7.3 Isogenies of abelian varieties
8 min read•august 21, 2024
Isogenies of abelian varieties are crucial tools in arithmetic geometry. They connect different abelian varieties, providing insights into their structure and relationships. Understanding isogenies helps us study arithmetic properties of these mathematical objects over various fields.
Isogenies are defined as surjective homomorphisms between abelian varieties with finite kernels. They preserve group structure and algebraic properties, generalizing multiplication maps on elliptic curves. The kernel's structure and order determine key characteristics of the isogeny, including its degree and type.
Definition of isogenies
- Isogenies play a crucial role in arithmetic geometry connecting different abelian varieties
- Understanding isogenies provides insights into the structure and relationships between abelian varieties
- Isogenies form a fundamental tool for studying arithmetic properties of abelian varieties over various fields
Homomorphisms between abelian varieties
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- Isogenies defined as surjective homomorphisms between abelian varieties with finite kernels
- Preserve group structure and algebraic properties of abelian varieties
- Can be viewed as generalizations of multiplication maps on elliptic curves
- Formalized using the language of schemes and group schemes in modern algebraic geometry
Finite kernels of isogenies
- consists of a finite
- Order of the kernel determines the degree of the isogeny
- Kernels can be cyclic, product of cyclic groups, or more complex finite group schemes
- Structure of the kernel provides important information about the isogeny and related abelian varieties
Degree of an isogeny
- Defined as the order of the kernel or the degree of the field extension induced by the isogeny
- Multiplicative property holds for composition of isogenies
- Relates to the index of the image of the isogeny in the codomain abelian variety
- Determines the "distance" between abelian varieties in the isogeny graph
Types of isogenies
- Classification of isogenies provides a framework for understanding their properties and behavior
- Different types of isogenies arise from various algebraic and geometric considerations
- Understanding isogeny types crucial for applications in cryptography and number theory
Separable vs inseparable isogenies
- Separable isogenies induce separable field extensions
- Inseparable isogenies occur in positive characteristic and involve purely inseparable field extensions
- Frobenius isogeny serves as a fundamental example of an inseparable isogeny
- Separability affects the structure of the kernel and properties of the induced maps on tangent spaces
Simple vs complex isogenies
- Simple isogenies connect simple abelian varieties (those with no proper abelian subvarieties)
- Complex isogenies involve abelian varieties with non-trivial decompositions
- Simple isogenies play a key role in the study of endomorphism rings of abelian varieties
- Complex isogenies arise in the decomposition of abelian varieties into simple factors
Cyclic isogenies
- Defined by kernels that are cyclic as group schemes
- Often arise from multiplication-by-n maps on elliptic curves
- Play a crucial role in the theory of modular curves and moduli spaces
- Cyclic isogenies form the basis for many cryptographic protocols based on isogeny problems
Properties of isogenies
- Isogenies exhibit various algebraic and geometric properties that make them powerful tools in arithmetic geometry
- Understanding these properties allows for deeper analysis of abelian varieties and their relationships
- Properties of isogenies often reflect underlying structures of the abelian varieties involved
Composition and factorization
- Isogenies can be composed yielding new isogenies
- Degree of composed isogenies multiplies
- Factorization of isogenies relates to the structure of the kernel and intermediate abelian varieties
- Composition and factorization play key roles in studying isogeny graphs and volcanoes
Isogeny classes
- Abelian varieties connected by isogenies form isogeny classes
- Isogeny classes preserve many arithmetic invariants (L-functions, endomorphism algebras)
- Study of isogeny classes crucial for understanding moduli spaces of abelian varieties
- Isogeny classes relate to Tate's isogeny theorem and the Honda-Tate theory
Dual isogenies
- Every isogeny has a unique dual isogeny going in the opposite direction
- Composition of an isogeny with its dual yields a multiplication map
- Dual isogenies preserve various duality properties of abelian varieties
- Play a crucial role in the theory of polarizations on abelian varieties
Isogeny theorems
- Isogeny theorems form cornerstones of modern arithmetic geometry
- These theorems connect isogenies to various aspects of abelian varieties including Galois representations and moduli theory
- Understanding isogeny theorems provides deep insights into the arithmetic of abelian varieties
Tate's isogeny theorem
- States that abelian varieties over finite fields are isogenous if and only if they have the same zeta function
- Provides a powerful tool for classifying abelian varieties over finite fields
- Relates isogeny classes to Frobenius eigenvalues and Weil numbers
- Has important applications in the study of abelian varieties in positive characteristic
Faltings' isogeny theorem
- Proves that abelian varieties over number fields with isomorphic l-adic Tate modules for all l are isogenous
