## April 19, 2024

### The Modularity Theorem as a Bijection of Sets

#### Posted by John Baez

guest post by Bruce Bartlett

John has been making some great posts on counting points on elliptic curves (Part 1, Part 2, Part 3). So I thought I’d take the opportunity and float my understanding here of the Modularity Theorem for elliptic curves, which frames it as an explicit bijection between sets. To my knowledge, it is not stated exactly in this form in the literature. There are aspects of this that I don’t understand (the explicit isogeny); perhaps someone can assist.

## Bijection statement

Here is the statement as I understand it to be, framed as a bijection of sets. My chief reference is the wonderful book Elliptic Curves, Modular Forms and their L-Functions by Álvaro Lozano-Robledo (and references therein), as well as the standard reference A First Course in Modular Forms by Diamond and Shurman.

I will first make the statement as succinctly as I can, then I will ask the question I want to ask, then I will briefly explain the terminology I’ve used.

Modularity Theorem (Bijection version). The following maps are well-defined and inverse to each other, and give rise to an explicit bijection of sets:

$\left\{\begin{array}{c} \text{Elliptic curves defined over}\: \mathbb{Q} \\ \text{with conductor}\: N \end{array} \right\} \: / \: \text{isogeny} \quad \leftrightarrows \quad \left\{ \begin{array}{c} \text{Integral normalized newforms} \\ \text{of weight 2 for }\: \Gamma_0(N) \end{array} \right\}$

• In the forward direction, given an elliptic curve $E$ defined over the rationals, we build the modular form $f_E(z) = \sum_{n=1}^\infty a_n q^n , \quad q=e^{2 \pi i z}$ where the coefficients $a_n$ are obtained by expanding out the following product over all primes as a Dirichlet series, $\prod_p \text{exp}\left( \sum_{k=1}^\infty \frac{|E(\mathbb{F}_{p^k})|}{k} p^{-k s} \right) = \frac{a_1}{1^s} + \frac{a_2}{2^s} + \frac{a_3}{3^s} + \frac{a_4}{4^s} + \cdots ,$ where $|E(\mathbb{F}_{p^k})|$ counts the number of solutions to the equation for the elliptic curve over the finite field $\mathbb{F}_{p^k}$ (including the point at infinity). So for example, as John taught us in Part 3, for good primes $p$ (which is almost all of them), $a_p = p + 1 - |\!E(\mathbb{F}_p)\!|.$ But the above description tells you how to compute $a_n$ for any natural number $n$. (By the way, the nontrivial content of the theorem is proving that $f_E$ is indeed a modular form for any elliptic curve $E$).

• In the reverse direction, given an integral normalized newform $f$ of weight $2$ for $\Gamma_0(N)$, we interpret it as a differential form on the genus $g$ modular surface $X_0(N)$, and then compute its period lattice $\Lambda \subset \mathbb{C}$ by integrating it over all the 1-cycles in the first homology group of $X_0(N)$. Then the resulting elliptic curve is $E_f = \mathbb{C}/\Lambda$.

## An explicit isogeny?

My question to the experts is the following. Suppose we start with an elliptic curve $E$ defined over $\mathbb{Q}$, then compute the modular form $f_E$, and then compute its period lattice $\Lambda$ to arrive at the elliptic curve $E' = \mathbb{C} / \Lambda$. The theorem says that $E$ and $E'$ are isogenous. What is the explicit isogeny?

## Explanations

• An elliptic curve is a complex curve $E \subset \mathbb{C}\mathbb{P}^2$ defined by a cubic polynomial $F(X,Y,Z)=0$ with rational coefficients, such that $E$ is smooth, i.e. the tangent vector $(\frac{\partial F}{\partial X}, \frac{\partial F}{\partial Y}, \frac{\partial F}{\partial Z})$ does not vanish at any point $p \in E$. If the coefficients are all rational, then we say that $E$ is defined over $\mathbb{Q}$. We can always make a transformation of variables and write the equation for $E$ in an affine chart in Weierstrass form, $y^2 = x^3 + A x + B.$ Importantly, every elliptic curve is isomorphic to one of the form $\mathbb{C} / \Lambda$ where $\Lambda$ is a rank 2 sublattice of $\mathbb{C}$. So, an elliptic curve is topologically a doughnut $S^1 \times S^1$, and it has an addition law making it into an abelian group.

• An isogeny from $E$ to $E'$ is a surjective holomorphic homomorphism. This is actually an equivalence relation on the class of elliptic curves.

