\[
\underline{V}_{C/S} = f^*\bigl(s^*(\underline{V}_{C/S})\bigr) = f^*(\mathcal{L}_{C/S})
\qquad [\,\mathcal{L} = \text{faisceau de Lie}\,]
\]

\[
s^*(\underline{L}) \simeq s^*(\underline{L}\otimes_{C}\underline{O}_{S'})
\simeq s^*(\underline{N}_{C/S'}) \simeq s^*(\underline{V}_{C/S}) = \mathcal{L}
\]

\[
s^*(\underline{L}^{\otimes n}) \simeq \mathcal{L}_{C/S}^{\otimes n}.
\]

\[
\sigma\colon \underline{O}_C \longrightarrow \underline{L}, \qquad
\sigma\in\Gamma(C,\underline{L})
\]

\[
\cdots \longrightarrow \underline{L}^{\otimes n} \longrightarrow
\underline{L}^{\otimes(n+1)} \longrightarrow \cdots \longrightarrow
\underline{L}^{\otimes(n+k)} \longrightarrow \cdots
\]

\[
\cdots \longrightarrow f_*(\underline{L}^{\otimes n}) \longrightarrow
f_*(\underline{L}^{\otimes(n+1)}) \longrightarrow \cdots
\]

\[
\cdots \longrightarrow \mathcal{E}_n \longrightarrow \mathcal{E}_{n+1} \longrightarrow \cdots
\]

\[
\mathcal{E}_n = f_*(\underline{L}^{\otimes n}).
\]

\[
\mathcal{E}_0 = \mathcal{E}_1 \subset \mathcal{E}_2 \subset \mathcal{E}_3 \subset \cdots
\]

\[
\mathcal{E}_0 = \underline{O}_S .
\]

\[
\mathcal{L}_n \simeq \mathcal{E}_{n}/\mathcal{E}_{n-1}
\]

\[
\mathcal{L}_n\otimes\mathcal{L}_m \simeq \mathcal{L}_{m+n} \quad \text{si } n,m
\underset{\neq 1}{\geqslant 0}
\]

\[
\mathcal{E} = \varinjlim \mathcal{E}_n = f_*(\varinjlim \underline{L}^{\otimes n})
\]

\[
\mathcal{E}_m\times\mathcal{E}_n \longrightarrow \mathcal{E}_{m+n}
\]

\[
\underline{\mathcal{L}}_n\otimes\underline{\mathcal{L}}_m \longrightarrow
\underline{\mathcal{L}}_{n+m}, \qquad n,m \underset{\neq 1}{\geqslant 0},
\]

\[
\underline{\mathcal{L}}\otimes\underline{\mathcal{L}}_n \longrightarrow
\underline{\mathcal{L}}_{n+1}, \qquad n\geqslant 2 .
\]

\[
\underline{\mathcal{L}}\otimes\underline{\mathcal{L}}_n \simeq
\underline{\mathcal{L}}_{n+1} \quad \text{pour } n\geqslant 2, \text{ donc}
\]

\[
\mathcal{L}_n \simeq \mathcal{L}^{n-2}\otimes\mathcal{L}_2, \qquad \text{donc pour }
n\geqslant 2,
\]

\[
\mathcal{L}_2 \simeq \mathcal{L}^{\otimes 2} .
\]

\[
\cdots\longrightarrow\underline{L}^{\otimes n}\longrightarrow
\underline{L}^{\otimes(n+1)}\longrightarrow
(\underline{V}_{C/S})^{\otimes(n+1)}\otimes_{\underline{O}_C}\underline{O}_{S'}
\longrightarrow 0
\]

\[
0\longrightarrow\mathcal{E}_n\longrightarrow\mathcal{E}_{n+1}\longrightarrow
\mathcal{L}^{\otimes(n+1)}\longrightarrow R^1f_*(\underline{L}^{\otimes n})
\]

\[
\boxed{\ \mathcal{L}_{n+1}\simeq\mathcal{L}^{\otimes(n+1)} \qquad n\geqslant 2\ }
\]

\[
C \longrightarrow \mathbf{P}(\mathcal{E}_3)
\]

\[
\mathbf{P}_3(\mathbf{P}(\mathcal{E}_3)) \simeq
\mathbf{P}(\operatorname{Symm}_3(\check{\mathcal{E}}_3)) .
\]

\[
a x^{?} + b y^3 + c xy + d y^2 + e x + f y + g = 0
\]

\[
1,\ y,\ x,\ y^2,\ xy,\ y^3
\]

\[
\varphi(x,y) = \boxed{\,x^2 + a y^3 + b xy + c y^2 + d x + e y + f = 0\,}
\]

\[
\Phi(x,y,z) = x^2 z + a y^3 + b xyz + c y^2 z + d x z^2 + f z^3 = 0
\]

\[
\delta(a,b,c,d,e,f)
\]

\[
\mathfrak{M} = \operatorname{Spec}\bigl(\mathbf{Z}[A,B,C,D,E,F][\delta^{-1}]\bigr)
\]

\[
\begin{pmatrix} 1 & \alpha & \gamma \\ 0 & \lambda & \beta \\ 0 & 0 & \mu \end{pmatrix}
\]

\[
X^2 + AY^3 + BXY^{2} + CY^2 + DX + EY + F
\]

\[
Y' = \lambda Y + \alpha, \qquad X' = \mu X + \beta Y + \gamma
\]

\[
\left(\lambda,\ \begin{pmatrix} 1 & \alpha & \beta \\ 0 & \lambda^2 & \gamma \\ 0 & 0 & \lambda^3 \end{pmatrix}\right)
\qquad
\begin{array}{l}
\lambda\in\Gamma(\mathbf{G}_{m,S}/S)=\Gamma(S,\underline{O}_S^*)\\
\alpha,\beta,\gamma\in\Gamma(S,\underline{O}_S)
\end{array}
\]

\[
\begin{pmatrix} 1 & \alpha & \beta \\ 0 & \lambda & \gamma \\ 0 & 0 & \mu \end{pmatrix}
\quad\text{pour}\quad
\lambda = \left(\frac{\mu}{\lambda}\right)^2, \quad
\mu = \left(\frac{\mu}{\lambda}\right)^3 \qquad\text{i.e.}\quad
\boxed{\mu^2 - \lambda^3 = 0}
\]

\[
x^2 - y^3 + bxy + cy^2 + dx + ey + f = 0
\]

\[
x^2 = y^3 + axy + by^2 + cx + dy + e
\]

\[
x^2 = y^3 - axy + by^2 - cx + dy + e
\]

\[
\boxed{\,x^2 = y^3 + ay^2 + by + c\,}
\]

\[
\begin{pmatrix} 1 & \alpha & 0 \\ 0 & \lambda^2 & 0 \\ 0 & 0 & \lambda^3 \end{pmatrix}
\quad\text{i.e.}\quad
\begin{pmatrix} 1 & \alpha & 0 \\ 0 & u & 0 \\ 0 & 0 & v \end{pmatrix}
\quad\text{avec}\quad u^3 - v^2 = 0 .
\]

\[
j\equiv 1 \quad \Bigl(\frac{t}{t-1}\Bigr)^3 \in O^6O^* \qquad
\Bigl(\frac{t-1}{t}\Bigr)^2 \in O^6O^* \qquad
\sqrt[6]{\frac{t}{t-1}}
\]

\[
\sqrt[2]{\frac{t}{t-1}} \qquad \sqrt[3]{\frac{t-1}{t}} \qquad \tfrac12 \qquad \tfrac13
\]

\[
\lambda^{3/6}\cdot\lambda^{-2/6} = \lambda^{1/6} \qquad
s^6 = \frac{t}{t-1}
\]

\[
\frac{-27c^2}{b^3-27c^2} = j \qquad 27c^2(j-1) = b^3 j \qquad
\frac{b^3}{c^2} = j
\]

\[
\begin{array}{ccc} L & L^2 & L^3 \\ & b & c \end{array}
\qquad
\boxed{\ \frac{b^3}{c^2} = 27\,\frac{j-1}{j}\ }
\]

\[
b = 27\,\frac{j-1}{j} \qquad c = 27\,\frac{j-1}{j}
\]

\[
\boxed{\ x^3 + 27\,\frac{j-1}{j}\,(x+1) = 0\ }
\]

\[
\underline{\mathrm{Hom}}_{\underline{A}}(E,\underline{A})\times
\underline{\mathrm{Hom}}_{\underline{O}}(\underline{A},\underline{O})
\]

\[
\prod_{i\neq j}(x_i-x_j) = \prod_i \Bigl(\frac{f(x)}{x-x_i}\Bigr)_{x\to x_i}
= \prod_i f'(x_i)
\]

\[
f(x_i) =
\]

\[
3x^2 + 2ax + b
\]

\[
(3x_1^2+2ax_1+b)(3x_2^2+2ax_2+b)(3x_3^2+2ax_3+b)
\]

\[
j = -\frac{27c^2}{\Delta}
\]

\[
\Delta = \prod(3x_i^2+b) = 27(x_1x_2x_3)^2 + 9b\bigl(\textstyle\sum(x_1x_2)^2
+ 3b^2\sum x_i^2\bigr) + b^3
\]

\[
(x_1x_2x_3)^2 = c^2
\]

\[
\textstyle\sum(x_1x_2)^2 = \bigl(\sum x_1x_2\bigr)^2 - 2\sum x_1^2x_2x_3
\]

