When Serre and Grothendieck were coming up with an extension of the usual cohomology theory to varieties in algebraic geometry, an important example was given by Serre which showed that you couldn’t have a Weil cohomology theory with coefficients in in characteristic . I will explain the example and extend it to show something about the ordinary case.
Suppose there were a cohomology theory with coefficients in . In particular, for a supersingular elliptic curve over a finite field, should exists as a vector space of dimension (by comparison theorems).
In particular, , which is a dimensional quaternion algebra over , should act on . In particular, this gives a map which is automatically injective since the source is a division algebra and therefore splits over .
Similarly, etale cohomology theory with coefficients in gives a splitting of over the other finite primes. On the other hand, is not globally split and therefore it is not split over infinity. However, this contradicts the Brauer group exact sequence from class field theory.
Of course, there do exist Weil cohomology theories over an extension of . In particular, crystalline cohomology is defined over the Witt vectors of the base field. The same argument as before now shows simply that no supersingular elliptic curve defined over can have all it’s endomorphosms defined over the base field since .
In fact, it is not hard to show that every supersingular elliptic curve is in fact defined . This shows that has order two in the corresponding Brauer group.
Note that we could replace with the Tate module (or more precisely, it’s dual) if . However, over , the Tate module is always lower dimension than one would expect and in the supersingular case, it is in fact trivial. This is also in fact implied by our above argument since otherwise we would have an injective map for which is evidently absurd.
On the other hand, for ordinary elliptic curves over a finite field, the Tate module is a one dimensional, free module while the endomorphism algebra is an imaginary quadratic number field .
In particular, we have an injective map . It’s injective because, for instance, the torsion points can have arbitrarily high degree. We can tensor up to to get a map .
Note that this map can no longer be injective simply for dimension reasons! In particular, has to split over and moreover, we can find a sequence of endomorphisms of that are non-zero in the -adic limit in but converge to the endomorphism of the Tate module. We can do this explicitly:
Let be the Frobenius endomorphism (over the base field) and consider the sequence of endomorphisms . This sequence is Cauchy in the p-adic topology because the successive difference is .
This fixes a large subgroup of rational points because of the factor and it fixes a large subgroup of points because of the inseparable factor. Therefore, it factors through a large power of .
The limit is however non-zero because acts on by and therefore it does not factor through the endomorphism.
For any finite field , there exists a large enough so that fixes it. Since all torsion points are algebraic, this shows that kills the Tate module in the limit.