• The conductor of an elliptic curve $E$ defined over the rationals is $N = \prod_p p^{f_p}$ where: $f_p = \begin{cases} 0 & \text{if}\:E\:\text{remains smooth over}\:\mathbb{F}_p \\ 1 & \text{if}\:E\:\text{gets a node over}\:\mathbb{F}_p \\ 2 & \text{if}\:E\:\text{gets a cusp over}\:\mathbb{F}_p\:\text{and}\: p \neq 2,3 \\ 2+\delta_p & \text{if}\:E\:\text{gets a cusp over}\:\mathbb{F}_p\:\text{and}\: p = 2\:\text{or}\:3 \end{cases}$ where $\delta_p$ is a technical invariant that describes whether there is wild ramification in the action of the inertia group at $p$ of $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the Tate module $T_p(E)$.

• The modular curve $X_0(N)$ is a certain compact Riemann surface which parametrizes isomorphism classes of pairs $(E,C)$ where $E$ is an elliptic curve and $C$ is a cyclic subgroup of $E$ of order $N$.The genus of $X_0(N)$ depends on $N$.

• A modular form $f$ for $\Gamma_0(N)$ of weight $k$ is a certain kind of holomorphic function $f : \mathbb{H} \rightarrow \mathbb{C}$. The number $N$ is called the level of the modular form.

• Every modular form $f(z)$ can be expanded as a Fourier series $f(z) = \sum_{n=0}^\infty a_n q^n, \quad q=e^{2 \pi i z}$ We say that $f$ is integral if all its Fourier coefficients $a_n$ are integers. We say $f$ is a cusp form if $a_0 = 0$. A cusp form is called normalized if $a_1 = 1$.

• Geometrically, a cusp form of weight $k$ can be interpreted as a holomorphic section of a certain line bundle $L_k$ over $X_0(N)$. Since $X_0(N)$ is compact, this implies that the vector space of cusp modular forms is finite-dimensional. (In particular, this means that $f$ is determined by only finitely many of its Fourier coefficients).

• In particular, $L_2$ is the cotangent bundle of $X_0(N)$. This means that the cusp modular forms for $\Gamma_0(N)$ of weight 2 can be interpreted as differential forms on $X_0(N)$. That is to say, they are things that can be integrated along curves on $X_0(N)$.

• If you have a modular form of level $M$ which divides $N$, then there is a way to build a new modular form of level $N$. We call level $N$ forms of this type old. They form a subspace of the vector space $S_2(\Gamma_0(N))$. If we’re at level $N$, then we are really interested in the new forms — these are the forms in $S_2(\Gamma_0(N))$ which are orthogonal to the old forms, with respect to a certain natural inner product.

• If you have a weight 2 newform $f$, and you interpret it as a differential form on $X_0(N)$, then the integrals of $f$ along 1-cycles $\gamma$ in $X_0(N)$ will form a rank-2 sublattice $\Lambda \subset \mathbb{C}$. (This may seem strange, since $X_0(N)$ has genus $g$, so you would expect the period integrals of $f$ to give a dense subset of $\mathbb{C}$, but that is the magic of being a newform: it only “sees” two directions in $H_1(X_0(N), \mathbb{Q})$).

• So, given a weight 2 newform $f$, we get a canonical integration map $I: X_0(N) \rightarrow \mathbb{C}/\Lambda$ obtained by fixing a basepoint $x_0 \in X_0(N)$ and then defining $I(x) = \int_{\gamma} f$ where $\gamma$ is any path from $x_0$ to $x$ in $X_0(N)$. The answer won’t depend on the choice of path, because different choices will differ by a 1-cycle, and we are modding out by the periods of 1-cycles!

• The Jacobian of a Riemann surface $X$ is the quotient group $\text{Jac}(X) = \Omega^1_\text{hol} (X)^\vee / H_1(X; \mathbb{Z})$ This is why one version of the Modularity Theorem says:

Modularity Theorem (Diamond and Shurman’s Version $J_C$). There exists a surjective holomorphic homomorphism of the (higher-dimensional) complex torus $\text{Jac}(X_0(N))$ onto $E$.

I would like to ask the same question here as I asked before: is there an explicit description of this map?

Posted at April 19, 2024 4:06 PM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3528

### Re: The Modularity Theorem as a Bijection of Sets

Great explanation! This really helps me see what I need to learn, to understand the Modularity Theorem and its proof.

