# Setup

- \(x=\mathbb{P}^{1} / k, \quad k=\mathbb{F}_{q}\), \(\quad\) char \(k=p\)

- \(F=K(X), \mathcal{O}_{x}\) valuation rings for \(x \in|X|\) \(S=\{0, \infty\} \subseteq|X|, \quad G=S_{p}(V)\)

---

- Fix a symplectic basis of \(V\) were \( \langle e_i,\, e_{-j}\rangle = 1\) and \( \langle e_{\pm i},\, e_{\pm j}\rangle = 0 \), so e.g. $\langle e_n,\, e_{-n} \rangle = \pm 1$:
\[
\left\{ 
\underbrace{e_{1}, \cdots, e_{n} }_{0},\,\, 
e_{-n}, \cdots, e_{-1}
\right\}
\]


- Choose the following level structures:

  - For $t$ a uniformizer at $x=0$:
  \[
  K_0 = \left\{
  \left[\begin{array}{c|c}
  \theta_{0} & t \theta_{0} \\
  \hline \theta_{0} & \theta_{0}
  \end{array}\right] \right\},\qquad \text{each block }2\times 2 \text{ in the above basis}
  \]
  - For $\tau$ a uniformizer at $x=0\infty$:
  \[
  K_\infty = 
  \left\{
  \left[\begin{array}{c|c}
  I_{2}+\tau \theta_{\infty} & \theta_{\infty} \\
  \hline \tau \theta_{\omega} & I_{2}+\tau \theta_{\infty}
  \end{array}\right]\right\}
  \]
  
Then define
\[
\operatorname{Bun}_{{\mathrm{Sp}_{2 n}}}\left(K_{0}, K_{\infty}\right)=\left\{
\begin{array}{r}
\left(V, \hspace{5.6em}\right. \text { Vect. bun. rk } 2n \\
\omega\hspace{6em} \text{a symplectic form}\\
L_{0} \subseteq V_{0}, \hspace{10.5em}\\
L_{\infty} \subseteq V_{\infty}, \hspace{4em}\text { Lagrangians } \\
\left.\left\{l_{1}, \cdots, l_{n}\right\}\right) 
\hspace{3em}
\text { a basis for }L_{\infty} \\
\end{array}\right\}
\]