\[
\textstyle\sum x_i^2x_2x_3 = \bigl(\sum x_i^2\bigr)\bigl(\sum x_2x_3
\]

\[
\boxed{\ \Delta = b^3 - 27c^2\ }\ ?
\qquad x_1^2x_2^2 + \qquad \textstyle\prod 3x_i^2+b
\]

\[
3\textstyle\sum(x_1x_2)^2 + \bigl(\sum x_1x_2\bigr)\bigl(\sum x_i^2\bigr)
\]

\[
\frac{(t+1)(2t-1)(t-2)}{t(t-1)} =
\frac{(2x_1-x_2-x_3)(2x_2-x_3-x_1)(2x_3-x_1-x_2)}{(x_1-x_2)(x_2-x_3)(x_3-x_1)} .
\]

\[
\boxed{\ j = -\frac{1}{27\Delta}\bigl[(2x_1-x_2-x_3)(2x_2-x_3-x_1)(2x_3-x_1-x_2)\bigr]^2\ }
\]

\[
\sum x_i = -a, \qquad \sum x_ix_j = b, \qquad x_1x_2x_3 = -c
\qquad\qquad
\Delta = \bigl((x_1-x_2)(x_2-x_3)(x_3-x_1)\bigr)^2
\]

\[
j = \frac{\bigl[(3x_1+a)(3x_2+a)(3x_3+a)\bigr]^2}{-27\,\Delta}
\]

\[
\begin{aligned}
\sqrt{-27\Delta j} &= a^3 + 3a^2(x_1+x_2+x_3) + 9a(x_1x_2+x_2x_3+x_3x_1) + 27\,x_1x_2x_3\\
&= a^3 - 3a^3 + 9ab - 27c\\
&= -2a^3 + 9ab - 27c
\end{aligned}
\]

\[
\boxed{\ j = \frac{(27c + 2a^3 - 9ab)^2}{-27\,\Delta}\ }
\]

\[
y^2 = 4x^3 - h(x+1) \qquad\qquad h = 27j\,(j - 2^6 3^3)^{-1}
\]

\[
y^2 = x^3 + ax^2 + bx + c
\]

\[
\begin{array}{cccc}
x_1 & x_2 & x_3 & x_4\\
\downarrow & \downarrow & \downarrow & \downarrow\\
\infty & 0 & 1 & t
\end{array}
\]

\[
((0,1,\infty,t)) = ((x_1,x_2,x_3,x_4))
\]

\[
\frac{x_1-x_3}{x_1-x_4} : \frac{x_2-x_3}{x_2-x_4}
= \frac{0-\infty}{0-t} : \frac{1-\infty}{1-t}
= \frac{1-t}{-t} = -\frac{1}{t} + 1
\]

\[
\frac{t-1}{t} \qquad\qquad \frac{t-\infty}{t-0} : \frac{1-\infty}{1-0}
\]

\[
t = \frac{1-\infty}{1-0} : \frac{t-\infty}{t-0}
= \frac{x_3-x_1}{x_3-x_2} : \frac{x_4-x_1}{x_4-x_2}
\]

\[
\boxed{\ t = \frac{x_3-x_1}{x_3-x_2}\ }
\qquad
t-1 = \frac{x_2-x_1}{x_3-x_2}
\qquad
\frac{c-a}{c-b} \;\Big|\; \frac{c-b}{c-a}
\]

\[
\frac{a-b}{a-c} \qquad \frac{a-b}{c-b} \cdot \frac{b-a}{b-c}
\]

\[
t+1 = \frac{2x_3-x_1-x_2}{x_3-x_2}
\qquad
2t-1 = \frac{x_3-2x_1+x_2}{x_3-x_2}
\qquad
t-2 = \frac{-x_3-x_1+2x_2}{x_3-x_2}
\]

\[
(1)\quad
\begin{cases}
\varphi(t) = 1 - \dfrac{1}{t}\\[2ex]
\psi(t) = \dfrac{1}{1-t}
\end{cases}
\]

\[
(2)\quad
\begin{cases}
\lambda t = \dfrac{1}{t}\\[1ex]
\mu t = 1-t\\[1ex]
\nu t = 1 + \dfrac{1}{t-1} = \dfrac{t}{t-1}
\end{cases}
\qquad\qquad
\begin{array}{c} 1\\ \infty\\ 0 \end{array}
\]

\[
3)\ \text{a)}\ \boxed{\begin{array}{ll} \varphi^2 = \psi & \psi^2 = \varphi\\
& \varphi\psi = \psi\varphi = 1 \end{array}}
\quad
\text{b)}\ \boxed{\lambda^2 = \mu^2 = \nu^2 = 1}
\quad
\text{c)}\ \boxed{\begin{array}{l} \lambda\mu = \mu\nu = \nu\lambda = \varphi\\
\mu\lambda = \nu\mu = \lambda\nu = \psi \end{array}}
\]

\[
\text{d)}\ \boxed{\begin{array}{l|l}
\varphi\lambda = \mu\varphi = \nu & \psi\nu = \mu\psi = \lambda\\
\varphi\mu = \nu\varphi = \lambda & \psi\mu = \lambda\psi = \nu\\
\varphi\nu = \lambda\varphi = \mu & \psi\lambda = \nu\psi = \mu
\end{array}}
\]

\[
4)\quad
\begin{cases}
{}^{\varphi}\lambda = \mu \quad {}^{\varphi}\mu = \nu \quad {}^{\varphi}\nu = \lambda\\
{}^{\psi}\nu = \mu \quad {}^{\psi}\mu = \lambda \quad {}^{\psi}\lambda = \nu
\end{cases}
\qquad
5)\quad
\begin{cases}
{}^{\lambda}\varphi = {}^{\mu}\varphi = {}^{\nu}\varphi = \psi\\
{}^{\lambda}\psi = {}^{\mu}\psi = {}^{\nu}\psi = \varphi
\end{cases}
\]

\[
6)\quad \sigma t = \frac{t-2}{2t-1} \qquad [\sigma^2 = \mathrm{id}]
\]

\[
7)\quad
\begin{cases}
\sigma 0 = 2\\
\sigma 1 = -1\\
\sigma\infty = \tfrac12
\end{cases}
\]

\[
1 - \frac1t = t \quad \ldots
\]

\[
8)\qquad t^2 - t + 1 = 0
\]

\[
(\mathbf{P}^1_S/(\mathbf{Z}/3))\times_S S' \simeq \mathbf{P}^1_{S'} .
\]

\[
\begin{array}{lllcccc}
t & \mathbf{P}^1 & & \{0,1,\infty\} & \{2,-1,\tfrac12\} & x' = \tfrac12(1+\sqrt{-3}) & x'' = \tfrac12(1-\sqrt{-3})\\
\mathbf{Z}/3 & \big\downarrow & f & \big| & \big| & \big\| & \big\|\\
s & \mathbf{P}^1 & & \infty & 0 & 1 & -1\\
\mathbf{Z}/2 & \big\downarrow & g & \big\| & \big\| & \big\downarrow & \swarrow\\
r & \mathbf{P}^1 & & \infty & 0 & 1 &
\end{array}
\]

\[
(9)\qquad r = g(s) = s^2
\]

\[
s = f(t) = \frac{1 + \Bigl(\dfrac{t-x'}{t-x''}\Bigr)^3}{1 - \Bigl(\dfrac{t-x'}{t-x''}\Bigr)^3}
= \frac{(t-x'')^3 + (t-x')^3}{(t-x'')^3 - (t-x')^3}
\]

\[
= \frac{2t^3 - 3t^2(x''+x') + 3t(x'^2+x''^2) - (x'^3+x''^3)}
{3t^2(x'-x'') + 3t(x''^2-x'^2) - (x''^3-x'^3)}
\]

\[
f = v^{-1}hu
\]

\[
\begin{aligned}
f^*(t) &= u^*h^*v^{-1*}(t) = u^*h^*\Bigl(\frac{1-t}{1+t}\Bigr)\\
&= u^*\,\frac{1-h^*t}{1+h^*t} = u^*\,\frac{1+t^3}{1-t^3}\\
&= \frac{1+u^*(t)^3}{1-u^*(t)^3} =
\end{aligned}
\]

\[
x'^3 = x''^3 = -1,
\]

\[
x''^3 - x'^3 = 0
\]

\[
\begin{cases} x'+x'' = 1\\ x'x'' = 1 \end{cases}
\]

\[
\begin{aligned}
x'^2 + x''^2 &= (x'+x'')^2 - 2x'x'' = -1\\
(x'-x'')^2 &= (x'+x'')^2 - 4x'x'' = -3\\
x'-x'' &= \sqrt{-3}\\
x''^2 - x'^2 &= (x''-x')(x''+x') = -\sqrt{-3}
\end{aligned}
\]

\[
f(t) = \frac{1}{3\sqrt{-3}}\,\frac{2t^3 - 3t^2 - 3t + 2}{t(t-1)}
= \frac{1}{3\sqrt{-3}}\,\frac{(t+1)(2t-1)(t-2)}{t(t-1)}
\]

\[
(10)\qquad f(t) = \frac{1}{3\sqrt{-3}}\,\frac{(t+1)(2t-1)(t-2)}{t(t-1)}
\]

\[
\boxed{\ (g f)^*(t) = j = \frac{1}{-27}\Bigl(\frac{(t+1)(2t-1)(t-2)}{t(t-1)}\Bigr)^2\ }
\qquad (11)
\]

