2.5 Elliptic curve isogenies
7 min read•august 21, 2024
Elliptic curve isogenies are crucial in arithmetic geometry, connecting different curves while preserving their group structure. They help us understand relationships between curves and their arithmetic properties, playing a key role in various mathematical and cryptographic applications.
Isogenies come in different types, each with unique properties. From separable to cyclic isogenies, these classifications help us analyze curve relationships. Fundamental theorems about isogenies form the backbone of elliptic curve theory, linking them to various aspects of curves and their arithmetic.
Definition of elliptic curve isogenies
- Elliptic curve isogenies form a fundamental concept in arithmetic geometry connecting different elliptic curves
- Isogenies preserve the group structure of elliptic curves while mapping between them, crucial for understanding relationships between curves
Morphisms between elliptic curves
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- Isogenies defined as surjective morphisms between elliptic curves preserving the identity element
- Morphisms represented by rational functions satisfying specific algebraic conditions
- Include both constant and non-constant maps between curves
- Non-constant isogenies preserve the group structure and point at infinity
Kernel of isogenies
- Kernel consists of points on the source curve mapped to the identity element of the target curve
- Finite subgroup of the source curve determines the up to isomorphism
- Size of the kernel relates to the degree of the isogeny
- often form kernels of isogenies (points of finite order)
Degree of isogenies
- Degree defined as the size of the kernel for separable isogenies
- For inseparable isogenies, degree involves both separable and inseparable parts
- Relates to the extension degree of function fields of the curves
- Isogenies of degree 1 are isomorphisms between elliptic curves
Types of isogenies
- Isogenies classified based on their properties and structures play crucial roles in arithmetic geometry
- Understanding different types of isogenies helps in analyzing relationships between elliptic curves and their applications
Separable vs inseparable isogenies
- Separable isogenies have kernels with order equal to their degree
- Inseparable isogenies occur in positive characteristic and involve Frobenius morphisms
- Separable isogenies preserve torsion structures more directly
- Inseparable isogenies relate to p-torsion points in characteristic p
Cyclic isogenies
- Cyclic isogenies have kernels generated by a single point
- Often arise from division polynomials and torsion points
- Play a key role in isogeny-based cryptography
- Degree of a cyclic isogeny equals the order of the generating point
Endomorphisms and automorphisms
- Endomorphisms are isogenies from a curve to itself
- Automorphisms are invertible endomorphisms (isomorphisms)
- Multiplication-by-n maps form important endomorphisms
- Complex multiplication provides additional endomorphisms for some curves
Isogeny theorems
- Fundamental theorems about isogenies form the backbone of elliptic curve theory in arithmetic geometry
- These theorems connect isogenies to various aspects of elliptic curves, from their structure to their arithmetic properties
Tate's isogeny theorem
- States that two abelian varieties over a finite field are isogenous if and only if their zeta functions are equal
- Provides a powerful tool for classifying elliptic curves over finite fields
- Extends to higher-dimensional abelian varieties
- Crucial in understanding the arithmetic of elliptic curves over finite fields
Poincaré's reducibility theorem
- Asserts that any is isogenous to a product of simple abelian varieties
- Applies to elliptic curves as one-dimensional abelian varieties
- Helps in decomposing complex abelian varieties into simpler components
- Fundamental in the study of higher-dimensional abelian varieties
Isogeny-torsion theorem
- Relates the to the structure of torsion points
- States that for an isogeny of degree n, the kernel is contained in the n-torsion points
- Provides insights into the relationship between isogenies and torsion subgroups
- Useful in analyzing the structure of isogeny classes
Computational aspects
- Computational methods for working with isogenies are essential in both theoretical and applied arithmetic geometry
- These techniques enable practical applications of isogeny theory in cryptography and other fields
Vélu's formulas
- Provide explicit equations for computing isogenies given their kernels
- Allow efficient calculation of the image curve and the isogeny map
- Crucial for implementing isogeny-based cryptographic protocols
- Extend to higher-degree isogenies through composition of lower-degree ones
Isogeny graphs
- Represent relationships between isogenous elliptic curves
- Vertices correspond to j-invariants of curves, edges to isogenies
- Supersingular isogeny graphs have special properties (expander graphs)
- Used in cryptographic protocols and for studying isogeny classes
Cryptographic applications
- Isogeny-based cryptography offers post-quantum security
- Supersingular Isogeny Diffie-Hellman () key exchange protocol
- Isogeny-based digital signatures and encryption schemes
- Rely on the hardness of computing isogenies between arbitrary elliptic curves
Dual isogenies
- Dual isogenies provide a way to reverse the direction of isogenies, crucial for understanding isogeny structures
- They play a fundamental role in the theory of abelian varieties and elliptic curves in arithmetic geometry
Definition and properties
- For every isogeny φ, there exists a unique dual isogeny φ̂
- Composition of an isogeny with its dual yields a multiplication-by-n map
