论文标题
$Δ$ - invariant的Hecke通信理论$ \ MATHCAL A_G $
The $δ$-invariant theory of Hecke correspondences on $\mathcal A_g$
论文作者
论文摘要
让$ p $为素数，让$ n \ geq 3 $为$ p $的整数素数，让$ r $是$ p $ p $ typical witt vectors的戒指，其系数为$ \ mathbb f_p $ of gengebraic of系数A_ {G，1，N，R} $通过将所有Prime的结合到Mumford的Moduli方案中的所有Prime汇总到$ r $ r $ schemes上的符号相似级别$ n $结构。众所周知，均等$ \ MATHCAL A_ {G，1，N，R}/\ MATHCAL A'_ {G，1，N，R} $存在于上述信件的，并且在方案类别中是微不足道的，即$ \ text {spec {spec}（r）$。我们在更精致的几何形状（类别）中详细构建和研究这种均衡器（分类商），称为{\ it $Δ$ - 几何}。本质上是从通常的代数几何形状获得的。特别是，我们证明了我们的替代品代替$ \ Mathcal a_ {g，1，n，r}/\ Mathcal a'_ {g，1，1，n，r} $中的$δ$ - 几何形式具有相同的“维度”，与$ \ nathcal a__ {g，1，1，n，n，r} $ solve a baru nove a nove a nove a nove a nove a norku n os a n o n o n of baru。我们还将应用于$ \ Mathcal a_ {g，1，n，r}的各种Zariski致密基因座的研究，例如等级类和具有复杂乘法的点。为了证明我们的结果，我们为任意属的{\ it siegel $δ$ - 模块化形式}开发了一种序列扩展理论，然后我们将其与多种二次形式和多种构态术的几何不变理论结合起来。
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 $δ$-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 $\mathcal A_{g,1,N,R}/\mathcal A'_{g,1,N,R}$ in $δ$-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 $δ$-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.
