\[
x_2,\ y_3,\ g_4,\ g_6 \qquad \boxed{y_3^2 = x_2^3 + g_4x_2 + g_6}
\]

\[
\begin{cases}
x = x_\omega = \omega^{-2}x_2\\
y = y_\omega = \omega^{-3}y_3\\
\gamma_4 = \gamma_{4\omega} = \omega^{-4}g_4\\
\gamma_6 = \gamma_{6\omega} = \omega^{-6}g_6
\end{cases}
\qquad \text{i.e.}\quad y^2 = x^3 + \gamma_4x + \gamma_6
\]

\[
\boxed{dx = 2y\,\omega}\quad \text{i.e.}\quad
\boxed{d(\omega^{-2}x_2) = 2\omega^{-2}y_3}
\]

\[
2y\,dy = (3x^2+\gamma_4)\,dx = 2y\,(3x^2+\gamma_4)\,\omega
\]

\[
\boxed{dy = (3x^2+\gamma_4)\,\omega}\qquad
\boxed{d(\omega^{-3}y_3) = (3x_2^2+g_4)\,\omega^{-3}}
\]

\[
\begin{cases}
\omega = \dfrac12\,\dfrac{dx}{y}\\[2mm]
\eta = \omega x = \dfrac12\,\dfrac{x\,dx}{y}
\end{cases}
\qquad [\,= \omega^{-1}x_2\,]
\]

\[
\begin{cases}
\int_{\gamma_1}\omega = \omega_1, & \omega_1 = \int_0^{\omega_1}dz\\[1mm]
\int_{\gamma_2}\omega = \omega_2, & \omega_2 = \int_0^{\omega_2}dz
\end{cases}
\qquad X^{\mathrm{an}} \simeq \mathbf{C}/E(\omega_1,\omega_2)
\]

\[
\begin{cases}
x = \dfrac{1}{z^2} + \sum'_{a\in E(\omega_1,\omega_2)}
\Bigl[\dfrac{1}{(z-a)^2} - \dfrac{1}{a^2}\Bigr]\\[3mm]
y = -\dfrac{1}{z^3} - \sum_{a\in E(\omega_1,\omega_2)}\dfrac{1}{(z-a)^3}
\end{cases}
\]

\[
\begin{cases}
\omega = dz\\
\eta = x\,dz
\end{cases}
\qquad \eta_1 = \int_u^{u+\omega_1}x\,dz,\quad
\eta_2 = \int_u^{u+\omega_2}x\,dz
\]

\[
\gamma_4 = \qquad \gamma_6 =
\]

\[
\begin{cases}
\eta + \alpha\omega = \beta\bar\omega\\[1mm]
\alpha = -\dfrac{\bar\omega_1\eta_2 - \bar\omega_2\eta_1}
{\bar\omega_1\omega_2 - \bar\omega_2\omega_1}
\end{cases}
\]

\[
\pi = \alpha\,\omega^2 \in \Gamma\,\mathrm{Hom}(\underline{\omega}^{-1},
\underline{\omega}) = \Gamma\,\underline{\omega}^2
\]

\[
\eta_1 = \Bigl[-\frac1z\Bigr]_u^{u+\omega_1}
+ \sum_{a\neq 0}\Bigl[-\frac{1}{z-a} - \frac{1}{a^2}\,z\Bigr]_u^{u+\omega_1}
\]

\[
= \Bigl(\frac1u - \frac{1}{u+\omega_1}\Bigr)
+ \sum_{a\neq 0}\Bigl[\frac{1}{u-a} - \frac{1}{u+\omega_1-a}
- \frac{\omega_1}{a^2}\Bigr]
\]

\[
= \omega_1\Bigl[\frac{1}{u(u+\omega_1)}
+ \sum_{a\neq 0}\Bigl(\frac{1}{(u-a)(u+\omega_1-a)} - \frac{1}{a^2}\Bigr)\Bigr]
\]

\[
= \omega_1\Bigl[\frac{1}{u(u+\omega_1)} + \frac{1}{(u-\omega_1)u}
- \frac{1}{\omega_1^2} + \sum_{\substack{a\neq 0\\ a\neq\omega_1}}\cdots\Bigr]
\]

\[
\eta_1 = \omega_1\Bigl[-\frac{3}{\omega_1^2}
+ \sum_{\substack{a\neq 0\\ a\neq\omega_1}}\frac{1}{a^3}\,
\frac{\omega_1}{1-\omega_1a^{-1}}\Bigr]
\]