- Degree of the dual isogeny equals the degree of the original isogeny
- Duals of separable isogenies are always separable
Composition of isogenies
- Composition of isogenies yields another isogeny
- Degree of the composition equals the product of degrees
- Dual of a composition is the composition of duals in reverse order
- Allows construction of higher-degree isogenies from lower-degree ones
Frobenius and Verschiebung
- Frobenius morphism is an inseparable isogeny in positive characteristic
- Verschiebung is the dual of the Frobenius morphism
- Play crucial roles in the theory of elliptic curves over finite fields
- Frobenius and Verschiebung compose to give multiplication-by-p maps (p = characteristic)
Isogeny classes
- Isogeny classes group together elliptic curves with similar arithmetic properties
- Understanding isogeny classes is crucial for classifying elliptic curves and studying their behavior
Isogeny classes of elliptic curves
- Consist of all elliptic curves isogenous to a given curve
- Characterized by having the same number of points over finite fields
- Relate to the Tate module and ℓ-adic representations
- Important in the study of elliptic curves over global fields
Tate-Shafarevich group
- Measures the failure of the Hasse principle for elliptic curves
- Related to isogeny classes through its finiteness conjecture
- Connects to the Birch and Swinnerton-Dyer conjecture
- Studied through isogeny descents and Selmer groups
Isogeny volcanoes
- Graph structures representing isogeny relationships within isogeny classes
- Levels correspond to the endomorphism rings of curves
- Surface consists of curves with maximal endomorphism rings
- Used in efficient algorithms for point counting and other computations
L-series and isogenies
- L-series provide deep connections between arithmetic properties of elliptic curves and analytic functions
- Isogenies play a crucial role in understanding these connections in arithmetic geometry
Relationship to L-functions
- Isogenous elliptic curves have the same L-function up to finite factors
- L-functions encode arithmetic information about elliptic curves
- Isogenies preserve important invariants related to L-functions (conductor, root number)
- Study of L-functions often involves analyzing isogeny classes
Birch and Swinnerton-Dyer conjecture
- Relates the rank of an elliptic curve to the behavior of its L-function at s=1
- Isogeny invariance of the BSD conjecture (up to rational factors)
- Connects algebraic, geometric, and analytic aspects of elliptic curves
- Partial results known for certain isogeny classes
Modularity theorem
- States that every elliptic curve over Q is modular (isogenous to a factor of J0(N))
- Proved for semistable elliptic curves by Wiles, leading to Fermat's Last Theorem
- Extended to all elliptic curves over Q by Breuil, Conrad, Diamond, and Taylor
- Relates elliptic curves to modular forms through isogenies
Isogenies in number theory
- Isogenies play a central role in connecting elliptic curves to various aspects of number theory
- They provide powerful tools for studying Galois representations and developing cryptographic systems
Galois representations
- Isogenies induce maps between Tate modules, preserving Galois action
- ℓ-adic representations of elliptic curves studied through isogeny classes
- Isogeny characters arise from studying Galois action on isogeny kernels
- Important in the study of the inverse Galois problem for elliptic curves
Serre's open image theorem
- States that for most elliptic curves, the image of the Galois representation is open in GL2(Zℓ)
- Isogeny classes play a role in exceptional cases (CM curves, Q-curves)
- Connects to the study of isogeny graphs and endomorphism rings
- Has implications for the distribution of Frobenius elements
Isogeny-based cryptography
- Utilizes the difficulty of finding isogenies between arbitrary elliptic curves
- Supersingular Isogeny Key Encapsulation (SIKE) as a post-quantum candidate
- Isogeny graphs used in constructing cryptographic hash functions
- Relies on computational hardness assumptions related to isogeny finding
Complex multiplication and isogenies
- Complex multiplication (CM) theory deeply intertwines with isogenies in the study of special elliptic curves
- CM provides a rich source of isogenies and connects elliptic curves to class field theory
CM theory and isogenies
- CM elliptic curves have extra endomorphisms, leading to more isogenies
- Isogenies between CM curves correspond to ideals in orders of imaginary quadratic fields
- CM curves have potentially large endomorphism rings, affecting their isogeny structures
- Isogeny classes of CM curves relate to ideal class groups of quadratic fields
Class field theory connections
- Isogenies of CM curves relate to class field theory of imaginary quadratic fields
- Hilbert class field realized as the field of j-invariants of isogenous CM curves
- Ring class fields correspond to isogeny classes with fixed endomorphism rings
- Provides concrete realizations of abelian extensions of quadratic fields
Hilbert class polynomials
- Minimal polynomials of j-invariants of CM curves with maximal orders
- Roots correspond to isomorphism classes within an isogeny class
- Used in explicit class field theory and the CM method for generating elliptic curves
- Degree relates to the class number of the imaginary quadratic field
Key Terms to Review (18)
Abelian variety: An abelian variety is a complete algebraic variety that has a group structure, meaning it allows for the addition of its points, and is defined over an algebraically closed field. This concept plays a crucial role in understanding the properties of elliptic curves, isogenies, and more complex structures like Jacobian varieties, connecting various areas of arithmetic geometry.