\[
(\mathbf{G}_a)_S - (0) - (1) = (\mathbf{G}_m)_S - 1 .
\]

\[
\begin{array}{ll}
\sigma & \mathbf{P}^1\\
& \big\downarrow f\\
\sigma' & \mathbf{P}^1\\
& \big\downarrow g\\
\sigma'' & \mathbf{P}^1
\end{array}
\]

\[
(12)\qquad \sigma'(t) = \sigma''(t) = 1/t
\]

\[
\mathbf{P}^1_S \xrightarrow{\ p\ } \mathbf{P}^1_S
\]

\[
X \xrightarrow{\ f\ } \mathbf{P}^1_S
\]

\[
\begin{array}{ccccc}
X & \longrightarrow & \mathbf{P}^1_S & \longrightarrow & \mathbf{P}^1_S \\
 & & \infty,0,1 & \longmapsto & \infty,0,j
\end{array}
\qquad t\mapsto jt^2 ;
\]

\[
S'\times_{(S,\mathfrak{S}_3)}\mathbf{P}^1_S
\]

\[
t\in\Gamma(S',\underline{O}_{S'}),
\]

\[
\sigma.t = \varphi_\sigma(t) \qquad \sigma\in\mathfrak{S}_3
\]

\[
\begin{cases} \lambda t = \dfrac{1}{t} \\[1ex] \mu t = 1-t \end{cases}
\]

\[
\Lambda \simeq \mathcal{L}
\]

\[
(D_\xi u_\xi)^2 = 4u_\xi^3 + b_\xi u_\xi + c_\xi
\]

\[
(D_{\lambda\xi}u_{\lambda\xi})^2 = 4u_{\lambda\xi}^3 + b_{\lambda\xi}u_{\lambda\xi}
+ c_{\lambda\xi}
\]

\[
u_\xi = 6x_{\xi^2} \qquad\qquad u_{\lambda\xi} = \rho\, u_\xi
\]

\[
C \,\big|\, \underline{O}\to\underline{O}(1) \qquad
f_*(\underline{O}(n)) = E_n \qquad\Big|\qquad
\lambda^2\rho^2(D_\xi u_\xi)^2 = 4\rho^3u_\xi^3 + \rho b_{\lambda\xi}u_\xi
+ c_{\lambda\xi}
\]

\[
\underline{O} = E_1 \subset E_2 \subset E_3 \subset E_4 \subset E_5 \subset E_6
\subset \cdots \subset E_n
\]

\[
\underline{O} = \Delta^0 \qquad L_2 \qquad L_3 \qquad\Big|\qquad L_2^2 \qquad
L_2L_3 \qquad L_2^3
\]

\[
y^2 = 4x^3 + bx + c \qquad\qquad \frac{y}{x} = D
\]

\[
(D_\xi u_\xi)^2 = 4\,\frac{\rho}{\lambda^2}\,u_\xi^3 +
\frac{1}{\rho\lambda^2}\,b_{\lambda\xi}u_\xi + \frac{c_{\lambda\xi}}{\lambda^2\rho^2}
\]

\[
\rho = \lambda^2 \qquad
\begin{cases}
u_{\lambda\xi} = \lambda^2 u_\xi\\
b_{\lambda\xi} = \lambda^4 b_\xi\\
c_{\lambda\xi} = \lambda^6 c_\xi
\end{cases}
\qquad
\begin{array}{l}
u_\xi \in L_2 \simeq \Delta^2\\
b_\xi \in \Delta^0\\
c_\xi \in \Delta^0
\end{array}
\]

\[
\Delta^4\longrightarrow\underline{O}_S \qquad \Delta^6\longrightarrow\underline{O}_S
\qquad \Delta^2\longrightarrow\Delta^2 \qquad\qquad
\begin{array}{l} \Delta^{?}\simeq\mathcal{L}^{18}\\ \Delta^2\simeq\mathcal{L}^3 \end{array}
\]

\[
\begin{cases}
b\in\Delta^{-4}\\
c\in\Delta^{-6}\\
u \ \text{scalaire}
\end{cases}
\qquad\qquad u_\xi = \xi^2
\]

\[
\begin{array}{c} X\\[1ex] \\ S \end{array} \qquad
X\times_S X \supset R \qquad\qquad
\begin{array}{ccc}
\mathcal{B} & & A\\
X & \longrightarrow & Y\\
\nearrow & & \nearrow\\
X\times_Y X & \longrightarrow & X
\end{array}
\qquad x_i :
\]

\[
B\otimes_A B
\]

\[
\begin{array}{c} B\\ | \\ A \end{array} \qquad
A \longrightarrow B\otimes_A B \qquad B
\]

\[
\sum\lambda_i x_i = 0 \qquad \lambda_i\in A
\]

\[
\Bigl(\sum\lambda_i x_i\Bigr)\otimes_A 1 =
\sum(x_i\otimes 1)(1\otimes\lambda_i) = 0
\]

\[
\lambda_i = 0 \ \text{dans}\ B
\]

\[
\wp z = \frac{1}{z^2} + c_1 z^2 + c_2 z^4 + \cdots
\]

\[
\wp'^2 z = 4\wp^3 z - g_2\wp z - g_3 \qquad g_2 = 20c_1 \quad g_3 = 28 c_2
\]

\[
z = \int_\infty^{Z} \frac{dZ}{\sqrt{4Z^3 - g_2 Z - g_3}}
\]

\[
\wp\omega_1 = e_1 \qquad \wp\omega_2 = e_2 \qquad \wp\omega_3 = e_3
\]

\[
\wp'\omega_1 = \wp'\omega_2 = \wp'\omega_3 = 0 .
\]

\[
\wp'' z = 6\wp^2 z - \frac{g_2}{2} \qquad \wp''' z = 12\,\wp z\,\wp' z
\]

\[
\zeta z = \frac{1}{z} - \int_0^z \Bigl(\wp z - \frac{1}{z^2}\Bigr)dz
\qquad \text{on a} \qquad \zeta' z = -\wp z
\]

\[
\zeta(z+2\omega_1) - \zeta z = \text{c}^{\text{te}} = 2\zeta\omega_1 = 2\eta_1
\]

\[
\omega_2\eta_1 - \omega_1\eta_2 = \frac{\pi i}{2} \quad \text{etc.} ;
\qquad \text{d'ailleurs} \quad \eta_1 + \eta_2 + \eta_3 = 0 .
\]

\[
\int \frac{Z\,dZ}{\sqrt{4Z^3 - g_2Z - g_3}}
\]

\[
\Bigl|\ \int \wp z\,dz = -\zeta z + \text{c}^{\text{te}}
\]

\[
\frac{A_\alpha}{(z-a)^\alpha} + \cdots + \frac{A_1}{z-a} \ ; \qquad
\frac{B_\beta}{(z-b)^\beta} + \cdots + \frac{B_1}{z-b} \ ; \qquad \ldots
\]

\[
\begin{aligned}
f(z) = \text{c}^{\text{te}} &+ \Bigl\lbrace A_1\zeta(z-a) + A_2\wp(z-a)
- \frac{A_3}{2!}\wp'(z-a) + \cdots
+ (-1)^\alpha \frac{A_\alpha}{(\alpha-1)!}\wp^{(\alpha-2)}(z-a) \Bigr\rbrace \\
&+ \Bigl\lbrace B_1\zeta(z-b) + \cdots \Bigr\rbrace + \cdots
\end{aligned}
\]

\[
\int \zeta z\,dz = \log \sigma z
\]

\[
\sigma z = e^{\int \zeta z\,dz}
\]

\[
\wp z = \frac{1}{z^2} + \sum\mathop{{\sum}'} \left[\frac{1}{(z-w)^2} - \frac{1}{w^2}\right]
\]

\[
c_1 = 3\sum\mathop{{\sum}'} \frac{1}{w^4} \qquad
c_2 = 5\sum\mathop{{\sum}'} \frac{1}{w^6} \qquad \cdots
\]

\[
\zeta z = \frac{1}{z} + \sum\mathop{{\sum}'} \left[\frac{1}{z-w} + \frac{1}{w}
+ \frac{z}{w^2}\right]
\]

\[
\sigma z = z \prod\mathop{{\prod}'} \Bigl(1 - \frac{z}{w}\Bigr)
e^{\frac{z}{w} + \frac{z^2}{2w^2}}
\]

\[
\frac{\sigma' z}{\sigma z} = \zeta z
\]

\[
\sigma(z+2\omega_1) = -e^{2\eta_1(z+\omega_1)}\sigma z \qquad
\sigma(z+2\omega_2) = -e^{2\eta_2(z+\omega_2)}\sigma z
\]

\[
a + b + \cdots + l = a_1 + b_1 + \cdots + l_1 + \Omega .
\]

\[
f(z) = \text{c}^{\text{te}} \times
\frac{\sigma(z-a_1)\sigma(z-b_1)\cdots\sigma(z-l_1-\Omega)}
{\sigma(z-a)\sigma(z-b)\cdots\sigma(z-l)}
\]

\[
f(z) = \text{c}^{\text{te}} \times
\frac{\sigma(z-a_1)\sigma(z-b_1)\cdots\sigma(z-l_1)}
{\sigma(z-a)\sigma(z-b)\cdots\sigma(z-l+\Omega)}
\]

\[
f(z) = \text{c}^{\text{te}} \times e^{2(n_1\eta_1+n_2\eta_2)z}\,
\frac{\sigma(z-a_1)\cdots\sigma(z-l_1)}{\sigma(z-a)\cdots\sigma(z-l)}
\]

