cong I_{K}^{1}\times M:\quad {\begin{cases}M\subset I_{K}{\text{ discrete and }}M\cong \mathbb {Z} &\operatorname {char} (K)>0\\M\subset I_{K}{\text{ closed and }}M\cong \mathbb {R} _{+}&\operatorname {char} (K)=0\end{cases}}\\C_{K}&\cong I_{K}^{1}/K^{\times }\times N:\quad {\begin{cases}N=\mathbb {Z} &\operatorname {char} (K)>0\\N=\mathbb {R} _{+}&\operatorname {char} (K)=0\end{cases}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9900251cf637db9473eb3903e2b3a59516a931)
Proof.
For each place
of
so that for all
belongs to the subgroup of
generated by
Therefore for each
is in the subgroup of
generated by
Therefore the image of the homomorphism
is a discrete subgroup of
generated by
Since this group is non-trivial, it is generated by
for some
Choose
so that
then
is the direct product of
and the subgroup generated by
This subgroup is discrete and isomorphic to
For
define:
![{\displaystyle z(\lambda )=(z_{v})_{v},\quad z_{v}={\begin{cases}1&v\notin P_{\infty }\\\lambda &v\in P_{\infty }\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/237276dd8924cacbd5a75165de6f5129c8d8c87a)
The map
is an isomorphism of
in a closed subgroup
of
and
The isomorphism is given by multiplication:
![{\displaystyle {\begin{cases}\phi :M\times I_{K}^{1}\to I_{K},\\((\alpha _{v})_{v},(\beta _{v})_{v})\mapsto (\alpha _{v}\beta _{v})_{v}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/210e57f0e3151ff6ffae55197f89dc2d934cff91)
Obviously,
is a homomorphism. To show it is injective, let
Since
for
it stands that
for
Moreover, it exists a
so that
for
Therefore,
for
Moreover
implies
where
is the number of infinite places of
As a consequence
and therefore
is injective. To show surjectivity, let
It is defined that
and furthermore,
for
and
for
Define
It stands, that
Therefore,
is surjective.
The other equations follow similarly.
Characterisation of the idele group
- Theorem.[25] Let
be a number field. There exists a finite set of places
such that: ![{\displaystyle I_{K}=\left(I_{K,S}\times \prod _{v\notin S}O_{v}^{\times }\right)K^{\times }=\left(\prod _{v\in S}K_{v}^{\times }\times \prod _{v\notin S}O_{v}^{\times }\right)K^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c47ac7fc64783cb7f03656eeccda35bf5f62b18)
Proof. The class number of a number field is finite so let
be the ideals, representing the classes in
These ideals are generated by a finite number of prime ideals
Let
be a finite set of places containing
and the finite places corresponding to
Consider the isomorphism:
![{\displaystyle I_{K}/\left(\prod _{v<\infty }O_{v}^{\times }\times \prod _{v|\infty }K_{v}^{\times }\right)\cong J_{K},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07bccc27a3ca19807697e741f0013fba9da685b4)
induced by
![{\displaystyle (\alpha _{v})_{v}\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e483ad51c3fa13e79eba212b7d69279d8cb1825a)
At infinite places the statement is immediate, so the statement has been proved for finite places. The inclusion ″
″ is obvious. Let
The corresponding ideal
belongs to a class
meaning
for a principal ideal
The idele
maps to the ideal
under the map
That means
Since the prime ideals in
are in
it follows
for all
that means
for all
It follows, that
therefore
Applications
Finiteness of the class number of a number field
In the previous section the fact that the class number of a number field is finite had been used. Here this statement can be proved:
- Theorem (finiteness of the class number of a number field). Let
be a number field. Then ![{\displaystyle |\operatorname {Cl} _{K}|<\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/204e41a2171d4a37ad2bfbf7b63c39c1f64caea0)
Proof. The map
![{\displaystyle {\begin{cases}I_{K}^{1}\to J_{K}\\\left((\alpha _{v})_{v<\infty },(\alpha _{v})_{v|\infty }\right)\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/387550728dab376a0174d229402b08986bd9b010)
is surjective and therefore
is the continuous image of the compact set
Thus,
is compact. In addition, it is discrete and so finite.