Some questions, starting with some hard ones, and then moving on to some easier ones that reveal the true depths of my ignorance:

• Is this theorem the decategorified version of some deeper theorem? That is: can we boost your statement up to an equivalence of categories? There is a category of elliptic curves over $\mathbb{Q}$ with conductor and isogenies between them. It’s not a groupoid, but it may be a dagger category or something: whenever there’s an isogeny $f \colon E \to E'$ there’s an isogeny $g \colon E' \to E$ going back the other way, so ‘having an isogeny between them’ is an equivalence relation on elliptic curve. The left hand side of your statement of the Modularity Theorem works with the set of elliptic curves mod this equivalence relation. The right hand side is a set that doesn’t look like it comes from a category. But could there secretly be some notion of morphism between integral normalized newforms of weight 2 for $\Gamma_0(N)$? Perhaps one approach could be to note that these modular forms are Dirichlet generating functions of certain species, and there’s a category of species.

• Is there a way to state the theorem as a bijection involving Jacobians? E.g., do the Jacobians of moduli spaces $X_0(N)$ form some class of abelian varieties that we can characterize in some more intrinsic way?

• I don’t see why the integrals of a weight 2 newform around cycles in $X_0(N)$ lie in a lattice in $\mathbb{C}$, but that’s certainly cool. The rank of $H_1(X_0(N), \mathbb{Z})$, i.e. the genus of $X_0(N)$, varies in a rather complicated way recorded at A001617 in the On-Line Encyclopedia of Integer Sequences:

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 0, 2, 1, 2, 2, 3, 2, 1, 3, 3, 3, 1, 2, 4, 3, 3, 3, 5, 3, 4, 3, 5, 4, 3, 1, 2, 5, 5, 4, 4, 5, 5, 5, 6, 5, 7, 4, 7, 5, 3, 5, 9, 5, 7, 7, 9, 6, 5, 5, 8, 5, 8, 7, 11, 6, 7, 4, 9, 7, 11, 7, 10, 9, 9, 7, 11, 7, 10, 9, 11, 9, 9, 7, 7, 9, 7, 8, 15, …

I guess when we see a 0 or 1 in the above list, there are no weight 2 newforms of this level. Is that right?

Posted by: John Baez on April 20, 2024 8:19 AM | Permalink | Reply to this

### Re: The Modularity Theorem as a Bijection of Sets

• Is this theorem the decategorified version of some deeper theorem?

I like this idea, but perhaps it’s more along the lines of “hearing the shape of a drum”. (Indeed, building a modular form $f_E$ from an elliptic curve $E$ is very analogous to this, because we are literally building the modular form by stipulating its Fourier coefficients. In other words, $f_E$ is a function which hears the solution sets of $E$ over finite fields.)

In that problem (hearing the shape of the drum), I’m not aware that people use the approach “Given the sound of the drum, reconstruct the drum in a canonical way” but perhaps they do. I guess what I’m saying is, it is a fairly common pattern in maths to establish a bijection between isomorphism classes of a certain kind of mathematical structure and a discrete set Y. We don’t always feel the need to upgrade Y into a category and phrase this as an equivalence of categories, although sometimes we do.

But, I love the idea of thinking categorically about the category of elliptic curves and their isogenies. I would like to understand that aspect more cleanly.

• On why integrating newforms over cycles in $X_0(N)$ gives out a lattice in $\mathbb{C}$ (and not a dense set, which we’d expect with a general form, since the genus of $X_0(N)$ can be greater than 1) - this is the magic property of being a newform, see here.
Posted by: Bruce Bartlett on April 20, 2024 10:43 AM | Permalink | Reply to this

### Re: The Modularity Theorem as a Bijection of Sets

• Is this theorem the decategorified version of some deeper theorem?

Actually I’m now thinking this is true, in the manner I had guessed. We ordinarily think of an elliptic curve as having a Hasse–Weil zeta function

$\prod_p \text{exp}\left( \sum_{k=1}^\infty \frac{|E(\mathbb{F}_{p^k})|}{k} p^{-k s} \right)$

where $|E(\mathbb{F}_{p^k})|$ is the number of points of the elliptic curve $E$ over $\mathbb{F}_{p^k}$. But this is just a decategorification of the Hasse–Weil species

$\prod_p \text{exp}\left( \sum_{k=1}^\infty \frac{E(\mathbb{F}_{p^k})}{k} p^{-k s} \right)$

where $E(\mathbb{F}_{p^k})$ is the set of points of the elliptic curve over $\mathbb{F}_{p^k}$, and the formula can be thought of as just a funny way of listing all these sets, one for each $p$ and $k$. All this is worked out in more detail in my paper. An isogeny of elliptic curves should give a map between their sets of points, and thus a map between their Hasse–Weil species.