\[
\wp u - \wp v = -\frac{\sigma(u+v)\,\sigma(u-v)}{\sigma^2 u\,\sigma^2 v}
\]

\[
\zeta(u+v) = \zeta u + \zeta v + \frac{1}{2}\,\frac{\wp' u - \wp' v}{\wp u - \wp v}
\]

\[
\wp(u+v) + \wp u + \wp v = \frac{1}{4}\left(\frac{\wp' u - \wp' v}{\wp u - \wp v}\right)^2
= \bigl[\zeta(u+v) - \zeta u - \zeta v\bigr]^2
\]

\[
\int \frac{dZ}{\sqrt{(1-Z^2)(1-k^2Z^2)}}
\]

\[
\Bigl(\frac{dZ}{dz}\Bigr)^2 = (1-Z^2)(1-k^2Z^2)
\]

\[
Z = \frac{12\wp(z-z_0) - (5-k^2)}{12\wp(z-z_0) - (5k^2-1)}
\qquad
\begin{array}{l}
e_1 = \frac{1}{6}(1+k^2)\\[2pt]
e_2 = -\frac{1}{12}(k^2+6k+1)\\[2pt]
e_3 = -\frac{1}{12}(k^2-6k+1)
\end{array}
\]

\[
(1)\quad \mathrm{sn}\,z = \frac{12\wp(z-z_0) - (5-k^2)}{12\wp(z-z_0) - (5k^2-1)}
\qquad \text{avec} \quad \wp z_0 = \frac{5-k^2}{12} \quad
\wp' z_0 = -\frac{1-k^2}{2} .
\]

\[
Z^2 = \frac{1}{k^2}\bigl[\wp(z-z_0) - e_1\bigr]
\qquad (2)\ \left|\
\begin{array}{l}
e_1 = -\frac{1+k^2}{3}\\[2pt]
e_3 = \frac{2k^2-1}{3}\\[2pt]
e_2 = \frac{2-k^2}{3}
\end{array}\right.
\]

\[
(3)\quad \boxed{\ \mathrm{sn}^2 z = \frac{1}{k^2}\bigl[\wp(z-\omega_1) - e_1\bigr]\ }
\]

\[
Z^2 = \frac{1}{\wp(z-z_0) - e_1}
\]

\[
(4)\quad \boxed{\ \mathrm{sn}^2 z = \frac{1}{\wp z - e_1}\ }
\]

\[
(\wp z - e_1)\bigl(\wp(z-\omega_1) - e_1\bigr) = k^2
\]

\[
\frac{e^{2\eta_1\omega_1}}{\sigma^4\omega_1} = 3e_1^2 - \frac{g_2}{4}
\]

\[
k^2 = \frac{e^{2\eta_1\omega_1}}{\sigma^4\omega_1} = 3e_1^2 - \frac{g_2}{4}
\]

\[
(5)\quad \boxed{\ \mathrm{sn}(z+\omega_1)\,\mathrm{sn}\,z = \frac{1}{k}\ }
\]

\[
(6)\quad \boxed{\ \mathrm{sn}\,z = \sigma\omega_1\, e^{\eta_1 z}\,
\frac{\sigma z}{\sigma(z+\omega_1)}
= -\sigma\omega_1\, e^{-\eta_1 z}\,\frac{\sigma z}{\sigma(z-\omega_1)}\ }
\]

\[
(7)\quad
\left\lbrace
\begin{array}{l}
\mathrm{sn}(z+2\omega_1) = \mathrm{sn}\,z\\
\mathrm{sn}(z+2\omega_2) = -\mathrm{sn}\,z\\
\mathrm{sn}(z+2\omega_3) = -\mathrm{sn}\,z\\[4pt]
\text{d'où}\quad \mathrm{sn}(z+4\omega_2) = \mathrm{sn}\,z\\
\phantom{\text{d'où}}\quad \mathrm{sn}(z+4\omega_3) = \mathrm{sn}\,z
\end{array}
\right.
\]

\[
k = -\frac{e^{\eta_1\omega_2}}{\sigma'\omega_2}\ \Bigr)
\]

\[
\boxed{\ \mathrm{sn}(-z) = -\mathrm{sn}\,z\ }
\]

\[
\mathrm{sn}\,\omega_2 = \frac{1}{?}\,,
\]

\[
\mathrm{sn}'\omega_2 = 0 \qquad \text{de même} \qquad \mathrm{sn}\,{-\omega_2} = -1
\quad \mathrm{sn}'\,{-\omega_2} = 0
\]

\[
\mathrm{sn}(\omega_1+\omega_2) = \frac{1}{k} \quad \text{soit} \quad
\mathrm{sn}\,\omega_3 = -\frac{1}{k} \qquad \mathrm{sn}'\omega_3 = 0 .
\]

\[
(8)\ \left\lbrace
\begin{aligned}
\mathrm{cn}^2 z &= 1 - \mathrm{sn}^2 z = \frac{\wp z - e_2}{\wp z - e_1}
= -\frac{1}{k^2}\bigl[\wp(z-\omega_1) - e_3\bigr]
&\qquad& \boxed{\mathrm{cn}\,0 = +1}\\
\mathrm{dn}^2 z &= 1 - k^2\mathrm{sn}^2 z = \frac{\wp z - e_3}{\wp z - e_1}
= -\bigl[\wp(z-\omega_1) - e_2\bigr]
&& \boxed{\mathrm{dn}\,0 = +1}
\end{aligned}
\right.
\]

\[
(9)\ \left\lbrace
\begin{aligned}
\mathrm{cn}\,z &= \frac{\sigma\omega_1}{\sigma\omega_2}\,e^{(\eta_2-\eta_1)z}\,
\frac{\sigma(z-\omega_2)}{\sigma(z-\omega_1)}
= \frac{\sigma\omega_1}{\sigma\omega_2}\,e^{-(\eta_2-\eta_1)z}\,
\frac{\sigma(z+\omega_2)}{\sigma(z+\omega_1)}\\
\mathrm{dn}\,z &= \frac{\sigma\omega_1}{\sigma\omega_3}\,e^{(\eta_3-\eta_1)z}\,
\frac{\sigma(z-\omega_3)}{\sigma(z-\omega_1)}
= \frac{\sigma\omega_1}{\sigma\omega_3}\,e^{-(\eta_3-\eta_1)z}\,
\frac{\sigma(z+\omega_3)}{\sigma(z+\omega_1)}
\end{aligned}
\right.
\]

\[
(10)\ \left\lbrace
\begin{array}{lll}
{}^{*}\ \mathrm{cn}(-z) = \mathrm{cn}\,z & \mathrm{dn}(-z) = \mathrm{dn}\,z &
\text{$\mathrm{cn}\,z$ et $\mathrm{dn}\,z$ sont paires}\\[4pt]
{}^{*}\ \mathrm{cn}(z+2\omega_1) = -\mathrm{cn}\,z & \mathrm{dn}(z+2\omega_3) = -\mathrm{dn}\,z & \\
\phantom{{}^{*}}\ \mathrm{cn}(z+2\omega_3) = -\mathrm{cn}\,z & \mathrm{dn}(z+2\omega_1) = -\mathrm{dn}\,z & \\[4pt]
{}^{*}\ \mathrm{cn}(z+2\omega_2) = \mathrm{cn}\,z & \mathrm{dn}(z+2\omega_2) = \mathrm{dn}\,z &
\text{$\mathrm{cn}$ admet pour périodes}\\
\phantom{{}^{*}}\ \mathrm{cn}(z+4\omega_1) = \mathrm{cn}\,z & \mathrm{dn}(z+4\omega_3) = \mathrm{dn}\,z &
\quad 2\omega_2,\ 4\omega_1,\ 4\omega_3\\
\phantom{{}^{*}}\ \mathrm{cn}(z+4\omega_3) = \mathrm{cn}\,z & \mathrm{dn}(z+4\omega_1) = \mathrm{dn}\,z &
\text{$\mathrm{dn}$ admet pour périodes}\\
& & \quad 2\omega_3,\ 4\omega_1,\ 4\omega_2
\end{array}
\right.
\]

\[
\mathrm{cn}'\omega_2 = -\frac{e^{\eta_2\omega_2}}{\sigma'\omega_2}
\qquad (9,\ \text{puis } 6)\,;
\]

\[
\mathrm{cn}'\omega_2 = \pm\sqrt{1-k^2}
\]

\[
\mathrm{cn}\,\omega_3 = \frac{i}{k}\,\frac{e^{\eta_2\omega_2}}{\sigma'\omega_2}
= -\frac{i}{k}\,\mathrm{cn}'\omega_2 = \pm\frac{i}{k}\sqrt{1-k^2}
\qquad \mathrm{cn}'\omega_3 = 0
\]

\[
\mathrm{dn}\,\omega_2 = -\mathrm{cn}'\omega_2 = \frac{e^{\eta_2\omega_2}}{\sigma'\omega_2}
= \pm\sqrt{1-k^2} \qquad \mathrm{dn}'\omega_2 = 0
\]

\[
(11)\ \left\lbrace
\begin{aligned}
\mathrm{sn}'z &= \mathrm{cn}\,z\;\mathrm{dn}\,z\\
\mathrm{cn}'z &= -\mathrm{sn}\,z\;\mathrm{dn}\,z\\
\mathrm{dn}'z &= -k^2\,\mathrm{sn}\,z\;\mathrm{cn}\,z
\end{aligned}
\right.
\]

