Over \(\CC\), every elliptic curve \(E\) is isomorphic to \(\CC/\Lambda\) for a lattice \(\Lambda\), unique up to homothety, and to a Legendre curve \(y^2=x(x-1)(x-\lambda)\). For \(\Lambda=\ZZ+\ZZ\tau\) the \(j\)-invariant is \(j(\tau)\), and for the Legendre curve it is \(2^8(\lambda^2-\lambda+1)^3/(\lambda^2(\lambda-1)^2)\).
The functions
Elliptic functions, the Weierstrass $\wp$-function, and $(\wp')^2 = 4\wp^3 - g_2\wp - g_3$ (Tag D-CRVWEIER)
Fix a lattice \(\Lambda = \gens{1, \tau}_\ZZ \subseteq \CC\) with \(\tau \notin \RR\). An elliptic function for \(\Lambda\) is a meromorphic \(f \colon \CC \to \PP^1\) with \(f(z + \omega) = f(z)\) for every \(\omega \in \Lambda\), that is, a meromorphic function on the torus \(\CC/\Lambda\). These form a field.
(The Weierstrass functions)
Write \(\Lambda' \da \Lambda \smz\). \[ \wp(z) \da \frac{1}{z^2} + \sum_{\omega \in \Lambda'} \qty{ \frac{1}{(z-\omega)^2} - \frac{1}{\omega^2} }, \qquad \wp'(z) = \sum_{\omega \in \Lambda} \frac{-2}{(z-\omega)^3} . \] The series converges locally uniformly on \(\CC \setminus \Lambda\), so \(\wp\) is elliptic with a double pole at each lattice point and no other poles; \(\wp\) is even and \(\wp'\) is odd. Set the Eisenstein series of the lattice, \[ g_2 \da 60 \sum_{\omega \in \Lambda'} \frac{1}{\omega^4}, \qquad g_3 \da 140 \sum_{\omega \in \Lambda'} \frac{1}{\omega^6}, \qquad \Delta \da g_2^3 - 27 g_3^2 . \]
\[ (\wp')^2 = 4\wp^3 - g_2 \wp - g_3 , \] and the field of elliptic functions for \(\Lambda\) is \(\CC(\wp, \wp')\).
For \(z\) in a fixed compact set, the bracket \(\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2}\) is \(O(\abs\omega^{-3})\), and \(\sum_{\omega\in\Lambda'} \abs{\omega}^{-3}\) converges, so the series for \(\wp\) converges; \(\sum_{\omega\in\Lambda'}\abs{\omega}^{-2}\) diverges. For the same reason the Eisenstein series \(G_{2k}\da\sum_{\omega \in \Lambda'} \omega^{-2k}\) converge absolutely for \(k\ge2\), and the odd-weight sums vanish because \(\Lambda = -\Lambda\).
Integrating an elliptic function \(f\) around the boundary of a fundamental parallelogram gives \(\sum \operatorname{Res} f = 0\) on \(\CC/\Lambda\), so no elliptic function has exactly one simple pole in a fundamental parallelogram. The function \(\wp\) has a single double pole there. Counting poles gives the degree of the induced map \(\CC/\Lambda\to\PP^1\): \(\wp\) has degree \(2\) and \(\wp'\) degree \(3\).
The differential equation is proved by subtraction: expand \[ \wp(z) = z^{-2} + \sum_{k \geq 1} (2k+1) G_{2k+2} z^{2k}, \] form \((\wp')^2 - 4\wp^3 + g_2\wp + g_3\), and check that its polar terms and its constant term at \(0\) vanish. The difference is an elliptic function with no poles, hence constant by Liouville on the compact torus, hence \(0\).
Every elliptic function lies in \(\CC(\wp,\wp')\): an even elliptic function is a rational function of \(\wp\) (match its zeros and poles in \(\wp\)-values), and a general \(f\) splits as \(\tfrac{1}{2}(f(z)+f(-z)) + \wp'(z) \cdot \tfrac{f(z)-f(-z)}{2\wp'(z)}\) with both brackets even.
