Dirichlet Density Of Reducible Locus In Moduli Of Polarized Abelian Varieties

Delving into the intricate world of algebraic geometry and number theory, a fascinating question arises: what is the Dirichlet density of the reducible locus within the moduli space of polarized abelian varieties in positive characteristic? This inquiry, deeply rooted in the study of arithmetic geometry and moduli spaces, invites us to explore the landscape of abelian varieties defined over finite fields and their remarkable properties.

Exploring the Moduli Space of Polarized Abelian Varieties

At the heart of this discussion lies the concept of the moduli space of polarized abelian varieties. An abelian variety is a projective algebraic variety that also possesses the structure of an algebraic group. These varieties, which generalize elliptic curves to higher dimensions, play a crucial role in various areas of mathematics, including number theory, cryptography, and mathematical physics. A polarization on an abelian variety is essentially an ample line bundle, providing a way to measure the 'size' of subvarieties. The moduli space, denoted as ${ \mathcal{A}_{g,d,n} }$, serves as a geometric object that parametrizes isomorphism classes of polarized abelian varieties of dimension ${ g }$, with polarization degree ${ d }$ and level structure ${ n }$. Think of it as a 'space' where each point corresponds to a unique abelian variety (up to isomorphism) with specified polarization and level structure.

The study of these moduli spaces is crucial because they provide a framework for understanding the behavior of abelian varieties as they vary within families. In particular, understanding the geometry and arithmetic properties of these moduli spaces can shed light on the distribution of various types of abelian varieties. When working in positive characteristic, say over a finite field ${ \mathbb{F}_p }$, the landscape becomes even richer due to the presence of Frobenius endomorphisms and the interplay between algebraic and arithmetic structures. Oort's work, particularly his paper "A stratification of a moduli space of abelian varieties," provides a foundational framework for studying these moduli spaces in positive characteristic. He introduces a stratification, a way of decomposing the space into smaller, more manageable pieces, based on certain invariants related to the abelian varieties. This stratification is a powerful tool for analyzing the geometry and arithmetic of ${ \mathcal{A}_{g,d,n} }$ over ${ \mathbb{F}_p }$.

Defining the Reducible Locus

The reducible locus within this moduli space is a subset consisting of points that correspond to reducible abelian varieties. An abelian variety is considered reducible if it is isogenous to a product of abelian varieties of smaller dimension. In simpler terms, a reducible abelian variety can be 'broken down' into smaller abelian varieties. This notion of reducibility is critical because it reflects the complexity and structure of the abelian variety. Reducible abelian varieties often exhibit different arithmetic and geometric properties compared to irreducible ones. For instance, their endomorphism rings, which encode the self-maps of the variety, can be more complex, and their behavior under reduction modulo primes can be significantly different.

Understanding the reducible locus is essential for several reasons. First, it provides insight into the distribution of abelian varieties with specific structural properties. Second, it helps in classifying abelian varieties and understanding their moduli. Third, the geometry of the reducible locus can reveal important information about the moduli space itself. The Dirichlet density, which we will discuss shortly, provides a quantitative measure of how 'common' reducible abelian varieties are within the moduli space. Determining this density is a challenging problem, but it offers a deep understanding of the arithmetic landscape of abelian varieties in positive characteristic.

Dirichlet Density: A Measure of Abundance

The Dirichlet density is a concept from number theory that provides a way to measure the 'size' or 'abundance' of a set of prime numbers or, in this context, a set of points in the moduli space. To understand Dirichlet density, it’s helpful to first consider the idea of natural density. The natural density of a set of primes is the limit, as ${ x }$ approaches infinity, of the proportion of primes less than ${ x }$ that belong to the set. However, natural density doesn't always exist, and even when it does, it may not capture the subtle distribution patterns of primes or other arithmetic objects. Dirichlet density offers a more refined measure.

Formally, the Dirichlet density of a set ${ S }$ of prime numbers is defined using a Dirichlet series. We consider the series ${ D(s) = \sum_{p \in S} \frac{1}{p^s} }$ where the sum is taken over all prime numbers ${ p }$ in the set ${ S }$. If the limit ${ \lim_{s \to 1^+} \frac{D(s)}{\log(\frac{1}{s-1})} }$ exists, then this limit is the Dirichlet density of ${ S }$. The Dirichlet density, when it exists, provides a measure of the 'proportion' of primes in the set ${ S }$ relative to the set of all primes. In the context of moduli spaces, we can adapt this notion to measure the proportion of points in the moduli space that satisfy a certain condition, such as corresponding to reducible abelian varieties. Thus, the Dirichlet density of the reducible locus quantifies how frequently we encounter reducible abelian varieties as we 'sample' points from the moduli space. This measure is particularly useful in positive characteristic, where the arithmetic properties of abelian varieties can exhibit subtle and complex behaviors.