\[
(12)\quad \boxed{\ \mathrm{cn}^2 z + \mathrm{sn}^2 z = 1 \qquad
\mathrm{dn}^2 z + k^2\mathrm{sn}^2 z = 1 \qquad
\mathrm{dn}^2 z - k^2\mathrm{cn}^2 z = 1 - k^2\ }
\]

\[
(13)\ \left\lbrace
\begin{aligned}
\mathrm{cn}'^2 z &= (1-\mathrm{cn}^2 z)(1-k^2+k^2\mathrm{cn}^2 z)
= (1-k^2)(1-\mathrm{cn}^2 z)\Bigl(1 + \frac{k^2}{1-k^2}\mathrm{cn}^2 z\Bigr)\\
\mathrm{dn}'^2 z &= (1-\mathrm{dn}^2 z)\bigl(\mathrm{dn}^2 z - (1-k^2)\bigr)
= -(1-k^2)(1-\mathrm{dn}^2 z)\Bigl(1 - \frac{1}{1-k^2}\mathrm{dn}^2 z\Bigr)
\end{aligned}
\right.
\]

\[
\Bigl(\frac{dZ}{dz}\Bigr)^2 = (1-Z^2)(1-\rho^2 Z^2) \qquad
\]

\[
(1)\quad \mathrm{sn}_1^2 z = \frac{1}{\wp z - e_1} \qquad (\mathrm{sn}_1'0 = +1)
\]

\[
(2)\quad \mathrm{sn}_1 z = \sigma\omega_1\,e^{\eta_1 z}\,\frac{\sigma z}{\sigma(z+\omega_1)}
= -\sigma\omega_1\,e^{-\eta_1 z}\,\frac{\sigma z}{\sigma(z-\omega_1)}
\]

\[
(3)\quad \mathrm{sn}_1 z\;\mathrm{sn}_1(z+\omega_1) = \frac{1}{k_1} \qquad
\mathrm{sn}_1^2 z = \frac{1}{k_1^2}\bigl(\wp(z-\omega_1) - e_1\bigr) \qquad
\begin{array}{l}
k_1 = -\frac{e^{\eta_1\omega_1}}{\sigma'\omega_1}\\[2pt]
k_1^2 = (e_3-e_1)(e_2-e_1)
\end{array}
\]

\[
(4)\ \left\lbrace
\begin{array}{l}
\mathrm{sn}_1(z+2\omega_1) = \mathrm{sn}_1 z\\
\mathrm{sn}_1(z+2\omega_2) = -\mathrm{sn}_1 z\\
\mathrm{sn}_1(z+2\omega_3) = -\mathrm{sn}_1 z\\[4pt]
\mathrm{sn}_1(z+4\omega_2) = \mathrm{sn}_1 z\\
\mathrm{sn}_1(z+4\omega_3) = \mathrm{sn}_1 z\\[4pt]
\mathrm{sn}_1(-z) = -\mathrm{sn}_1 z\\[4pt]
\mathrm{sn}_1(2\omega_2 - z) = \mathrm{sn}_1 z\\
\mathrm{sn}_1(2\omega_3 - z) = \mathrm{sn}_1 z
\end{array}
\right.
\]

\[
\left\|\
\begin{array}{ll}
\mathrm{sn}_1\omega_2 = \frac{\sigma\omega_1\,\sigma\omega_2}{\sigma(\omega_1+\omega_2)}\,
e^{\eta_1\omega_2} = \pm\frac{1}{\sqrt{e_2-e_1}} & \mathrm{sn}_1'\omega_2 = 0\\[6pt]
\mathrm{sn}_1\omega_3 = \frac{\sigma\omega_1\,\sigma\omega_3}{\sigma(\omega_1+\omega_2)}\,
e^{\eta_1\omega_3} = \pm\frac{1}{\sqrt{e_3-e_1}} & \mathrm{sn}_1'\omega_3 = 0
\end{array}
\right.
\]

\[
(5)\ \left\lbrace
\begin{aligned}
\mathrm{cn}_1^2 z &= 1 - (e_2-e_1)\,\mathrm{sn}_1^2 z = \frac{\wp z - e_2}{\wp z - e_1}
&\qquad& (\mathrm{cn}_1 0 = 1)\\
\mathrm{dn}_1^2 z &= 1 - (e_3-e_1)\,\mathrm{sn}_1^2 z = \frac{\wp z - e_3}{\wp z - e_1}
&& (\mathrm{dn}_1 0 = 1)
\end{aligned}
\right.
\]

\[
(6)\ \left\lbrace
\begin{aligned}
\mathrm{cn}_1 z &= \frac{\sigma\omega_1}{\sigma\omega_2}\,e^{(\eta_2-\eta_1)z}\,
\frac{\sigma(z-\omega_2)}{\sigma(z-\omega_1)}
= \frac{\sigma\omega_1}{\sigma\omega_2}\,e^{-(\eta_2-\eta_1)z}\,
\frac{\sigma(z+\omega_2)}{\sigma(z+\omega_1)}\\[4pt]
\mathrm{dn}_1 z &= \frac{\sigma\omega_1}{\sigma\omega_3}\,e^{(\eta_3-\eta_1)z}\,
\frac{\sigma(z-\omega_3)}{\sigma(z-\omega_1)}
= \frac{\sigma\omega_1}{\sigma\omega_3}\,e^{-(\eta_3-\eta_1)z}\,
\frac{\sigma(z+\omega_3)}{\sigma(z+\omega_1)}
\end{aligned}
\right.
\]

\[
(7)\ \left\lbrace
\begin{array}{ll}
\mathrm{cn}_1(-z) = \mathrm{cn}_1 z & \mathrm{dn}_1(-z) = \mathrm{dn}_1 z\\[4pt]
\mathrm{cn}_1(z+2\omega_1) = -\mathrm{cn}_1 z & \mathrm{dn}_1(z+2\omega_1) = -\mathrm{dn}_1 z\\
\mathrm{cn}_1(z+2\omega_2) = -\mathrm{cn}_1 z & \mathrm{dn}_1(z+2\omega_3) = -\mathrm{dn}_1 z\\[4pt]
\mathrm{cn}_1(z+2\omega_3) = \mathrm{cn}_1 z & \mathrm{dn}_1(z+2\omega_2) = \mathrm{dn}_1 z\\
\mathrm{cn}_1(z+4\omega_1) = \mathrm{cn}_1 z & \mathrm{dn}_1(z+4\omega_3) = \mathrm{dn}_1 z\\
\mathrm{cn}_1(z+4\omega_2) = \mathrm{cn}_1 z & \mathrm{dn}_1(z+4\omega_1) = \mathrm{dn}_1 z
\end{array}
\right.
\]

\[
\text{résidu en } \omega_1 : \frac{i}{k_1\,\mathrm{sn}_1\omega_2} \ ;\ = \pm i\sqrt{e_3-e_1}
\]

\[
\text{résidu en } \omega_1 : \frac{-i}{k_1\,\mathrm{sn}_1\omega_3} = \pm i\sqrt{e_2-e_1}
\]

\[
(1)\quad \boxed{\mathrm{sn}'^2 z = (1-z^2)(1-k^2z^2)} \qquad
\mathrm{sn}\,0 = 0 \quad \mathrm{sn}'0 = 1
\]

\[
(2)\quad \boxed{\mathrm{sn}^2 z = \frac{1}{\wp z - e_1}
= \frac{1}{k^2}\bigl[\wp(z-\omega_1) - e_1\bigr]}
\qquad
\left\lbrace
\begin{array}{l}
e_1 = -\frac{1+k^2}{3}\\[4pt]
e_2 = \frac{2-k^2}{3}\\[4pt]
e_3 = \frac{2k^2-1}{3}
\end{array}
\right.
\qquad
\begin{array}{l}
e_2 - e_1 = 1\\
e_3 - e_1 = k^2
\end{array}
\]

\[
(3)\quad \boxed{\mathrm{sn}\,z = \sigma\omega_1\,e^{\eta_1 z}\,\frac{\sigma z}{\sigma(z+\omega_1)}
= -\sigma\omega_1\,e^{-\eta_1 z}\,\frac{\sigma z}{\sigma(z-\omega_1)}}
\]

\[
\boxed{\mathrm{sn}\,z\;\mathrm{sn}(z+\omega_1) = \frac{1}{k}}
\]

\[
(5)\ \left\lbrace
\begin{array}{l}
\mathrm{sn}(z+2\omega_1) = \mathrm{sn}\,z\\
\mathrm{sn}(z+2\omega_2) = \mathrm{sn}\,z\\
\mathrm{sn}(z+2\omega_3) = -\mathrm{sn}\,z\\[4pt]
\mathrm{sn}(z+4\omega_2) = \mathrm{sn}\,z\\
\mathrm{sn}(z+4\omega_3) = \mathrm{sn}\,z\\[4pt]
\mathrm{sn}(-z) = -\mathrm{sn}\,z
\end{array}
\right.
\]

\[
\underline{\ \mathrm{sn}\,\omega_2 = 1 \quad \mathrm{sn}'\omega_2 = 0\ } \qquad
\underline{\ \mathrm{sn}\,\omega_3 = -\frac{1}{k} \quad \mathrm{sn}'\omega_3 = 0\ }
\]