The residues of an elliptic function sum to zero on \(\CC/\Lambda\), so a nonconstant elliptic function has degree at least \(2\), and \(\wp\), with a double pole at \(0\), has degree exactly \(2\). A holomorphic elliptic function is bounded on a fundamental parallelogram, hence constant; so two elliptic functions with the same principal parts differ by a constant, and comparing principal parts at \(0\) proves \((\wp')^2=4\wp^3-g_2\wp-g_3\).
From a lattice to a cubic
Every elliptic curve over $\CC$ is a torus $\CC/\Lambda$ (Tag T-CRVUNIF)
Let \(\Lambda \subseteq \CC\) be a lattice and \(E_\Lambda \da V(y^2 = 4x^3 - g_2 x - g_3) \subseteq \PP^2_{/\CC}\). Then \(\Delta = g_2^3 - 27g_3^2 \neq 0\), so \(E_\Lambda\) is a smooth cubic, and \[ \phi \colon \CC/\Lambda \to E_\Lambda, \qquad z \mapsto \tv{\wp(z) : \wp'(z) : 1}, \quad 0 \mapsto \tv{0:1:0} , \] is an isomorphism of Riemann surfaces which is also a group isomorphism onto \((E_\Lambda, p_0)\) with \(p_0 = \phi(0)\). Conversely every elliptic curve over \(\CC\) is isomorphic to some \(E_\Lambda\): given \(c_2, c_3\) with \(c_2^3 - 27c_3^2 \neq 0\) there is a lattice with \(g_2 = c_2\) and \(g_3 = c_3\).
(Abel's theorem on the torus)
A divisor \(\sum n_i (a_i)\) on \(\CC/\Lambda\) is the divisor of an elliptic function exactly when \[ \sum n_i = 0 \qquad\text{and}\qquad \sum n_i a_i = 0 \text{ in } \CC/\Lambda . \]
The two halves of the first statement are proved by opposite methods and it is worth keeping them apart.
That \(\phi\) lands in the cubic is the differential equation and nothing else. That it is bijective is degree counting: \(\wp\) has degree \(2\) on the torus, \(\wp'\) degree \(3\), and \(\wp\) is even, so \(\wp(z) = \wp(z')\) forces \(z' = \pm z\) and the sign is then pinned by the odd function \(\wp'\); the point at infinity is hit once, by \(z=0\). Holomorphic and bijective gives an isomorphism of Riemann surfaces, and GAGA upgrades it to an isomorphism of algebraic curves.
Surjectivity of \(\Lambda \mapsto (g_2,g_3)\) is the one step that is not formal, and it is really a statement about the modular function: \(J = g_2^3/\Delta\) takes every complex value, and the scaling \(\Lambda \mapsto \alpha\Lambda\) sends \((g_2,g_3) \mapsto (\alpha^{-4}g_2, \alpha^{-6}g_3)\), which moves \((g_2,g_3)\) anywhere on the fibre of \(J\) over a fixed value. So: \(J\) surjective supplies the shape, homothety supplies the scale.
The second theorem is why the analytic and the algebraic group laws agree, and it is the exact analogue of Abel’s theorem for a general curve. The condition \(\sum n_i = 0\) is degree zero; the condition \(\sum n_i a_i = 0\) is the Abel–Jacobi map, which for genus \(1\) is the identity \(\CC/\Lambda \to \CC/\Lambda\). Specialising to \(a_1 + a_2 - b - 0\): three points of the torus sum to \(0\) exactly when the corresponding points of \(E_\Lambda\) are cut out by a line, which is the chord-and-tangent law. The torsion subgroup is immediate from the lattice model: \(E[n] = \tfrac{1}{n}\Lambda / \Lambda \cong (\ZZ/n)^2\), generated by \(\tfrac{a}{n} + \tfrac{b}{n}\tau\), and topologically \(E \cong (\RR/\ZZ)^2\).
