(The modular function)
For \(\tau\) in the upper half plane \(\HH\) write \(\Lambda_\tau \da \gens{1, \tau}_\ZZ\) and define \[ J(\tau) \da \frac{g_2(\Lambda_\tau)^3}{\Delta(\Lambda_\tau)}, \qquad \Delta = g_2^3 - 27 g_3^2, \qquad j \da 1728 \, J . \] This \(j\) is the \(j\)-invariant of \(E_{\Lambda_\tau}\). Then \(J(\tau) = J(\tau')\) if and only if \[ \tau' = \frac{a\tau + b}{c\tau + d} \qquad\text{for some}\qquad \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in \SL_2(\ZZ) , \] and every \(\SL_2(\ZZ)\)-orbit has exactly one representative in the fundamental region \[ F = \ts{ \tau \in \HH \st -\tfrac{1}{2} \leq \Re \tau < \tfrac{1}{2}, \ \abs{\tau} \geq 1, \text{ and } \abs\tau > 1 \text{ when } \Re\tau > 0 } . \] So \(J \colon \HH/\SL_2(\ZZ) \to \CC\) is a bijection.
(Normalisation, stated once)
Conventions differ between sources and the difference is the factor \(1728 = 2^6 \cdot 3^3\), so fix it before computing: here \(J = g_2^3/\Delta\) and \(j = 1728 J\). The two checkable consequences, which pin the convention without appeal to any book, are \[ J(i) = 1, \quad j(i) = 1728 , \qquad J(\rho) = 0, \quad j(\rho) = 0 , \qquad \rho \da e^{2\pi i/3} . \] The first holds because multiplication by \(i\) preserves \(\gens{1,i}\), so \(g_3 = \sum \omega^{-6} \cdot 140 = i^{-6} g_3 = -g_3\) forces \(g_3 = 0\) and \(\Delta = g_2^3\). The second holds because multiplication by \(\rho\) preserves \(\gens{1,\rho}\), so \(g_2 = \rho^{-4} g_2\) forces \(g_2 = 0\). Many sources instead write \(j = 1728 g_2^3/\Delta\) with no intermediate \(J\), and analytic number theory rescales \(\Delta\) by \((2\pi)^{12}\) to make \(q\)-expansions integral; none of that moves \(j(i)\) or \(j(\rho)\).
\(J\) is a function of \(\tau\) rather than of \(\Lambda\) because numerator and denominator have the same weight: \(\Lambda \mapsto \alpha\Lambda\) sends \(g_2 \mapsto \alpha^{-4}g_2\) and \(g_3 \mapsto \alpha^{-6}g_3\), so both \(g_2^3\) and \(\Delta\) scale by \(\alpha^{-12}\) and the ratio does not move. Every lattice is homothetic to some \(\Lambda_\tau\), by dividing through by one generator, so nothing is lost.
The \(\SL_2(\ZZ)\) statement is then just a change of basis: \(\Lambda_\tau\) and \(\Lambda_{\tau'}\) are homothetic exactly when \((1,\tau)\) and \((1,\tau')\) are related by an integral matrix of determinant \(\pm 1\), and demanding that both \(\tau\) and \(\tau'\) lie in the upper half plane forces determinant \(+1\), since \(\Im\qty{\tfrac{a\tau+b}{c\tau+d}} = \tfrac{(ad-bc)\Im\tau}{\abs{c\tau+d}^2}\). That identity explains why preserving the upper half-plane forces determinant \(+1\), hence the acting group is \(\SL_2(\ZZ)\) rather than \(\GL_2(\ZZ)\).
The uniqueness clause is what makes this a moduli statement rather than an invariance statement. \(\SL_2(\ZZ)\) is generated by \(\tau \mapsto \tau+1\) and \(\tau \mapsto -1/\tau\), which identify the two vertical edges of \(F\) and fold the bottom arc onto itself; the half-open description above picks one point from each identified pair. Combined with surjectivity of \(J\) this gives the analytic proof that \(\AA^1\) is a coarse moduli space, matching the algebraic proof by Legendre normal form. The two fixed points of the folding are the two curves with extra automorphisms: \(\tau = i\) fixed by \(\tau \mapsto -1/\tau\), and \(\tau = \rho\) fixed by an order-three element — exactly the \(j = 1728\) and \(j = 0\) cases.
Dependencies
D-CRVWEIERElliptic functions, the Weierstrass $\wp$-function, and $(\wp')^2 = 4\wp^3 - g_2\wp - g_3$usesT-CRVUNIFEvery elliptic curve over $\CC$ is a torus $\CC/\Lambda$uses
Backlinks
T-CRVCMComplex multiplication, and $\operatorname{End}(E,p_0)$ as an order in an imaginary quadratic fieldrelated-toT-CRVCMCFTFor a CM curve, $j$ is an algebraic integer and $K(j)/K$ is abelian of degree $h(R)$related-toE-HM213A17-143Derive the modular function from the cross-ratio of the Weierstrass rootsuses