Dirichlet Density Of The Reducible Locus In Moduli Of Abelian Varieties

Introduction

In the fascinating intersection of algebraic geometry and number theory lies the study of abelian varieties, particularly their behavior in positive characteristic. Understanding the distribution of these varieties within their moduli spaces is a central question, with deep connections to arithmetic geometry. This article delves into a specific aspect of this question: the Dirichlet density of the reducible locus within the moduli space of polarized abelian varieties in positive characteristic. This exploration is motivated by recent work, notably section 2 of the Karemaker Notes from the 2024 Arizona Winter School, which provides a foundation for our discussion. We aim to unpack the key concepts, relevant theorems, and the significance of the results in this area, making it accessible to both experts and those new to the field. This field intertwines complex mathematical structures and theoretical frameworks, so this article will attempt to make the material accessible while maintaining accuracy.

Background on Abelian Varieties and Moduli Spaces

To fully grasp the concept of the Dirichlet density of the reducible locus, it’s essential to first establish a solid understanding of the fundamental objects involved: abelian varieties and their moduli spaces.

Abelian Varieties

An abelian variety is, in simple terms, a projective algebraic variety that also forms a group. More formally, it's a complete algebraic variety equipped with a group structure such that the group operations (multiplication and inverse) are morphisms of varieties. These objects are central to many areas of algebraic geometry and number theory due to their rich geometric and arithmetic properties. Examples of abelian varieties include elliptic curves (abelian varieties of dimension 1) and Jacobians of algebraic curves. Their group structure allows for the study of endomorphisms, which are homomorphisms from the variety to itself, and the structure of these endomorphism rings provides significant insights into the variety's properties. The study of abelian varieties often involves understanding their torsion points, which are points of finite order under the group operation, and their behavior over different fields, especially finite fields.

Moduli Spaces

A moduli space is a geometric object that parameterizes a family of algebraic objects, such as curves, surfaces, or, in our case, abelian varieties. Each point in the moduli space corresponds to an isomorphism class of the object being parameterized. The construction of moduli spaces is a delicate process, often involving sophisticated techniques from algebraic geometry. A key challenge is ensuring that the moduli space itself has a reasonable geometric structure, such as being a scheme or an algebraic space. For abelian varieties, the moduli space typically parameterizes polarized abelian varieties of a fixed dimension and polarization degree. A polarization is essentially an ample line bundle on the abelian variety, and it plays a crucial role in the moduli problem. The moduli space of polarized abelian varieties, denoted as ${ A_{g,d} }$, where ${ g }$ is the dimension and ${ d }$ is the degree of polarization, is a fundamental object in algebraic geometry. The study of these moduli spaces involves understanding their geometry, topology, and arithmetic properties, including their behavior over different fields and their compactifications.

Fine and Coarse Moduli Spaces

In the context of moduli spaces, it is important to distinguish between fine and coarse moduli spaces. A fine moduli space is one where there exists a universal family of objects parameterized by the space. This means that every object in the family corresponds to a unique point in the moduli space, and every point in the moduli space corresponds to an object in the family. However, fine moduli spaces do not always exist, due to the presence of automorphisms in the objects being parameterized. A coarse moduli space, on the other hand, is a weaker notion. It is a space that represents the moduli functor, meaning that there is a natural bijection between the points of the space and the isomorphism classes of the objects being parameterized. Coarse moduli spaces always exist, but they may not have a universal family. In practice, the construction of fine moduli spaces often involves introducing additional structures, such as level structures, which eliminate automorphisms and allow for the existence of a universal family. The Karemaker Notes mention a stratified fine moduli space, which suggests the use of level structures to achieve a fine moduli space.

The Reducible Locus

Having established the basics of abelian varieties and their moduli spaces, we can now turn our attention to the reducible locus. This is a specific subset of the moduli space 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 dimensions. In other words, if an abelian variety ${ A }$ can be related via an isogeny (a surjective homomorphism with finite kernel) to a product ${ A_1 \times A_2 }$, where ${ A_1 }$ and ${ A_2 }$ are abelian varieties of positive dimensions, then ${ A }$ is reducible. Understanding the reducible locus is crucial for understanding the structure of the moduli space, as it provides insights into how abelian varieties decompose and how they are related to each other. The geometry and arithmetic properties of the reducible locus are complex and depend on the characteristic of the base field, the dimension of the abelian varieties, and the degree of polarization. The study of the reducible locus often involves techniques from representation theory, as the endomorphism rings of abelian varieties play a key role in determining their reducibility.

