The $\delta$-invariant theory of Hecke correspondences on $\mathcal A_g$

TL;DR

This paper develops a new geometric framework called $oldsymbol{ extit{ extbf{$oldsymbol{ extdelta}}}$-geometry} to study Hecke correspondences on moduli spaces of abelian varieties, revealing new structural insights and applications to dense loci and modular forms.

Contribution

It constructs a $ extit{ extdelta}$-geometric quotient of Hecke correspondences, solving a key open problem and introducing a Serre--Tate expansion theory for Siegel $ extit{ extdelta}$-modular forms.

Findings

The $ extit{ extdelta}$-geometry quotient has the same dimension as the original moduli space.

Established a Serre--Tate expansion theory for Siegel $ extit{ extdelta}$-modular forms.

Applications to Zariski dense loci, isogeny classes, and CM points.

Abstract

Let $p$ be a prime, let $N\geq 3$ 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 $\mathcal A'_{g,1,N,R}\rightrightarrows \mathcal A_{g,1,N,R}$ 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 $\mathcal A_{g,1,N,R}/\mathcal A'_{g,1,N,R}$ 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 $\mathcal A_{g,1,N,R}/\mathcal A'_{g,1,N,R}$ in $\delta$-geometry has the same `dimension' as $\mathcal A_{g,1,N,R}$, thus solving a main open problem in the work of Barc\u{a}u--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.