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Chapter 4: Problem 9

We say two elliptic curves \(X, X^{\prime}\) are isogenous if there is a finite morphism \(f: X \rightarrow X^{\prime}\) (a) Show that isogeny is an equivalence relation. (b) For any elliptic curve \(X\), show that the set of elliptic curves \(X^{\prime}\) isogenous to \(X\) up to isomorphism, is countable. [Hint: \(X^{\prime}\) is uniquely determined by \(X\) and \(\operatorname{ker} f .]\)

Short Answer

Expert verified
Part (a): Isogeny is reflexive because the identity morphism is always an isogeny, it is symmetric because there exists a dual isogeny, and it is transitive because the composition of two isogenies is another isogeny. Therefore, isogeny is an equivalence relation. Part (b): The set of all elliptic curves \(X'\) isogenous to a given curve \(X\), up to isomorphism, is countable because each isogeny defines a unique curve \(X'\) which in turn is characterized by its kernel, a finite subgroup of \(X\), and the set of all finite subgroups of \(X\) itself is countable.

Step by step solution

01

Define Terms

Firstly, it is important to define the terms involved. An elliptic curve refers to a type of cubic curve. An isogeny is a non-constant morphism between two elliptic curves that respects the group structure, i.e., it is a finite morphism. The morphism is said to have a kernel, elements of the domain that are mapped to the neutral element of the codomain.
02

Proving Reflexivity

The identity function \(f: X \rightarrow X\) defined by \(f(P)=P\) for all \(P \in X\) is an isogeny, which shows the reflexivity of isogeny since every elliptic curve is isogenous to itself.
03

Proving Symmetry

If there is an isogeny \(f: X \rightarrow X'\), then there exists a dual isogeny \( \hat{f}: X' \rightarrow X\). Therefore, if \(X\) is isogenous to \(X'\), then \(X'\) is isogenous to \(X\), showing the symmetry of isogeny.
04

Proving Transitivity

If we have two isogenies \(f: X \rightarrow X'\) and \(g: X' \rightarrow X''\) then the composition of \(f\) and \(g\) is also an isogeny \(g \circ f: X \rightarrow X''\). That proves the transitivity of isogeny.
05

Countability of Isogenous Curves

Given an elliptic curve \(X\), and the set of all elliptic curves isogenous to \(X\). Each isogeny \(f: X \rightarrow X'\) defines a unique curve \(X'\) as well as being defined by its kernel, which is a finite subgroup of \(X\). The number of finite subgroups of \(X\) is countable, hence the set of all elliptic curves \(X'\) isogenous to \(X\) (up to isomorphism) is countable.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isogeny
Isogeny is a central concept in the study of elliptic curves. It refers to a special type of function that maps one elliptic curve to another while preserving the complex group structure inherent in these curves.
This function, or morphism, must be non-constant and continuous over finite fields which makes it quite significant in algebraic geometry. The existence of an isogeny between two elliptic curves implies a deep, intrinsic connection between them.
  • Provides a morphism compatible with the group law.
  • Isogenies are finite morphisms, meaning they interact with finite sets in algebraic patterns.
  • They allow us to compare and contrast different elliptic curves, understanding them as part of a broader, interconnected landscape.
Finite Morphism
A finite morphism, like isogeny, is a function that maps elements between algebraic structures in a controlled manner. In the context of elliptic curves, it means that an elliptic curve can be transformed into another in a predictable and limited way.
Finite morphisms hold importance because they help preserve and utilize the properties of elliptic curves when exploring their multitude of interrelations.
  • Preserves finite nature, creating an understandable relationship between distinct curves.
  • Ensures that each part of an elliptic structure contributes to the mapping.
  • Carries a crucial role in understanding cryptographic applications due to its well-defined nature.
Equivalence Relation
Understanding isogeny as an equivalence relation allows us to see the symmetry and reflexivity of elliptic curves on an equal footing. An equivalence relation like this must encompass reflexivity, symmetry, and transitivity among its properties.
  • Reflexivity: Every elliptic curve is isogenous to itself.
  • Symmetry: If one elliptic curve is isogenous to another, the reverse is equally true.
  • Transitivity: If one curve is linked to a second, and the second to a third, then the first curve is linked directly to the third.
This triangular relationship illustrates the vast, interconnected network of elliptic curves through the process of isogenies.
Countability
Countability in mathematics refers to the idea that a set can be enumerated or matched with the natural numbers. When dealing with elliptic curves, countability comes into play as we consider the various curves isogenous to a given curve.
Each unique isogeny creates a corresponding unique elliptic curve, building upon a potentially infinite set.
  • The kernel of each isogeny is a finite subgroup and contributes to the countable set of isogenous curves.
  • By understanding these subgroups, mathematicians can, in theory, list all potential outcomes.
  • This concept underlines the systematic nature of relationships between elliptic curves.
ker(f)
The kernel of a morphism, noted as ker(f), plays a crucial role in understanding the nature of isogenies between elliptic curves. It is composed of all elements within the domain elliptic curve that map to the neutral element on the codomain.
This kernel is not just a technical detail, but a key element that dictates the properties and potential of the isogeny.
  • Defines specific aspects of the morphism's structure.
  • Its finiteness ensures a well-defined number of possible isogenous curves.
  • Knowing the kernel helps in categorizing elliptic curves in a countable manner, driving deeper geometrical insights.