Significance of the Reducible Locus

The reducible locus plays a pivotal role in understanding the architecture of moduli spaces of abelian varieties. An abelian variety's reducibility—whether it decomposes, via isogeny, into lower-dimensional abelian varieties—sheds light on its intrinsic structure and its relationships within the broader moduli space. Points within the reducible locus correspond to abelian varieties with particular decomposition properties, offering a lens through which to examine the moduli space's stratification and the interactions among its components. This is crucial for classifying abelian varieties and discerning their complex behavior. Studying the reducible locus helps us understand how abelian varieties decompose and how they relate to each other, giving us a deeper understanding of the moduli space's structure.

Geometric and Arithmetic Implications

The geometry of the reducible locus reveals fundamental properties of the moduli space, such as its stratification and connectivity. The reducible locus can be viewed as a union of subvarieties, each corresponding to a specific type of decomposition. Understanding these subvarieties and their intersections provides a detailed picture of the moduli space's geometric structure. Arithmetically, the reducible locus offers insights into the number theory of abelian varieties. For instance, the distribution of points in the reducible locus over finite fields is related to the arithmetic properties of the abelian varieties they represent. The density of these points, as measured by the Dirichlet density, is a key indicator of the prevalence of reducible abelian varieties in certain contexts. The reducible locus allows us to study the number-theoretic aspects of abelian varieties, such as their distribution over finite fields. This link between geometry and arithmetic is a hallmark of the study of moduli spaces.

Dirichlet Density: A Measure of Distribution

To quantify the “size” or prevalence of the reducible locus within the moduli space, we employ the concept of Dirichlet density. This measure, borrowed from number theory, provides a way to assess the distribution of a set of primes or, in our case, points in the moduli space that satisfy certain conditions.

Definition and Significance

The Dirichlet density is a way to measure how common a set of prime numbers (or, more generally, a set of points in a space) is. It's especially useful when the set doesn't have a natural density in the usual sense. For a set ${ S }$ of prime numbers, the Dirichlet density is defined as the limit (if it exists) of a certain function involving a sum over primes in ${ S }$ divided by a similar sum over all primes, as a complex variable approaches 1. This definition might seem technical, but it has a crucial intuitive meaning: it tells us the “proportion” of primes that belong to the set ${ S }$. A Dirichlet density of 0 means the set is relatively sparse, while a density of 1 means the set is quite abundant. In our context, we're interested in the Dirichlet density of the reducible locus, which will tell us how often we can expect to find reducible abelian varieties within the moduli space as we vary the characteristic of the base field. It helps us understand how often we find reducible abelian varieties in the moduli space by giving us a measure of their frequency.

Application to Moduli Spaces

In the context of moduli spaces, the Dirichlet density allows us to quantify the prevalence of certain types of abelian varieties, such as the reducible ones. We consider the set of prime powers ${ q }$ for which the reduction of an abelian variety modulo a prime dividing ${ q }$ lies in the reducible locus. The Dirichlet density of this set of prime powers then gives us a measure of how often we encounter reducible abelian varieties as we vary ${ q }$. This is particularly relevant in positive characteristic, where the behavior of abelian varieties can be quite different from characteristic zero. Understanding the Dirichlet density of the reducible locus can provide insights into the distribution of isogeny classes of abelian varieties and the structure of their endomorphism rings. It helps us understand how common reducible abelian varieties are when we change the characteristic of the field we're working over.

Reducibility in Positive Characteristic

The behavior of abelian varieties in positive characteristic is significantly different from their behavior in characteristic zero. In positive characteristic, the Frobenius endomorphism plays a central role, and the structure of the endomorphism ring can be much more intricate. This has profound implications for the reducibility of abelian varieties and the structure of the reducible locus.

Frobenius Endomorphism

The Frobenius endomorphism is a fundamental tool in the study of algebraic varieties over fields of positive characteristic ${ p }$. For an abelian variety ${ A }$ defined over a field of characteristic ${ p }$, the Frobenius endomorphism ${ F }$ is a morphism from ${ A }$ to itself that raises coordinates to the ${ p ext{th} }$ power. This seemingly simple map has deep consequences for the structure of ${ A }$. For instance, the endomorphism ring of ${ A }$ often contains ${ F }$ and its powers, leading to a rich algebraic structure. The Frobenius endomorphism also plays a key role in determining the zeta function of ${ A }$, which encodes important arithmetic information about the variety. The properties of the Frobenius endomorphism, such as its characteristic polynomial, are closely related to the isogeny class of ${ A }$ and its reducibility. This seemingly simple map is crucial because it greatly affects the structure and properties of abelian varieties.

