Exercises for Maryna’s course at Bielefeld

\(\DeclareMathOperator{\Sp}{Sp}\)

Here are some exercises from Maryna’s course at Bielefeld.

Lecture 1

Jacobi’s theorem

Complete the steps in the proof of Jacobi’s theorem.

Siegel theta series

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 \(\Theta(h_{\tau},g_{\tau}, \Theta) = \Theta_{\Lambda}(\tau)\).

Schwartz-Bruhat functions

Read about Schwartz-Bruhat functions.

How would you define the space \(\mathcal{S}(V_\mathbb{A}^n)\) in Maryna’s lecture?

Lecture 2

Schwartz-Bruhat functions

Find a more practical definition for \(\mathcal{S}(k_{\mathbb{A}})\) from Maryna’s lecture.

Symplectic group

Show that how \(\Sp(X)\) is defined in the lecture is a good definition.

Lecture 3

Here are the two exercises I copied during the lectures. See Lecture 3 for more context.

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.
Main Page