Linnik’s Theorem for Sato-Tate Laws on CM Elliptic Curves

\title{A Variant of Linnik for Elliptic Curves} \maketitle

Here I talk about my first project at the Emory REU. Prerequisites for this post: some familiarity with number fields.

1. Motivation: Arithemtic Progressions

Given a property {P} about primes, there’s two questions we can ask:

  1. How many primes {\le x} are there with this property?
  2. What’s the least prime with this property?

As an example, consider an arithmetic progression {a}, {a+d}, \dots, with {a < d} and {\gcd(a,d) = 1}. The strong form of Dirichlet’s Theorem tells us that basically, the number of primes {\equiv a \pmod d} is {\frac 1d} the total number of primes. Moreover, the celebrated Linnik’s Theorem tells us that the first prime is {O(d^L)} for a fixed {L}, with record {L = 5}.

As I talked about last time on my blog, the key ingredients were:

  • Introducing Dirichlet characters {\chi}, which are periodic functions modulo {q}. One uses this to get the mod {q} into the problem.
  • Introducing an {L}-function {L(s, \chi)} attached to {\chi}.
  • Using complex analysis (Cauchy’s Residue Theorem) to boil the proof down to properties of the zeros of {L(s, \chi)}.

With that said, we now move to the object of interest: elliptic curves.

2. Counting Primes

Let {E} be an elliptic curve over {\mathbb Q}, which for our purposes we can think of concretely as a curve in Weirestrass form

\displaystyle y^2 = x^3 + Ax + B

where the right-hand side has three distinct complex roots (viewed as a polynomial in {x}). If we are unlucky enough that the right-hand side has a double root, then the curve ceases to bear the name “elliptic curve” and instead becomes singular.

elliptic-curve

Here’s a natural number theoretic question: for any rational prime {p}, how many solutions does {E} have modulo {p}?

To answer this it’s helpful to be able to think over an arbitrary field {F}. While we’ve written our elliptic curve {E} as a curve over {\mathbb Q}, we could just as well regard it as a curve over {\mathbb C}, or as a curve over {\mathbb Q(\sqrt 2)}. Even better, since we’re interested in counting solutions modulo {p}, we can regard this as a curve over {\mathbb F_p}. To make this clear, we will use the notation {E/F} to signal that we are thinking of our elliptic curve over the field {F}. Also, we write {\#E(F)} to denote the number of points of the elliptic curve over {F} (usually when {F} is a finite field). Thus, the question boils down to computing {\#E(\mathbb F_p)}.

Anyways, the question above is given by the famous Hasse bound, and in fact it works over any number field!

Theorem 1 (Hasse Bound)

Let {K} be a number field, and let {E/K} be an elliptic curve. Consider any prime ideal {\mathfrak p \subseteq \mathcal O_K} which is not ramified. Then we have

\displaystyle \#E(\mathbb F_\mathfrak p) = \mathrm{N}\mathfrak p + 1 - a_\mathfrak p

where {\left\lvert a_\mathfrak p \right\rvert \le 2\sqrt{\mathrm{N}\mathfrak p}}.

Here {\mathbb F_\mathfrak p = \mathcal O_K / \mathfrak p} is the field of {\mathrm{N}\mathfrak p} elements. The extra “{+1}” comes from a point at infinity when you complete the elliptic curve in the projective plane.

Here, the ramification means what you might guess. Associated to every elliptic curve over {\mathbb Q} is a conductor {N}, and a prime {p} is ramified if it divides {N}. The finitely many ramified primes are the “bad” primes for which something breaks down when we take modulo {p} (for example, perhaps the curve becomes singular).

In other words, for the {\mathbb Q} case, except for finitely many bad primes {p}, the number of solutions is {p + 1 + O(\sqrt p)}, and we even know the implied {O}-constant to be {2}.

Now, how do we predict the error term?

3. The Sato-Tate Conjecture

For elliptic curves over {\mathbb Q}, we the Sato-Tate conjecture (which recently got upgraded to a theorem) more or less answers the question. But to state it, I have to introduce a new term: an elliptic curve {E/\mathbb Q}, when regarded over {\mathbb C}, can have complex multiplication (abbreviated CM). I’ll define this in just a moment, but for now, the two things to know are

  • CM curves are “special cases”, in the sense that a randomly selected elliptic curve won’t have CM.
  • It’s not easy in general to tell whether a given elliptic curve has CM.

Now I can state the Sato-Tate result. It is most elegantly stated in terms of the following notation: if we define {a_p = p + 1 - \#E(\mathbb F_p)} as above, then there is a unique {\theta_p \in [0,\pi]} which obeys

\displaystyle a_p = 2 \sqrt p \cos \theta_p.

