

???????
\begin{theorem}
Such a pairing is isometric to a pairing:
\[
\begin{bmatrix}
1
\end{bmatrix}
\times 
\begin{bmatrix}
1
\end{bmatrix}
\rightarrow
\frac{\epsilon}{p^k_{\xi}}, 
\: \epsilon \in {\pm 1}
\]
\end{theorem}
?????????????
\[ 
\begin{bmatrix}
1
\end{bmatrix} = 1 \in \quot{\Lambda}{p^k_{\xi}  \Lambda }
\]
????????
\begin{theorem}
The jump of the signature function at $\xi$ is equal to 
$2 \sum\limits_{k_i \odd} \epsilon_i$. \\
The peak of the signature function is equal to $\sum\limits_{k_i \even} \epsilon_i$.
\[
(\quot{\Lambda}{p^{k_1} \Lambda}, \epsilon_1) \oplus \dots \oplus (\quot{\Lambda}{p^{k_n} \Lambda}, \epsilon_n)
\]
%$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$
\end{theorem}

\begin{definition}
A matrix $A$ is called Hermitian if 
$\overline{A(t)} = {A(t)}^T$
\end{definition}

\begin{theorem}[Borodzik-Friedl 2015, Borodzik-Conway-Politarczyk 2018]
A square Hermitian matrix $A(t)$ of size $n$ with coefficients in $\mathbb{Z}[t, t^{-1}]$ 
(or $\mathbb{R}[t, t^{-1}]$ ) represents 
the Blanchfield pairing if:
\begin{eqnarray*}
H_1(\bar{X}, \Lambda) = \quot{\Lambda^n }{A\Lambda^n },\\
(x, y) \mapsto {\overline{x}}^T A^{-1} y \in \quot{\Omega}{\Lambda}\\
H_1(\widetilde{X}, \Lambda) \times 
H_1(\widetilde{X}, \Lambda) \longrightarrow
\quot{\Omega}{\Lambda},
\end{eqnarray*}
where $\Lambda = \mathbb{Z}[t, t^{-1}]$ or $\mathbb{R}[t, t^{-1}]$, $\Omega = \mathbb{Q}(t)$ or $\mathbb{R}(t)$
\end{theorem}
????????\\field of fractions ??????
\begin{eqnarray*}
H_1(\Sigma(K), \mathbb{Z}) = \quot{\mathbb{Z}^n}{(V + V^T) \mathbb{Z}^n}\\
H_1(\Sigma(K), \mathbb{Z}) 
\times
H_1(\Sigma(K), \mathbb{Z}) 
\longrightarrow
= \quot{\mathbb{Q}}{\mathbb{Z}}\\
(a, b) \mapsto a{(V + V^T)}^{-1} b
\end{eqnarray*}
???????????????????\\
\begin{eqnarray*}
y \mapsto y + Az \\
\overline{x^T} A^{-1}(y + Az) = \overline{x^T} A^{-1} + \overline{x^T} \mathbb{1} z
\end{eqnarray*}
