Dirichlet Density Of Reducible Locus In Moduli Space Of Polarized Abelian Varieties
Introduction
In the realm of algebraic geometry and number theory, the study of moduli spaces holds a central position. Moduli spaces, in essence, are geometric objects that parameterize families of algebraic objects, such as curves, surfaces, or abelian varieties. These spaces provide a framework for understanding the variation and classification of these objects, offering deep insights into their underlying structure and properties. Among the various moduli spaces, those associated with abelian varieties are particularly rich and intricate, exhibiting connections to diverse areas of mathematics.
Abelian varieties, which are projective algebraic varieties equipped with a group structure, play a crucial role in number theory, cryptography, and algebraic geometry. They generalize the notion of elliptic curves to higher dimensions and serve as fundamental building blocks in the study of algebraic cycles and motives. The moduli spaces of abelian varieties, denoted by , parameterize principally polarized abelian varieties of dimension , where the polarization is a geometric structure that endows the abelian variety with a notion of duality. These moduli spaces are fascinating objects in their own right, possessing a rich geometric structure and exhibiting deep connections to arithmetic.
When considering abelian varieties over fields of positive characteristic, the landscape becomes even more intriguing. The reduction of abelian varieties modulo a prime number introduces new phenomena and complexities, leading to a deeper understanding of their arithmetic properties. In positive characteristic, the moduli spaces of abelian varieties exhibit a stratification, as demonstrated by Oort's work, which provides a way to decompose these spaces into locally closed subvarieties based on the p-rank and other invariants of the abelian varieties. This stratification allows for a more refined analysis of the moduli spaces and the abelian varieties they parameterize.
One particularly interesting aspect of the moduli spaces of abelian varieties is the reducible locus, which consists of points corresponding to abelian varieties that decompose as a product of lower-dimensional abelian varieties. Understanding the distribution and density of the reducible locus within the moduli space provides valuable information about the prevalence of decomposable abelian varieties and their significance in the overall landscape. The Dirichlet density is a measure of the asymptotic proportion of prime numbers for which a certain property holds, and in this context, it can be used to quantify the density of the reducible locus within the moduli space as the characteristic varies.
This article delves into the question of the Dirichlet density of the reducible locus in the moduli of polarized abelian varieties in positive characteristic. We will explore the concepts of moduli spaces, abelian varieties, and their behavior in positive characteristic. Further, we will delve into the stratification of moduli spaces and the notion of the reducible locus, aiming to shed light on the arithmetic properties of abelian varieties and their moduli spaces. By examining the Dirichlet density of the reducible locus, we seek to gain insights into the distribution of decomposable abelian varieties and their significance in the overall landscape of moduli spaces.
Moduli Spaces and Abelian Varieties: A Foundation
To understand the Dirichlet density of the reducible locus, it's crucial to first establish a solid understanding of moduli spaces and abelian varieties. Moduli spaces, at their core, are geometric spaces that serve as parameter spaces for families of algebraic objects. These objects can range from curves and surfaces to vector bundles and, most relevant to our discussion, abelian varieties. The key idea is that each point on the moduli space corresponds to a specific algebraic object, and the geometric structure of the moduli space reflects the relationships and variations among these objects.
Abelian varieties themselves are a fundamental concept in algebraic geometry and number theory. They are projective algebraic varieties that are also equipped with a group structure, meaning that there is a well-defined addition operation between points on the variety. This combination of geometric and algebraic structure makes abelian varieties particularly rich and interesting objects of study. Elliptic curves, which are abelian varieties of dimension one, are perhaps the most well-known examples, but abelian varieties exist in higher dimensions as well. They play a critical role in various areas of mathematics, including cryptography, number theory, and the study of algebraic cycles.
The moduli space of abelian varieties, often denoted as , parameterizes principally polarized abelian varieties of dimension . The term