There is stuff to check, but I’m optimistic.

One thing I’m worried about is that in your blog article you’re using the Hasse–Weil zeta function of the elliptic curve, while I would have guessed the Modularity Theorem used the $L$-function of the elliptic curve. I explained how these two functions are related here. Briefly:

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

This matters for the Modularity Theorem because if one of these two functions is a modular form of weight $2$ and level $N$ the other is probably not. But once we get that straightened out, it may not matter so much in the categorification game, since the formula relating $L(E,s)$ and $\zeta_E(s)$ gives a bijection, so categorifying one of these functions (regarding it as encoding an isomorphism class of objects in some nondiscrete category) should be a way of categorifying the other.

Posted by: John Baez on April 20, 2024 7:01 PM | Permalink | Reply to this

### Re: The Modularity Theorem as a Bijection of Sets

John wrote:

• I don’t see why the integrals of a weight 2 newform around cycles in $X_0(N)$ lie in a lattice in $\mathbb{C}$, but that’s certainly cool. The rank of $H_1(X_0(N), \mathbb{Z})$, i.e. the genus of $X_0(N)$, varies in a rather complicated way recorded at A001617 in the On-Line Encyclopedia of Integer Sequences:

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 0, 2, 1, 2, 2, 3, 2, 1, 3, 3, 3, 1, 2, 4, 3, 3, 3, 5, 3, 4, 3, 5, 4, 3, 1, 2, 5, 5, 4, 4, 5, 5, 5, 6, 5, 7, 4, 7, 5, 3, 5, 9, 5, 7, 7, 9, 6, 5, 5, 8, 5, 8, 7, 11, 6, 7, 4, 9, 7, 11, 7, 10, 9, 9, 7, 11, 7, 10, 9, 11, 9, 9, 7, 7, 9, 7, 8, 15, …

I guess when we see a 0 or 1 in the above list, there are no weight 2 newforms of this level. Is that right?

It looks right! The first 1 on the list happens after 10 zeros, and my old friend Steve Finch says

The first nonzero weight 2 cusp form has level 11:

$f(z) = q \prod_{n = 1}^\infty (1 - q^n)^2 (1 - q^{11 n})^2$

where $q = exp(2\pi i z)$ as usual.

Posted by: John Baez on April 21, 2024 10:01 AM | Permalink | Reply to this

### Re: The Modularity Theorem as a Bijection of Sets

Yes - that’s right. I like your species idea! See my comment below. Your $f$ above corresponds to the one I wrote down.

Posted by: Bruce Bartlett on April 21, 2024 7:02 PM | Permalink | Reply to this

### Re: The Modularity Theorem as a Bijection of Sets

John Cremona’s book Algorithms for Modular Elliptic Curves gives explicit methods for obtaining an elliptic curve from the action of the Hecke algebra on $H_1(\Gamma_0(N)/\mathbf{Z})$, for which the cusp forms appear as eigenforms. Elliptic curves over $\mathbf{Q}$ come from dimension 1 eigenspaces. The method yields a specific elliptic curve within the isogeny class, the so-called “strong Weil curve”.

Posted by: Richard Pinch on April 21, 2024 12:56 PM | Permalink | Reply to this

### Re: The Modularity Theorem as a Bijection of Sets

Thanks for reminding me about this great book. Let’s see if I understand his tables correctly. It should all tie in with what John is saying (making the correspondence functorial) in a nice way.

Table 1 in John Cremona’s book is a list of all isomorphism classes (over $\mathbb{Q}$, I think) of elliptic curves defined over $\mathbb{Q}$. They are ordered by their conductor.

We start at $N=11$. He is saying that for this conductor there is only 1 isogeny class, and in this isogeny class there are 3 isomorphism classes of elliptic curves:

$\text{A1:} \quad y^2 + y = x^3 - x^2 -10x -20$

$\text{A2:} \quad y^2 + y = x^3 - x^2 -7820x -263580$

$\text{A3:} \quad y^2 + y = x^3 - x^2$

All 3 of these curves correspond, under the Modularity Theorem, to a single newform $f$ (see pg 55 in his book):

$f = q - 2q^2 - q^3 + 2q^4 + q^5 + 2q^6 + \cdots$

The curve A1 is called the ‘strong Weil curve’, which I believe means it is, by definition, the one that is isomorphic to $\mathbb{C}/Lambda$ where $\Lambda$ are the periods of $f$.

