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March 23, 2024

Counting Points on Elliptic Curves (Part 3)

Posted by John Baez

In Part 1 of this little series I showed you Wikipedia’s current definition of the LL-function of an elliptic curve, and you were supposed to shudder in horror. In this definition the LL-function is a product over all primes pp. But what do we multiply in this product? There are 4 different cases, each with its own weird and unmotivated formula!

In Part 2 we studied the 4 cases. They correspond to 4 things that can happen when we look at our elliptic curve over the finite field 𝔽 p\mathbb{F}_{p}: it can stay smooth, or it can become singular in 3 different ways. In each case we got a formula for number of points the resulting curve over the fields 𝔽 p k\mathbb{F}_{p^k}.

Now I’ll give a much better definition of the LL-function of an elliptic curve. Using our work from last time, I’ll show that it’s equivalent to the horrible definition on Wikipedia. And eventually I may get up the nerve to improve the Wikipedia definition. Then future generations will wonder what I was complaining about.

I want to explain the LL-function of an elliptic curve as simply as possible — thus, with a minimum of terminology and unmotivated nonsense.

The LL-function of an elliptic curve is a slight tweak of something more fundamental: its zeta function. So we have to start there.

The zeta function of an elliptic curve

You can define the zeta function of any gadget SS that assigns a finite set S(R)S(R) to any finite commutative ring RR. It goes like this:

ζ S(s)= n=1 |Z S(n)|n!n s \zeta_S(s) = \sum_{n = 1}^\infty \frac{|Z_S(n)|}{n!} n^{-s}

where ss is a complex number and the sum will converge if Re(s)Re(s) is big enough.

What’s Z S(n)Z_S(n)? A ring that’s a finite product of finite fields is called a finite semisimple commutative ring. An element of Z S(n)Z_S(n) is a way to make the set {1,,n}\{1, \dots, n\} into a finite semisimple commutative ring, say RR, and choose an element of S(R)S(R).

So, to define the zeta function of an elliptic curve, we just need a way for an elliptic curve EE to assign a finite set E(R)E(R) to any finite commutative ring RR. This is not hard. By an elliptic curve I simply mean an equation

y 2=P(x) y^2 = P(x)

where PP is a cubic equation with integer coefficients and distinct roots. For any finite commutative ring RR, this equation will have a finite set of solutions in RR. We take those and one extra ‘point at infinity’ to be the points of our set E(R)E(R).

Then the zeta function of an elliptic curve EE is

ζ E(s)= n=1 |Z E(n)|n!n s \zeta_E(s) = \sum_{n = 1}^\infty \frac{|Z_E(n)|}{n!} n^{-s}

The L-function of an elliptic curve

Later today we will calculate the zeta function of an elliptic curve. And we’ll see that it always has a special form:

ζ E(s)=ζ(s)ζ(s1)somerationalfunctionofs \zeta_E(s) = \frac{ \zeta(s) \zeta(s - 1)}{some \; rational \; function \; of \; s}

where ζ\zeta is the Riemann zeta function. The denominator here is called the L-function of our elliptic curve, L(E,s)L(E,s). That’s all there is to it!

In short:

L(E,s)=ζ(s)ζ(s1)ζ E(s) L(E,s) = \frac{ \zeta(s) \zeta(s - 1)}{\zeta_E(s)}

You should think of the LL-function as the ‘interesting part’ of the zeta function of the elliptic curve — but flipped upside down, just to confuse amateurs. That’s also why we write n sn^{-s} in the formula for the zeta function instead of n sn^s: it’s a deliberately unnatural convention designed to keep out the riff-raff.

Arbitrary conventions aside, I hope you see the LL-function of an elliptic curve is a fairly simple thing. You might wonder why the zeta function is defined as it is, and why the zeta function of the elliptic curve has a factor of ζ(s)ζ(s1)\zeta(s) \zeta(s-1) in it. Those are very good questions, with good answers. But my point is this: all the gory complexity of the LL-function arises when we actually try to compute it more explicitly.

Now let’s do that.

