The Modularity Theorem as a Bijection of Sets

April 19, 2024

The Modularity Theorem as a Bijection of Sets

Posted by John Baez

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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.

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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

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Re: The Modularity Theorem as a Bijection of Sets

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

Re: The Modularity Theorem as a Bijection of Sets

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Did you mean to write $a_n=n + 1- \frac{|\{n\text{-points in }E\}|}{n!}$ ?

Posted by: Gregor on August 2, 2024 4:23 PM | Permalink | Reply to this

Re: The Modularity Theorem as a Bijection of Sets

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That would be for the $L$ -function of an elliptic curve; I think Bruce was giving the formula for its zeta function.

Posted by: John Baez on February 8, 2025 8:32 PM | Permalink | Reply to this

Re: The Modularity Theorem as a Bijection of Sets

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Thank you for this formulation! Could you elaborate upon/try to motivate how $n$ -points might intuitively be seen to relate to different types of reduction? I couldn’t immediately find something directly on this upon skimming the paper.

A trivial point is that your re-formulation looks like it must be positive, whilst you gave $-1$ as the value originally in the case of non-split reduction.

But really it is spelling out how $n$ -points are related to smoothness that would be interesting to see!

Posted by: reduction on February 10, 2025 9:32 AM | Permalink | Reply to this

Re: The Modularity Theorem as a Bijection of Sets

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I wrote a series of articles explaining how the number of points of an elliptic curve over a finite field $\mathbb{F}_p$ depends on whether the curve has

good reduction,
additive reduction,
split multiplicative reduction

non-split multiplicative split reduction,

and how this impacts the zeta function and $L$ -function of the elliptic curve. If you’re in a rush, you may want to start at Part 2, where I go through the four cases. This answers your main question. But if you’re really interested in this stuff, start with Part 1 and continue all the way to Part 3.

As I mentioned in my reply to Gregor above, Bruce’s formula

$a_n = \frac{|\{n-\text{points in } \; E\}|}{n!}$

is wrong if $a_n$ is supposed to be the $n$ th coefficient of the $L$ -function of the elliptic curve. (When people talk about $a_p$ , that’s what they usually mean.) Bruce’s formula is correct when $a_n$ is the $n$ th coefficient of the zeta function of an elliptic curve.

The relation between the zeta function and the $L$ -function is explained in Part 3 of my posts. It’s

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

where plain old $\zeta(s)$ is the Riemann zeta function.

The point of my blog series was to demystify the $L$ -function of an elliptic curve. I start by giving the terribly mysterious definition I found on Wikipedia. In the end I point out that the zeta function $\zeta_E(s)$ of an elliptic curve $E$ is easy to define in a way that doesn’t require considering the 4 special cases listed above: its $n$ th coefficient is simply

$a_n = \frac{|\{n-\text{points in } \; E\}|}{n!}$

and only when you try to actually compute those coefficients do you start needing to think about cases. Once you have defined $\zeta_E(s)$ , you can define the $L$ -function $L_E(s)$ using the formula above. There’s a good reason for that formula, too.

Posted by: John Baez on February 11, 2025 11:18 PM | Permalink | Reply to this

Re: The Modularity Theorem as a Bijection of Sets

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Thank you very much for the reply! As far as I see, though, the proof itself is computational, also relying on a non-trivial result of Hasse. What I was hoping for was a conceptual argument to believe in the result, i.e. some kind of rough idea as to how the result might not be a mere coincidence, but that even a priori one might be able to guess that $n$ -points are related to reduction mod $p$ or something.

Is there any evidence for instance that something similar goes through for other kinds of varieties than elliptic curves? In Proposition 1.3 of this, for instance, there is some reformulation in general of the zeta function in terms of symmetric products, which seems rather close to what you have observed for elliptic curves?

Posted by: reduction on February 12, 2025 2:03 AM | Permalink | Reply to this

Re: The Modularity Theorem as a Bijection of Sets

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reduction wrote:

What I was hoping for was a conceptual argument to believe in the result, i.e. some kind of rough idea as to how the result might not be a mere coincidence…

Unfortunately you didn’t say what “the result” is. There are several results floating around here. Earlier you asked me to “motivate how $n$ -points are related to different kinds of reduction”. So I thought maybe you were interested in that. But now it seems maybe not.

The number of $n$ -points of an elliptic curve, or indeed any scheme over $\mathbb{Z}$ , is simply the product of the number of $p^k$ -points of that scheme where $p$ runs over all primes dividing $n$ , and $p^k$ is the largest power of that prime dividing $n$ . Bruce stated that here. And it’s not a coincidence: it follows from the fact that everything in Bruce’s definition of ‘ $n$ -point’ can be understood ‘one prime at a time’.

For example, a semisimple commutative ring of cardinality $n$ is a product of finite fields of cardinality $p^k$ where $p$ runs over primes dividing $n$ , and $p^k$ for a particular prime $p$ is the largest power of that prime dividing $n$ . And so on. This is simple and conceptual. It has nothing to do with whether the scheme has good or bad reduction mod $p$ .

But then there’s another bunch of results: the specific rather complicated formulas for the numbers of $p$ -points of an elliptic curve. These formulas inevitably depend on whether the curve has

good reduction,
additive reduction,
split multiplicative reduction

non-split multiplicative split reduction

mod $p$ . Luckily, we don’t need these results at all to see the simple conceptual stuff!

Posted by: John Baez on February 12, 2025 3:31 AM | Permalink | Reply to this

Re: The Modularity Theorem as a Bijection of Sets

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Apologies, by “result” I meant that $a_{n} = \frac{\left| n-points \right|}{n!}$ , and mainly the case $n=p$ . This very nice result certainly appears conceptual, but it would be a bit disappointing if the only reason for it to be true were a ‘combinatorial coincidence’, i.e. one just counts two different things and observes that they are the same.

In particular, with regard to…

“motivate how $n$ -points are related to different kinds of reduction”. So I thought maybe you were interested in that.

…I was indeed definitely interested in that, but more in the sense of when $n=p$ , why ‘on a higher level’ could I expect numbers of $p$ -points to break up into cases akin to those of reduction?

It seems to me that if we, instead of looking at the definition of the zeta function directly, we look at the proposition in the notes I linked to where it is defined in terms of $Sym(X)$ , then a conceptual explanation may be possible. E.g. $n!$ is the same as the cardinality of the symmetric group, and the symmetric algebra $Sym(V)$ on a vector space $V$ is the universal way to build a commutative ring out of $V$ , and it is conceivable to me that points of it over $\mathbb{F}_{p}$ are the same as $n$ -points of $X$ : equipping $\{ 1, \ldots, p \}$ with the structure of a ring and looking at homomorphisms into it from $\mathbb{Z}[x,y] / E$ might well be adjoint in some sense to mapping into $Sym(E)$ .

Apparently the reformulation of the zeta function in terms of symmetric powers is not a hard result, boiling down to Galois theory. Thus, putting all this together, maybe one actually has a really conceptual story that is true more generally than for elliptic curves…

Posted by: reduction on February 12, 2025 10:08 AM | Permalink | Reply to this

Re: The Modularity Theorem as a Bijection of Sets

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reduction wrote:

Apologies, by “result” I meant that $a_{n} = \frac{\left| n-points \right|}{n!}$ , and mainly the case $n=p$ . This very nice result certainly appears conceptual, but it would be a bit disappointing if the only reason for it to be true were a ‘combinatorial coincidence’, i.e. one just counts two different things and observes that they are the same.

As far as I’m concerned, this formula for the coefficients of the zeta function is the definition, and the results are that it matches other more well-known, but more clumsy definitions. You can see proofs of some of key results in Dirichlet species and the arithmetic zeta function — like Theorem 16, which equates two formulas for the zeta function:

$\sum_{n \ge 1} \frac{|Z_X(n)|}{n!} n^{-s} = \prod_{p \in \P} \exp \left( \sum_{n \ge 1} \frac{|X(\F_{p^n})|}{n} p^{-n s} \right)$

Here $X \colon \mathsf{CommRing} \to \mathsf{Set}$ is any product-preserving functor that happens to send finite fields to finite sets. For example $X$ could be an elliptic curve defined using polynomial equations with integer coefficients; then $X(k)$ is the set of $k$ -points of that elliptic curve. At left, $Z_X(n)$ is the set of ways of making an $n$ -element set into a commutative ring $R$ that’s a product of finite fields and choosing an element of $X(R)$ . The proof is very nice and conceptual, with no mucking about. I could have made it shorter if I weren’t trying to categorify the identity above.

It seems to me that if we, instead of looking at the definition of the zeta function directly, we look at the proposition in the notes I linked to where it is defined in terms of $Sym(X)$ , then a conceptual explanation may be possible.

Beware: Proposition 1.3 in Bejleri’s notes is talking about a different zeta function, namely the zeta function of a variety over a finite field $\mathbb{F}_q$ where $q = p^n$ .