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Most popular questions from this chapter

If \(X\) is a curve of genus \(\geqslant 2\) which is a complete intersection (II, Ex. 8.4 ) in some \(\mathbf{P}^{n},\) show that the canonical divisor \(K\) is very ample. Conclude that a curve of genus 2 can never be a complete intersection in any \(\mathbf{P}^{n}\). Cf. (Ex. 5.1 ).

Let \(X\) be a plane curve of degree 4 (a) Show that the effective canonical divisors on \(X\) are exactly the divisors \(X . L\) where \(L\) is a line in \(\mathbf{P}^{2}\) (b) If \(D\) is any effective divisor of degree 2 on \(X\), show that \(\operatorname{dim}|D|=0\) (c) Conclude that \(X\) is not hyperelliptic (Ex. 1.7 ).

Let \(X, P_{0}\) be an elliptic curve having an endomorphism \(f: X \rightarrow X\) of degree 2 (a) If we represent \(X\) as a \(2-1\) covering of \(\mathbf{P}^{1}\) by a morphism \(\pi: X \rightarrow \mathbf{P}^{1}\) ramified at \(P_{0},\) then as in \((4.4),\) show that there is another morphism \(\pi^{\prime}: X \rightarrow \mathbf{P}^{1}\) and a morphism \(g: \mathbf{P}^{1} \rightarrow \mathbf{P}^{1},\) also of degree \(2,\) such that \(\pi \quad f=g \quad \pi^{\prime}\) (b) For suitable choices of coordinates in the two copies of \(\mathbf{P}^{1}\), show that \(y\) can be taken to be the morphism \(x \rightarrow x^{2}\) (c) Now show that \(g\) is branched over two of the branch points of \(\pi\), and that \(g^{-1}\) of the other two branch points of \(\pi\) consists of the four branch points of \(\pi\). Deduce a relation involving the invariant , of \(X\). (d) Solving the above, show that there are just three values of \(j\) corresponding to elliptic curves with an endomorphism of degree \(2,\) and find the corresponding values of \(\lambda\) and \(j.\)

We say a (singular) integral curve in \(\mathbf{P}^{n}\) is strange if there is a point which lies on all the tangent lines at nonsingular points of the curve. (a) There are many singular strange curves, e.g., the curve given parametrically by \(x=t, y=t^{p}, z=t^{2 p}\) over a field of characteristic \(p>0\). (b) Show, however, that if char \(k=0,\) there aren't even any singular strange curves besides \(\mathbf{P}^{1}\).

For any curve \(X,\) the algebraic fundamental group \(\pi_{1}(X)\) is defined as \(\lim \operatorname{Gal}\left(K^{\prime} / K\right),\) where \(K\) is the function field of \(X,\) and \(K^{\prime}\) runs over all Galois extensions of \(K\) such that the corresponding curve \(X^{\prime}\) is 茅tale over \(X(\mathrm{III}, \mathrm{Ex} .10 .3)\) Thus, for example, \(\pi_{1}\left(\mathbf{P}^{1}\right)=1(2.5 .3) .\) Show that for an elliptic curve \(X\) \(\pi_{1}(X)=\prod_{\text {prime }} \mathbf{Z}_{l} \times \mathbf{Z}_{l} \quad \text { if char } k=0\) \(\pi_{1}(X)=\prod_{l \neq p} \mathbf{Z}_{l} \times \mathbf{Z}_{l}\) if char \(k=p\) and Hasse \(X=0\) \(\pi_{1}(X)=\mathbf{Z}_{p} \times \prod_{l \neq p} \mathbf{Z}_{l} \times \mathbf{Z}_{l} \quad\) if char \(k=p\) and Hasse \(X \neq 0\) where \(\mathbf{Z}_{l}=\lim \mathbf{Z} / l^{n}\) is the \(l\) -adic integers. [Hints: Any Galois 茅tale cover \(X^{\text {' }}\) of an elliptic curve is again an elliptic curve If the degree of \(X\) ' over \(X\) is relatively prime to \(p\), then \(X\) ' can be dominated by the cover \(n_{X}: X \rightarrow X\) for some integer \(n\) with \((n, p)=1 .\) The Galois group of the covering \(n_{X}\) is \(\mathbf{Z}\) in \(\times \mathbf{Z}\),n. Etale covers of degree divisible by \(p\) can occur only if the Hasse invariant of \(X\) is not zero. Note: More generally, Grothendieck has shown [SGA 1, X, 2.6, p. 272] that the algebraic fundamental group of any curve of genus \(g\) is isomorphic to a quotient of the completion, with respect to subgroups of finite index, of the ordinary topological fundamental group of a compact Riemann surface of genus \(g,\) i.e., a group with \(2 g\) generators \(a_{1}, \ldots, a_{q}, b_{1}, \ldots, b_{q}\) and the relation \(\left(a_{1} b_{1} a_{1}^{-1} b_{1}^{-1}\right) \cdots\) \(\left(a_{q} b_{q} a_{q}^{-1} b_{q}^{-1}\right)=1.\)
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