The Euler product formula

An elliptic curve EE gives a finite set E(R)E(R) for each finite semisimple commutative ring RR. We need to count these sets to compute the zeta function or LL-function of our elliptic curve. But it’s not hard to see that

E(R×R)E(R)×E(R) E(R \times R') \cong E(R) \times E(R')

Since every finite semisimple commutative ring is a product of finite fields, this lets us focus on counting E(R)E(R) when RR is a finite field. And since every finite field has a prime power number of elements, we can tackle this counting problem ‘one prime at a time’.

If we carry this through, we get an interesting formula for the zeta function of an elliptic curve. In fact it’s a very general thing:

Euler Product Formula. Suppose SS is any functor from finite commutative rings to finite sets such that S(R×R)S(R)×S(R)S(R \times R') \cong S(R) \times S(R'). Then

ζ S(s)= pexp( k=1 |S(𝔽 p k)|kp ks) \zeta_S(s) = \prod_p \exp \left( \sum_{k = 1}^\infty \frac{|S(\mathbb{F}_{p^k})|}{k} p^{-k s} \right)

where we take the product over all primes pp, and 𝔽 p k\mathbb{F}_{p^k} is the field with p kp^k elements.

I wrote up a proof here:

so check it out if you want. I was not trying to make the argument look as simple as possible, but it’s really quite easy given what I’ve said: you can probably work it out yourself.

So: the zeta function of an elliptic curve EE is a product over primes. The factor for the prime pp is called the local zeta function

Z p(E,s)=exp( k=1 |E(𝔽 p k)|kp ks) Z_p(E,s) = \exp \left( \sum_{k = 1}^\infty \frac{|E(\mathbb{F}_{p^k})|}{k} p^{-k s} \right)

To compute this, we need to know the numbers |E(𝔽 p k)||E(\mathbb{F}_{p^k})|. Luckily we worked these out last time! But there are four cases.

In every case we have

|E(𝔽 p k)|=p k+1+c(p,k) |E(\mathbb{F}_{p^k})| = p^k + 1 + c(p,k)

where c(p,k)c(p,k) is some sort of ‘correction’. If the correction c(p,k)c(p,k) is zero, we get

Z p(E,s) = exp( k=1 p k+1kp ks) = exp(ln(1p s+1)ln(1p s)) = 1(1p s+1)(1p s) \begin{array}{ccl} Z_p(E,s) &=& \displaystyle{ \exp \left(\sum_{k = 1}^\infty \frac{p^k + 1}{k} p^{-k s} \right) } \\ \\ &=& \displaystyle{ \exp \left( -\ln(1 - p^{-s + 1}) - \ln(1 - p^{-s}) \right) } \\ \\ &=& \displaystyle{ \frac{1}{(1 - p^{-s + 1})(1 - p^{-s}) } } \end{array}

I did the sum pretty fast, but not because I’m good at sums — merely to keep you from getting bored. To do it yourself, all you need to know is the Taylor series for the logarithm.

To get the zeta function of our elliptic curve we multiply all the local zeta functions Z p(E,s)Z_p(E,s). So if all the corrections c(p,k)c(p,k) were zero, we’d get

Z(E,s)= p11p s+1 p11p s=ζ(s1)ζ(s) Z(E,s) = \prod_p \frac{1}{1 - p^{-s + 1}} \prod_p \frac{1}{1 - p^{-s} } = \zeta(s-1) \zeta(s)

Here I used the Euler product formula for the Riemann zeta function.

This is precisely why folks define the LL-function of an elliptic curve to be

L(E,s) 1=ζ E(s)ζ(s)ζ(s1) L(E,s)^{-1} = \frac{\zeta_E(s)}{ \zeta(s) \zeta(s - 1)}

It lets us focus on the effect of the corrections!. Well, it doesn’t explain that stupid reciprocal on the left-hand side, which is just a convention — but apart from that, we’re taking the zeta function of the elliptic curve and dividing out by what we’d get if all the corrections c(p,k)c(p,k) were zero. So, if you think about it a bit, we have

L(E,s) 1= pexp( k=1 c(p,k)kp ks) L(E,s)^{-1} = \prod_p \exp \left( \sum_{k = 1}^\infty \frac{c(p,k)}{k} p^{-k s} \right)

It’s like the Euler product formula for the zeta function, but using only the corrections c(p,k)c(p,k) instead of the full count of points |E(𝔽 p k)||E(\mathbb{F}_{p^k})|.

