Isogeny: Difference between revisions
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In [[algebraic geometry]], an '''isogeny''' between [[abelian variety|abelian varieties]] is a [[rational map]] which is also a [[group homomorphism]], with finite kernel. | In [[algebraic geometry]], an '''isogeny''' between [[abelian variety|abelian varieties]] is a [[rational map]] which is also a [[group homomorphism]], with finite kernel. | ||
==Elliptic curves== | ==Elliptic curves== | ||
As 1-dimensional abelian varieties, [[elliptic curve]]s provide a convenient introduction to the theory. If <math>\phi: E_1 \rightarrow E_2</math> is a non-trivial rational map which maps the zero of ''E''<sub>1</sub> to the zero of ''E''<sub>1</sub>, then it is necessarily a group homomorphism. The kernel of φ is a proper subvariety of ''E''<sub>1</sub> and hence a finite set of order ''d'', the ''degree'' of φ. There is a ''dual isogeny'' <math>\hat\phi: E_2 \rightarrow E_1</math> defined by | As 1-dimensional abelian varieties, [[elliptic curve]]s provide a convenient introduction to the theory. If <math>\phi: E_1 \rightarrow E_2</math> is a non-trivial rational map which maps the zero of ''E''<sub>1</sub> to the zero of ''E''<sub>1</sub>, then it is necessarily a group homomorphism. The kernel of φ is a proper subvariety of ''E''<sub>1</sub> and hence a finite set of order ''d'', the ''degree'' of φ. Conversely, every finite subgroup of ''E''<sub>1</sub> is the kernel of some isogeny. | ||
There is a ''dual isogeny'' <math>\hat\phi: E_2 \rightarrow E_1</math> defined by | |||
:<math>\hat\phi : Q \mapsto \sum_{P: \phi(P)=Q} P ,\,</math> | :<math>\hat\phi : Q \mapsto \sum_{P: \phi(P)=Q} P ,\,</math> | ||
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The curves ''E''<sub>1</sub> and ''E''<sub>2</sub> are said to be ''isogenous'': this is an [[equivalence relation]] on isomorphism classes of elliptic curves. | The curves ''E''<sub>1</sub> and ''E''<sub>2</sub> are said to be ''isogenous'': this is an [[equivalence relation]] on isomorphism classes of elliptic curves. | ||
=== | ===Examples=== | ||
Let ''E''<sub>1</sub> be an elliptic curve over a field ''K'' of [[characteristic of a field|characteristic]] not 2 or 3 in Weierstrass form. | Let ''E''<sub>1</sub> be an elliptic curve over a field ''K'' of [[characteristic of a field|characteristic]] not 2 or 3 in Weierstrass form. | ||
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is an isogeny from ''E''<sub>1</sub> to the isogenous curve ''E''<sub>2</sub> with equation <math>Y^2 + XY + 3mY = X^3 - 6mX - (m+9m^2)</math>. | is an isogeny from ''E''<sub>1</sub> to the isogenous curve ''E''<sub>2</sub> with equation <math>Y^2 + XY + 3mY = X^3 - 6mX - (m+9m^2)</math>. | ||
===Elliptic curves over the complex numbers=== | |||
An elliptic curve over the [[complex number]]s is isomorphic to a quotient of the complex numbers by some [[lattice (geometry)|lattice]]. If ''E''<sub>1</sub> = '''C'''/''L''<sub>1</sub>, and ''L''<sub>1</sub> is a sublattice of ''L''<sub>2</sub> of index ''d'', then ''E''<sub>2</sub> = '''C'''/''L''<sub>2</sub> is an isogenous curve. Representing the homothety class of a lattice by a point τ in the upper half-plane, the isogenous curves correspond to the lattices with [[moduli]] | |||
:<math> \frac{a\tau + b}{c} \,</math> | |||
with ''a''.''c'' = ''d'' and ''b''=0,1,...,''c''-1. | |||
===Elliptic curves over finite fields=== | ===Elliptic curves over finite fields=== | ||
Isogenous elliptic curves over a finite field have the same number of points (although not necessarily the same group structure). The converse is also true: this is the ''Honda-Tate theorem''. | Isogenous elliptic curves over a finite field have the same number of points (although not necessarily the same group structure). The converse is also true: this is the ''Honda-Tate theorem''. | ||
==References== | |||
* {{cite book | author=J.W.S. Cassels | authorlink=J. W. S. Cassels | title=Lectures on Elliptic Curves | series=LMS Student Texts | volume=24 | publisher=[[Cambridge University Press]] | year=1991 | isbn=0-521-42530-1 | pages=58-65 }} | |||
* {{cite book | author=Dale Husemöller | title=Elliptic curves | series=[[Graduate Texts in Mathematics]] | volume=111 | publisher=Springer-Verlag | year=1987 | isbn=0-387-96371-5 | pages=91-96,163 }} | |||
* {{cite book | author=Joseph H. Silverman | title=The Arithmetic of Elliptic Curves | series=[[Graduate Texts in Mathematics]] | volume=106 | publisher=Springer-Verlag | year=1986 | isbn=0-387-96203-4 | pages=70-79,84-90 }}[[Category:Suggestion Bot Tag]] |
Latest revision as of 07:01, 3 September 2024
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In algebraic geometry, an isogeny between abelian varieties is a rational map which is also a group homomorphism, with finite kernel.
Elliptic curves
As 1-dimensional abelian varieties, elliptic curves provide a convenient introduction to the theory. If is a non-trivial rational map which maps the zero of E1 to the zero of E1, then it is necessarily a group homomorphism. The kernel of φ is a proper subvariety of E1 and hence a finite set of order d, the degree of φ. Conversely, every finite subgroup of E1 is the kernel of some isogeny.
There is a dual isogeny defined by
the sum being taken on E1 of the d points on the fibre over Q. This is indeed an isogeny, and the composite is just multiplication by d.
The curves E1 and E2 are said to be isogenous: this is an equivalence relation on isomorphism classes of elliptic curves.
Examples
Let E1 be an elliptic curve over a field K of characteristic not 2 or 3 in Weierstrass form.
Degree 2
A subgroup of order 2 on E1 must be of the form where P = (e,0) with e being a root of the cubic in X. Translating so that e=0 and the curve has equation , the map
is an isogeny from E1 to the isogenous curve E2 with equation .
Degree 3
A subgroup of order 3 must be of the form where x is in K but y need not be. We shall assume that (by taking a quadratic twist if necessary). Translating, we can put E in the form . The map
is an isogeny from E1 to the isogenous curve E2 with equation .
Elliptic curves over the complex numbers
An elliptic curve over the complex numbers is isomorphic to a quotient of the complex numbers by some lattice. If E1 = C/L1, and L1 is a sublattice of L2 of index d, then E2 = C/L2 is an isogenous curve. Representing the homothety class of a lattice by a point τ in the upper half-plane, the isogenous curves correspond to the lattices with moduli
with a.c = d and b=0,1,...,c-1.
Elliptic curves over finite fields
Isogenous elliptic curves over a finite field have the same number of points (although not necessarily the same group structure). The converse is also true: this is the Honda-Tate theorem.
References
- J.W.S. Cassels (1991). Lectures on Elliptic Curves. Cambridge University Press, 58-65. ISBN 0-521-42530-1.
- Dale Husemöller (1987). Elliptic curves. Springer-Verlag, 91-96,163. ISBN 0-387-96371-5.
- Joseph H. Silverman (1986). The Arithmetic of Elliptic Curves. Springer-Verlag, 70-79,84-90. ISBN 0-387-96203-4.