The n-Category Café

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?