So, if we take $q = p$ , then Bejleri’s zeta function is not this zeta function Bruce Bartlett and I are talking about:

$Z_X(s) = \sum_{n \ge 1} \frac{|Z_X(n)|}{n!} n^{-s} = \prod_{p \in \P} \exp \left( \sum_{n \ge 1} \frac{|X(\F_{p^n})|}{n} p^{-n s} \right)$

Instead, up to a change of variables, it’s a single factor in the product at right, namely

$\exp \left( \sum_{n \ge 1} \frac{|X(\F_{p^n})|}{n} p^{-n s} \right)$

for one particular prime $p$ . This is sometimes called a $p$ -local zeta function.

If you’re interested in the $p$ -local zeta function, you can find a bunch of equivalent definitions of it here:

Niranjan Ramachandran, Zeta functions, Grothendieck groups, and the Witt ring.

starting on page 7 in the section “Schemes over finite fields and their zeta functions”. The symmetric power definition you like is mentioned in Remark 3.2i. I believe it’s not hard to show this is equivalent to all the rest.

I agree that the symmetric power definition is nice.

Posted by: John Baez on February 12, 2025 11:28 PM | Permalink | Reply to this

Re: The Modularity Theorem as a Bijection of Sets

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1). Can one somehow treat breaking into cases reduction mod $p$ already at the categorified level, i.e. for $n$ -points, on the ‘zeta side’ of things?

I’m not sure I completely understand this so I’ll give two answers. The first answer is stuff you may already know; the second may be the answer to the question you’re really asking.

1a). My paper does break the zeta function into pieces, one for each prime, at the categorified level.

More precisely it shows how the zeta species of any scheme of finite type over $\mathbb{Z}$ is a product of ‘ $p$ -local’ species, one for each prime. A species is any type of structure that can be put on finite sets, and a ‘ $p$ -local species’ is one that can only be put on finite sets whose cardinality is a power of $p$ .

This stuff shows how ‘ $n$ -points’, in Bruce’s sense, are related to ‘ $p$ -points’ for the various prime factors of $n$ .

1b) One can go further in the case of the zeta function of an elliptic curve $E$ . One can then collect the primes into four kinds and separately discuss these four cases:

good reduction
additive reduction
split multiplicative reduction
non-split multiplicative split reduction

You can do all this at the categorified level because these four cases are four kinds of structure you can put on a finite set. The zeta species of an elliptic curve is the Dirichlet product of four species $X_1, X_2, X_3, X_4$ , one for each of the four cases.

For example there’s a species $X_3$ where to put an $X_3$ -structure on a finite set you make that set into a field $k$ for which $E$ has split multiplicative reduction and then choose a $k$ -point of $E$ .

It might be fun to analyze these four cases and categorify the four formulas for $p$ -local zeta functions given in Theorems 1, 2, 3 and 4 here. Ironically, the hardest case is actually ‘good reduction’, because the formula in this case involves an irrational number $\alpha$ . I think in that case we really need to get more serious and figure out how to categorify the zeta function of a motive. That’s a general project I am slowly working on in my spare time.

2) Could one ‘categorify’/give a more conceptual proof of the equivalence of the symmetric power version of the $p$ -local zeta-function with the one which is a factor in the product you give?

Yes. There’s a paper I want to write, in which this might fit nicely. This seems easier than the ‘categorify the zeta function of a motive’ project.

Your other comments are interesting too, but I’m worn out now!

Posted by: John Baez on February 17, 2025 1:34 AM | Permalink | Reply to this

Re: The Modularity Theorem as a Bijection of Sets

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Thank you very much for the elaboration and the reference! My cursory pattern-matching of the blog, paper, and Bruce’s post had led me to think that you had proven a general categorified result involving $n$ -points, and then shown by a computation that in the case of an elliptic curve, one side of the isomorphism involved recovered its $L$ -function. Whereas in fact I see now that these computations were a pedagogical aside: your categorified result recovers the arithmetic zeta function in complete generality, and the relationship to the $L$ -function is just the standard one using the zeta function!

But I still sit with the question and the suggestion I put forward:

Can one somehow treat breaking into cases reduction mod p already at the categorified level, i.e. for $n$ -points, on the ‘zeta side’ of things?
Could one ‘categorify’/give a more conceptual proof of the equivalence of the symmetric power version of the $p$ -local zeta-function with the one which is a factor in the product you give (I was aware they were not the same, apologies for confusion, and thank you for clarifying for anybody else reading!)?

I don’t know exactly what a treatment of reduction mod p would look like for $n$ -points, but one can describe reduction mod p for commutative rings as pushout along $\mathbb{Z} \rightarrow \mathbb{Z}/p$ , and by 2-categorical stuff one should then I imagine be able to cook up a tame, multiplicative functor $Comm / \left(\mathbb{Z}/p\right) \rightarrow FinSet$ from one $Comm \rightarrow FinSet$ . I’ve not actually seen reduction mod p described like that from a functor of points perspective, but I suppose it ends up like that. And then one could attempt various things: maybe one can glue together the mod-p functors to recover the original one, or in a different direction maybe one could give a conceptual explanation of definitions (3), (4), and (5) in Bruce’s post (maybe something at the categorified level can distinguish good reduction from the other cases…).