\[
\eta_1 = -\frac{3}{\omega_1} + \omega_1\sum_{\substack{a\neq 0\\ a\neq\omega_1}}
\Bigl(\frac{1}{a^3}\,\frac{1}{\omega_1^{-1}-a^{-1}}\Bigr)
= -\frac{3}{\omega_1} + \omega_1^2\sum_{\substack{a\neq 0\\ a\neq\omega_1}}
\frac{1}{a^2}\cdot\frac{1}{a-\omega_1}
\]

\[
a = m\omega_1 + n\omega_2
\]

\[
\eta_2 = \qquad -\frac{3}{\omega_2} + \omega_2^2\sum_{\substack{a\neq 0\\ a\neq\omega_2}}
\frac{1}{a^2}\,\frac{1}{a-\omega_2}
\]

\[
\eta_1 = -\frac{3}{\omega_1} + \omega_1^2
\sum_{\substack{(m,n)\neq(0,0)\\ (m,n)\neq(1,0)}}
\frac{1}{(m\omega_1+n\omega_2)^2}\,\frac{1}{(m-1)\omega_1+n\omega_2}
\]

\[
\eta_2 = -\frac{3}{\omega_2} + \omega_2^2
\sum_{\substack{(m,n)\neq(0,0)\\ (m,n)\neq(0,1)}}
\frac{1}{(m\omega_1+n\omega_2)^2}\,\frac{1}{m\omega_1+(n-1)\omega_2}
\]

\[
\bar\omega_1\eta_2 - \bar\omega_2\eta_1
= -3\Bigl(\frac{\bar\omega_1}{\omega_2} - \frac{\bar\omega_2}{\omega_1}\Bigr)
+ \sum_{(m,n)\neq(0,0),(1,0),(0,1)} \frac{N_{m,n}}{D_{m,n}}
\]

\[
N_{m,n} = |\omega_1|^2\Bigl((m-1)\omega_2^2 + n\,\frac{\omega_2^3}{\omega_1}\Bigr)
- |\omega_2|^2\Bigl((n-1)\omega_1^2 + m\,\frac{\omega_1^3}{\omega_2}\Bigr)
\]

\[
D_{m,n} = (m\omega_1+n\omega_2)^2\,\bigl((m-1)\omega_1+n\omega_2\bigr)
\bigl(m\omega_1+(n-1)\omega_2\bigr)
\]

\[
+\ \bar\omega_1\omega_2^2\,\frac{1}{\omega_1^2(\omega_1-\omega_2)}
- \bar\omega_2\omega_1^2\,\frac{1}{\omega_2^2(\omega_2-\omega_1)}
\]

\[
= \frac{\bar\omega_1\omega_2^4+\bar\omega_2\omega_1^4}{\omega_1^2\omega_2^2(\omega_1-\omega_2)}
- 3\,\frac{|\omega_1|^2-|\omega_2|^2}{\omega_1\omega_2}
+ \sum_{(m,n)\neq(0,0),(1,0),(0,1)}\cdots
\]

\[
= \frac{|\omega_1|^2}{\omega_1^2}\,\frac{\tau^4+\bar\tau}{\tau^2(1-\tau)}
- \frac{|\omega_1|^2}{\omega_1^2}\,\frac{1-|\tau|^2}{\tau}
\]

\[
+ \sum_{(m,n)\neq(0,0),(0,1),(1,0)}\frac{|\omega_1|^2}{\omega_1^2}\,
\frac{(m-1)\tau^2+n\tau^3-(n-1)|\tau|^2+m|\tau|^2\tau^{-1}}
{(m+n\tau)^2\bigl((m-1)+n\tau\bigr)\bigl(m+(n-1)\tau\bigr)}
\]

\[
= \frac{|\omega_1|^2}{\omega_1^2}\Bigl(\frac{\tau^5+|\tau|^2}{\tau^3(1-\tau)}
- \frac{1-|\tau|^2}{\tau}
+ \sum_{(m,n)\neq(0,0),(0,1),(1,0)}
\frac{n\tau^4+(m-1)\tau^3+(n-1)|\tau|^2\tau+m|\tau|^2}
{\tau(m+n\tau)^2\bigl((m-1)+n\tau\bigr)\bigl(m+(n-1)\tau\bigr)}\Bigr)
\]

\[
\alpha = -\frac{\bar\omega_1\eta_2-\bar\omega_2\eta_1}{2i\,|\omega_1|^2\,\mathrm{Im}\,\tau}
= -\frac{1}{2i\,\omega_1^2\,\mathrm{Im}(\tau)}
\Bigl(\frac{\tau^5+|\tau|^2}{\tau^3(1-\tau)} - \frac{1-|\tau|^2}{\tau}
+ \sum_{(m,n)\neq(0,0),(0,1),(1,0)}\cdots\Bigr)
\]