As you can see, the LL-function is a product of local L-functions

L p(E,s) 1=exp( k=1 c(p,k)kp ks) L_p(E,s)^{-1} = \exp \left( \sum_{k = 1}^\infty \frac{c(p,k)}{k} p^{-k s}\right)

So let’s work those out! There are four cases.

The local zeta function of an elliptic curve: additive reduction

If our elliptic curve gets a cusp over 𝔽 p\mathbb{F}_p, we say it has additive reduction. In this case we saw in Theorem 2 last time that

|E(𝔽 p k)|=p k+1 |E(\mathbb{F}_{p^k})| = p^k + 1

So in this case the correction vanishes:

c(p,k)=0 c(p,k) = 0

This makes the local LL-function very simple:

L p(E,s) 1=exp( k=1 c(p,k)kp ks)=1 L_p(E,s)^{-1} = \exp \left( \sum_{k = 1}^\infty \frac{c(p,k)}{k} p^{-k s}\right) = 1

The local zeta function of an elliptic curve: split multiplicative reduction

If our elliptic curve gets a node over 𝔽 p\mathbb{F}_p and the two lines tangent to this node have slopes defined in 𝔽 p\mathbb{F}_p, we say our curve has split multiplicative reduction. In this case we saw in Theorem 3 last time that

|E(𝔽 p k)|=p k |E(\mathbb{F}_{p^k})| = p^k

So in this case, the correction is 1-1:

c(p,k)=1 c(p,k) = -1

This gives

L p(E,s) 1 = exp( k=1 1kp ks) = exp(ln(1p s)) = 1p s \begin{array}{ccl} L_p(E,s)^{-1} &=& \displaystyle{ \exp \left( \sum_{k = 1}^\infty \frac{1}{k} p^{-k s}\right) } \\ \\ &=& \displaystyle{ \exp \left( \ln(1 - p^{-s}) \right) } \\ \\ &=& 1 - p^{-s} \end{array}

Again I used my profound mastery of Taylor series of the logarithm to do the sum.

The local zeta function of an elliptic curve: split multiplicative reduction

If our elliptic curve gets a node over 𝔽 p\mathbb{F}_p and the two lines tangent to this node have slopes that are not defined in 𝔽 p\mathbb{F}_p, we say our curve has nonsplit multiplicative reduction. In this case we saw in Theorem 4 last time that

|E(𝔽 p k)|=p k+1(1) k |E(\mathbb{F}_{p^k})| = p^k + 1 - (-1)^k

In this case the correction is more interesting:

c(p,k)=(1) k c(p,k) = -(-1)^k

This gives

L p(E,s) 1 = exp( k=1 (1) kkp ks) = exp(ln(1+p s)) = 1+p s \begin{array}{ccl} L_p(E,s)^{-1} &=& \displaystyle{ \exp \left( -\sum_{k = 1}^\infty \frac{(-1)^k}{k} p^{-k s}\right) } \\ \\ &=& \displaystyle{ \exp \left( \ln(1 + p^{-s}) \right) } \\ \\ &=& 1 + p^{-s} \end{array}

Again, I just used the Taylor series of the log function.

The local zeta function of an elliptic curve: good reduction

If our elliptic curve stays smooth over 𝔽 p\mathbb{F}_p, we say it has good reduction. Ironically this gives the most complicated local zeta function. In Theorem 1 last time we saw

|E(𝔽 p k)|=p kα kα¯ k+1 |E(\mathbb{F}_{p^k})| = p^k - \alpha^k - \overline{\alpha}^k + 1

where α\alpha is a complex number with αα¯=p\alpha \overline{\alpha} = p. We didn’t prove this, we literally just saw it: it’s a fairly substantial result due to Hasse.

So, in this case the correction is

c(p,k)=α k+α¯ k c(p,k) = \alpha^k + \overline{\alpha}^k

This gives

L p(E,s) 1 = exp( k=1 α k+α¯ kkp ks) = exp(ln(1αp s)+ln(1α¯p s)) = (1αp s)(1α¯p s) \begin{array}{ccl} L_p(E,s)^{-1} &=& \displaystyle{ \exp \left( -\sum_{k = 1}^\infty \frac{\alpha^k + \overline{\alpha}^k}{k} p^{-k s}\right) } \\ \\ &=& \displaystyle{ \exp \left( \ln\left(1 - \alpha p^{-s}\right) \; + \; \ln\left(1 - \overline{\alpha} p^{-s}\right) \right) } \\ \\ &=& (1 - \alpha p^{-s})(1 - \overline{\alpha} p^{-s}) \end{array}

Again I just used the Taylor series of the log function. I’m sure glad I went to class that day.