The Significance of Positive Characteristic

Working in positive characteristic introduces unique challenges and opportunities in the study of abelian varieties. In characteristic zero (e.g., over the complex numbers), the theory of abelian varieties is well-developed, with powerful tools from complex analysis and algebraic topology at our disposal. However, in positive characteristic (e.g., over a finite field ${ \mathbb{F}_p }$), new phenomena emerge due to the presence of the Frobenius endomorphism, which raises elements to the power of ${ p }$. This endomorphism plays a crucial role in the arithmetic of abelian varieties in positive characteristic, influencing their structure, endomorphism rings, and behavior under reduction modulo primes.

One of the key differences in positive characteristic is the existence of isogenies, which are surjective homomorphisms between abelian varieties with finite kernel. Isogenies can drastically alter the structure of abelian varieties, and their behavior is often more intricate in positive characteristic. For instance, the notion of supersingularity, which is specific to positive characteristic, describes abelian varieties with particularly rich endomorphism rings. Supersingular abelian varieties play a vital role in the geometry of moduli spaces and the arithmetic of modular forms. Furthermore, the interplay between the algebraic structure of abelian varieties and their arithmetic properties, such as their zeta functions and L-functions, is significantly influenced by the characteristic of the base field. Understanding the Dirichlet density of the reducible locus in positive characteristic requires a careful analysis of these characteristic-specific phenomena.

Determining the Dirichlet Density: A Complex Challenge

Calculating the Dirichlet density of the reducible locus in the moduli space of polarized abelian varieties in positive characteristic is a formidable task. It requires a deep understanding of the geometry and arithmetic of both the moduli space and the abelian varieties themselves. The challenge stems from several factors. First, the moduli space ${ \mathcal{A}_{g,d,n} }$ is a complex object, and its geometry is not fully understood, especially in positive characteristic. Second, characterizing reducible abelian varieties is itself a non-trivial problem. One must understand the possible decompositions of abelian varieties into products of lower-dimensional varieties and the conditions under which such decompositions occur.

Third, the Dirichlet density, as a measure of asymptotic behavior, requires a global understanding of the distribution of reducible abelian varieties within the moduli space. This necessitates counting arguments and the analysis of arithmetic invariants, such as the zeta functions of abelian varieties. Furthermore, the interaction between the polarization and the reducibility condition adds another layer of complexity. The polarization influences the geometry of the abelian variety and its endomorphism ring, which in turn affects its reducibility. Techniques from algebraic geometry, number theory, and representation theory are often employed to tackle this problem. One approach involves studying the stratification of the moduli space introduced by Oort, which provides a way to decompose the problem into smaller, more manageable pieces. Another approach involves analyzing the Frobenius endomorphism and its action on the abelian variety, as this often reveals information about the variety's reducibility.

Potential Applications and Further Research

The determination of the Dirichlet density of the reducible locus has significant implications and potential applications in various areas of mathematics. First, it provides a quantitative understanding of the structure of the moduli space of polarized abelian varieties. This knowledge is crucial for further investigations into the geometry and arithmetic of these spaces. Second, it sheds light on the distribution of abelian varieties with specific properties, which is relevant to the study of L-functions and modular forms. The arithmetic of abelian varieties is intimately connected to these analytic objects, and understanding their distribution can provide valuable insights into their behavior.

Third, this research has connections to cryptography, particularly in the context of elliptic curve cryptography and higher-dimensional analogs. The security of these cryptographic systems often relies on the difficulty of certain computational problems related to abelian varieties, and understanding the distribution of reducible varieties can have implications for the design and analysis of these systems. Further research in this area could explore the Dirichlet density of other interesting loci within the moduli space, such as the supersingular locus or the locus of abelian varieties with specific endomorphism rings. Additionally, investigating the behavior of these densities as the characteristic ${ p }$ varies can lead to a deeper understanding of the arithmetic of abelian varieties in positive characteristic. The quest to understand the Dirichlet density of the reducible locus is a challenging but rewarding endeavor, offering a glimpse into the rich and intricate world of arithmetic geometry.