- Resolves the isogeny conjecture formulated by Tate
- Has profound implications for the study of Galois representations attached to abelian varieties
- Plays a crucial role in various aspects of the Langlands program
Serre-Tate theorem
- Establishes a correspondence between deformations of an abelian variety in characteristic p and deformations of its p-divisible group
- Provides a powerful tool for studying moduli spaces of abelian varieties in mixed characteristic
- Relates the local deformation theory of abelian varieties to formal group theory
- Has applications in the study of canonical lifts and CM theory
Applications of isogenies
- Isogenies find applications across various areas of mathematics and cryptography
- Understanding isogeny-based techniques crucial for modern research in arithmetic geometry
- Applications of isogenies often leverage their computational aspects and algebraic properties
Cryptography and isogenies
- Isogeny-based cryptography emerges as a post-quantum cryptographic candidate
- Supersingular isogeny Diffie-Hellman (SIDH) protocol uses walks in supersingular isogeny graphs
- Security of isogeny-based cryptosystems relies on the hardness of computing isogenies between given abelian varieties
- Isogeny-based signatures and zero-knowledge proofs actively researched
Moduli spaces of abelian varieties
- Isogenies play a crucial role in understanding the structure of moduli spaces
- Hecke correspondences on moduli spaces defined using isogenies
- Isogeny classes form important subsets of moduli spaces with arithmetic significance
- Study of isogeny classes relates to questions about special points on Shimura varieties
Isogeny volcanoes
- Graph structures formed by isogenies between elliptic curves or abelian varieties
- Volcano structure arises from the interplay between isogeny degrees and endomorphism rings
- Used in efficient algorithms for computing isogenies and classifying abelian varieties
- Provide insights into the distribution of isogenous abelian varieties over finite fields
Computational aspects
- Computational techniques for isogenies crucial for applications in cryptography and number theory
- Efficient algorithms for computing isogenies enable practical implementations of isogeny-based systems
- Computational complexity of isogeny problems underlies the security of various cryptographic protocols
Algorithms for computing isogenies
- Vélu's formulas provide efficient methods for computing isogenies between elliptic curves
- Generalized algorithms for higher-dimensional abelian varieties based on theta functions and algebraic theta functions
- Algorithms for computing isogenies in positive characteristic using p-adic methods
- Quantum algorithms for isogeny computations actively researched (Childs-Jao-Soukharev algorithm)
Complexity of isogeny computations
- Computing arbitrary isogenies between abelian varieties believed to be a hard problem
- Supersingular isogeny problem forms the basis for post-quantum cryptographic schemes
- Complexity varies depending on the dimension of abelian varieties and characteristics of the base field
- Subexponential algorithms known for certain special cases (ordinary elliptic curves over finite fields)
Isogeny graphs
- Graphs with vertices representing classes of abelian varieties and edges representing isogenies
- Structure of isogeny graphs depends on the endomorphism rings of the abelian varieties
- Regular structure of supersingular isogeny graphs exploited in cryptographic applications
- Isogeny graphs provide a visual and computational tool for studying relationships between abelian varieties
Isogenies in characteristic p
- Isogenies in positive characteristic exhibit unique features due to the presence of inseparable morphisms
- Understanding isogenies in characteristic p crucial for applications in finite field cryptography and coding theory
- Interplay between isogenies and p-adic theory provides powerful tools for studying abelian varieties
p-divisible groups
- Formal groups associated to abelian varieties that capture p-power torsion behavior
- Isogenies between abelian varieties induce isogenies between their p-divisible groups
- Dieudonné modules provide a linear algebraic description of p-divisible groups
- Serre-Tate theory relates deformations of abelian varieties to deformations of p-divisible groups
Frobenius and Verschiebung isogenies
- Frobenius isogeny arises from the in characteristic p
- Verschiebung isogeny serves as the dual of the Frobenius isogeny
- Interplay between Frobenius and Verschiebung crucial for understanding the structure of abelian varieties in positive characteristic
- Slopes of Frobenius and Verschiebung determine important invariants of abelian varieties (p-rank, a-number)
Supersingular abelian varieties
- Abelian varieties with maximal possible inseparable degree for the Frobenius isogeny
- Form an important class of abelian varieties with special arithmetic properties
- All supersingular abelian varieties in a given dimension are isogenous over an algebraically closed field
- Endomorphism rings of supersingular abelian varieties have a rich structure related to quaternion algebras
Isogenies and L-functions
- Isogenies preserve various arithmetic invariants including L-functions
- Understanding the relationship between isogenies and L-functions crucial for studying arithmetic properties of abelian varieties
- Isogeny invariance of L-functions plays a key role in formulating and studying important conjectures in arithmetic geometry
