Elliptic Curve Quotients A Detailed Discussion

Understanding the nuances of quotient constructions in algebraic geometry, particularly concerning elliptic curves and their subgroups, is crucial for advanced studies in the field. This article delves into the intricacies of forming quotients of elliptic curves by subgroup schemes, addressing common points of confusion and providing a detailed exploration of the concepts involved. We will draw upon fundamental principles from Mumford's "Abelian Varieties" and other relevant literature to offer a comprehensive understanding.

Clarifying Quotient Constructions in Elliptic Curves

When exploring elliptic curves, a key concept involves understanding different types of quotients, especially in relation to subgroup schemes. These quotients reveal how an elliptic curve behaves when divided by certain subgroups, providing insights into its structure and properties. This discussion often arises in the context of questions like, "Is an elliptic curve canonically isomorphic to its quotient by its own n-torsion?" To address this, we must first clarify the different ways a quotient can be constructed and interpreted.

The Notion of a Quotient

The term "quotient" in algebraic geometry can refer to several distinct constructions, each with its own implications. Generally, a quotient involves dividing a geometric object (in this case, an elliptic curve) by an equivalence relation defined by a subgroup scheme. The resulting object retains information about the original curve while also reflecting the structure of the subgroup by which we are dividing. The quotient operation is fundamental in understanding the symmetries and underlying structures of elliptic curves.

Types of Quotients

There are primarily two types of quotients that we need to distinguish:

1. Quotients in the Category of Schemes: This involves constructing a new scheme that represents the quotient of the elliptic curve by the subgroup scheme. This construction relies on the general machinery of quotient schemes in algebraic geometry. A scheme is a fundamental concept in modern algebraic geometry, providing a framework to study algebraic varieties using the language of commutative algebra. This type of quotient provides a rigorous way to understand the geometric structure resulting from the division.

2. Quotients as Isomorphism Classes: Here, the quotient is understood as the set of isomorphism classes of elliptic curves that are isogenous to the original curve via a specific isogeny (a surjective homomorphism with finite kernel). Isogenies play a crucial role in the study of elliptic curves, linking curves with related structures. This perspective emphasizes the relationships between different elliptic curves and their modular properties.

Understanding these distinctions is vital. When we talk about the quotient of an elliptic curve by its n-torsion subgroup, we must specify which type of quotient we are considering. The scheme-theoretic quotient provides a concrete geometric object, while the quotient as an isomorphism class focuses on the broader landscape of elliptic curves and their interconnections.

Scheme-Theoretic Quotients: A Deep Dive

The scheme-theoretic quotient is a fundamental construction in algebraic geometry. Given an elliptic curve E and a subgroup scheme G, the quotient E/G is a new scheme that represents the geometric object resulting from dividing E by G. This construction involves advanced algebraic techniques and provides a rigorous framework for understanding quotients.

Construction of Scheme-Theoretic Quotients

The construction of the quotient scheme E/G typically involves the following steps:

1. Defining the Action: The subgroup scheme G acts on the elliptic curve E. This action is defined by a morphism G x E → E, where G x E represents the fiber product of the schemes G and E. Understanding this action is crucial, as it dictates how the subgroup scheme transforms the points on the elliptic curve. The action must satisfy certain compatibility conditions to ensure it behaves like a group action.

2. Quotient in the Étale Topology: The quotient is first constructed in the étale topology. Étale morphisms are a generalization of covering maps in topology and are crucial in algebraic geometry for studying local properties of schemes. This step involves finding a scheme E/G and a morphism E → E/G that satisfy a universal property with respect to morphisms that are invariant under the action of G. In essence, we are looking for a scheme that captures the quotient in a way that respects the local structure of the schemes involved.

3. Descent Theory: Descent theory is then used to show that this quotient descends to a quotient in the Zariski topology, which is the usual topology considered in algebraic geometry. Descent theory provides conditions under which objects defined in a covering space (in this case, the étale cover) can be "descended" to the base space (the Zariski topology). This step ensures that the quotient we have constructed is a well-behaved scheme in the standard sense.

