### 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?

## 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: