isogeny

The two curves are called isogenous if there is an isogeny between them.

For general abelian varieties, still over the complex numbers, A is in the same isogeny class as its dual.

This comes down to asking that 'J' is a product of elliptic curves, up to an isogeny.

Therefore, isogeny is an equivalence relation on the category of tori.

Criteria to be a product of two elliptic curves (up to isogeny) were a popular study in the nineteenth century.

This is an equivalence relation, symmetry being due to the existence of the dual isogeny.

An isogeny is an isomorphism if and only if its degree is one.

In mathematics, isogeny theorem may refer to:

An isogeny is a finite-to-one morphism.

An isogeny is a surjective morphism of tori whose kernel is a finite flat group scheme.

An isogeny from an abelian variety A to another one B is a surjective morphism with finite kernel.

Dual isogeny (elliptic curve case)

As above, every isogeny induces homomorphisms of the groups of the k-valued points of the elliptic curves.

Every isogeny is an algebraic homomorphism and thus induces homomorphisms of the groups of the elliptic curves for k-valued points.

In mathematics, Raynaud's isogeny theorem, proved by , relates the Faltings heights of two isogeneous elliptic curves.

In arithmetic geometry, the Selmer group, named in honor of the work of by , is a group constructed from an isogeny of abelian varieties.

In mathematics, an isogeny is a morphism of algebraic groups between two abelian varieties (e.g. elliptic curves) that is surjective and has a finite kernel.

Such an isogeny f then provides a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined.

In characteristic p, an isogeny of degree p of abelian varieties must, for their function fields, give either an Artin-Schreier extension or a purely inseparable extension.

The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible Abelian varieties, corresponding to Hecke eigenforms of weight 2.

In mathematics, Tate's isogeny theorem, proved by , states that two abelian varieties over a finite field are isogeneous if and only if their Tate modules are isomorphic (as Galois representations).

The degree of the isogeny is defined to be the order of the kernel, i.e., the rank of its structure sheaf as a locally free -module, and it is a locally constant function on the base.

Given an isogeny f of degree n, one can prove using linear algebra on weights and faithfully flat descent that there exists a dual isogeny g such that gf is the nth power map on the source torus.

It states that the isogeny classes of simple abelian varieties over a finite field of order q correspond to algebraic integers all of whose conjugates (given by eigenvalues of the Frobenius endomorphism on the first cohomology group or Tate module) have absolute value q.

Let E and D be elliptic curves over a field k. An isogeny between E and D is a finite morphism f : E D of varieties that preserves basepoints (in other words, maps the given point on E to that on D).