Implications for Reducibility

In positive characteristic, the Frobenius endomorphism significantly impacts the reducibility of abelian varieties. The eigenvalues of the Frobenius endomorphism acting on the Tate module of the abelian variety determine its isogeny class and, consequently, its reducibility. For instance, if the characteristic polynomial of the Frobenius endomorphism factors in a certain way, the abelian variety may be isogenous to a product of abelian varieties of smaller dimensions. This connection between the Frobenius endomorphism and reducibility is a key feature of abelian varieties in positive characteristic. The structure of the endomorphism ring, which is heavily influenced by the Frobenius endomorphism, also plays a crucial role. Abelian varieties with large endomorphism rings are more likely to be reducible, as their endomorphisms can provide the necessary maps to decompose them into products. The interplay between the Frobenius endomorphism and the endomorphism ring is thus a central theme in the study of reducibility in positive characteristic. It affects how abelian varieties can be broken down into simpler components, making their behavior unique in positive characteristic.

Main Results and Conjectures

The study of the Dirichlet density of the reducible locus in the moduli of polarized abelian varieties in positive characteristic is an active area of research. While there are no definitive, universally accepted results that provide a complete picture, several important results and conjectures shed light on this topic. Specifically, the notes from the 2024 Arizona Winter School, referenced in the initial query, likely delve into some of these findings, providing a valuable starting point for further investigation. This part will go through some of the main ideas and predictions in the field, acknowledging that it's a work in progress.

Existing Results

Some existing results focus on specific cases, such as the moduli space of elliptic curves or abelian surfaces. For example, it is known that the Dirichlet density of the supersingular locus (a special case of the reducible locus) in the moduli space of elliptic curves is 0. This means that, in a certain sense, supersingular elliptic curves are rare. However, the situation becomes more complex for higher-dimensional abelian varieties. There are results that provide lower bounds for the Dirichlet density of the reducible locus in certain situations, suggesting that reducible abelian varieties are not always rare. These results often depend on the specific parameters of the moduli space, such as the dimension and polarization degree, as well as the characteristic of the base field. The focus on particular examples, like elliptic curves and surfaces, shows the variety of behaviors we see in different cases. Results may give estimates for how often reducible abelian varieties appear under certain conditions, but these often depend on the details of the moduli space and the field we're looking at.

Conjectures and Open Questions

Several conjectures and open questions remain in this area. One major question is to determine the exact value of the Dirichlet density of the reducible locus for general moduli spaces of abelian varieties. This is a challenging problem, as it requires a deep understanding of the arithmetic and geometry of these spaces. Another important question is to understand how the Dirichlet density behaves as the dimension and polarization degree of the abelian varieties vary. There are conjectures that relate the Dirichlet density to certain arithmetic invariants of the moduli space, such as the number of points over finite fields. These conjectures, if proven, would provide a powerful tool for studying the distribution of abelian varieties in positive characteristic. Many open questions are still being investigated, such as determining the precise density of the reducible locus in general cases and understanding how this density changes with different parameters. Conjectures aim to connect this density to other properties of the moduli space, which would be a significant step forward in the field.

Conclusion

The investigation into the Dirichlet density of the reducible locus in the moduli of polarized abelian varieties in positive characteristic represents a fascinating and challenging area of research at the intersection of algebraic geometry and number theory. While much remains to be discovered, the existing results and ongoing work provide valuable insights into the distribution and behavior of abelian varieties. The concepts discussed here—abelian varieties, moduli spaces, the reducible locus, Dirichlet density, and the Frobenius endomorphism—are fundamental to this field. By understanding these concepts and the relationships between them, we can continue to unravel the intricate structure of moduli spaces and the arithmetic properties of abelian varieties. The study of the Dirichlet density of the reducible locus not only enriches our theoretical understanding but also has potential applications in cryptography and other areas. This field continues to evolve, offering exciting opportunities for future research and discovery. This article has aimed to explore these complex topics, providing a foundation for further study and hopefully inspiring more interest in this rich and active area of mathematics.