Barry Mazur: Barry Mazur is a prominent mathematician known for his influential work in number theory and arithmetic geometry, particularly in the study of elliptic curves and their isogenies. His contributions have significantly advanced the understanding of the connections between algebraic geometry, arithmetic, and the theory of L-functions, which are key components in the study of elliptic curves and their properties.
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.
Endomorphism Ring: The endomorphism ring is a mathematical structure that consists of all endomorphisms of an object, such as an elliptic curve, along with the operations of addition and composition. This ring captures the symmetries of the object and provides important insights into its structure, particularly in the context of group laws, isogenies, and complex multiplication, which can lead to a deeper understanding of the underlying algebraic geometry.
Horizontal Isogeny: A horizontal isogeny is a morphism between elliptic curves that preserves the structure of the curve while mapping points from one curve to another in a way that maintains their respective j-invariants. This type of isogeny reflects the relationship between curves that are defined over the same field, enabling a deeper understanding of their geometric properties and how they interact within the broader context of elliptic curves.
Isogenous Curves: Isogenous curves are elliptic curves that are linked through a special kind of morphism called an isogeny, which is a non-constant, algebraic map between them that preserves the group structure. This relationship implies that there is a way to relate the points on one curve to the points on another while maintaining their elliptic properties. Isogenies play a crucial role in understanding the structure of elliptic curves and their associated abelian varieties.
Isogeny: Isogeny is a morphism between two elliptic curves that preserves the group structure, meaning it is a surjective homomorphism with a finite kernel. This concept is crucial because it allows for the study of relationships between elliptic curves and their respective properties, such as their endomorphism rings and how they relate to modular forms and Jacobian varieties. Understanding isogenies helps bridge various areas in arithmetic geometry, linking them through their algebraic and geometric structures.
J-invariant: The j-invariant is a complex number that serves as a key invariant for classifying elliptic curves over the complex numbers. It plays an essential role in connecting the geometry of elliptic curves with the theory of complex tori and isogenies. The j-invariant captures the moduli space of elliptic curves, meaning that it helps to understand how different elliptic curves relate to each other through isogenies and complex multiplication.
John Tate: John Tate is a prominent mathematician known for his contributions to number theory and arithmetic geometry, particularly in the study of elliptic curves, isogenies, and the Tate module. His work laid foundational insights into the structure of algebraic varieties and local fields, significantly influencing modern developments in these areas.
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.
Montgomery Form: Montgomery form is a specific representation of an elliptic curve that simplifies certain computations, especially in the context of elliptic curve isogenies. This form allows for more efficient arithmetic operations and is particularly useful when dealing with isogenies, as it reduces the complexity of the calculations involved in mapping between different elliptic curves.
Morrison's Algorithm: Morrison's Algorithm is an efficient method used to compute isogenies between elliptic curves, particularly focusing on the case of isogenies of degree 2. It allows for the construction of isogenies by exploiting the properties of the kernel and facilitates computations in the context of elliptic curves over finite fields, making it an essential tool in arithmetic geometry and cryptography.
Sea Algorithm: The Sea Algorithm is a method used for computing isogenies between elliptic curves, which are fundamental objects in arithmetic geometry. It focuses on finding efficient ways to compute the action of isogenies, which are morphisms between elliptic curves that preserve their group structure. The algorithm employs a combination of mathematical structures and techniques, enabling effective calculations essential for applications in cryptography and number theory.
Separability: Separability refers to a property of algebraic structures, particularly in the context of fields and polynomials, where a polynomial is said to be separable if it does not have repeated roots. This concept is crucial when discussing the nature of extensions in algebraic number fields and the behavior of elliptic curves under isogenies, as it helps to characterize the relationships and morphisms between different algebraic structures.
SIDH: SIDH, or Supersingular Isogeny Diffie-Hellman, is a cryptographic protocol based on the mathematics of isogenies between supersingular elliptic curves. It enables secure key exchange over an insecure channel by using the properties of these isogenies, which are mappings between elliptic curves that preserve their structure. The security of SIDH relies on the difficulty of finding isogenies between supersingular elliptic curves, making it an interesting candidate for post-quantum cryptography.
Supersingular Isogeny Problem: The supersingular isogeny problem is a mathematical challenge that involves finding isogenies between supersingular elliptic curves, particularly in characteristic p. It plays a critical role in understanding the structure of these curves and has applications in cryptography, especially in constructing secure systems against certain types of attacks.
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.
Weierstrass Form: The Weierstrass form is a specific equation used to represent elliptic curves, typically given by the equation $$y^2 = x^3 + ax + b$$ where $a$ and $b$ are constants. This form is crucial in studying the properties of elliptic curves, including their group structure, isogenies, and rational points. It serves as a standard representation that simplifies the analysis of elliptic curves and their applications in number theory and algebraic geometry.