\[
(8)\ \left\lbrace
\begin{aligned}
\mathrm{sn}_1'z &= \mathrm{cn}_1 z\;\mathrm{dn}_1 z\\
\mathrm{cn}_1'z &= -(e_2-e_1)\,\mathrm{sn}_1 z\;\mathrm{dn}_1 z\\
\mathrm{dn}_1'z &= -(e_3-e_1)\,\mathrm{sn}_1 z\;\mathrm{cn}_1 z
\end{aligned}
\right.
\]

\[
(9)\quad \mathrm{sn}_1'^2 z = \bigl(1-(e_2-e_1)\,\mathrm{sn}_1^2 z\bigr)
\bigl(1-(e_3-e_1)\,\mathrm{sn}_1^2 z\bigr)
\]

\[
(1)\quad f(z+2\omega_1) = f(z+2\omega_2) = f(z)
\]

\[
(2)\quad \wp(z) = \frac{1}{z^2} + \mathop{{\sum}'}_{m,n}\left(
\frac{1}{(z-2m\omega_1-2n\omega_2)^2} - \frac{1}{(2m\omega_1+2n\omega_2)^2}\right)
\]

\[
(3)\quad \wp'(z) = -\frac{2}{z^3} - 2\mathop{{\sum}'}_{m,n}
\frac{1}{(z-2m\omega_1-2n\omega_2)^3} = -2\sum
\frac{1}{(z-2m\omega_1-2n\omega_2)^3}
\]

\[
\wp z = \frac{1}{z^2} + c_1 z^2 + c_2 z^4 + \cdots
\]

\[
f(z) = \frac{c_k}{z^{2k}} + \cdots, \qquad f(z) - c_k\,\wp(z)^k
\]

\[
g(z) = f(z)\prod_{1\leqslant i\leqslant n}\bigl[\wp(z) - \wp(a_i)\bigr]^{l_i}
\]

\[
(3)\quad f = R_1(\wp) + \wp' R_2(\wp)
\]

\[
(1)\ \left\lbrace
\begin{aligned}
\wp &= \frac{1}{z^2} + \mathop{{\sum}'}_{m,n}\left(
\frac{1}{(z-2m\omega_1-2n\omega_2)^2} - \frac{1}{(2m\omega_1+2n\omega_2)^2}\right)\\
\wp' &= -2\sum_{m,n}\frac{1}{(z-2m\omega_1-2n\omega_2)^3}
\end{aligned}
\right.
\]

\[
(2)\quad \wp z = \frac{1}{z^2} + c_1 z^2 + c_2 z^4 + \cdots
\]

\[
(3)\ \left\lbrace
\begin{array}{l}
c_1 = 3\displaystyle\mathop{{\sum}'}_{m,n}\frac{1}{(2m\omega_1+2n\omega_2)^4}\\
\quad -\ -\ -\\
c_k = (2k+1)\displaystyle\mathop{{\sum}'}_{m,n}\frac{1}{(2m\omega_1+2n\omega_2)^{2k+2}}
\end{array}
\right.
\]

\[
\boxed{\ \wp\omega_i = e_i\ }
\]

\[
\wp'^2 = 4\wp^3 - 20c_1\,\wp - 28c_2
\]

\[
(?)\quad \boxed{\ \wp'^2 = 4\wp^3 - g_2\,\wp - g_3\ } \qquad
(g_2 = 20c_1,\ g_3 = 28c_2)
\]

\[
\frac{\wp'^2}{(\wp-e_1)(\wp-e_2)(\wp-e_3)}
\]

\[
(?)\quad \boxed{\ \wp'^2 = 4(\wp-e_1)(\wp-e_2)(\wp-e_3)\ }
\]

\[
(6)\quad \boxed{\ \wp'' = 6\wp^2 - \tfrac{1}{2}g_2 \qquad \wp''' = 12\,\wp\wp'\ }
\]

\[
\boxed{\ c_3 = \frac{c_1^2}{3}\,, \qquad c_4 = \frac{3}{11}\,c_1 c_2\ } \qquad (7)
\]

\[
(7)\quad \zeta u - \frac{1}{u} = \int_0^u -\Bigl(\wp z - \frac{1}{z^2}\Bigr)dz
\qquad \zeta' u = -\wp u
\]

\[
(8)\quad \zeta u = \frac{1}{u} - \frac{c_2}{3}u^3 - \cdots - \frac{c_n}{2n+1}u^{2n+1} - \cdots
\]

\[
(9)\quad \zeta z = \frac{1}{z} + \sum_{\omega\in\mathfrak{P}^*}
\Bigl(\frac{1}{z-\omega} + \frac{1}{\omega} + \frac{z}{\omega^2}\Bigr)
\]

\[
(10)\quad \boxed{\ \zeta(z+2\omega) - \zeta(z) = 2\eta\ } \]

\[
(11)\quad \eta = \zeta(\omega)\,;
\]

\[
\eta_1\omega_2 - \eta_2\omega_1 = \frac{i\pi}{2}
\]

\[
\frac{A_{i,1}}{z-a_i} + \frac{A_{i,2}}{(z-a_i)^2} + \cdots +
\frac{A_{i,k_i}}{(z-a_i)^{k_i}}
\]

\[
(12)\quad f(z) = c^{te} + \sum_i \Bigl(A_{i,1}\,\zeta(z-a_i) + A_{i,2}\,\wp(z-a_i)
+ \cdots + \frac{(-1)^{k_i}}{(k_i-1)!}\,A_{i,k_i}\,\wp^{(?)}(z-a_i)\Bigr)
\]

\[
(13)\quad \sigma z = z \prod_{\omega\in\mathfrak{P}^*}
\Bigl(1 - \frac{z}{\omega}\Bigr) e^{\frac{z}{\omega} + \frac{z^2}{2\omega^2}}
\]

\[
(14)\quad \frac{\sigma' z}{\sigma z} = (\log\sigma z)' = \zeta z \qquad
\sigma z = z\, e^{\int_0^z (\zeta u - \frac{1}{u})\,du}
\]

\[
(15)\quad \sigma u = u - c_1\frac{u^5}{12} - c_2\frac{u^7}{30}
\]

\[
(16)\quad \sigma(u+2\omega_i) = \sigma u\, e^{2\eta_i(u+\omega_i) + 2i\pi}
\]

\[
(17)\quad f(z) = c^{te}\times\frac{\prod\sigma(u-a_i)}{\prod\sigma(u-b_i)}
\]

\[
(18)\quad \wp u - \wp v = -\frac{\sigma(u+v)\,\sigma(u-v)}{\sigma^2 u\,\sigma^2 v}
\]

\[
(19)\quad \left\lbrace
\begin{aligned}
\frac{\wp' u}{\wp u - \wp v} &= \zeta(u+v) + \zeta(u-v) - 2\zeta u\\
-\frac{\wp' v}{\wp u - \wp v} &= \zeta(u+v) - \zeta(u-v) - 2\zeta v
\end{aligned}
\right.
\]

\[
(20)\quad \zeta(u+v) - \zeta u - \zeta v = \frac{1}{2}\,\frac{\wp' u - \wp' v}{\wp u - \wp v}
\]

\[
(20\ \mathrm{bis})\quad \zeta(2u) - 2\zeta u = \frac{1}{2}\,\frac{\wp'' u}{\wp' u}
\]

\[
(21)\quad \wp(u+v) + \wp u + \wp v = \frac{1}{4}\Bigl(\frac{\wp' u - \wp' v}{\wp u - \wp v}\Bigr)^2
\]

\[
(21\ \mathrm{bis})\quad \wp(2u) + 2\wp u = \frac{1}{4}\Bigl(\frac{\wp'' u}{\wp' u}\Bigr)^2
\]

\[
(22)\quad 4\wp(2u) = \wp u + \wp(u+\omega_1) + \wp(u+\omega_2) + \wp(u+\omega_3)
\]

\[
(23)\quad \frac{1}{\wp u - e_i} = \frac{\wp(u+\omega_i) - e_i}{(e_j - e_i)(e_k - e_i)}
\qquad (i,j,k \ \text{permutation de}\ 1,2,3)
\]

\[
(24)\quad \wp u - e_i = -\frac{\sigma(u+\omega_i)\,\sigma(u-\omega_i)}{\sigma^2 u\,\sigma^2\omega_i}
= \Bigl(\frac{\sigma_i u}{\sigma u}\Bigr)^2 \quad\text{où}
\]

\[
(25)\quad \sigma_i u = \frac{1}{\sigma\omega_i}\,\sigma(u+\omega_i)\,e^{-\eta_i u}
\]

\[
(26)\quad \left\lbrace
\begin{aligned}
\sigma_i(u+2\omega_i) &= \sigma_i u\, e^{2\eta_i(u+\omega_i) + i\pi}\\
\sigma_i(u+2\omega_j) &= \sigma_i u\, e^{2\eta_j(u+\omega_j)} \qquad (j\neq i)
\end{aligned}
\right.
\]

\[
\sigma u = e^{\frac{\eta_1 u^2}{2\omega_1}} \sum_{-\infty}^{+\infty}
A_n\, e^{(2n-1)i\pi\frac{u}{2\omega_1}}
\]

\[
A_{n+1} = -h^{2n}A_n, \quad\text{où}\quad h = e^{i\pi\frac{\omega_2}{\omega_1}}
\qquad \Bigl(\Im\frac{\omega_2}{\omega_1} > 0\Bigr),
\]

\[
(1)\quad \Theta(v|t) = -\frac{1}{i}\sum_{-\infty}^{+\infty}(-1)^n
e^{(n-\frac{1}{2})^2 i\pi t + (2n-1)i\pi v}
\]

