## March 10, 2024

### Counting Points on Elliptic Curves (Part 1)

#### Posted by John Baez

You’ve probably heard that there are a lot of deep conjectures about $L$-functions. For example, there’s the Langlands program. And I guess the Riemann Hypothesis counts too, because the Riemann zeta function is the grand-daddy of all $L$-functions. But there’s also a million-dollar prize for proving the Birch-Swinnerton–Dyer conjecture about $L$-functions of elliptic curves. So if you want to learn about this stuff, you may try to learn the definition of an $L$-function of an elliptic curve.

But in many expository accounts you’ll meet a big roadblock to understanding.

The $L$-function of elliptic curve is often written as a product over primes. For most primes the factor in this product looks pretty unpleasant… but worse, for a certain finite set of ‘bad’ primes the factor looks completely different, in one of 3 different ways. Many authors don’t explain why the $L$-function has this complicated appearance. Others say that tweaks must be made for bad primes to make sure the $L$-function is a modular form, and leave it at that.

I don’t think it needs to be this way.

For example, Wikipedia defines the $L$-function of an elliptic curve $E$ this way:

$L(E,s) = \prod_p L_p(E,s)^{-1}$ where $L_p(E,s) = \left\{ \begin{array}{lc} 1-a_p p^{-s}+p\cdot p^{-2s} & p \nmid N\\ 1 - a_p p^{-s} & p\mid N \; and \; p^2 \nmid N \\ 1 & p^2|N \end{array} \right.$ where in the case of good reduction $a_p$ is $p + 1 -$ (number of points of $E$ mod $p$), and in the case of multiplicative reduction $a_p$ is ±1 depending on whether E has split (plus sign) or non-split (minus sign) multiplicative reduction at $p$. A multiplicative reduction of curve $E$ by the prime $p$ is said to be split if $-c_6$ is a square in the finite field with $p$ elements.

I won’t list all the ways this is confusing and unpleasant. The article doesn’t say what $c_6$ is! I won’t either. I’ll just say that $p$ ranges over primes and $N$ is a number called the conductor of the elliptic curve. If you follow the link and read Wikipedia’s explanation of that, you’ll enter a whole new world of pain.

Surely this can’t be the fundamental definition of the $L$-function of an elliptic curve! How could such a messy, complicated thing be fundamental? Surely this should be a theorem, where we take some simpler definition and grind out its consequences in gory detail.

I won’t actually give you what I think is a better definition of the $L$-function of an elliptic curve. Sorry! I might someday, but I still need to check that it matches this definition.

As a small step in this direction, let’s unpack the mysteries of the ‘conductor’ and transform the above definition into something easier to understand. The definition will still be extremely unpleasant. But it will involve some nice concepts about curves.

For our purposes let’s start with a lowbrow definition of an elliptic curve. Take an equation

$y^2 = P(x)$

where $P$ is any cubic with integer coefficients. If $P$ does not have repeated complex roots, we call this an elliptic curve. The solutions $(x,y) \in \mathbb{C}^2$ will form a smooth manifold, and in fact a smooth affine variety. If we tack on one extra point, a so-called ‘point at infinity’, we get a smooth projective variety that’s shaped like a torus.

What if $P$ has repeated roots? Then the solutions will not form a smooth variety. Two things can go wrong: we can get a cusp, or a node. For example,

$y^2 = x^3$

has a cusp at the origin:

The cusp is the sharp pointy thing.

On the other hand,

$y^2 = x^3 + x^2$

has a node at the origin:

There’s a node where the curve crosses itself. Near the node the curve looks like a little X. In other words, there are two different tangent lines at a node.

So far I’ve been talking about complex solutions of

$y^2 = P(x)$

and drawing you real solutions. But because $P$ has integer coefficients, we can look at solutions in any field whatsoever! The $L$-function of an elliptic curve is a clever way of keeping track of how many solutions there are in every finite field. There’s one finite field $F_q$ for each prime power $q = p^n$.

The horrible definition of $L$-function given above only talks about the finite fields $F_p$, not the other finite fields involving powers of primes. That’s one reason it’s forced to be tricky and complicated. It’s trying to be ‘economical’, but premature economizing is one of the main causes of suffering in mathematics.

Okay, now let me unpack the above definition of $L$-function:

The $L$-function of an elliptic curve is defined to be $L(E,s) = \prod_p L_p(E,s)^{-1}$ where:

1) $L_p(E,s) = 1 - a_p p^{-s} + p^{1-2s}$ if $E$ has good reduction at $p$. This means that our elliptic curve $E$ remains smooth if we treat it as a curve over $\mathbb{F}_p$.

2) $L_p(E,s) = 1$ if $E$ has additive reduction at $p$. This means that $E$ gets a cusp if we treat $E$ as a curve over $\mathbb{F}_p$.

