# 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.

Written on November 13, 2015