Math notes dirichlet-theoremchebotarev-density-theoremsupersingular-elliptic-curveselkies-theoremnumber-theory 13 Nov 2015

A question on primes

I recently encountered a result which seems to be analogous to the following result of Dirichlet, which I wrote in a few common forms to be more suggestive.

Theorem (Dirichlet on Arithmetic Progressions). Let a and b be relatively prime positive integers, then there are infinitely many primes p \(\equiv\) a mod b (i.e., the progression \(a + b \mathbb{N}\) contains infinitely many primes).

Corollary 1. There are infinitely many primes such that a is not a square and quadratic residue (mod p) and infinitely many primes such that a is a quadratic non-residue (mod p). In nicer terms, given n, there are infinitely many primes such that the Legendre symbol \((\frac{-n}{p})\) takes on each value for infinitely many p (for p ≠ 2).

Corollary 2. Something like: Given a prime \(p\), there’s a fifty-fifty chance that the image of the prime splits (through an order two extension) [remains inert/ramify when passed to the extension]. This is saying half are quadratic non-residues and half are quadratic residues, respectively, that is, half are squares mod p, and the other half aren’t. For example, \(2 \mod 5\) is not a square, but \(1 \mod 5\) and \(3 mod 5\) are squares.

A result that tastes like Corollary 2 is the existence of infinitely many super singular primes for every elliptic curve over \(\mathbb{Q}\):

Theorem. (Elkies) Given an elliptic curve E/\(\mathbb{Q}\), there are infinitely many primes p such that E/Fp is supersingular and infinitely many primes q such that E/Fq is ordinary.

Here is the next step.

Questions:

Dirichlet’s theorem on primes in progressions is apparently a very special case of Chebotarev’s density theorem. Is Elkies result also a special case of Chebortarev density?
Given a hyper elliptic curve H over \(\mathbb{Q}\), are there infinitely many primes such that passing to H/Fp is a supersingular elliptic curve, an ordinary elliptic curve, or still a hyperelliptic curve? Is there a similar density result for this case?
Viewing Dirichlet’s result as a statement about primes as 0-dimensional varieties, and Elkies’s result as a statement about 1-dimension varieties, is there an analogous statement for 2-dimensional varieties?

Thanks to Dr. Ngo for discussing this with me.

Edit: Jonah Sinick sent me this rather nice article which gives the result of Elkies a bit more context.

Comments

Comments from the old blog (1)

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Jack November 18, 2015

Hi Rin! A few comments:
1) The existence of infinitely many ordinary primes is fairly easy; Elkies result is really about the supersingular ones, which are really rather sparse.
2) Elkies’ result is definitely not a special chase of Chebotarev. Chebotarev can answer the following kind of question: fix an elliptic curve, and a prime number l. For every other prime number p (where the curve has good reduction), compute 1 + p – #E(F_p) modulo l. Which residue classes occur and how often? Elkies, on the other hand, answers the question: for all primes p such that E has good reduction, compute 1 + p – #E(F_p) mod p. Does 0 occur infinitely often (open question: how often?). Note that p here is playing two quite different roles.
As a side remark, Sato-Tate is also about the first kind of question, and knowing it shows little about the second kind of question in general.
3) The second question is a bit confusing; for a given hyperelliptic curve over Q, its reduction will be hyperelliptic for every prime p where it has good reduction (in some sense, the geometry can’t change at primes of good reduction). However, you can define a notion of supersingular (in fact there are varying degrees of supersingularity) and ask the same questions, and as far as I know little is known.

Jack