# 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.

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.

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.