This is an incredibly lazy post, but why not!
Matt is running a seminar this quarter on the Weil conjectures. It came up that one possible way to prove the Weil conjectures for elliptic curves over finite fields is to lift them to CM elliptic curves using Deuring’s theorem. But after some discussions we couldn’t quite work out whether this was circular or not.
Certainly if you can lift to a CM elliptic curve and lift Frobenius to an endomorphism \(\phi\) of the lift you get Weil immediately; the degree of \(\phi\) is \(p\) which implies the norm of \(\phi\) is \(p\), but for imaginary quadratic fields the norm coincides with the absolute value. But how did Deuring prove his theorem?
The most obvious way to lift an (ordinary, say) elliptic curve \(E/\mathbf{F}_p\) to characteristic zero is to note that, by the Weil conjectures, the order \(\mathcal{O} = \mathbf{Z}[\phi]\) generated by Frobenius lies inside an imaginary quadratic field \(K\) (this is equivalent to the Weil conjectures), and so one can consider \(\mathbf{C}/\mathbf{Z}[\phi]\). To make things simple, if the order is maximal, then this is defined over the Hilbert class field \(H\) of \(K\), and since \(p\) splits principally in \(K\) (since \(\phi\) has norm \(p\)) it follows that \(p\) splits principally in \(H\) as well by class field theory, and so the CM elliptic curve is also defined over \(\mathbf{Z}_p\) and gives a lift. Of course, this argument uses the Weil conjectures! Without that, the ring \(\mathcal{O}\) lives inside a real quadratic field and it’s not clear what one can do.
One approach is to prove the existence of the canonical lift, which automatically will have extra endomorphisms and thus be CM since it lives in characteristic zero. This doesn’t depend on the Weil conjectures. But the canonical lift is a construction I associate more with Serre-Tate than with Deuring. But it’s certainly possible that Deuring’s argument was via the canonical lift.
Some might say that the easy way to solve this is simply to look in one of Deuring’s papers. But instead I will try to call upon my readers (possibly either number theorists who speak German or Brian Conrad) to save me the work and tell reveal all in the comments!
Frank,
You may still need to brush up on your German, but I believe that the first proof of the Riemann hypothesis (Artin’s conjecture) for elliptic curves over finite fields via lifting to characteristic zero is due to Hasse:
Hasse, Helmut Zur Theorie der abstrakten elliptischen Funktionenkörper III. Die Struktur des Meromorphismenrings. Die Riemannsche Vermutung. J. Reine Angew. Math. 175 (1936), 193–208.
Surely you are not too lazy to have a look!
What Deuring did was to refine these arguments to describe all possible endomorphism rings of elliptic curves in char p > 0. They are:
Z (which cannot occur over finite fields),
an order of conductor prime to p in an imaginary quadratic field where p splits,
a maximal order in the quaternion algebra over Q ramified at p and \infty.
(I can’t help but adding that the story of how Hasse came to work on the problem is an amusing one —- see e.g. Roquette’s https://www.mathi.uni-heidelberg.de/~roquette/rv2.pdf , particularly the letter excerpts on the bottom of pages 9 and 11, and also the story mentioned at the bottom of page 23!)
By an argument similar to Hasse, one can show that for any integer a with
-2.\sqrt(p) < a < 2.\sqrt(p)
there is an elliptic curve E = E(a) over the field of p elements with p + 1 – a points. The proof uses the theory of complex multiplication for elliptic curves in characteristic zero, so E is obtained by reduction (mod \frak p) of a curve over the Hilbert class field of the imaginary quadratic field defined by the equation x^2 – a.x + p, when a is prime to p.
This makes the calculation of E somewhat of a challenge. The crypto people would like a simpler construction of E, perhaps working purely over Z/pZ, but we don't yet know how to do this. More generally, one would like a proof of the theorem of Honda-Tate without lifting to characteristic zero and using the general theory of complex multiplication.