3) $L_p(E,s) = 1 - p^{-s}$ if $E$ has split multiplicative reduction at $p$. This means that we gets a node if we treat $E$ as a curve over $\mathbb{F}_p$, and the two tangent lines to this node have slopes that are defined in $\mathbb{F}_p$.

4) $L_p(E,s) = 1 + p^{-s}$ if $E$ has nonsplit multiplicative reduction at $p$. This means that we gets a node if we treat it as a curve over $\mathbb{F}_p$, but the two tangent lines to this node have slopes that are not defined in $\mathbb{F}_p$.

It’s still an annoying definition by cases, where each case involves an arbitrary-looking formula. Whenever a definition involves different cases, you should assume someone hasn’t precisely put their finger on the concept they’re after… or else they don’t want to tell you about it.

And of course we need purely algebraic definitions of ‘smooth’, ‘cusp’ and ‘node’ that work for algebraic curves over finite fields, for this definition to even make sense. But that’s what algebraic geometers are paid to do, so we can assume that’s been done.

More confusing to me was case 4). What the heck is a line whose slope is not defined? Here’s the deal: when you try to work out the slopes of the tangent lines to your node, you need to solve some polynomial equations. Since $\mathbb{F}_p$ is not algebraically closed, you may be unable to solve these equations in $\mathbb{F}_p$. That’s what happens in case 4). But you can always solve these equations in some algebraic extension of $\mathbb{F}_p$: that is, some bigger field $\mathbb{F}_q$ where $q = p^n$.

Next time I want to give some examples of reducing elliptic curves mod $p$. I want to illustrate some things that can happen, and how the different cases above affect patterns in the count of points not just over $\mathbb{F}_p$ but also over $\mathbb{F}_q$ where $q$ is any power of $p$.

I also want to say what this ‘additive’ and ‘multiplicative’ jargon is all about!

Posted at March 10, 2024 9:08 PM UTC

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### Another form of the definition

Let $a_p := p+1-|E(\mathbb{F}_p)|$ and let $\Delta$ denote the discriminant. The L-function of $E/\mathbb{Q}$ is $L(s,E) := \prod_{p|\Delta} \frac{1}{1-a_p/p^s} \prod_{p \nmid \Delta} \frac{1}{1-a_p/p^s + p^{1-2s}}.$ (See (10.9) of https://doi.org/10.2307/j.ctv346st5.16.) This seems pretty clean to me given the fact that it’s (ugh) number theory. Taniyama-Shimura boils down to the existence of a newform of weight two for $\Gamma_0(N)$ with coefficients equal to $a_p$ for $p \mid N$. Also pretty clean IMO considering.

Posted by: Steve Huntsman on March 11, 2024 4:06 PM | Permalink | Reply to this

### Re: Another form of the definition

Thanks, Steve! I really enjoyed the early chapters of Knapp’s book Elliptic Curves in earlier stages of learning about the subject. I wrote about it in “week13” of This Week’s Finds. I see now it’s time for me to delve into the later chapters! He’s generally very clear so I’m not shocked that he gives a cleaner definition of the $L$-function of an elliptic curve than some. Thanks for pointing it out — time to start reading.

Nonetheless, I believe this is still the result of taking a simpler definition and working out the consequences. I shouldn’t count my chickens before they’ve hatched, but I’m getting ready to check my guess about how this works.

I’m also waiting for some super-expert to descend from the clouds and say “everyone knows there’s a simpler-looking definition; it’s just harder to do computations with it.”

Posted by: John Baez on March 11, 2024 5:33 PM | Permalink | Reply to this

### Re: Another form of the definition

The preface of a math book I once read recounted an old joke that the typical theorem says that something you don’t understand is equal to something else that you can’t compute. I find there’s a lot of truth to that, and if you reformulate it contrapositively, then it helps to understand the point of a lot of theorems.

That viewpoint has also encouraged me to approach exposition and teaching by starting with definitions that you can understand, and then proving afterward that they’re equivalent to something that’s easier to compute with but less well-motivated. I can get really sidetracked when I’m trying to learn something new and I can’t see why the definitions are what they are.

Posted by: Mark Meckes on March 11, 2024 8:39 PM | Permalink | Reply to this
Read the post Counting Points on Elliptic Curves (Part 2)
Weblog: The n-Category Café
Excerpt: Four things can happen when you take an elliptic curve with integer coefficients and look at it over a finite field. There's good reduction, bad reduction, ugly reduction and weird reduction. Let's see examples of these four cases, and how they affec...
Tracked: March 13, 2024 9:28 PM
Read the post The Modularity Theorem as a Bijection of Sets
Weblog: The n-Category Café
Excerpt: Bruce Bartlett floats a version of the Modularity Theorem for elliptic curves that frames it as an explicit bijection between sets, and has a question for the experts.
Tracked: April 20, 2024 6:17 PM

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