More generally than 1. or 2., using the categorified point of view to prove or shed light on anything known and usually proven at the decategorified level would be very interesting.

I really like the fact that your results are expressed just in terms of a tame, multiplicative functor. I wonder what happens if one tweaks aspects, e.g. the Clausen-Scholze point of view is that it is fruitful to think of real numbers as profinite sets; could one tweak the range to profinite sets to obtain a notion of $\infty$ -point? Or what if one simply looks at the set $\mathbb{N}$ and imposes the structure of a semisimple ring on it?

It strikes me as well that the notion of $n$ -point has a kind of $\mathbb{F}_1$ aspect. One is just looking at the set $\{ 1, \ldots, n \}$ and imposing structure on it. What if one works with monoids instead, say? Maybe one can no longer prove something like Lemma 13 in your paper, but one could just use the resulting notion of $n$ -point (particularly as Bruce formulated it) to define a zeta-function in that setting.

Posted by: reduction on February 16, 2025 3:03 AM | Permalink | Reply to this

Re: The Modularity Theorem as a Bijection of Sets

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reduction wrote:

1). Can one somehow treat breaking into cases reduction mod $p$ already at the categorified level, i.e. for $n$ -points, on the ‘zeta side’ of things?

I’m not sure I completely understand this so I’ll give two answers. The first answer is stuff you may already know; the second may be the answer to the question you’re really asking.

1a). My paper does break the zeta function into pieces, one for each prime, at the categorified level.

This stuff shows how ‘ $n$ -points’, in Bruce’s sense, are related to ‘ $p$ -points’ for the various prime factors of $n$ .

1b) One can go further in the case of the zeta function of an elliptic curve $E$ . One can then collect the primes into four kinds and separately discuss these four cases:

good reduction
additive reduction
split multiplicative reduction
non-split multiplicative split reduction

2) Could one ‘categorify’/give a more conceptual proof of the equivalence of the symmetric power version of the $p$ -local zeta-function with the one which is a factor in the product you give?

Yes. There’s a paper I want to write, in which this might fit nicely. This seems easier than the ‘categorify the zeta function of a motive’ project.

Your other comments are interesting too, but I’m worn out now!

Posted by: John Baez on February 22, 2025 1:30 AM | Permalink | Reply to this

Re: The Modularity Theorem as a Bijection of Sets

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Thank you very much! This is all very nice! You are completely correct that 1b), especially the direction you hint at with “it might be fun”, is exactly the kind of thing I was looking for! Great that you are able to do 2)! And thank you for pointing out 1a); I was thinking a little more globally, but it’s very good to think about it this way too.

I thought I’d experiment a little with my suggestion that there is something $\mathbb{F}_{1}$ -like here: trying the ‘obvious $\mathbb{F}_1$ thing’ of working with commutative monoids instead of commutative rings. The first observation is that a simple commutative monoid is the same as a cyclic group of prime order. Then, if I’m not making a mistake, I think there is only one way up to isomorphism to put a semisimple commutative monoid structure on any finite set, due to the uniqueness of prime factorisation, but any permutation of the sets in the factors, as well as permutation of the factors, gives an automorphism, and thus the analogue of the Riemann zeta function over $\mathbb{F}_1$ becomes I think $\sum_{n \geq 1} \frac{r_n! p^1_n! \cdots p^{r_n}_n!}{n!} n^{-s}$ , where $p^1_n \cdots p^{r_{n}}_n$ is the prime factorisation of $n$ . In other words, the coefficients in the terms are the same as the Riemann zeta function when $n$ is prime, but different when not (smaller I think).

For the case of $\mathbb{Z}$ (maybe persuasively thought as $\mathbb{N}[x]/x+1=0$ ), we’d like to view it as a curve over $\mathbb{F}_1$ , and a nice thing about your approach is that we could bypass the question of building algebraic geometry over $\mathbb{F}_1$ and defining what a curve is and just work with it as functor of points, but alas it is not of finite type over $\mathbb{F}_1$ . Still, if we try to somehow generalise the formalism so that we view $\mathbb{Z}$ in some ‘profinite’ way, maybe one then can recover the Riemann zeta function in some way.

There are some predicted formulas for the zeta functions of projective spaces over $\mathbb{F}_1$ ; it would be fantastic if one obtained these! But I’ve not attempted to make the calculation, yet at least.

Posted by: reduction on February 22, 2025 1:34 AM | Permalink | Reply to this

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