(Normalisations)
The model \(y^2 = 4x^3 - g_2x - g_3\) is what \(\wp\) produces; the Legendre model \(y^2 = x(x-1)(x-\lambda)\) is what the branch-point normalisation produces. Passing between them is a linear change of variable together with \(x \mapsto \alpha^2 x\), \(y \mapsto \alpha^3 y\), which is exactly the homothety \(\Lambda \mapsto \alpha \Lambda\) seen on the curve. Any statement that a formula for \(j\) or \(g_2\) “is” a number depends on which model is in force, so say which one before computing.
By the differential equation, \(z\mapsto[\wp(z):\wp'(z):1]\), with \(0\mapsto[0:1:0]\), maps \(\CC/\Lambda\) into the cubic \(y^2=4x^3-g_2x-g_3\). The map is bijective: \(\wp\) is even of degree \(2\), so \(\wp(z)=\wp(w)\) exactly when \(w\equiv\pm z\), and \(\wp'(-z)=-\wp'(z)\) separates \(z\) from \(-z\) unless \(2z\in\Lambda\).
By Abel’s theorem on the torus, the chord-and-tangent law on the cubic corresponds to addition on \(\CC/\Lambda\), so the uniformization \(\CC/\Lambda\to E\) is a group isomorphism. In particular \(E[n]\cong\tfrac{1}{n}\Lambda/\Lambda\cong(\ZZ/n)^2\).
The modular function
The modular function $J(\tau) = g_2^3/\Delta$ and the $\SL_2(\ZZ)$-action on $\HH$ (Tag T-CRVMODJ)
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.
Under \(\Lambda\mapsto\alpha\Lambda\), both \(g_2^3\) and \(\Delta\) scale by \(\alpha^{-12}\), so \(J\) depends only on the homothety class of \(\Lambda\). The group is \(\SL_2(\ZZ)\) because \(\Im\qty{\tfrac{a\tau+b}{c\tau+d}}=(ad-bc)\Im\tau/\abs{c\tau+d}^2\): a matrix of determinant \(-1\) sends \(\HH\) to the lower half plane.
Since each \(\SL_2(\ZZ)\)-orbit meets the fundamental region once, \(J\) is a bijection from isomorphism classes of elliptic curves over \(\CC\) onto \(\CC\). The points of the fundamental region with nontrivial stabilizer in \(\PSL_2(\ZZ)\) are \(\tau=i\), of order \(2\), and \(\tau=\rho\), of order \(3\); these are the curves with \(j=1728\) and \(j=0\), the only ones with automorphism group larger than \(\{\pm1\}\).
Complex multiplication
Complex multiplication, and $\operatorname{End}(E,p_0)$ as an order in an imaginary quadratic field (Tag T-CRVCM)
Let \(E = \CC/\Lambda\) with \(\Lambda = \Lambda_\tau = \gens{1,\tau}_\ZZ\). Then \[ \operatorname{End}(E, p_0) \iso R_\Lambda \da \ts{ \alpha \in \CC \st \alpha \Lambda \subseteq \Lambda } , \] the isomorphism carrying an endomorphism to the scalar it is, and carrying \([n]\) to the integer \(n\). Say \(E\) has complex multiplication when \(R_\Lambda \supsetneq \ZZ\). This happens exactly when \(\tau\) is imaginary quadratic, that is \(\tau \in K \da \QQ(\sqrt{-d})\) for some squarefree \(d > 0\), and in that case \[ \ZZ \subsetneq R_\Lambda \subseteq \OO_K \] is an order: a subring of finite index in the ring of integers. Concretely, if \(A\tau^2 + B\tau + C = 0\) with \(A, B, C \in \ZZ\), \(A > 0\) and \(\gcd(A,B,C) = 1\), then \[ R_{\Lambda_\tau} = \ZZ[A\tau], \qquad \disc = B^2 - 4AC = f^2 d_K , \] where \(f = [\OO_K : R_{\Lambda_\tau}]\) is the conductor of the order.
