\(\DeclareMathOperator{\Sp}{Sp}\)
Here are some exercises from Maryna’s course at Bielefeld.
Complete the steps in the proof of Jacobi’s theorem.
Recall the definitions from Maryna’s first lecture and show the following
For \(\tau \in \mathbb{H}_{n}\), and \(\Lambda\) a quadratic lattice, Define \(h_{\tau},g_{\tau}\) and \(\Theta_{\tau, \Lambda}\) set \(\Theta(h_{\tau},g_{\tau}, \Theta) = \Theta_{\Lambda}(\tau)\).
Read about Schwartz-Bruhat functions.
How would you define the space \(\mathcal{S}(V_\mathbb{A}^n)\) in Maryna’s lecture?
Find a more practical definition for \(\mathcal{S}(k_{\mathbb{A}})\) from Maryna’s lecture.
Show that how \(\Sp(X)\) is defined in the lecture is a good definition.
Elements \(\sigma \in \Sp(X/A)\) preserve the alternating \(k\)-linear form \[\begin{equation} [(x_{1},x_{1}^{*}), (x_{2},x_{2}^{*}) ] = \tau (\{x_{1},x_{2}^{*}\} - \{x_{2},x_{1}^{*}\}) \end{equation}\]
What should be \((A,\iota,\tau)\) for this to correspond to Jan’s lectures?
This page was updated on October 2, 2025.
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