Thus an element of is a real number along with a p-adic rational for each p 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 K=Fq(P1)=Fq(t) of the projective line over a finite field. Its valuations correspond to points x of X=P1, i.e. maps over Spec Fq For instance, there are q+1 points of the form SpecFq → P1. In this case Oν=ÔX,x is the completed stalk of the structure sheaf at x (i.e. functions on a formal neighbourhood of x) and Kν=KX,x is its fraction field. Thus The same holds for any smooth proper curve X/Fq over a finite field, the restricted product being over all points of x∈X. Related notions The group of units in the adele ring is called the idele group The quotient of the ideles by the subgroup K×⊆IK is called the idele class group The integral adeles are the subring Applications Stating Artin reciprocity The Artin reciprocity law says that for a global field K, where Kab is the maximal abelian algebraic extension of K and means the profinite completion of the group. Giving adelic formulation of Picard group of a curve If X/Fq is a smooth proper curve then its Picard group is and its divisor group is Div(X)=AK×/OK×. Similarly, if G is a semisimple algebraic group (e.g. SLn, it also holds for GLn) then Weil uniformisation says that Applying this to G=Gm gives the result on the Picard group. Tate's thesis There is a topology on AK for which the quotient AK/K 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 X is a smooth proper curve over the complex numbers, one can define the adeles of its function field C(X) exactly as the finite fields case. John Tate proved that Serre duality on X can be deduced by working with this adele ring AC(X). Here L is a line bundle on X. Notation and basic definitions Global fields Throughout this article,