(The three standard computations)
\(\tau = i\). Here \(\tau^2 + 1 = 0\), so \(A = 1\) and \(R = \ZZ[i] = \OO_K\) for \(K = \QQ(i)\), the maximal order, conductor \(1\). Its unit group is \(\ts{\pm 1, \pm i} \cong C_4\), so \(\abs{\Aut(E,p_0)} = 4\), which is the \(j = 1728\) row of the automorphism count; the curve is \(y^2 = x^3 - ax\) for some \(a\).
\(\tau = \rho \da e^{2\pi i/3}\). Here \(\tau^2 + \tau + 1 = 0\), so \(A = 1\) and \(R = \ZZ[\rho] = \OO_K\) for \(K = \QQ(\sqrt{-3})\), conductor \(1\). Its unit group is the sixth roots of unity, \(\cong C_6\), so \(\abs{\Aut(E,p_0)} = 6\), which is the \(j = 0\) row; the curve is \(y^2 = x^3 - b\) for some \(b\).
\(\tau = 2i\). Here \(\tau^2 + 4 = 0\), so \(A = 1\) and \(R = \ZZ[2i] = \ZZ + 2\ZZ[i]\), of index \(2\) in \(\ZZ[i]\): an order of conductor \(2\) in \(K = \QQ(i)\), with discriminant \(-16 = 2^2 \cdot (-4)\). Its unit group is only \(\ts{\pm 1}\), so this curve has just the two automorphisms every elliptic curve has, and \(j(2i) \neq 0, 1728\). Thus the endomorphism ring need not be the full ring of integers of the imaginary quadratic field.
The realization inside \(\CC\) identifies endomorphisms with lattice-preserving scalars in both directions. Given \(\alpha\) with \(\alpha\Lambda \subseteq \Lambda\), multiplication by \(\alpha\) descends to a holomorphic endomorphism of \(\CC/\Lambda\) fixing \(0\), which is algebraic by GAGA. Conversely a holomorphic map \(\CC/\Lambda \to \CC/\Lambda\) fixing \(0\) lifts to an entire map of \(\CC\) fixing \(0\) and commuting with \(\Lambda\)-translation, whose derivative is \(\Lambda\)-periodic and therefore constant; so the lift is \(z \mapsto \alpha z\). Addition of endomorphisms becomes addition of scalars and composition becomes multiplication, so the identification is a ring isomorphism, not just a bijection.
Why the dichotomy is sharp: \(R_\Lambda \subseteq \Lambda\) because \(\alpha = \alpha \cdot 1 \in \Lambda\), so \(R_\Lambda\) is a rank-one or rank-two subgroup of \(\Lambda \cong \ZZ^2\) closed under multiplication. Rank one is \(\ZZ\). Rank two means some \(\alpha \notin \RR\) satisfies a monic-after-clearing integral quadratic, and since \(\alpha^2 \in R_\Lambda\) is an integer combination of \(1\) and \(\alpha\), that quadratic has no real root; writing \(\alpha = a + b\tau\) with \(b \neq 0\) then puts \(\tau\) in the same imaginary quadratic field. There is no intermediate case, and in particular the endomorphism ring of an elliptic curve over \(\CC\) is never an order in a quaternion algebra — that happens only for supersingular curves in characteristic \(p\).
Complex multiplication is decided by the lattice parameter \(\tau\), and only countably many \(\tau\) are imaginary quadratic, so only countably many \(j \in \CC\) are CM \(j\)-invariants; the generic curve has \(\operatorname{End} = \ZZ\). That still leaves it hard to decide for a given equation, because passing from an equation to \(\tau\) requires transcendental data. The route that does work is class field theory.
The endomorphism ring \(R_\Lambda=\{\alpha\in\CC\st\alpha\Lambda\subseteq\Lambda\}\) lies in \(\Lambda\), so it is either \(\ZZ\) or an order in an imaginary quadratic field. The order need not be maximal: \(\tau=2i\) gives \(\ZZ[2i]\), of conductor \(2\) in \(\ZZ[i]\).
