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Richelot Isogenies

Let k be a field of characteristic different from 2. We consider a curve of genus 2, given by an equation C:y2=f(x) where f(x) is a square-free polynomial of degree 5 or 6. Let J be the Jacobian of C. In this section we mean by a Richelot isogeny a polarized isogeny Φ: J to A between principally polarized abelian surfaces, such that the kernel of Φover the algebraic closure has group structure Z/2Z x Z/2Z. We have that J[Φ]⊂J[2] is maximal isotropic with respect to the Weil-pairing on J[2].

We can represent the points of J[Φ] over the algebraic closure as divisors in the following way. We write f(x)=cQ1(x)Q2(x)Q3(x), where the Qi are degree 2 polynomials if deg(f)=6. If deg(f)=5 then Q2, Q3 are of degree 2 and Q1 is of degree 1 and is considered to represent a degree 2 with a root at x=∞. Then {0, [Q1(x)=0] - [Q2(x)=0], [Q1(x)=0] - [Q3(x)=0], [Q2(x)=0] - [Q3(x)=0] } is the kernel of some Richelot-isogeny and, conversely, any Richelot kernel can be represented in this way.

The Qi do not have to be defined over the ground field individually. One way of specifying such a kernel is to write f(x)=c( Norm)L[x]/k[x] Q(x) where L=k[t]/(h(t)) for some square free cubic polynomial h and Q(x)∈L[x]. If L is totally split and A is the Jacobian of a genus 2 curve then a description the genus 2 curve D such that A=( Jac)(D) is classically known. See [Smi05], Chapter 8 for an exposition that is relatively close to the description given here. See [BD09] for a description of D for general L.

In special cases, the codomain A can be a product of elliptic curves or the Weil-restriction of an elliptic curve with respect to a quadratic extension of k. In that case, the curve C has extra automorphisms that respects the representation f(x)=c( Norm)L[x]/k[x] Q(x). and one can find the relevant elliptic curves as quotients of C.

RichelotIsogenousSurfaces(J) : JacHyp -> List, List
RichelotIsogenousSurfaces(C) : CrvHyp -> List, List
    Kernels: BoolElt                    Default: true
abelian varieties and returns a list of objects representing the codomains. If the codomain is the Jacobian of a genus 2 curve, then that Jacobian is returned or, if a curve is given instead of a Jacobian, the corresponding curve.

If the codomain is a product of elliptic curves, a Cartesian product of elliptic curves is returned. If the codomain is the Weil restriction of an elliptic curve relative to a quadratic extension, then the elliptic curve over the quadratic extension is returned.

If Kernels is specified then a second list is returned, consisting of quadratic polynomials over cubic algebras. Each describes the kernel of the relevant isogeny.

RichelotIsogenousSurface(J, kernel) : JacHyp, RngUPolElt[RngUPolRes] -> .
RichelotIsogenousSurface(C, kernel) : CrvHyp, RngUPolElt[RngUPolRes] -> .
Given a genus 2 Jacobian and a Richelot kernel, return the codomain. The genus 2 curve must be given by a model of the form C:y2=f(x) and the kernel must be a quadratic polynomial Q(x) over a cubic algebra L such that ( Norm)L[x]/k[x] Q(x)= cf(x). The elements of the second list returned by RichelotIsogenousSurfaces when given Kernels:=true are valid kernel descriptions. The codomain is returned using the same conventions as for RichelotIsogenousSurfaces.

Example CrvHyp_richelot_isogeny (H125E14)

We will determine the Richelot isogenies on the Jacobian of y2=x5 + x. This Jacobian has the amusing property that there are 3 such isogenies and that each of the types of codomain (Jacobian, Weil restriction, product of elliptic curves) is represented.

> R<x>:=PolynomialRing(Rationals());
> C:=HyperellipticCurve(x^5+x);
> J:=Jacobian(C);
> RichelotIsogenousSurfaces(J);
[*
    Cartesian Product<Elliptic Curve defined by y^2 = x^3 + 5/32*x^2 -
    5/1024*x - 1/32768 over Rational Field, Elliptic Curve defined by 
    y^2 = x^3 - 5/32*x^2 - 5/1024*x + 1/32768 over Rational Field>,
    Jacobian of Hyperelliptic Curve defined by y^2 = -2*x^5 - 2*x over
    Rational Field,
    Elliptic Curve defined by y^2 = x^3 + 5/32*$.1*x^2 + 5/1024*x + 
        1/32768*$.1 over Number Field with defining polynomial x^2 + 1
    over the Rational Field
*]

We now illustrate how the kernels are represented.

> codomains,kernels:=RichelotIsogenousSurfaces(J:Kernels);
> Q:=kernels[1];
> LX<X>:=Parent(Q);
> L<alpha>:=BaseRing(LX);
> Q;
(-1/2*alpha^2 + 2*alpha)*X^2 + (-1/2*alpha^2 + alpha + 1)*X - 
    1/2*alpha^2 + 2*alpha
> L;
Univariate Quotient Polynomial Algebra in alpha over Rational Field
with modulus alpha^3 - 4*alpha^2 + 2*alpha
Let us check that the norm of Q gives us x5 + x again and that calling RichelotIsogenousSurface allows us to recreate the corresponding codomain.

> _,swp:=SwapExtension(LX);
> Norm(swp(Q));
x^5 + x
We can use Q to recreate the corresponding codomain.

> codomains[1] eq RichelotIsogenousSurface(J,Q);
true
Finally, to verify that the computed abelian surfaces are all isogenous, we verify that their L-series over Q are equal. For each type of return value we have to create the L-Series in a slightly different way, but once done, we can easily check that their coefficients agree.

> LC:=LSeries(C : ExcFactors:="Ogg");
> myL:=func< A|  
>     case<Type(A) | SetCart : LSeries(A[1])*LSeries(A[2]),
>                    JacHyp : LSeries(Curve(A) : ExcFactors:="Ogg"),
>                    CrvEll : LSeries(A),
>                    default : false>>;
> cfs:=[c: c in LGetCoefficients(LC,1000)];
> [[c: c in LGetCoefficients(myL(A),1000)] eq cfs : A in codomains];
[ true, true, true ]

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013