Isogeny invariance of L-functions
- L-functions of isogenous abelian varieties are identical
- Provides a powerful tool for studying arithmetic properties of abelian varieties within isogeny classes
- Relates to the Sato-Tate conjecture and equidistribution of Frobenius elements
- Isogeny invariance extends to various refinements of L-functions (p-adic L-functions, motivic L-functions)
Birch and Swinnerton-Dyer conjecture
- Relates arithmetic properties of abelian varieties to the behavior of their L-functions at s=1
- Isogeny invariance of L-functions implies BSD conjecture is isogeny invariant
- Study of isogeny classes crucial for understanding the arithmetic of elliptic curves and higher-dimensional abelian varieties
- Isogeny descent techniques used in attempts to prove cases of the BSD conjecture
Isogeny characters
- Characters associated to isogenies that appear in the functional equation of L-functions
- Relate to the arithmetic of the kernel of the isogeny
- Play a role in the study of twists of L-functions and their special values
- Isogeny characters provide a refined tool for studying arithmetic properties preserved by isogenies
Advanced topics
- Advanced aspects of isogeny theory connect to deep areas of arithmetic geometry and representation theory
- Understanding these topics crucial for current research directions in the field
- Advanced isogeny theory often involves sophisticated techniques from algebraic geometry, number theory, and representation theory
Isogeny decomposition of abelian varieties
- Decomposition of abelian varieties into simple factors up to isogeny
- Relates to the structure of the endomorphism algebra of the abelian variety
- Poincaré's complete reducibility theorem guarantees the existence of such decompositions
- Isogeny decompositions play a crucial role in the study of Shimura varieties and special points
Isogenies and Galois representations
- Galois representations attached to abelian varieties preserved up to isomorphism by isogenies
- Tate modules and their l-adic realizations provide a key tool for studying Galois representations
- Isogeny invariance of Galois representations underlies many deep results in arithmetic geometry
- Connects to the Serre-Tate theorem and the theory of motives
Isogeny estimates and bounds
- Estimates on the degrees of isogenies between abelian varieties with given properties
- Masser-Wüstholz isogeny theorem provides bounds on minimal isogeny degrees
- Isogeny estimates play a crucial role in effective versions of the Mordell conjecture
- Relate to questions about the distribution of on abelian varieties
Key Terms to Review (18)
Abelian isogeny: An abelian isogeny is a morphism between two abelian varieties that is surjective, has a finite kernel, and respects the group structure. This concept is significant in the study of the relationships between abelian varieties and provides insights into their structure, leading to deeper results in arithmetic geometry and the study of their endomorphisms.
David Mumford: David Mumford is a prominent mathematician known for his work in algebraic geometry, particularly in the areas of modular forms and algebraic curves. His contributions have significantly advanced the understanding of complex tori, modular curves, and other structures relevant to arithmetic geometry.
Degree of an isogeny: The degree of an isogeny is a numerical value that measures the 'size' or 'complexity' of the map between two elliptic curves or abelian varieties. It indicates the number of points in the fiber of a morphism, and thus reflects how many times one curve wraps around another. This concept is vital for understanding how these curves relate to each other, especially when considering properties like rational points and their behavior under morphisms.
Dual abelian variety: A dual abelian variety is a concept in algebraic geometry that refers to a specific construction associated with an abelian variety, essentially serving as its counterpart or 'dual'. It captures important properties of the original abelian variety, particularly relating to its morphisms and isogenies, and provides insights into the structure of its points and the relationships between them. The dual abelian variety allows for a deeper understanding of the isogenies of abelian varieties by linking them to their dual counterparts.
Finite isogeny: A finite isogeny is a morphism between two abelian varieties that is both finite and surjective, meaning it has finite kernel and induces an algebraic correspondence. This concept is crucial in the study of the structure and classification of abelian varieties, as it allows for the exploration of relationships between different varieties through their morphisms. Finite isogenies help establish connections in the context of the arithmetic of abelian varieties and can reveal important properties like their endomorphism rings.
Frobenius Endomorphism: The Frobenius endomorphism is a fundamental operation in algebraic geometry and number theory, particularly relating to the structure of varieties over finite fields. It maps a point in an algebraic variety to its 'p-th power,' where 'p' is the characteristic of the field, thereby providing insights into the properties of the variety and its points. This endomorphism plays a critical role in understanding group laws on elliptic curves, l-adic representations, and the relationships between different algebraic structures.
