The $\delta$-invariant theory of Hecke correspondences on ${\mathcal{A}}_g$

Let $p$ be a prime, let $N\ge 3$ be an integer prime to $p$, let $R$ be the ring of $p$-typical Witt vectors with coefficients in an algebraic closure of ${\mathbb{F}}_p$, and consider the correspondence ${\mathcal{A}}_{g,1,N,R}^{\mathrm{\prime }}\rightrightarrows {\mathcal{A}}_{g,1,N,R}$ obtained by taking the union of all prime to $p$ Hecke correspondences on Mumford’s moduli scheme of principally polarized abelian schemes of relative dimension $g$ endowed with symplectic similitude level-$N$ structure over $R$-schemes. It is well-known that the coequalizer ${\mathcal{A}}_{g,1,N,R}\mathrm{/}{\mathcal{A}}_{g,1,N,R}^{\mathrm{\prime }}$ of the above correspondence exists and is trivial in the category of schemes, i.e., is $\text{Spec}(R)$. We construct and study in detail such a coequalizer (categorical quotient) in a more refined geometry (category) referred to as {\it $\delta$-geometry}. This geometry is in essence obtained from the usual algebraic geometry by equipping all $R$-algebras with {\it $p$-derivations}. In particular, we prove that our substitute of ${\mathcal{A}}_{g,1,N,R}\mathrm{/}{\mathcal{A}}_{g,1,N,R}^{\mathrm{\prime }}$ in $\delta$-geometry has the same `dimension’ as ${\mathcal{A}}_{g,1,N,R}$, thus solving a main open problem in the work of Barcău–Buium. We also give applications to the study of various Zariski dense loci in ${\mathcal{A}}_{g,1,N,R}$ such as of isogeny classes and of points with complex multiplication. To prove our results we develop a Serre–Tate expansion theory for {\it Siegel $\delta$-modular forms} of arbitrary genus which we then combine with old and new results from the geometric invariant theory of multiple quadratic forms and of multiple endomorphisms.