Level Structures on Finite Group Schemes and Applications

The notion of level structures originates from the study of the moduli of elliptic curves. In this thesis, we consider generalizing the notion of level structures and make explicit calculations on different moduli spaces. The first moduli space we consider is the moduli of finite flat (commutative) group schemes. We give a definition of $\Gamma(p)$-level structure (also called the ``full level structure") over group schemes of the form $G\times G$, where $G$ is a group scheme or rank $p$ over a $\Z_p$-scheme. The full level structure over $G\times G$ is flat over the base of rank $|\GL_2(\F_p)|$. We also observe that there is no natural notion of full level structures over the stack of all finite flat commutative group schemes. The second moduli space we consider is the moduli of principally polarized abelian surfaces in characteristic $p>0$ with symplectic level-$n$ structure ($n\ge 3$), which is known as the Siegel threefold. By decomposing the Siegel threefold using the Ekedahl--Oort stratification, we analyze the $p$-torsion group scheme of the universal abelian surface over each stratum. To do this, we establish a machinery to produce group schemes from their Dieudonn\'e modules using a version of Dieudonn\'e theory due to de Jong. By using this machinery, we give explicit local equations of the Hopf algebras over the superspecial locus, the supersingular locus and ordinary locus. Using these local equations, we calculate explicit equations of the $\Gamma_1(p)$-covers over these strata using Kottwitz--Wake primitive elements. These equations can be used to prove geometric and arithmetic properties of the $\Gamma_1(p)$-cover over the Siegel threefold. In particular, we prove that the $\Gamma_1(p)$-cover over the Siegel threefold is not normal.