\[
(2)\quad \Theta(v|t) = 2\sum_{1}^{\infty}(-1)^{n-1} e^{(n-\frac{1}{2})^2 i\pi t}
\sin(2n-1)\pi v
\]

\[
(3)\quad \sigma(u|\omega_1,\omega_2) = \frac{2\omega_1}{\Theta'_v(0|\frac{\omega_2}{\omega_1})}\,
e^{\frac{\eta_1 u^2}{2\omega_1}}\,\Theta\Bigl(\frac{u}{2\omega_1}\Big|\frac{\omega_2}{\omega_1}\Bigr)
\]

\[
(4)\quad \left\lbrace
\begin{aligned}
&\Theta(v+1|t) = -\Theta(v|t) \qquad \Theta(v+2|t) = \Theta(v|t)\\
&\Theta(v+t|t) = \Theta(v|t)\, e^{-2i\pi v - i\pi t - i\pi}
\end{aligned}
\right.
\]

\[
(5)\quad \Theta(v|t+4) = -\Theta(v|t), \quad\text{d'où}\quad \Theta(v|t+8) = \Theta(v|t)
\]

\[
(6)\quad \left\lbrace
\begin{aligned}
\Theta(v|t) &= -\frac{1}{i}\sum_{-\infty}^{\infty}(-1)^n\, h(t)^{(n-\frac12)^2}
e^{(2n-1)i\pi v}\\
&= 2\sum_{1}^{\infty}(-1)^{n-1}\, h(t)^{(n-\frac12)^2}\sin(2n-1)\pi v\\
&= 2\bigl[h(t)^{\frac14}\sin\pi v - h(t)^{\frac94}\sin 3\pi v + \cdots\bigr]
\end{aligned}
\right.
\]

\[
\Theta(v|iT) = 2\sum_{1}^{+\infty}(-1)^{n-1} e^{-(n-\frac12)^2\pi T}\sin(2n-1)\pi v
\]

\[
(7)\quad \sum_{-\infty}^{+\infty} e^{-an^2+bn+c} = -e^{c+\frac{b}{2}-\frac{a}{4}}\,
\Theta\Bigl(\frac{b-a-i\pi}{2i\pi}\Big|\frac{ia}{\pi}\Bigr)
\]

\[
(8)\quad \Theta(v|t) = e^{-\frac{i\pi v^2}{t}}\,
\Theta\Bigl(\frac{v}{t}\Big|-\frac{1}{t}\Bigr) \times
\frac{\Theta'_v(0|t)}{\Theta'_v(0|-\frac{1}{t})}
\]

\[
(9)\quad \frac{\partial^2\Theta}{\partial v^2} = 4i\pi\,\frac{\partial\Theta}{\partial t}
\]

\[
\Theta(v|t) = e^{-\frac{i\pi v^2}{t}}\,\Theta\Bigl(\frac{v}{t}\Big|-\frac{1}{t}\Bigr)\frac{a}{\sqrt t},
\]

\[
(10)\quad \boxed{\ \Theta(v|t) = e^{-\frac{i\pi v^2}{t}}\,
\Theta\Bigl(\frac{v}{t}\Big|-\frac{1}{t}\Bigr)\, i\sqrt{\frac{i}{t}}\ }
\]

\[
(11)\quad \Theta'_v(0|t) = \frac{i}{t}\sqrt{\frac{i}{t}}\;\Theta'_v\Bigl(0\Big|-\frac{1}{t}\Bigr)
\]

\[
(12)\quad \sum_{1}^{\infty}(-1)^{n-1}(2n-1)\,e^{-(n-\frac12)^2\pi T}
= \frac{1}{T^{\frac32}}\sum_{1}^{\infty}(-1)^{n-1}(2n-1)\,e^{-(n-\frac12)^2\frac{\pi}{T}}
\]

\[
(12\ \mathrm{bis})\quad \Theta'_v(v|t) = 2\pi\sum_{1}^{\infty}(-1)^{n-1}(2n-1)\,
e^{(n-\frac12)^2 i\pi t}\cos(2n-1)\pi v
\]

\[
\Theta'_v(0|iT) = 2\pi\sum_{1}^{\infty}(-1)^{n-1}(2n-1)\,e^{-(n-\frac12)^2\pi T}
\]

\[
\Theta(v|iT) = 2\sum_{1}^{\infty}(-1)^{n-1}h^{(n-\frac12)^2}\sin(2n-1)\pi v,
\quad\text{où}\quad h = e^{-\pi T}\ \text{ou}\ e^{-\frac{\pi}{T}},
\]

\[
(13)\quad \sigma_i u = e^{\frac{\eta_1 u^2}{2\omega_1}}\,
\frac{\Theta_i(\frac{u}{2\omega_1}|\frac{\omega_2}{\omega_1})}{\Theta_i(0|\frac{\omega_2}{\omega_1})}
\]

\[
(14)\quad \left\lbrace
\begin{aligned}
\Theta_1(v|t) &= \sum_{-\infty}^{\infty} e^{(n-\frac12)^2 i\pi t + (2n-1)i\pi v}\\
&= 2\sum_{1}^{\infty} e^{(n-\frac12)^2 i\pi t}\cos(2n-1)\pi v = \vartheta_2(v)\\
\Theta_2(v|t) &= \sum_{-\infty}^{+\infty}(-1)^n e^{n^2 i\pi t + 2ni\pi v}\\
&= 1 + 2\sum_{1}^{\infty} e^{n^2 i\pi t}\cos 2n\pi v = \vartheta_0(v)
\end{aligned}
\right.
\]

\[
\left\lbrace
\begin{aligned}
\Theta_3(v|t) &= \sum_{-\infty}^{+\infty} e^{n^2 i\pi t + 2ni\pi v}
= 1 + 2\sum_{1}^{\infty} e^{n^2 i\pi t}\cos 2n\pi v = \vartheta_3(v)
\end{aligned}
\right.
\]

\[
e_i = \Bigl(\frac{1}{2\omega_1}\Bigr)^2\Bigl(\frac{1}{3}\,\frac{\Theta'''0}{\Theta'0}
- \frac{\Theta''_i 0}{\Theta_i 0}\Bigr)
\]

\[
(15)\quad \left\lbrace
\begin{aligned}
e_1 &= \frac13\Bigl(\frac{\pi}{2\omega_1}\Bigr)^2\bigl[(\Theta_2 0)^4 + (\Theta_3 0)^4\bigr]\\
e_2 &= \frac13\Bigl(\frac{\pi}{2\omega_1}\Bigr)^2\bigl[-(\Theta_1 0)^4 - (\Theta_3 0)^4\bigr]\\
e_3 &= \frac12\Bigl(\frac{\pi}{2\omega_1}\Bigr)^2\bigl[(\Theta_1 0)^4 - (\Theta_? 0)^4\bigr]
\end{aligned}
\right.
\]

\[
2\eta_1\omega_1 = -\frac{\Theta'''0}{6\,\Theta'0}\ ; \qquad
\Theta'0 = \pi\,\Theta_1 0\;\Theta_2 0\;\Theta_3 0
\]

\[
\omega_1\ \ \omega_2 \qquad \Bigl[\frac{\omega_1}{\omega_2}\notin\mathbf{R}\Bigr]
\]

\[
g_2\ \ g_3 \qquad \boxed{\ g_2 = 20c_1 \quad g_3 = 28c_2\ } \qquad
\bigl(g_2^3 - 27g_3^2 \neq 0\bigr)
\]

\[
c_1\ \ c_2
\]

\[
e_1\ \ e_2\ \ e_3 \qquad 4z^3 - g_2 z - g_3 = 4(z-e_1)(z-e_2)(z-e_3)
\quad (\text{les } e_i \text{ distincts})
\]

\[
c_1 = 3\sum_{\omega\in\mathfrak{P}^*}\Bigl(\frac{1}{2\omega}\Bigr)^4 \qquad
c_2 = 5\sum_{\omega\in\mathfrak{P}^*}\Bigl(\frac{1}{2\omega}\Bigr)^6
\]

\[
g_2 = 20c_1 \qquad g_3 = 28c_2
\]

\[
e_i = \wp\omega_i = \frac{1}{\omega_i^2} + \sum_{\omega\in\mathfrak{P}^*}
\frac{1}{(\omega_i + 2\omega)^2}
\]

\[
c_1 = \frac{1}{20}\,g_2\,, \qquad c_2 = \frac{1}{28}\,g_3
\]

\[
\omega_i = \int_\infty^{e_i}\frac{dz}{\sqrt{4z^3 - g_2 z - g_3}}
\qquad (i,j,k\ \uncertain{distincts})
\]

\[
g_2 = 4(e_2e_3 + e_3e_1 + e_1e_2) \qquad g_3 = 4e_1e_2e_3
\]

\[
g_2 = 20c_1\,, \qquad g_? = 20\,c_?\,,
\]

\[
\frac{g_2^3}{g_2^3 - 27g_3^2}
\]

\[
J(\tau) = \frac{g_2^3}{\Delta} = \frac{1}{1 - 27\frac{g_3^2}{g_2^3}}
\]

\[
(1)\quad \left.
\begin{aligned}
&\mathrm{cn}^2u + \mathrm{sn}^2u = 1\\
&\mathrm{dn}^2u + k^2\mathrm{sn}^2u = 1\\
&(\mathrm{dn}^2u - k^2\mathrm{cn}^2u = 1 - k^2 = k'^2)
\end{aligned}
\right\rbrace \quad
\begin{aligned}
&\mathrm{sn}\,0 = 0\\
&\mathrm{cn}\,0 = \mathrm{dn}\,0 = 1
\end{aligned}
\]

