Elliptic Curves, Lattices, and the Upper Half-Plane

This post is a rough sketch at writing down my understanding of the correspondence found on page 36 of Silverman’s Advanced Topics in the Arithmetic of Elliptic Curves (ATAEC), following material there and in The Arithmetic of Elliptic Curves (AEC): \(\DeclareMathOperator{\Ell}{Ell}\)

\[ \begin{array} \mathrm{Ell}_\mathbb{C} & \longleftrightarrow & \mathcal{L}/\mathbb{C}^\times & \longleftrightarrow & \Gamma(1)\backslash \mathfrak{h} & \longleftrightarrow & \mathbb{C} \\ \left\{ E_\Lambda \right\} & & \left\{ \Lambda_\tau \right\} & & \tau & & j(\tau) \end{array} \]

We go from right to left.

Elliptic curves and lattices

Here, we write \(\Ell_\mathbb{C}\) for the set of elliptic curves defined over \(\mathbb{C}\), up to \(\mathbb{C}\)-isomorphism, and \(\mathcal{L}\) for the set of lattices in \(\mathbb{C}\).

Fix a lattice \(\Lambda\). Consider the Weierstrass \(\wp\)-function of the lattice, defined as \[ \wp(z;\Lambda) = \frac{1}{z^2} + \sum_{\lambda \in \Lambda}^{*} \left( \frac{1}{(z+\lambda)^2} - \frac{1}{\lambda^2} \right). \] This is evidently \(\Lambda\)-periodic, so we can also consider it as a function on \(\mathbb{C}/\Lambda\) instead of just on \(\mathbb{C}\), and it is a fact that it satisfies the differential equation \[ (\wp'(z))^2 = 4(\wp(z))^3 + g_2\wp(z) + g_3, \] where \[ g_2 = 60G_4 = 60\sum_{\lambda \in \Lambda}^{*} \lambda^{-4} \\ g_3 = 140G_6 = 140\sum_{\lambda \in \Lambda}^{*} \lambda^{-6} \] are the modular invariants of the lattice, and the asterisk in the sum denotes the omission of the zero term. We want to identify the elliptic curve \(E : y^2 = 4x^3 + g_2x + g_3\) with the torus \(\mathbb{C}/\Lambda\). To do this we need two facts: first, that that curve is actually an elliptic curve, and second, that the map \(z \mapsto (\wp(z),\wp'(z))\) is an isomorphism of Riemann surfaces that preserves the group structure. These facts are provided in AEC.VI.3.6.

Now, we have associated to each lattice an elliptic curve over \(\mathbb{C}\). Suppose we have two lattices, \(\Lambda_1\) and \(\Lambda_2\). Then their associated elliptic curves \(E_1\) and \(E_2\) are isomorphic over \(\mathbb{C}\) if and only if the lattices \(\Lambda_1\) and \(\Lambda_2\) are homothetic, i.e., \(\alpha\Lambda_1 \subseteq \Lambda_2\) for some \(\alpha \in \mathbb{C}^\times\). This is AEC.VI.4.1.1. Finally, we have

Theorem (Uniformization Theorem)

Let \(A, B \in \mathbb{C}\) be complex numbers with \(4A^3 - 27B^2 \neq 0\). Then there exists a lattice \(\Lambda \subset \mathbb{C}\) with \(g_2(\Lambda) = A\), \(g_3(\Lambda) = B\).

Thus, for every elliptic curve \(E\) over \(\mathbb{C}\), there is a lattice so that \(E \cong \mathbb{C}/\Lambda\). Putting everything together, we see that there is a bijection between \(\Ell_\mathbb{C}\), elliptic curves over \(\mathbb{C}\), and \(\mathcal{L}/\mathbb{C}^\times\), the set of lattices up to homothety.

Lattices and the upper half-plane

We now wish to describe the correspondence between \(\mathcal{L}/\mathbb{C}^\times\) and the upper half-plane \(\mathfrak{h}\). Pick a lattice \(\Lambda\), and some basis \(\Lambda = \omega_1 \mathbb{Z} \oplus \omega_2 \mathbb{Z}\). We can choose the basis to be oriented, i.e., so that \(\tau = \omega_1 / \omega_2\) has positive imaginary part. Now since the lattices are only determined up to homothety, we may normalize the basis, multiplying by \(\frac{1}{\omega_2}\) to get \(\Lambda = \mathbb{Z} \oplus \tau \mathbb{Z}\).

Fact (ATAEC.I.1.2):

If \(\psi_1, \psi_2\) and \(\omega_1, \omega_2\) are two bases for the same lattice \(\Lambda\), then they differ by an element of \(SL_2(\mathbb{Z})\). Also, for \(\tau_1, \tau_2 \in \mathfrak{h}\), we have that the associated lattices \(\Lambda_{\tau_1}, \Lambda_{\tau_2}\) are homothetic if and only if \(\tau_1 = \gamma\tau_2\) for some \(\gamma \in \Gamma(1) = PSL_2(\mathbb{Z})\). (Here we use the usual action by Möbius transformations).

From this fact and the preceding paragraph, we see that there is a bijection between \(\mathcal{L}/\mathbb{C}^\times\) and \(\Gamma(1)\backslash \mathfrak{h}\), giving the second bijection.

The last bijection is given by the \(j\)-invariant, \(j(\tau) = 1728g_2^3/\Delta\), where \(\Delta = g_2^3 - 27g_3^2\). The fact that this is a bijection can be seen in e.g., Koblitz’s Modular Forms and Elliptic Curves, III.2.11.

Posted October 31, 2012 under math.