Remark. There is a similar result for the case of a global function field. In this case, the so-called divisor group is defined. It can be shown that the quotient of the set of all divisors of degree
by the set of the principal divisors is a finite group.[26]
Group of units and Dirichlet's unit theorem
Let
be a finite set of places. Define
![{\displaystyle {\begin{aligned}\Omega (P)&:=\prod _{v\in P}K_{v}^{\times }\times \prod _{v\notin P}O_{v}^{\times }=(\mathbb {A} _{K}(P))^{\times }\\E(P)&:=K^{\times }\cap \Omega (P)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ceadcea8d23377f8487c483e1b8ea4bb9916262b)
Then
is a subgroup of
containing all elements
satisfying
for all
Since
is discrete in
is a discrete subgroup of
and with the same argument,
is discrete in
An alternative definition is:
where
is a subring of
defined by
![{\displaystyle K(P):=K\cap \left(\prod _{v\in P}K_{v}\times \prod _{v\notin P}O_{v}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89937d9c07d3950a5e3b86b0e786ba0322b4e9e6)
As a consequence,
contains all elements
which fulfil
for all
- Lemma 1. Let
The following set is finite: ![{\displaystyle \left\{\eta \in E(P):\left.{\begin{cases}|\eta _{v}|_{v}=1&\forall v\notin P\\c\leq |\eta _{v}|_{v}\leq C&\forall v\in P\end{cases}}\right\}\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d0804c8c9c5ea4c1798acbbee28af67adf0f298)
Proof. Define
![{\displaystyle W:=\left\{(\alpha _{v})_{v}:\left.{\begin{cases}|\alpha _{v}|_{v}=1&\forall v\notin P\\c\leq |\alpha _{v}|_{v}\leq C&\forall v\in P\end{cases}}\right\}\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e884cb927ef00a9a5e9361ffd33f3f40cb0e238)
is compact and the set described above is the intersection of
with the discrete subgroup
in
and therefore finite.
- Lemma 2. Let
be set of all
such that
for all
Then
the group of all roots of unity of
In particular it is finite and cyclic.
Proof. All roots of unity of
have absolute value
so
For converse note that Lemma 1 with
and any
implies
is finite. Moreover
for each finite set of places
Finally suppose there exists
which is not a root of unity of
Then
for all
contradicting the finiteness of
- Unit Theorem.
is the direct product of
and a group isomorphic to
where
if
and
if
[27]
- Dirichlet's Unit Theorem. Let
be a number field. Then
where
is the finite cyclic group of all roots of unity of
is the number of real embeddings of
and
is the number of conjugate pairs of complex embeddings of
It stands, that ![{\displaystyle [K:\mathbb {Q} ]=r+2s.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4a6794fb048a2f632ee764850bb88ae743a79e)
Remark. The Unit Theorem generalises Dirichlet's Unit Theorem. To see this, let
be a number field. It is already known that
set
and note
Then there is:
![{\displaystyle {\begin{aligned}E\times \mathbb {Z} ^{r+s-1}=E(P_{\infty })&=K^{\times }\cap \left(\prod _{v|\infty }K_{v}^{\times }\times \prod _{v<\infty }O_{v}^{\times }\right)\\&\cong K^{\times }\cap \left(\prod _{v<\infty }O_{v}^{\times }\right)\\&\cong O^{\times }\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d72b8dd5c7f14a0f5e8e232a1540197f3919c709)
Approximation theorems
- Weak Approximation Theorem.[28] Let
be inequivalent valuations of
Let
be the completion of
with respect to
Embed
diagonally in
Then
is everywhere dense in
In other words, for each
and for each
there exists
such that: ![{\displaystyle \forall n\in \{1,\ldots ,N\}:\quad |\alpha _{n}-\xi |_{n}<\varepsilon .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d27fd3593d707d73265f71b0b35899bcf06a8e63)
- Strong Approximation Theorem.[29] Let
be a place of
Define ![{\displaystyle V:={\prod _{v\neq v_{0}}}^{'}K_{v}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c72400798a890f73602861fcb5013f5e2308b018)
- Then
is dense in ![{\displaystyle V.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978)
Remark. The global field is discrete in its adele ring. The strong approximation theorem tells us that, if one place (or more) is omitted, the property of discreteness of
is turned into a denseness of
Hasse principle
- Hasse-Minkowski Theorem. A quadratic form on
is zero, if and only if, the quadratic form is zero in each completion ![{\displaystyle K_{v}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/686b2b59e6937d3c11fd4bafcd98ca8ba3f6a12f)
Remark. This is the Hasse principle for quadratic forms. For polynomials of degree larger than 2 the Hasse principle isn't valid in general. The idea of the Hasse principle (also known as local–global principle) is to solve a given problem of a number field
by doing so in its completions
and then concluding on a solution in
Characters on the adele ring
Definition. Let
be a locally compact abelian group. The character group of
is the set of all characters of
and is denoted by
Equivalently
is the set of all continuous group homomorphisms from
to
Equip
with the topology of uniform convergence on compact subsets of
One can show that
is also a locally compact abelian group.