\[
(2)\quad \left\lbrace
\begin{aligned}
\mathrm{sn}'u &= \sqrt{(1-\mathrm{sn}^2u)(1-k^2\mathrm{sn}^2u)} = \mathrm{cn}\,u\,\mathrm{dn}\,u\\
\mathrm{cn}'u &= \sqrt{(1-\mathrm{cn}^2u)(k'^2+k^2\mathrm{cn}^2u)} = -\mathrm{sn}\,u\,\mathrm{dn}\,u\\
\mathrm{dn}'u &= \sqrt{(1-\mathrm{dn}^2u)(\mathrm{dn}^2u-k'^2)} = -k^2\mathrm{sn}\,u\,\mathrm{cn}\,u
\end{aligned}
\right.
\]

\[
(3)\quad \left\lbrace
\begin{aligned}
\mathrm{sn}\,u &= u - \frac{2\alpha k}{3!}u^3 + \frac{4k^2(\alpha^2+3)}{5!}u^5
- \frac{8k^3(\alpha^3+33\alpha)}{7!}u^7 + \cdots\\
\mathrm{cn}\,u &= 1 - \frac{u^2}{2!} + \frac{1+4k^2}{4!}u^4
- \frac{1+44k^2+16k^4}{6!}u^6 + \cdots\\
\mathrm{dn}\,u &= 1 - \frac{k^2u^2}{2!} + \frac{k^2(4+k^2)}{4!}u^4
- k^2\,\frac{16+44k^2+k^4}{6!}u^6 + \cdots
\end{aligned}
\right.
\]

\[
\Bigl(\alpha = \frac12\Bigl(k + \frac1k\Bigr)\Bigr)
\]

\[
\left\lbrace
\begin{aligned}
&k = 0 \quad \mathrm{sn} = \sin \quad \mathrm{cn} = \cos, \quad \mathrm{dn} = 1,\\
&k = 1 \quad \mathrm{sn} = \mathrm{th} \quad \mathrm{cn} = \frac{1}{\mathrm{ch}} \quad
\mathrm{dn} = \frac{1}{\mathrm{th}}
\end{aligned}
\right.
\]

\[
K = \int_0^1\frac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}} \qquad
K' = \int_0^1\frac{dx}{\sqrt{(1-x^2)(1-k'^2x^2)}}
\]

\[
\begin{array}{llll}
\mathrm{sn}\,u & \text{périodes} & 4K\quad 2iK' & \\
 & \text{zéros} & 2mK + 2m'iK' & \\
 & \text{pôles} & iK' + 2mK + 2m'iK' & \\[1ex]
\mathrm{cn}\,u & \text{périodes} & 4K,\ -2K'' & \\
 & \text{zéros} & K + 2mK + 2m'iK & \\
 & \text{pôles} & iK' + 2mK + 2m'iK' & \\[1ex]
\mathrm{dn}\,u & \text{périodes} & 2K,\ 4iK' & \\
 & \text{zéros} & K'' + 2mK + 2m'iK & \\
 & & iK' + 2mK + 2m'iK' &
\end{array}
\]

\[
\left\lbrace
\begin{aligned}
&\mathrm{sn}(u+2K) = -\mathrm{sn}\,u && \mathrm{cn}(u+2K) = -\mathrm{cn}\,u
&& \mathrm{dn}(u+2K) = \mathrm{dn}\,u\\
&\mathrm{sn}(u+2iK') = \mathrm{sn}\,u && \mathrm{cn}(u+2iK') = -\mathrm{cn}\,u
&& \mathrm{dn}(u+2iK') = -\mathrm{dn}\,u\\
&\mathrm{sn}(u+2K'') = -\mathrm{sn}\,u && \mathrm{cn}(u+2K'') = \mathrm{cn}\,u
&& \mathrm{dn}(u+2K'') = -\mathrm{dn}\,u
\end{aligned}
\right.
\]

\[
\left\lbrace
\begin{aligned}
&\mathrm{sn}(u+K) = \frac{\mathrm{cn}\,u}{\mathrm{dn}\,u}
&& \mathrm{cn}(u+K) = -k'\frac{\mathrm{sn}\,u}{\mathrm{dn}\,u}
&& \mathrm{dn}(u+K) = \frac{k'}{\mathrm{dn}\,u}\\
&\mathrm{sn}(u+iK') = \frac{1}{k\,\mathrm{sn}\,u}
&& \mathrm{cn}(u+iK') = -\frac{i}{k}\frac{\mathrm{dn}\,u}{\mathrm{sn}\,u}
&& \mathrm{dn}(u+iK') = -i\,\frac{\mathrm{cn}\,u}{\mathrm{sn}\,u}\\
&\mathrm{sn}(u+K'') = -\frac{1}{k}\frac{\mathrm{dn}\,u}{\mathrm{cn}\,u}
&& \mathrm{cn}(u+K'') = -\frac{ik'}{k\,\mathrm{cn}\,u}
&& \mathrm{dn}(u+K'') = ik'\frac{\mathrm{sn}\,u}{\mathrm{cn}\,u}
\end{aligned}
\right.
\]

\[
\left\lbrace
\begin{aligned}
\mathrm{sn}(iu|k) &= i\,\frac{\mathrm{sn}(u|k')}{\mathrm{cn}(u|k')}\\
\mathrm{cn}(iu|k) &= \frac{1}{\mathrm{cn}(u|k')}\\
\mathrm{dn}(iu|k) &= \frac{\mathrm{dn}(u|k')}{\mathrm{cn}(u|k')}
\end{aligned}
\right.
\]

\[
\begin{array}{c|c|c|c|c|c|c|c|c|c|c}
 & 0 & K & 2K & 3K & iK' & 2iK' & 3iK' & K'' & 2K'' & 3K''\\
\hline
\mathrm{sn}\,u & 0 & 1 & 0 & -1 & \infty & 0 & \infty & -\frac1k & 0 & \frac1k\\
\mathrm{cn}\,u & 1 & 0 & -1 & 0 & \infty & -1 & \infty & -i\frac{k'}{k} & 1 & -i\frac{k'}{k}\\
\mathrm{dn}\,u & 1 & k' & 1 & k' & \infty & -1 & \infty & 0 & -1 & 0
\end{array}
\]

\[
\left\lbrace
\begin{aligned}
\mathrm{sn}(u|k) &= M\,\frac{\sigma(\frac{u}{M}|\omega_1,\omega_2)}{\sigma_2(\frac{u}{M}|\omega_1,\omega_2)}\\
\mathrm{cn}(u|k) &= \frac{\sigma_1(\frac{u}{M}|\omega_1,\omega_2)}{\sigma_2(\frac{u}{M}|\omega_1,\omega_2)}\\
\mathrm{dn}(u|k) &= \frac{\sigma_3(\frac{u}{M}|\omega_1,\omega_2)}{\sigma_?(\frac{u}{M}|\omega_1,\omega_2)}
\end{aligned}
\right.
\]

\[
\wp(u|\omega_1,\omega_2) = e_2 + \frac{M^2}{\mathrm{sn}^2(Mu|k)}
\]

\[
K = M\omega_1,\quad iK' = M\omega_2,\quad K'' = M\omega_3
\]

\[
3e_1 = (1+k'^2)M^2 \qquad 3e_2 = (?+k'^2)M^2 \qquad 3e_3 = (1-2k'^2)M^2
\]

\[
\left\lbrace
\begin{aligned}
\mathrm{sn}(u|k) &= \frac{1}{\sqrt k}\,
\frac{\Theta(\frac{u}{2K}|\frac{iK'}{K})}{\Theta_2(\frac{u}{2K}|\frac{iK'}{K})}\\
\mathrm{cn}(u|k) &= \sqrt{\frac{k'}{k}}\,
\frac{\Theta_1(\frac{u}{2K},\frac{iK'}{K})}{\Theta_2(\frac{u}{2K},\frac{iK'}{K})}\\
\mathrm{dn}(u|k) &= \sqrt{k'}\,
\frac{\Theta_3(\frac{u}{2K},\frac{iK'}{K})}{\Theta_2(\frac{u}{2K},\frac{iK'}{K})}
\end{aligned}
\right.
\]

\[
\mathrm{sn}(u+v) = \frac{\mathrm{sn}\,u\,\mathrm{cn}\,v\,\mathrm{dn}\,v
+ \mathrm{sn}\,v\,\mathrm{cn}\,u\,\mathrm{dn}\,u}{1 - k^2\mathrm{sn}^2u\,\mathrm{sn}^2v}
\]

\[
\mathrm{cn}(u+v) = \frac{\mathrm{cn}\,u\,\mathrm{cn}\,v
- \mathrm{sn}\,u\,\mathrm{sn}\,v\,\mathrm{dn}\,u\,\mathrm{dn}\,v}{1 - k^2\mathrm{sn}^2u\,\mathrm{sn}^2v}
\]

\[
\mathrm{dn}(u+v) = \frac{\mathrm{dn}\,u\,\mathrm{dn}\,v
- k^2\mathrm{sn}\,u\,\mathrm{sn}\,v\,\mathrm{cn}\,u\,\mathrm{cn}\,v}{1 - k^2\mathrm{sn}^2u\,\mathrm{sn}^2v}
\]