field is discrete and cocompact in its adèles

For brevity, we write Pf for the set of finite places of K, and P for the set of infinite places. We also write for a restricted direct product. Then

Theorem 1.

K is discrete as a subgroup of AK.

Proof. Since 𝔸K is a topological ring, it suffices to show that there is a neighborhood in 𝔸K meeting K in only 0.



Since 𝔬v is open in Kv for v finite, this is an open set. (Note that 𝔬v=𝒪Kv, the ring of algebraic integers of Kv).

Now consider an element xUK𝔸K. If x=(xv), then for v finite, xv𝔬v, and for v infiniteMathworldPlanetmath, xvB(0,12). Assume x0. Then

|xv|v{1v finite12<1v infinite

but then


in contradictionMathworldPlanetmathPlanetmath to the product formula. Thus x=0 and we are done.

The above theorem is very sensitive to the fact that all places are included in 𝔸K. For example, it is clear that the image of 𝔸K in vPKv is dense, since Kv is characterized by an embeddingMathworldPlanetmathPlanetmath KKv,,p. Then by an argument familiar from Minkowski’s theorem, 𝒪K is a full-rank latticeMathworldPlanetmathPlanetmath in the image of K. But K is the -span of that lattice, so is dense in Kv.

Furthermore, the same is true for the finite places:

Proposition 2.

The image of K in vPfov is dense.

Proof. Suppose x=(xv)vvPf𝔬v. We show that x can be approximated as closely as desired by an element of K by showing that for any ideal I𝒪K, there is yK such that y-xvI𝔬v for each vPf.

First multiply through by some z so that everything is in 𝒪K: choose z0 such that zxv𝔬v for all vPf. This is possible since all but finitely many xv are already in 𝔬v. Thus y-xvI𝔬v is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to zy-zxv(zI)𝔬vI𝔬v. So assume wlog that xv𝔬v for all v; in the end simply divide by z to recover the general case. But then the existence of y is guaranteed by the Chinese Remainder TheoremMathworldPlanetmathPlanetmathPlanetmath, since if I=𝔭iei, then I𝔬v=𝔭iei for some i.

It is true, though somewhat harder to prove, that K is in fact dense in 𝔸K if even one place is missing from the productPlanetmathPlanetmathPlanetmath!

Theorem 3.

𝔸K/K is compactPlanetmathPlanetmath.

Proof. The set


is open in 𝔸K.

Claim first that K+U=𝔸K. Choose (xv)𝔸K. There is a finite setMathworldPlanetmath S of finite places v such that xv𝔬v for vS. Using an argument identical to the approximation argument above, choose yK such that y-xv𝔬v,vS and y𝔬v,vS. Then (xv-y)v is in 𝔬v for vS, is in 𝔬v for vS but finite, and is in Kv for v infinite. Thus (xv-y)vU and we are done.

Claim next that KU=𝒪K. is obvious. To see , note that an element of KU is an element of K that is integralDlmfPlanetmath at every finite place, so it is integral and is in 𝒪K.

Thus we get a natural map U𝔸K𝔸K/K. This map is surjectivePlanetmathPlanetmath since K+U=𝔸K, and its kernel is KU. So it suffices to show that U/(KU)=U/𝒪K is compact. There is obviously an exact sequence induced by the projection UvPKv,


The left-hand side is compact since each 𝔬v is, and the right-hand side is


which know is compact since 𝒪K forms a full-rank lattice in n. Thus U/𝒪K is also compact and we are done.

So we have shown that 𝔸K is a locally compact ring, and that K𝔸K is discrete and cocompact. This is analogous to two other situations with which we are familiar:

 is locally compact,  is discrete and cocompact
vPKv is locally compact, 𝒪KvPKv is discrete and cocompact

This is a useful concept because in such a situation one can do Fourier analysis. For example, if f: is a C functionMathworldPlanetmath with exponential decay (or at least integrable on all of ), then we can define its Fourier transformDlmfMathworldPlanetmath f^, and the Poisson summation formula


relates the two. The same theory thus exists for appropriately defined functions f:𝔸K, and the Poisson formula again holds with the sum over K rather than over . This can be used to show that the L-functions have analytic continuations, just as the real Poisson formula is used to show this for ζ.

Title field is discrete and cocompact in its adèles
Canonical name FieldIsDiscreteAndCocompactInItsAdeles
Date of creation 2013-03-22 18:00:05
Last modified on 2013-03-22 18:00:05
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Theorem
Classification msc 11R56