Properties of Scheme-Theoretic Quotients

The resulting quotient scheme E/G has several important properties:

• Universal Property: The quotient E/G satisfies a universal property: any morphism from E that is invariant under the action of G factors uniquely through the quotient morphism E → E/G. This property ensures that the quotient construction is canonical and captures the essence of dividing by the subgroup scheme.
• Geometric Structure: The geometric structure of E/G depends on the properties of the subgroup scheme G. For example, if G is a finite subgroup scheme, then the quotient E/G is another elliptic curve, and the quotient morphism E → E/G is an isogeny. This isogeny relates the two elliptic curves, providing a powerful tool for studying their relationship.
• Kernel of Isogeny: If G is the n-torsion subgroup E[n], the quotient E/E[n] is isogenous to E, and the kernel of the isogeny E → E/E[n] is precisely E[n]. This specific case is critical in understanding the modular properties of elliptic curves and their endomorphism rings.

Example: Quotient by the n-torsion Subgroup

A particularly important example is the quotient of an elliptic curve E by its n-torsion subgroup E[n]. The n-torsion subgroup consists of all points P on E such that nP = 0, where 0 is the identity element on the elliptic curve. The quotient E/E[n] is isogenous to E, meaning there is an isogeny (a surjective homomorphism with finite kernel) between them. This isogeny has degree n², reflecting the size of the n-torsion subgroup.

Understanding this quotient is crucial for studying the arithmetic of elliptic curves, especially in the context of modular forms and the modularity theorem. The isogeny E → E/E[n] provides a way to relate different elliptic curves and study their common properties. The construction of this quotient involves significant algebraic machinery, but the resulting insights are invaluable.

Quotients as Isomorphism Classes: Isogeny and Moduli Spaces

Another way to understand quotients is in terms of isomorphism classes of elliptic curves. This perspective is closely related to the concept of isogeny and the construction of moduli spaces. In this context, the quotient of an elliptic curve E by a subgroup G is seen as representing a class of elliptic curves that are isogenous to E via an isogeny with kernel G.

Isogenies and Their Role

An isogeny is a non-constant morphism between elliptic curves that is also a group homomorphism. If φ: E → E' is an isogeny, then E' is said to be isogenous to E. Isogenies preserve the group structure of the elliptic curves and provide a way to relate different elliptic curves. The degree of an isogeny is the size of its kernel, which is a finite subgroup scheme.

Quotient via Isogeny

If G is a finite subgroup scheme of an elliptic curve E, there exists an isogeny φ: E → E' such that the kernel of φ is G. The elliptic curve E' is then considered a quotient of E by G in the sense of isomorphism classes. This perspective shifts the focus from a specific geometric object (the scheme-theoretic quotient) to a class of curves that are equivalent under isogeny.

Connection to Moduli Spaces

This viewpoint is particularly relevant when considering moduli spaces of elliptic curves. A moduli space is a geometric object that parameterizes isomorphism classes of certain objects (in this case, elliptic curves). The moduli space of elliptic curves, often denoted M, is a fundamental object in algebraic geometry and number theory. Points on M correspond to isomorphism classes of elliptic curves.

When we consider quotients in terms of isomorphism classes, we are essentially looking at how the isogeny E → E/G maps the point corresponding to E in the moduli space M to a different point corresponding to E/G. This perspective allows us to study the geometry of the moduli space and the relationships between different elliptic curves.

Example: Quotient by n-torsion and Moduli

Consider again the quotient of E by its n-torsion subgroup E[n]. The isogeny E → E/E[n] maps the point in the moduli space corresponding to E to a new point corresponding to an elliptic curve isogenous to E. This mapping provides insights into the structure of the moduli space and the modular properties of elliptic curves.

In this context, the question of whether E is canonically isomorphic to E/E[n] can be reframed in terms of the moduli space. If there is a canonical isomorphism, it would imply a certain symmetry or involution in the moduli space. However, in general, there is no such canonical isomorphism, and the isogeny E → E/E[n] represents a non-trivial transformation in the moduli space.

