69 lines
1.8 KiB
TeX
69 lines
1.8 KiB
TeX
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???????
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\begin{theorem}
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Such a pairing is isometric to a pairing:
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\[
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\begin{bmatrix}
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1
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\end{bmatrix}
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\times
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\begin{bmatrix}
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1
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\end{bmatrix}
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\rightarrow
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\frac{\epsilon}{p^k_{\xi}},
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\: \epsilon \in {\pm 1}
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\]
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\end{theorem}
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?????????????
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\[
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\begin{bmatrix}
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1
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\end{bmatrix} = 1 \in \quot{\Lambda}{p^k_{\xi} \Lambda }
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\]
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????????
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\begin{theorem}
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The jump of the signature function at $\xi$ is equal to
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$2 \sum\limits_{k_i \odd} \epsilon_i$. \\
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The peak of the signature function is equal to $\sum\limits_{k_i \even} \epsilon_i$.
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\[
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(\quot{\Lambda}{p^{k_1} \Lambda}, \epsilon_1) \oplus \dots \oplus (\quot{\Lambda}{p^{k_n} \Lambda}, \epsilon_n)
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\]
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%$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$
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\end{theorem}
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\begin{definition}
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A matrix $A$ is called Hermitian if
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$\overline{A(t)} = {A(t)}^T$
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\end{definition}
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\begin{theorem}[Borodzik-Friedl 2015, Borodzik-Conway-Politarczyk 2018]
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A square Hermitian matrix $A(t)$ of size $n$ with coefficients in $\mathbb{Z}[t, t^{-1}]$
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(or $\mathbb{R}[t, t^{-1}]$ ) represents
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the Blanchfield pairing if:
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\begin{eqnarray*}
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H_1(\bar{X}, \Lambda) = \quot{\Lambda^n }{A\Lambda^n },\\
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(x, y) \mapsto {\overline{x}}^T A^{-1} y \in \quot{\Omega}{\Lambda}\\
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H_1(\widetilde{X}, \Lambda) \times
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H_1(\widetilde{X}, \Lambda) \longrightarrow
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\quot{\Omega}{\Lambda},
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\end{eqnarray*}
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where $\Lambda = \mathbb{Z}[t, t^{-1}]$ or $\mathbb{R}[t, t^{-1}]$, $\Omega = \mathbb{Q}(t)$ or $\mathbb{R}(t)$
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\end{theorem}
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????????\\field of fractions ??????
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\begin{eqnarray*}
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H_1(\Sigma(K), \mathbb{Z}) = \quot{\mathbb{Z}^n}{(V + V^T) \mathbb{Z}^n}\\
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H_1(\Sigma(K), \mathbb{Z})
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\times
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H_1(\Sigma(K), \mathbb{Z})
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\longrightarrow
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= \quot{\mathbb{Q}}{\mathbb{Z}}\\
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(a, b) \mapsto a{(V + V^T)}^{-1} b
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\end{eqnarray*}
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???????????????????\\
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\begin{eqnarray*}
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y \mapsto y + Az \\
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\overline{x^T} A^{-1}(y + Az) = \overline{x^T} A^{-1} + \overline{x^T} \mathbb{1} z
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\end{eqnarray*}
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