field is discrete and cocompact in its adèles
is discrete as a subgroup of .
The above theorem is very sensitive to the fact that all places are included in . For example, it is clear that the image of in is dense, since is characterized by an embedding . Then by an argument familiar from Minkowski’s theorem, is a full-rank lattice in the image of . But is the -span of that lattice, so is dense in .
Furthermore, the same is true for the finite places:
The image of in is dense.
Proof. Suppose . We show that can be approximated as closely as desired by an element of by showing that for any ideal , there is such that for each .
First multiply through by some so that everything is in : choose such that for all . This is possible since all but finitely many are already in . Thus is equivalent to . So assume wlog that for all ; in the end simply divide by to recover the general case. But then the existence of is guaranteed by the Chinese Remainder Theorem, since if , then for some .
It is true, though somewhat harder to prove, that is in fact dense in if even one place is missing from the product!
Proof. The set
is open in .
Claim first that . Choose . There is a finite set of finite places such that for . Using an argument identical to the approximation argument above, choose such that and . Then is in for , is in for but finite, and is in for infinite. Thus and we are done.
Claim next that . is obvious. To see , note that an element of is an element of that is integral at every finite place, so it is integral and is in .
Thus we get a natural map . This map is surjective since , and its kernel is . So it suffices to show that is compact. There is obviously an exact sequence induced by the projection ,
The left-hand side is compact since each is, and the right-hand side is
which know is compact since forms a full-rank lattice in . Thus is also compact and we are done.
So we have shown that is a locally compact ring, and that is discrete and cocompact. This is analogous to two other situations with which we are familiar:
This is a useful concept because in such a situation one can do Fourier analysis. For example, if is a function with exponential decay (or at least integrable on all of ), then we can define its Fourier transform , and the Poisson summation formula
relates the two. The same theory thus exists for appropriately defined functions , and the Poisson formula again holds with the sum over rather than over . This can be used to show that the -functions have analytic continuations, just as the real Poisson formula is used to show this for .
|Title||field is discrete and cocompact in its adèles|
|Date of creation||2013-03-22 18:00:05|
|Last modified on||2013-03-22 18:00:05|
|Last modified by||rm50 (10146)|