Let \(\wp\) be the Weierstrass function of a lattice with invariants \(g_2,g_3\). Find a rational function \(R(x)\), allowed to depend on \(g_2,g_3\), such that \[ \wp(2z)=R(\wp(z)). \]
Solution
The required rational function is \[ \boxed{ R(x)= \frac{(x^2+g_2/4)^2+2g_3x} {4x^3-g_2x-g_3}. } \] Equivalently, \[ R(x) =-2x+ \frac{(12x^2-g_2)^2} {16(4x^3-g_2x-g_3)}. \]
1. On the Weierstrass cubic\[ E:\quad y^2=4x^3-g_2x-g_3, \]the slope of the tangent at\[ (x,y)=(\wp(z),\wp'(z)) \]is\[ m=\frac{6x^2-g_2/2}{y}. \]
Differentiate the cubic equation implicitly: \[ 2y\frac{dy}{dx}=12x^2-g_2. \] Hence, wherever \(y\ne0\), \[ \frac{dy}{dx} =\frac{12x^2-g_2}{2y} =\frac{6x^2-g_2/2}{y}. \] The resulting identity is rational on the curve and therefore extends across the exceptional points in the usual limiting sense.
2. If the tangent at \((x,y)\) meets \(E\) again at a point with \(x\)-coordinate \(x_3\), then\[ x_3=\frac{m^2}{4}-2x. \]
Write the tangent line as \[ Y=m(X-x)+y. \] Substituting into the cubic gives a cubic polynomial equation in \(X\): \[ 4X^3-g_2X-g_3-\bigl(m(X-x)+y\bigr)^2=0. \] Its leading coefficient is \(4\), and its \(X^2\) coefficient is \(-m^2\). Thus the sum of its three roots is \[ \frac{m^2}{4}. \] The tangency means that \(X=x\) is a double root, while the third root is \(x_3\). Hence \[ 2x+x_3=\frac{m^2}{4}, \] which gives the formula.
3. The third intersection of the tangent represents \(-2z\), so its \(x\)-coordinate is \(\wp(2z)\).
Under the isomorphism \(z\mapsto(\wp(z),\wp'(z))\) from \(\mathbb C/\Lambda\) to \(E\), the chord–tangent group law holds: the three intersection points of a line with the cubic sum to zero in the group \(\mathbb C/\Lambda\). For a tangent at the point corresponding to \(z\), the three points are \[ z, \qquad z, \qquad -2z. \] Negation on the Weierstrass cubic changes \((x,y)\) to \((x,-y)\) and leaves the \(x\)-coordinate unchanged. Since \(\wp\) is even, \[ x_3=\wp(-2z)=\wp(2z). \] Combining with 2 and 1 gives \[ \wp(2z) =-2x +\frac14 \frac{(6x^2-g_2/2)^2}{y^2}. \] Using \[ y^2=4x^3-g_2x-g_3 \] gives \[ \wp(2z) =-2x+ \frac{(12x^2-g_2)^2} {16(4x^3-g_2x-g_3)}. \]
4. Simplifying gives the displayed \(R(x)\).
Put \[ D=4x^3-g_2x-g_3. \] Then \[ \begin{aligned} -2x+\frac{(12x^2-g_2)^2}{16D} &=\frac{-32xD+(12x^2-g_2)^2}{16D}\\ &=\frac{16x^4+8g_2x^2+32g_3x+g_2^2}{16D}\\ &=\frac{(x^2+g_2/4)^2+2g_3x}{D}. \end{aligned} \] Thus \[ \wp(2z)=R(\wp(z)) \] as an identity of meromorphic functions.