But we can get a bit further using αα¯=p\alpha \overline{\alpha} = p:

L p(E,s) 1 = (1αp s)(1α¯p s) = 1(α+α¯)p s+p 12s \begin{array}{ccl} L_p(E,s)^{-1} &=& (1 - \alpha p^{-s})(1 - \overline{\alpha} p^{-s}) \\ &=& 1 - (\alpha + \overline{\alpha})p^{-s} + p^{1-2s} \end{array}

At this point people usually notice that

|E(𝔽 p)|=pαα¯+1 |E(\mathbb{F}_{p})| = p - \alpha - \overline{\alpha} + 1

so

α+α¯=p+1|E(𝔽 p)| \alpha + \overline{\alpha} = p + 1 - |E(\mathbb{F}_{p})|

Thus, you can compute this number using just the number of points of our curve over 𝔽 p\mathbb{F}_p. And to be cute, people call this number something like a p(E)a_p(E). So in the end, for elliptic curves of good reduction over the prime pp we have

L p(E,s) 1=1a p(E)p s+p 12s L_p(E,s)^{-1} = 1 - a_p(E) p^{-s} + p^{1-2s}

Whew, we’re done!

The L-function of an elliptic curve, revisited

Okay, now we can summarize all our work in an explicit formula for the LL-function of an elliptic curve.

Theorem. The LL-function of an elliptic curve EE equals

L(E,s)= pL p(E,s) 1 L(E,s) = \prod_p L_p(E,s)^{-1}

where:

1) L p(E,s)=1a p(E)p s+p 12sL_p(E,s) = 1 - a_p(E) p^{-s} + p^{1-2s} if EE remains smooth over 𝔽 p\mathbb{F}_p. Here a p(E)a_p(E) is p+1p + 1 minus the number of points of EE over 𝔽 p\mathbb{F}_p.

2) L p(E,s)=1L_p(E,s) = 1 if EE gets a cusp over 𝔽 p\mathbb{F}_p.

3) L p(E,s)=1p sL_p(E,s) = 1 - p^{-s} if EE gets a node over 𝔽 p\mathbb{F}_p, and the two tangent lines to this node have slopes that are defined in 𝔽 p\mathbb{F}_p.

4) L p(E,s)=1+p sL_p(E,s) = 1 + p^{-s} if EE we gets a node over 𝔽 p\mathbb{F}_p, but the two tangent lines to this node have slopes that are not defined in 𝔽 p\mathbb{F}_p.

My god! This is exactly what I showed you in Part 1. So this rather elaborate theorem is what some people run around calling the definition of the LL-function of an elliptic curve!

Posted at March 23, 2024 1:00 AM UTC

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Re: Counting Points on Elliptic Curves (Part 3)

Is this really called a “simple ring”? Wouldn’t semisimple be a better name?

Great stuff by the way.

Posted by: Allen Knutson on March 23, 2024 4:38 AM | Permalink | Reply to this

Re: Counting Points on Elliptic Curves (Part 3)

Whoops, I meant semisimple. Of course a semisimple ring is defined as one for which every ideal has a complementary ideal. But you can show finite semisimple commutative rings are just finite products of finite fields, so I took that as a definition for today.

Posted by: John Baez on March 23, 2024 5:52 AM | Permalink | Reply to this

Re: Counting Points on Elliptic Curves (Part 3)

Lovely followup to that 2005 TWF article!

Posted by: Mike Stay on March 24, 2024 6:43 AM | Permalink | Reply to this

Re: Counting Points on Elliptic Curves (Part 3)

… and its predecessor.

Posted by: Mike Stay on March 24, 2024 6:45 AM | Permalink | Reply to this

Re: Counting Points on Elliptic Curves (Part 3)

Thanks!

Yes, this is a slow-burning project. Back in 2005, in “week218” of This Week’s Finds I introduced the zeta species of a \mathbb{Z}-set — meaning a set XX equipped with an action of the group \mathbb{Z}, or in other words a bijection f:XXf: X \to X.

