Conjectured invariants of the Shtuka surface for level 9amma_0(4(0))

Let $q$ be a prime power, let $F:=\mathbb{F}_q(P,Q)$, and let

$$S_q:=\overline{\operatorname{Sht}}^{2,\operatorname{tr}}(\Gamma_0(4(0)))$$

be the associated surface over $F$. Write $p_a(S_q)$ for its arithmetic genus, $b_2(S_q)$ for its second Betti number, and call the lattice generated by the zero section and the components of singular fibers its trivial lattice.

Conjectured invariants of $S_q$. If $q$ is even, then

$$p_a(S_q)=\left\lfloor\frac q4\right\rfloor+1,\qquad b_2(S_q)=12\left\lfloor\frac q4\right\rfloor+10.$$

The trivial lattice has rank $8$ when $q=2$ and rank $2q+10$ for $q>2$. The surface has two singular fibers: one of type $\operatorname{I}_{2q+2}$ at $1$, and one of type $\operatorname{III}^*$ at $\infty$, except that for $q=2$ the fiber at $\infty$ is of type $\operatorname{III}$. If $q$ is odd, then

$$p_a(S_q)=\frac{q+1}{2},\qquad b_2(S_q)=6q+4,$$

the trivial lattice has rank $4q+5$, and over an algebraic closure $\overline F$ there are $2q+2$ singular fibers: one of type $\operatorname{I}_{2q-2}^*$ at $\infty$, one of type $\operatorname{I}_{2q+2}$ at $1$, and the remaining fibers arise from two bad fibers of type $\operatorname{I}_1$ of degree $q$ over $F$.

These formulas and fiber descriptions are conjectured from computations for small prime powers. The cases $q=2$ and $q=3,4$ yield, respectively, a rational elliptic surface and K3 elliptic surfaces; the general assertions remain to be established.