Géorgy Shafarevich: Géorgy Shafarevich was a prominent Soviet mathematician known for his significant contributions to number theory, algebraic geometry, and the theory of algebraic varieties. His work on isogenies of abelian varieties has had a lasting impact on the understanding of the structure of abelian varieties and their interrelations, especially in the context of arithmetic geometry.
Group scheme: A group scheme is a mathematical structure that generalizes the notion of groups to the category of schemes, allowing the concepts of algebraic groups to be studied in a more flexible setting. This concept facilitates the understanding of isogenies and Néron models by providing a framework to analyze group actions and morphisms within the realm of algebraic geometry. Group schemes can be viewed as schemes that also possess the structure of a group, equipped with both multiplication and inverse operations defined in a compatible way.
Isogenous Varieties: Isogenous varieties are algebraic varieties that are connected by a morphism with specific properties, meaning they share the same dimension and can be thought of as having similar geometric structures. In the context of abelian varieties, isogenies play a crucial role as they allow for the classification of these varieties and provide insights into their arithmetic properties. Understanding isogenous varieties helps in exploring deeper connections within the framework of algebraic geometry and number theory.
Isogeny Class: An isogeny class is a collection of abelian varieties that are all related by isogenies, which are morphisms that preserve the group structure and have finite degree. This concept highlights how these varieties share significant properties, such as having the same number of points over finite fields and being defined over the same field. Isogeny classes play a crucial role in understanding the structure and classification of abelian varieties, revealing deep connections between different geometric objects.
Isogeny Factors: Isogeny factors refer to the components that make up an isogeny, which is a morphism between two abelian varieties that preserves the group structure. These factors can be understood as a series of simpler isogenies whose composition results in the original isogeny, and they play a crucial role in the study of the structure and classification of abelian varieties. Understanding isogeny factors helps in decomposing complex isogenies into manageable parts, which can reveal important properties of the varieties involved.
Isomorphism: Isomorphism is a mathematical concept that describes a structural similarity between two objects, indicating that they can be transformed into each other through a reversible process. In various branches of mathematics, including algebra and geometry, isomorphisms highlight the idea that different representations or structures can be fundamentally the same, allowing for a deeper understanding of their properties and behaviors. This concept plays a crucial role in analyzing equivalence classes and morphisms between algebraic and geometric structures.
Kernel of an isogeny: The kernel of an isogeny is a specific set of points on an elliptic curve (or more generally, on an abelian variety) that maps to the identity element under the isogeny. This kernel is crucial for understanding the structure of the isogeny itself, as it reflects the symmetries and properties of the elliptic curve or abelian variety involved. It plays a significant role in determining the degree of the isogeny and reveals important information about the relationship between different curves or varieties.
Morphism: A morphism is a structure-preserving map between two mathematical objects, typically in the context of algebraic geometry or algebraic structures. It provides a way to relate different objects while maintaining their properties and relations, allowing for the study of their interconnections. In arithmetic geometry, morphisms are crucial for understanding how various algebraic varieties interact, especially when examining concepts like isogenies, Néron models, and Jacobian varieties.
Néron-Severi group: The Néron-Severi group is an important algebraic invariant associated with a variety, defined as the group of divisor classes modulo numerical equivalence. It captures significant information about the geometry of the variety, particularly in relation to the properties of polarizations, isogenies of abelian varieties, and cycle class maps.
Rational Points: Rational points are solutions to equations that can be expressed as fractions of integers, typically where the coordinates are in the form of rational numbers. These points are crucial in the study of algebraic varieties, especially in understanding the solutions over different fields, including the rational numbers, which can reveal deeper properties of the geometric objects involved.
Theorem of the Kernel: The theorem of the kernel states that for any homomorphism of abelian groups, the kernel of this homomorphism is a subgroup of the domain. In the context of isogenies of abelian varieties, this theorem provides important insights into the structure and properties of morphisms between abelian varieties, allowing us to understand their behavior and interrelations through the kernels involved.
Torsion Points: Torsion points are points on an algebraic group, such as an elliptic curve, that have finite order, meaning they generate a subgroup of the group that repeats after a certain number of additions. They play a crucial role in understanding the structure of elliptic curves, their isogenies, and the behavior of rational points on these curves. Torsion points also relate to the study of complex tori and can influence the properties of abelian varieties and Jacobian varieties.