Given a \mathbb{Z}-set XX, a ZX-structure on a finite set SS is a way of making SS into a \mathbb{Z}-set over XX — that is, making it into a \mathbb{Z}-set and choosing a map of \mathbb{Z}-sets SXS \to X. This structure on finite sets defines a species, the zeta species Z XZ_X. And I computed the generating function of this species to be

ζ X(x)= k=1 |fix(f k)|x kk\zeta_X(x) = \sum_{k = 1}^\infty |fix(f^k)| \frac{x^k}{k}

Here fix(f k)fix(f^k) is the set of fixed points of f k:XXf^k: X \to X. There’s no need for ff to be a bijection here, any endofunction will do, so I really should have talked about the zeta function of an \mathbb{N}-set, but the alliteration was irresistible.

This sort of zeta function is general enough to include the ‘local zeta functions’ ζ p(V,x)\zeta_p(V,x), one for each prime, of an algebraic variety or more generally a scheme VV. The idea is that a scheme over \mathbb{Z} has a set SS of points over the algebraic closure 𝔽¯ p\overline{\mathbb{F}}_p, and the Frobenius defines an endofunction f:SSf: S \to S. So we just plug that into the above definition of zeta function.

We can then multiply all these local zeta functions in an appropriate way to get the so-called Hasse–Weil zeta function of the scheme. And a special case is the zeta function of an elliptic curve, which I discussed here.

But in 2010, James Dolan figured out a nice species whose generating function gives the Hasse–Weil zeta function of a scheme directly, in one fell swoop — rather than as a product over primes. I explained this in “week300” and wrote it up more carefully here:

This includes a proof that the Hasse–Weil species factors into a product of species, one for each prime.

The idea is simple: given a scheme VV over \mathbb{Z}, we define a species Z VZ_V such that a Z VZ_V-structure on a finite set is a way of making it into a semisimple commutative ring RR and choosing an RR-point of VV. Z VZ_V is the Hasse–Weil species of VV.

But I was still a bit worried, because the usual definition of the zeta function of an elliptic curve writes it as a product of factors, one for each prime, and — much worse! — defines the factors in different ways depending on whether the elliptic curve has good reduction, additive reduction, split multiplicative reduction or non-split multiplicative reduction over that prime. So I was afraid that my simple uniform approach was not catching some subtleties.

I shelved the project until recently, when I decided to bite the bullet and think about all 4 kinds of reduction. And it turned out that my simple uniform approach works just fine.

Why did I get more interested in doing these calculations? Mainly because learned that the formula for the number of points on a elliptic curve over a finite field is a baby version of the Weil conjectures, which can seen as a baby version of the Riemann hypothesis. I wrote some expository stuff about that here on the nn-Café in 2019:

Then I gave a talk about it in 2022:

  • Motivating motives, in “Grothendieck’s Approach to Mathematics”, Chapman University.

and wrote an expository paper for the conference proceedings:

This paper summarize all 3 blog articles on the Riemann Hypothesis and also my talk, so there’s less here than meets the eye.

In short: all this stuff is part of the same story! Much of it is just the record of me trying to understand known stuff, but the material on species seems to be original, and I want to write it up properly now.

Posted by: John Baez on March 24, 2024 10:32 PM | Permalink | Reply to this

Re: Counting Points on Elliptic Curves (Part 3)

In case you are interested in typos: on p. 3 of “Motivating motives”, there is “approximates π(n)\pi(n) fairly well well”.

On p. 4, “There is thus is ample reason” and “So far, numerical computations far suggest”. On p. 7, “the the first cohomology group has dimension 2g2g.” On p. 11, “The picture … was drawn for this pape”.

The link on p. 12 to Manin’s paper seems to be one link, but shows up as two separate ones.

Posted by: L Spice on March 29, 2024 2:59 PM | Permalink | Reply to this

Re: Counting Points on Elliptic Curves (Part 3)

Yes, I’m always seeking to root out typos. Alas, this paper is already published. But I can still improve the arXiv version, and the version on my website. So thanks! I’ll do it right now, since it’s Friday afternoon and I’m just goofing off.

Posted by: John Baez on March 30, 2024 12:51 AM | Permalink | Reply to this

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