- Theorem. The adele ring is self-dual:
![{\displaystyle \mathbb {A} _{K}\cong {\widehat {\mathbb {A} _{K}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55be9c8a6f05cb015e6307aba59b749744dc6e61)
Proof. By reduction to local coordinates, it is sufficient to show each
is self-dual. This can be done by using a fixed character of
The idea has been illustrated by showing
is self-dual. Define:
![{\displaystyle {\begin{cases}e_{\infty }:\mathbb {R} \to \mathbb {T} \\e_{\infty }(t):=\exp(2\pi it)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae29e802c6ab778e1181a6dbaa112331af2c2817)
Then the following map is an isomorphism which respects topologies:
![{\displaystyle {\begin{cases}\phi :\mathbb {R} \to {\widehat {\mathbb {R} }}\\s\mapsto {\begin{cases}\phi _{s}:\mathbb {R} \to \mathbb {T} \\\phi _{s}(t):=e_{\infty }(ts)\end{cases}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43e99fd0b78539835215d5a7c1a98ad580c5eb5b)
- Theorem (algebraic and continuous duals of the adele ring).[30] Let
be a non-trivial character of
which is trivial on
Let
be a finite-dimensional vector-space over
Let
and
be the algebraic duals of
and
Denote the topological dual of
by
and use
and
to indicate the natural bilinear pairings on
and
Then the formula
for all
determines an isomorphism
of
onto
where
and
Moreover, if
fulfils
for all
then ![{\displaystyle e^{\star }\in E^{\star }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c257f19174dfa39ac48325de8f8bd8f1ed1c92)
Tate's thesis
With the help of the characters of
Fourier analysis can be done on the adele ring.[31] John Tate in his thesis "Fourier analysis in Number Fields and Hecke Zeta Functions" proved results about Dirichlet L-functions using Fourier analysis on the adele ring and the idele group. Therefore, the adele ring and the idele group have been applied to study the Riemann zeta function and more general zeta functions and the L-functions. Adelic forms of these functions can be defined and represented as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. Functional equations and meromorphic continuations of these functions can be shown. For example, for all
with
![{\displaystyle \int _{\widehat {\mathbb {Z} }}|x|^{s}d^{\times }x=\zeta (s),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20045a8dd30ffb0a7e83b3496f7f80069948168d)
where
is the unique Haar measure on
normalised such that
has volume one and is extended by zero to the finite adele ring. As a result, the Riemann zeta function can be written as an integral over (a subset of) the adele ring.[32]
The theory of automorphic forms is a generalisation of Tate's thesis by replacing the idele group with analogous higher dimensional groups. To see this note:
![{\displaystyle {\begin{aligned}I_{\mathbb {Q} }&=\operatorname {GL} (1,\mathbb {A} _{\mathbb {Q} })\\I_{\mathbb {Q} }^{1}&=(\operatorname {GL} (1,\mathbb {A} _{\mathbb {Q} }))^{1}:=\{x\in \operatorname {GL} (1,\mathbb {A} _{\mathbb {Q} }):|x|=1\}\\\mathbb {Q} ^{\times }&=\operatorname {GL} (1,\mathbb {Q} )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/910936373a7cf6fd5caab939dd536867812159e1)
Based on these identification a natural generalisation would be to replace the idele group and the 1-idele with:
![{\displaystyle {\begin{aligned}I_{\mathbb {Q} }&\leftrightsquigarrow \operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} })\\I_{\mathbb {Q} }^{1}&\leftrightsquigarrow (\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }))^{1}:=\{x\in \operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }):|\det(x)|=1\}\\\mathbb {Q} &\leftrightsquigarrow \operatorname {GL} (2,\mathbb {Q} )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e311856b5cb4fdbdfa83c1b50d0faa871a4ad97e)
And finally
![{\displaystyle \mathbb {Q} ^{\times }\backslash I_{\mathbb {Q} }^{1}\cong \mathbb {Q} ^{\times }\backslash I_{\mathbb {Q} }\leftrightsquigarrow (\operatorname {GL} (2,\mathbb {Q} )\backslash (\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }))^{1}\cong (\operatorname {GL} (2,\mathbb {Q} )Z_{\mathbb {R} })\backslash \operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e82cd05d294b432a27287031d8700ea1315bd464)
where
is the centre of
Then it define an automorphic form as an element of
In other words an automorphic form is a function on
satisfying certain algebraic and analytic conditions. For studying automorphic forms, it is important to know the representations of the group
It is also possible to study automorphic L-functions, which can be described as integrals over
[33]
Generalise even further is possible by replacing
with a number field and
with an arbitrary reductive algebraic group.