Distinguishing Between the Two Quotients

The key difference between scheme-theoretic quotients and quotients as isomorphism classes lies in their focus:

• Scheme-Theoretic Quotients: Provide a concrete geometric object obtained by dividing the elliptic curve by the subgroup scheme. This construction is rigorous and involves the machinery of quotient schemes in algebraic geometry. It focuses on the explicit construction of a new scheme that represents the quotient.
• Quotients as Isomorphism Classes: Represent a class of elliptic curves that are isogenous to the original curve via an isogeny. This perspective is closely tied to the theory of isogenies and the construction of moduli spaces. It emphasizes the relationships between different elliptic curves and their modular properties.

To summarize the distinction, consider the following table:

Feature	Scheme-Theoretic Quotient	Quotient as Isomorphism Class
Focus	Concrete geometric object (scheme)	Class of isogenous elliptic curves
Construction	Quotient schemes in algebraic geometry	Isogenies and their kernels
Relevance	Explicit geometric structures	Moduli spaces, modularity, relationships between elliptic curves
Key Concept	Quotient morphism E → E/G	Isogeny φ: E → E', kernel of φ
Example	Quotient by n-torsion subgroup E/E[n]	Elliptic curves isogenous to E via an isogeny with kernel E[n]

Practical Implications

Understanding these distinctions is crucial for correctly interpreting results and applying them in various contexts. For instance, when working with modular forms and the modularity theorem, the perspective of quotients as isomorphism classes is more relevant. In contrast, when studying the explicit geometry of elliptic curves and their subgroups, the scheme-theoretic quotient provides the necessary framework.

Addressing the Canonical Isomorphism Question

Now, let's revisit the question: "Is an elliptic curve canonically isomorphic to its quotient by its own n-torsion?" The answer depends on the type of quotient being considered.

Scheme-Theoretic Perspective

From a scheme-theoretic perspective, the quotient E/E[n] is isogenous to E, but there is generally no canonical isomorphism between them. The isogeny E → E/E[n] has degree n², and its kernel is E[n]. While E/E[n] is an elliptic curve, it is not canonically the same as E. The construction of the quotient involves choices, and there is no natural way to identify E/E[n] with E without introducing additional data.

Isomorphism Class Perspective

From the perspective of isomorphism classes, E/E[n] represents a class of elliptic curves isogenous to E. Again, there is no canonical isomorphism between E and E/E[n] in this context. The isogeny E → E/E[n] defines a map in the moduli space, but this map does not generally have a canonical inverse that would correspond to an isomorphism.

Counterexamples and Further Considerations

To illustrate this point, consider specific examples of elliptic curves and their n-torsion subgroups. For instance, consider the case where n = 2. The quotient E/E[2] is isogenous to E via an isogeny of degree 4. The elliptic curve E/E[2] has a different j-invariant than E unless E has complex multiplication, indicating that they are not generally isomorphic.

Furthermore, the endomorphism ring of E and E/E[n] may differ, providing another way to distinguish them. The endomorphism ring of an elliptic curve is the ring of homomorphisms from the curve to itself. The structure of the endomorphism ring provides deep insights into the arithmetic and geometric properties of the elliptic curve.

Conclusion: Navigating the Landscape of Quotients

In conclusion, understanding the quotients of elliptic curves by subgroup schemes requires careful consideration of the type of quotient being discussed. The scheme-theoretic quotient provides a concrete geometric construction, while the quotient as an isomorphism class emphasizes the relationships between isogenous curves. The question of canonical isomorphism depends on the perspective and the specific properties of the elliptic curve and its subgroup.

By distinguishing between these two types of quotients, we gain a more nuanced understanding of the structure and properties of elliptic curves. This understanding is essential for advanced studies in algebraic geometry, number theory, and related fields. The exploration of these concepts not only clarifies the nature of quotients but also enriches our appreciation of the intricate connections within the world of elliptic curves.

This exploration has highlighted the importance of carefully specifying the context when discussing quotients in algebraic geometry. The distinction between scheme-theoretic quotients and quotients as isomorphism classes is not merely a technicality; it reflects fundamentally different ways of understanding the relationship between an elliptic curve and its subgroups. By grasping these nuances, we can navigate the complex landscape of elliptic curves with greater clarity and precision.