For a CM curve, $j$ is an algebraic integer and $K(j)/K$ is abelian of degree $h(R)$ (Tag T-CRVCMCFT)
Let \(E/\CC\) have complex multiplication by the order \(R\) of discriminant \(D\) in the imaginary quadratic field \(K\), and let \(h(R) \da \abs{\Pic(R)}\) be the class number of that order. Then:
\(j(E)\) is an algebraic integer;
\([K(j(E)) : K] = h(R)\), and \(K(j(E))/K\) is abelian, with \[ \Gal\big( K(j(E)) / K \big) \iso \Pic(R) ; \]
\(j(E) \in \ZZ\) if and only if \(h(R) = 1\).
There are exactly thirteen orders with \(h(R) = 1\), so exactly thirteen values of \(j\) arise this way, namely those of discriminant \[ D \in \ts{ -3,\ -4,\ -7,\ -8,\ -11,\ -12,\ -16,\ -19,\ -27,\ -28,\ -43,\ -67,\ -163 } . \]
(Thirteen orders, nine fields)
The two counts refer to different objects: fields versus orders.
Nine is the number of imaginary quadratic fields of class number one: \(d_K \in \ts{-3,-4,-7,-8,-11,-19,-43,-67,-163}\). That is the Baker–Heegner–Stark theorem, and it is a statement about maximal orders only.
Thirteen is the number of imaginary quadratic orders of class number one, and an order need not be maximal. Beyond the nine maximal ones there are four more, each a non-maximal order of conductor \(f\) inside a field already on the list: \[ D = -12 \ (f=2 \text{ in } \QQ(\sqrt{-3})), \quad -16 \ (f=2 \text{ in } \QQ(i)), \quad -27 \ (f=3 \text{ in } \QQ(\sqrt{-3})), \quad -28 \ (f=2 \text{ in } \QQ(\sqrt{-7})) . \] Endomorphism rings of elliptic curves range over all orders, not just maximal ones, so the relevant count for CM \(j\)-invariants is thirteen. The \(\tau = 2i\) curve is exactly the \(D = -16\) entry, and its \(j\)-invariant is the rational integer \(66^3 = 287496\). For example, \(D=-4\) gives \(j=1728\), \(D=-3\) gives \(j=0\), and \(D=-163\) gives \(j=-640320^3\), which explains the near-integrality of \(e^{\pi\sqrt{163}}\).
What the theorem buys is a bridge in the direction that was blocked. CM is immediate from an imaginary-quadratic lattice parameter \(\tau\), while recovering such a parameter from an algebraic equation is transcendental. The class field theory statement replaces that passage by an arithmetic test on \(j\) itself: a transcendental or non-integral \(j\) rules out CM by a class-number-one order immediately. For \(y^2=x(x-1)(x-2)\), the value \(j=\tfrac{2^6\cdot7^3}{3^2}\) is not an algebraic integer, hence cannot be a CM \(j\)-invariant.
The Galois statement is the reason the subject is called complex multiplication and not just “curves with extra endomorphisms”: \(K(j)\) is the ring class field of \(R\), and for \(R = \OO_K\) it is the Hilbert class field, the maximal unramified abelian extension of \(K\). So the \(j\)-invariants of CM curves generate the abelian extensions of an imaginary quadratic field, which is Kronecker’s Jugendtraum for that case: the values of a transcendental function at special points play the role that roots of unity play for \(\QQ\). The isomorphism \(\Gal \iso \Pic(R)\) is induced by the action of the class group on lattices — a class \([\mathfrak{a}]\) sends \(\CC/\Lambda\) to \(\CC/\mathfrak{a}^{-1}\Lambda\), which is another curve with the same \(R\), and this action is simply transitive on the isomorphism classes. That is also why \(j\) has exactly \(h(R)\) conjugates: they are the \(j\)-invariants of the \(h(R)\) curves in one orbit.
Nine imaginary quadratic fields have class number one, and thirteen imaginary quadratic orders have class number one; endomorphism rings of CM elliptic curves range over all orders. The four non-maximal orders among the thirteen have discriminants \(-12\), \(-16\), \(-27\), and \(-28\).
If \(j(E)\) is not an algebraic integer, then \(E\) has no complex multiplication.