Further applications
A generalisation of Artin reciprocity law leads to the connection of representations of
and of Galois representations of
(Langlands program).
The idele class group is a key object of class field theory, which describes abelian extensions of the field. 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. The Artin reciprocity law, which is a sweeping generalisation of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field. Thus, the global reciprocity map of the idele class group to the abelian part of the absolute Galois group of the field will be obtained.
The self-duality of the adele ring of the function field of a curve over a finite field easily implies the Riemann–Roch theorem and the duality theory for the curve.
References
- ^ Groechenig, Michael (August 2017). "Adelic Descent Theory". Compositio Mathematica. 153 (8): 1706–1746. arXiv:1511.06271. doi:10.1112/S0010437X17007217. ISSN 0010-437X. S2CID 54016389.
- ^ Sutherland, Andrew (1 December 2015). 18.785 Number theory I Lecture #22 (PDF). MIT. p. 4.
- ^ "ring of adeles in nLab". ncatlab.org.
- ^ Geometric Class Field Theory, notes by Tony Feng of a lecture of Bhargav Bhatt (PDF).
- ^ Weil uniformization theorem, nlab article.
- ^ Tate, John (1968), "Residues of differentials on curves" (PDF), Annales Scientifiques de l'École Normale Supérieure, 1: 149–159, doi:10.24033/asens.1162.
- ^ This proof can be found in Cassels & Fröhlich 1967, p. 64.
- ^ The definitions are based on Weil 1967, p. 60.
- ^ See Weil 1967, p. 64 or Cassels & Fröhlich 1967, p. 74.
- ^ For proof see Deitmar 2010, p. 124, theorem 5.2.1.
- ^ See Cassels & Fröhlich 1967, p. 64, Theorem, or Weil 1967, p. 64, Theorem 2.
- ^ The next statement can be found in Neukirch 2007, p. 383.
- ^ See Deitmar 2010, p. 126, Theorem 5.2.2 for the rational case.
- ^ This section is based on Weil 1967, p. 71.
- ^ A proof of this statement can be found in Weil 1967, p. 71.
- ^ A proof of this statement can be found in Weil 1967, p. 72.
- ^ For a proof see Neukirch 2007, p. 388.
- ^ This statement can be found in Cassels & Fröhlich 1967, p. 69.
- ^
is also used for the set of the
-idele but
is used in this example. - ^ There are many proofs for this result. The one shown below is based on Neukirch 2007, p. 195.
- ^ For a proof see Cassels & Fröhlich 1967, p. 66.
- ^ This proof can be found in Weil 1967, p. 76 or in Cassels & Fröhlich 1967, p. 70.
- ^ a b Part of Theorem 5.3.3 in Deitmar 2010.
- ^ The general proof of this theorem for any global field is given in Weil 1967, p. 77.
- ^ For more information, see Cassels & Fröhlich 1967, p. 71.
- ^ A proof can be found in Weil 1967, p. 78 or in Cassels & Fröhlich 1967, p. 72.
- ^ A proof can be found in Cassels & Fröhlich 1967, p. 48.
- ^ A proof can be found in Cassels & Fröhlich 1967, p. 67
- ^ A proof can be found in Weil 1967, p. 66.
- ^ For more see Deitmar 2010, p. 129.
- ^ A proof can be found Deitmar 2010, p. 128, Theorem 5.3.4. See also p. 139 for more information on Tate's thesis.
- ^ For further information see Chapters 7 and 8 in Deitmar 2010.
Sources
- Cassels, John; Fröhlich, Albrecht (1967). Algebraic number theory: proceedings of an instructional conference, organized by the London Mathematical Society, (a NATO Advanced Study Institute). Vol. XVIII. London: Academic Press. ISBN 978-0-12-163251-9. 366 pages.
- Neukirch, Jürgen (2007). Algebraische Zahlentheorie, unveränd. nachdruck der 1. aufl. edn (in German). Vol. XIII. Berlin: Springer. ISBN 9783540375470. 595 pages.
- Weil, André (1967). Basic number theory. Vol. XVIII. Berlin; Heidelberg; New York: Springer. ISBN 978-3-662-00048-9. 294 pages.
- Deitmar, Anton (2010). Automorphe Formen (in German). Vol. VIII. Berlin; Heidelberg (u.a.): Springer. ISBN 978-3-642-12389-4. 250 pages.
- Lang, Serge (1994). Algebraic number theory, Graduate Texts in Mathematics 110 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-94225-4.
External links
- What problem do the adeles solve?
- Some good books on adeles