Now, John’s species technology is hopefully going to allow us to resolve these 3 curves at the level of the `right hand side’ of the Modularity Theorem… in other words we should be able to distinguish them using functoriality. (They have the same solution set counts over each $p$, but they can hopefully be distinguished by functoriality…)

Posted by: Bruce Bartlett on April 21, 2024 6:57 PM | Permalink | Reply to this

### Re: The Modularity Theorem as a Bijection of Sets

You seem to have upped the ante, Bruce! I never intended my categorification of the Hasse–Weil zeta function (or L-function) of an elliptic curve to be a finer invariant than the usual Hasse–Weil zeta function. I wasn’t trying to do anything truly impressive. I was humbly seeking to reinterpret the Modularity Theorem as coming from some sort of functor between categories, preferably an equivalence but really just any functor.

But anyway, let’s think about this. Say we have a category $\mathsf{C}$ of

• elliptic curves over $\mathbb{Q}$
• isogenies between these

and a category $\mathsf{S}$ of

• species, i.e., functors from the groupoid of finite sets to $\mathsf{Set}$
• natural transformations between these

Suppose there’s a functor

$Z: \mathsf{C} \to \mathsf{S}$

sending each elliptic curve $E$ to the multiplicative species $Z(E)$ such that for any finite set $X$, $Z(E)(X)$ is the set of ways of making $X$ into a commutative semisimple ring $R$ and then choosing a point of $E(R)$. Here $E(R)$ is the set of $R$-points of the elliptic curve $E$.

(This is my hoped-for description of the Hasse–Weil species. The main problem is that the Hasse-Weil species works for schemes over $\mathbb{Z}$, but $E$ is an elliptic curve over $\mathbb{Q}$, so I don’t really know what an $R$-point of $E$ is! This problem also afflicts your description of the zeta function of an elliptic curve over $\mathbb{Q}$. I think the standard solution is to use a canonical ‘model’ of $E$ which is a scheme over $\mathbb{Z}$, called the Nerón model. However, I am very fuzzy about the details of this. For example, does an isogeny between elliptic curves over $\mathbb{Q}$ induce a map between their Nerón models? I think we may need that for my functor $F$ to be well-defined on morphisms.)

If this works, and if an isogeny $f \colon E \to E'$ induces a map on $R$-points, say

$R(f) \colon R(E) \to R(E'),$

then typically this map will not be a bijection, since an isogeny is a kind of covering. So I expect that that my hoped-for functor $Z$ will give a map of species

$Z(f) \colon Z(E) \to Z(E')$

that is not an isomorphism. Remember, $Z(E)$ knows the $R$-points $R(E)$ for all finite fields $R$ (these being the finite commutative semisimple rings with a prime power number of elements), so $Z(f)$ knows all the maps $R(f) \colon R(E) \to R(E')$.

This does not yet do what you want. I haven’t yet given an example of isogenous but not isomorphic elliptic curves $E, E'$ for which $Z(E) \ncong Z(E')$. I’ve just given a plausible argument that if $f$ is an isogeny that is not an isomorphism, $Z(f) \colon Z(E) \to Z(E')$ is not an isomorphism. But that’s a first step.

(In the process I’ve pointed out a more fundamental worry, which I want to address first!)

Posted by: John Baez on April 22, 2024 1:15 PM | Permalink | Reply to this

### Re: The Modularity Theorem as a Bijection of Sets

I think what you are saying will work out nicely! There isn’t a problem of going from $\mathbb{Q}$ to $\mathbb{Z}$. In fact, I prefer to think phrase everything in terms of $\mathbb{Z}$ right from the start.

To ease your concerns, look at Chapter III of Cremona. He very nicely and explicitly explains in the opening two pages that an elliptic curve over $\mathbb{Q}$ has a unique reduced minimal model over $\mathbb{Z}$. I’m sure that’s the one we want to work with.

