(G, F,  E) = 1 .$$ One deduces that $\zeta^{1/2} ( g'g E, g'E, E) =1 $ for $(h, h') = ( {\operatorname{id}}, {\operatorname{id}})$, $( {\operatorname{id}}, k )$, $( k , {\operatorname{id}})$ and $(k,k)$ where $k $ is any involution of $\Lambda$ with negative determinant.

For any $(h,z ) \in {\operatorname{Ml}}( \Lambda )$ such that $h - {\operatorname{id}}$ is invertible, the index $m$ of $j_\Lambda ( h , z)$ is determined by $ z =  i ^m |z| $.

One has to prove that $ \operatorname{det}^{1/2} \bigl( \tfrac{1}{2}
{\operatorname{id}}+ i A(g) \bigr)$ is positive for any $g \in {\operatorname{Sp}}( S , \Lambda, j \Lambda)$. If the matrix of $g$ is (\[eq:3\]), the one of $A(g)$ is $$-\tfrac{1}{2} \left( \begin{matrix} 0   & D \\ D^t &  0 \end{matrix} \right)
\qquad \text{ with }  D =( 1 +B) ( 1- B)^{-1}$$ So $$\det \bigl( \tfrac{1}{2} {\operatorname{id}}+ i A(g) \bigr) = \det \bigl( \tfrac{1}{2} (
{\operatorname{id}}+ D^t D) \bigr).$$ Deforming $A(g)$ to $0$ through a radial homothety we obtain that the square root of $\operatorname{det}
\bigl( \frac{1}{2} {\operatorname{id}}+  i A (g) \bigr )$ is positive.

Generalized half-form lines {#sec:gener-half-form}
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Let $p$ be a positive integer. A generalized half-form line of a symplectic 