Editing Adele ring (section)

==Applications==

=== Stating Artin reciprocity ===
The [[Artin reciprocity law]] says that for a global field <math>K</math>,
:<math>\widehat{C_K} = \widehat{\mathbf{A}_K^\times/K^\times} \ \simeq \ \text{Gal}(K^\text{ab}/K)</math>
where <math>K^{ab}</math> is the maximal [[Abelian extension|abelian algebraic extension]] of <math>K</math> and <math>\widehat{(\dots)}</math> means the [[Profinite group|profinite]] completion of the group.

=== Giving adelic formulation of Picard group of a curve ===
If ''<math>X/\mathbf{F_{\mathit{q}}}</math>'' is a smooth proper curve then its [[Picard group]] is<ref>{{Citation |title=Geometric Class Field Theory, notes by Tony Feng of a lecture of Bhargav Bhatt |url=https://math.berkeley.edu/~fengt/2GeometricCFT.pdf}}.</ref>
:<math>\text{Pic}(X) \ = \ K^\times\backslash \mathbf{A}^\times_X/\mathbf{O}_X^\times</math>
and its divisor group is <math>\text{Div}(X)=\mathbf{A}^\times_X/\mathbf{O}_X^\times</math>. Similarly, if <math>G</math> is a [[Semi-simplicity|semisimple]] algebraic group (e.g. <math display="inline">SL_n</math>, it also holds for <math>GL_n</math>) then Weil uniformisation says that<ref>{{Citation | title=Weil uniformization theorem, nlab article | url=https://ncatlab.org/nlab/show/Weil+uniformization+theorem}}.</ref>
:<math>\text{Bun}_G(X) \ = \ G(K)\backslash G(\mathbf{A}_X)/G(\mathbf{O}_X).</math>
Applying this to <math>G=\mathbf{G}_m</math> gives the result on the Picard group.

=== Tate's thesis ===
There is a topology on <math>\mathbf{A}_K</math> for which the quotient <math>\mathbf{A}_K/K</math> is compact, allowing one to do harmonic analysis on it. [[John T. Tate|John Tate]] in his thesis "Fourier analysis in number fields and Hecke Zeta functions"{{sfn|Cassels|Fröhlich|1967}} proved results about [[Dirichlet L-function|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 <math>X</math> is a smooth proper curve ''over the complex numbers'', one can define the adeles of its function field <math>\mathbf{C}(X)</math> exactly as the finite fields case. [[John T. Tate|John Tate]] proved<ref>{{Citation | title=Residues of differentials on curves | year=1968| doi=10.24033/asens.1162| url=http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1968_4_1_1/ASENS_1968_4_1_1_149_0/ASENS_1968_4_1_1_149_0.pdf| last1=Tate| first1=John| journal=Annales Scientifiques de l'École Normale Supérieure| volume=1| pages=149–159}}.</ref> that [[Serre duality]] on ''<math>X</math>''
:<math>H^1(X,\mathcal{L})\ \simeq \ H^0(X,\Omega_X\otimes\mathcal{L}^{-1})^*</math>
can be deduced by working with this adele ring <math>\mathbf{A}_{\mathbf{C}(X)}</math>. Here ''L'' is a [[line bundle]] on ''<math>X</math>''.