Theorem 2 (Sato-Tate)

Fix an elliptic curve {E/\mathbb Q} which does not have CM (when regarded over {\mathbb C}). Then as {p} varies across unramified primes, the asymptotic probability that {\theta_p \in [\alpha, \beta]} is

\displaystyle \frac{2}{\pi} \int_{[\alpha, \beta]} \sin^2\theta_p.

In other words, {\theta_p} is distributed according to the measure {\sin^2\theta}.

Now, what about the CM case?

4. CM Elliptic Curves

Consider an elliptic curve {E/\mathbb Q} but regard it as a curve over {\mathbb C}. It’s well known that elliptic curves happen to have a group law: given two points on an elliptic curve, you can add them to get a third point. (If you’re not familiar with this, Wikipedia has a nice explanation). So elliptic curves have more structure than just their set of points: they form an abelian group; when written in Weirerstrass form, the identity is the point at infinity.

elliptic-add

Letting {A = (A, +)} be the associated abelian group, we can look at the endomorphisms of {E} (that is, homomorphisms {A \rightarrow A}). These form a ring, which we denote {\text{End }(E)}. An example of such an endomorphism is {a \mapsto n \cdot a} for an integer {n} (meaning {a+\dots+a}, {n} times). In this way, we see that {\mathbb Z \subseteq \text{End }(E)}.

Most of the time we in fact have {\text{End }(E) \cong \mathbb Z}. But on occasion, we will find that {\text{End }(E)} is congruent to {\mathcal O_K}, the ring of integers of a number field {K}. This is called complex multiplication by {K}.

Intuitively, this CM is special (despite being rare), because it means that the group structure associated to {E} has a richer set of symmetry. For CM curves over any number field, for example, the Sato-Tate result becomes very clean, and is considerably more straightforward to prove.

Here’s an example. The elliptic curve

\displaystyle E : y^2 = x^3 - 17 x

of conductor {N = 2^6 \cdot 17^2} turns out to have

\displaystyle \text{End }(E) \cong \mathbb Z[i]

i.e. it has complex multiplication has {\mathbb Z[i]}. Throwing out the bad primes {2} and {17}, we compute the first several values of {a_p}, and something bizarre happens. For the {3} mod {4} primes we get

\displaystyle \begin{aligned} a_{3} &= 0 \\ a_{7} &= 0 \\ a_{11} &= 0 \\ a_{19} &= 0 \\ a_{23} &= 0 \\ a_{31} &= 0 \end{aligned}

and for the {1} mod {4} primes we have

\displaystyle \begin{aligned} a_5 &= 4 \\ a_{13} &= 6 \\ a_{29} &= 4 \\ a_{37} &= 12 \\ a_{41} &= -8 \end{aligned}

Astonishingly, the vanishing of {a_p} is controlled by the splitting of {p} in {\mathbb Z[i]}! In fact, this holds more generally. It’s a theorem that for elliptic curves {E/\mathbb Q} with CM, we have {\text{End }(E) \cong \mathcal O_K} where {K} is some quadratic imaginary number field which is also a PID, like {\mathbb Z[i]}. Then {\mathcal O_K} governs how the {a_p} behave:

Theorem 3 (Sato-Tate Over CM)

Let {E/\mathbb Q} be a fixed elliptic curve with CM by {\mathcal O_K}. Let {\mathfrak p} be a unramified prime of {\mathcal O_K}.

  1. If {\mathfrak p} is inert, then {a_\mathfrak p = 0} (i.e. {\theta_\mathfrak p = \frac{1}{2}\pi}).
  2. If {\mathfrak p} is split, then {\theta_\mathfrak p} is uniform across {[0, \pi]}.

I’m told this is much easier to prove than the usual Sato-Tate.

But there’s even more going on in the background. If I look again at {a_p} where {p \equiv 1 \pmod 4}, I might recall that {p} can be written as the sum of squares, and construct the following table:

\displaystyle \begin{array}{rrl} p & a_p & x^2+y^2 \\ 5 & 4 & 2^2 + 1^2 \\ 13 & 6 & 3^2 + 2^2 \\ 29 & 4 & 2^2 + 5^2 \\ 37 & 12 & 6^2 + 1^2 \\ 41 & -8 & 4^2 + 5^2 \\ 53 & 14 & 7^2 + 2^2 \\ 61 & 12 & 6^2 + 5^2 \\ 73 & -16 & 8^2 + 3^2 \\ 89 & -10 & 5^2 + 8^2 \\ \end{array}

Each {a_p} is double one of the terms! There is no mistake: the {a_p} are also tied to the decomposition of {p = x^2+y^2}. And this works for any number field.