\[
V = L + L^{-1}
\]

\[
\bar L \to L + L^{-1} \qquad \alpha\in \bar L^{-1}\otimes L,\quad
\beta\in(\bar L L)^{-1}
\]

\[
V = L + \varphi(L)
\]

\[
\alpha\,\bar z = y + \bar x
\]

\[
\varphi(x) = \bar x = \alpha(\tilde x) + \beta(\tilde x)
\]

\[
y = a + \varphi(b) \quad a\in L,\ b\in L;\qquad
= \bigl(a+\alpha(b)\bigr) + \beta(b)
\]

\[
y = -\alpha\beta^{-1}(y) + \overline{\beta^{-1}(y)}, \quad\text{d'où}
\]

\[
\bar y = \beta^{-1}(y) - \overline{\alpha\beta^{-1}(y)}
= \bigl(\beta^{-1}(y) - \alpha^2\beta^{-1}(y)\bigr) - \beta\alpha\beta^{-1}(y)
\]

\[
\langle\bar x,\bar y\rangle = \langle\underbrace{\alpha(x)}_{L}
+ \underbrace{\beta(x)}_{L^{-1}},\ \bigl(\beta^{-1}(y)-\alpha^2\beta^{-1}(y)\bigr)
- \beta\alpha\beta^{-1}(y)\rangle
\]

\[
= -\langle\alpha(x),\beta\alpha\beta^{-1}(y)\rangle
+ \langle\beta^{-1}y-\alpha^2\beta^{-1}y,\ \beta(x)\rangle
\]

\[
\beta(\omega,\omega) = \int_X\omega\bar\omega = \int_X(dx+i\,dy)(dx-i\,dy)
= -2i\int_X dx\wedge dy = -2i\,|\omega_1|^2\,\mathrm{Im}\,\tau
= |\omega_1|^2(\bar\tau-\tau)
\]

\[
\beta = -2i\rho \quad\text{avec}\quad \rho>0, \qquad \rho = \int_X dx\wedge dy
\]

\[
\rho = -\frac{1}{2i}\int\omega\bar\omega = \mathrm{Vol}(X),
\]

\[
\sigma = -\frac{1}{2i}\int\eta_0\bar\eta_0
\]

\[
4\rho\sigma = (1-\alpha\bar\alpha)
\]

\[
0\leq\alpha\bar\alpha\leq 1
\]

\[
1-\alpha\bar\alpha = 4\rho\sigma, \quad\text{i.e.}\quad
\alpha\bar\alpha = 1-4\rho\sigma
\]

\[
(\arg\alpha)^6\in\mathcal{U}.
\]

\[
\begin{cases}
\operatorname{div}(j)^{+} = 3\operatorname{div}(c_4) & \tfrac13\operatorname{div}(j)^{+} = \operatorname{div} c_4\\
\operatorname{div}(j)^{-} = M^{(\infty)} & \\
\operatorname{div}(j-12^3)^{+} = 2\operatorname{div}(c_6) & \tfrac12\operatorname{div}(j-12^3)^{+} = \operatorname{div} c_6\\
\operatorname{div}(j-12^3)^{-} = M^{(\infty)} & \\
M^{(\infty)} = \operatorname{div}\Delta &
\end{cases}
\]

\[
\begin{cases}
\mathrm{cl}\bigl(\tfrac13\operatorname{div}(j)^{+}\bigr) = \omega^{4}\\
\mathrm{cl}\bigl(\tfrac12\operatorname{div}(j-12^3)^{+}\bigr) = \omega^{6}\\
\mathrm{cl}(M^{\infty}) = \omega^{12}
\end{cases}
\]

\[
H^1(\operatorname{Spec}\mathbf{Z}[\tfrac16], \mu_6) \simeq
\bigl(\mathbf{Z}[\tfrac16]\bigr)^{*}_{6} \simeq \mathbf{Z}/6\mathbf{Z}
\times \mathbf{Z}/6\mathbf{Z} \times \mathbf{Z}/2\mathbf{Z}
\]

\[
\mapsto \mathbf{F}_4 \qquad \begin{pmatrix} * & * \\ 0 & 1 \end{pmatrix}
\]

\[
\struck{\ill{}}\ W[[t]] \qquad (t=a_1 \text{ en car.\ 2},\ t=b_2 \text{ en car.\ 3})
\]

\[
\mu_3^{*}(\tfrac16) \overset{B}{\times} G
\]