2022/07/30 by Alexandru Buium, Buium, Alexandru, Adrian Vasiu +1
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.2208.00286
Let p be a prime, let N≥ 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 Fp, and consider the correspondence \mathcal A'g,1,N,R\rightrightarrows \mathcal Ag,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 Ag,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 Spec(R). We construct and study in detail such a coequalizer (categorical quotient) in a more refined geometry (category) referred to as \it δ-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 Ag,1,N,R/\mathcal A'g,1,N,R in δ-geometry has the same `dimension' as \mathcal Ag,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 Ag,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 δ-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.