Posted by: Bruce Bartlett on April 22, 2024 8:13 PM | Permalink | Reply to this

### Re: The Modularity Theorem as a Bijection of Sets

Let me update the above after studying John and James’ article on Dirichlet species and the Hasse-Weil zeta function. The big insight we get there is that they give a direct conceptual understanding of the coefficients $a_n$ (for any $n$) of the $q$-series

(1)$f(z) = \sum_{n=1}^\infty a_n q^n$

that we attach to an elliptic curve $E$. You see, as John explained, normally this is a bit messy. People will define these coefficients when $n$ is a prime number $p$ first, namely

(2)$a_p = \begin{cases} p + 1 - |E(\mathbb{F}_p)| & \: \text{if E has good reduction at p} \\ 1 & \: \text{if E has split reduction at p } \\ -1 & \: \text{if E has non-split reduction at p} \\ 0 & \: \text{if E has additive reduction at p} \end{cases}$

Then, if $n=p^r$ for some $r \geq 1$, we define $a_{p^r}$ recursively using the relation

(3)$a_p a_{p^r} = a_{p^{r+1}} + p a_{p^{r-1}} \: \text{if E has good reduction at} \: p$

and

(4)$a_{p^r} = (a_p)^r \: \text{otherwise.}$

Finally, if $(m,n)=1$ then we define

(5)$a_{m n} = a_m a_n \, .$

Quite a mouthful! It seems quite technical. But, the insight from their article is that in fact we have a direct ‘highbrow’ canonical interpretation of $a_n$, namely

(6)$a_n = \frac{|\{n\text{-points in}\:E\}|}{n!}.$

Doesn’t that look better? Here, I’m using my own terminology that an ‘$n$-point in $E$’ is a pair consisting of a way of making the finite set $\underline{n} = \{1, 2, \ldots, n\}$ into a semisimple ring (i.e. equipping it with addition and multiplication operations) together with a homomorphism from $R_E$ into $\underline{n}$, where

(7)$R_E = \mathbb{Z}[x,y] / \langle \text{equation of}\: E \rangle$

is the integral ring of functions on $E$. Once you’ve equipped $\underline{n}$ with the structure of a semisimple ring, then a homomorphism from $R_E$ into $\underline{n}$ is nothing but a pair $(x,y)$ of elements in $\underline{n}$ satisfying the equation of $E$, i.e. it’s literally a point in $E$ over the ring $\underline{n}$. (There is a question mark here about the point at infinity?)

Anyway it’s a very nice conceptual insight. Of course, to actually compute $a_n$, we will factorize $n$ as a product of primes and then do the same steps as above, but at least we have a nice conceptual meaning for it now.

Posted by: Bruce Bartlett on April 22, 2024 8:48 PM | Permalink | Reply to this

### Re: The Modularity Theorem as a Bijection of Sets

Bruce wrote:

(There is a question mark here about the point at infinity?)

Due to this “point at infinity” issue, your description of “$n$-points of $E$” isn’t right. This is not your fault, because I screwed it up in Part 3 of my post on counting points on elliptic curves. But now I’ve fixed that.

The right description is:

An $n$-point of the elliptic curve $E$ is a way of making the finite set $\underline{n}$ into a commutative ring $A$ and then choosing a point of $E(A)$.

What’s $E(A)$?

Well, an elliptic curve $E$ is a scheme, and any scheme gives a functor from commutative rings to sets, called the functor of points. So $E$ assigns a set of points to any commutative ring $A$, and that’s how we get the set $E(A)$

But if you’re not into schemes this may sound obscure. What’s really going on?

If $A$ is a finite field, we can say what’s going on very simply. We just take the solutions in $A$ of that polynomial equation you mentioned, and then we tack on a single extra point at infinity, just like you said! That gives the set $E(A)$.

But for other finite semisimple commutative rings here’s what we do. Such a ring $A$ is always a product of finite fields $\mathbb{F}_{q_1} \times \cdots \times \mathbb{F}_{q_k}$, and we set

$E(A) \cong E(\mathbb{F}_{q_1}) \times \cdots \times E(\mathbb{F}_{q_k})$

This may look ad hoc, but that’s only because I’m trying to sidestep some stuff about schemes.

(Above I’m shelving my concerns and treating elliptic curves as schemes over $\mathbb{Z}$, since you have assured me that any elliptic curve over $\mathbb{Q}$ has a unique reduced minimal model over $\mathbb{Z}$. But I need to read about this.)

Posted by: John Baez on April 23, 2024 10:26 PM | Permalink | Reply to this

Post a New Comment