What’s happening? The main idea is that looking at a prime ideal {\mathfrak p = (x+yi)}, {a_\mathfrak p} is related to the argument of the complex number {x+yi} in some way. Of course, there are lots of questions unanswered (how to pick the {\pm} sign, and which of {x} and {y} to choose) but there’s a nice way to package all this information, as I’ll explain momentarily.

(Aside: I think the choice of having {x} be the odd or even number depends precisely on whether {p} is a quadratic residue modulo {17}, but I’ll have to check on that.)

5. {L}-Functions

I’ll just briefly explain where all this is coming from, and omit lots of details (in part because I don’t know all of them). Let {E/\mathbb Q} be an elliptic curve with CM by {\mathcal O_K}. We can define an associated {L}-function

\displaystyle L(s, E/K) = \prod_\mathfrak p \left( 1 - \frac{a_\mathfrak p}{(\mathrm{N}\mathfrak p)^{s+\frac{1}{2}}} + \frac{1}{(\mathrm{N}\mathfrak p)^{2s}} \right)

(actually this isn’t quite true actually, some terms change for ramified primes {\mathfrak p}).

At the same time there’s a notion of a Hecke Grössencharakter {\xi} on a number field {K} — a higher dimensional analog of the Dirichlet charaters we used on {\mathbb Z} to filter modulo {q}. For our purposes, think of it as a multiplicative function which takes in ideals of {\mathcal O_K} and returns complex numbers of norm {1}. Like Dirichlet characters, each {\xi} gets a Hecke {L}-function

\displaystyle L(s, \xi) = \prod_\mathfrak p \left( 1 - \frac{\xi(\mathfrak p)}{(\mathrm{N}\mathfrak p)^s} \right)

which again extends to a meromorphic function on the entire complex plane.

Now the great theorem is:

Theorem 4 (Deuring)

Let {E/\mathbb Q} have CM by {\mathcal O_K}. Then

\displaystyle L(s,E/K) = L(s, \xi)L(s, \overline{\xi})

for some Hecke Grössencharakter {\xi}.

Using the definitions given above and equating the Euler products at an unramified {\mathfrak p} gives

\displaystyle 1 - \frac{a_\mathfrak p}{(\mathrm{N}\mathfrak p)^{s+\frac{1}{2}}} + \frac{1}{(\mathrm{N}\mathfrak p)^{2s}} = \left( 1 - \frac{\xi(\mathfrak p)}{(\mathrm{N}\mathfrak p)^s} \right) \left( 1 - \frac{\overline{\xi(\mathfrak p)}}{(\mathrm{N}\mathfrak p)^s} \right)

Upon recalling that {a_\mathfrak p = 2 \sqrt{\mathrm{N}\mathfrak p} \cos \theta_\mathfrak p}, we derive

\displaystyle \xi(\mathfrak p) = \exp(\pm i \theta_\mathfrak p).

This is enough to determine the entire {\xi} since {\xi} is multiplicative.

So this is the result: let {E/\mathbb Q} be an elliptic curve of conductor {N}. Given our quadratic number field {K}, we define a map {\xi} from prime ideals of {\mathcal O_K} to the unit circle in {\mathbb C} by

\displaystyle \mathfrak p \mapsto \begin{cases} \exp(\pm i \theta_\mathfrak p) & \gcd(\mathrm{N}\mathfrak p, N) = 1 \\ 0 & \gcd(\mathrm{N}\mathfrak p, N) > 1. \end{cases}

Thus {\xi} is a Hecke Grössencharakter for some choice of {\pm} at each {\mathfrak p}.

It turns out furthermore that {\xi} has frequency {1}, which roughly means that the argument of {\xi\left( (\pi) \right)} is related to {1} times the argument of {\pi} itself. This fact is what explains the mysterious connection between the {a_p} and the solutions above.

6. Linnik-Type Result

With this in mind, I can now frame the main question: suppose we have an interval {[\alpha, \beta] \subset [0,\pi]}. What’s the first prime {p} such that {\theta_p \in [\alpha, \beta]}? We’d love to have some analog of Linnik’s Theorem here.

This was our project and the REU, and Ashvin, Peter and I proved that

Theorem 5

If a rational {E} has CM then the least prime {p} with {\theta_p \in [\alpha,\beta]} is

\displaystyle \ll \left( \frac{N}{\beta-\alpha} \right)^A.

I might blog later about what else goes into the proof of this. . . but Deuring’s result is one key ingredient, and a proof of an analogous theorem for non-CM curves would have to be very different.

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