Richelot Isogenies#
Let \(k\) be a field of characteristic different from \(2\). We consider a curve of genus \(2\), given by an equation
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 \(\Phi: J \rightarrow A\) between principally polarized abelian surfaces, such that the kernel of \(\Phi\) over the algebraic closure has group structure \({\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}\). We have that \(J[\Phi]\subset J[2]\) is maximal isotropic with respect to the Weil-pairing on \(J[2]\).
We can represent the points of \(J[\Phi]\) over the algebraic closure as divisors in the following way. We write
where the \(Q_i\) are degree \(2\) polynomials if \(\deg(f)=6\). If \(\deg(f)=5\) then \(Q_2,Q_3\) are of degree \(2\) and \(Q_1\) is of degree \(1\) and is considered to represent a degree \(2\) with a root at \(x=\infty\). Then
is the kernel of some Richelot-isogeny and, conversely, any Richelot kernel can be represented in this way.
The \(Q_i\) do not have to be defined over the ground field individually. One way of specifying such a kernel is to write
where \(L=k[t]/(h(t))\) for some square free cubic polynomial \(h\) and \(Q(x)\in L[x]\). 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 \(A={\rm Jac}(D)\) is classically known. See [Smith, 2005], Chapter \(8\) for an exposition that is relatively close to the description given here. See [Bruin and Doerksen, 2009] 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 \(f(x)=c{\rm Norm}_{L[x]/k[x]} Q(x)\). and one can find the relevant elliptic curves as quotients of \(C\).
A double Richelot isogeny \(\varphi\) between two principally polarized abelian surfaces \(A\) and \(A'\) (over some base field \(k\)) is the composition of two Richelot isogenies \(A \to A'' \to A'\) that are individually not defined over \(k\) (but \(\varphi\) is). We exclude the trivial case that \(\varphi\) is multiplication by 2 (which occurs when the two Richelot isogenies are duals of each other). Then the kernel \(K\) of \(\varphi\) is isomorphic to \({\mathbb{Z}}/2{\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}}\times {\mathbb{Z}}/4{\mathbb{Z}}\) as an abelian group. Twice any of the points of order 4 in the kernel is a point \(P\) of order 2 on \(A\), which must be \(k\)-rational (it is also the non-trivial element in the annihilator of \(K \cap A[2]\) under the Weil pairing). There are three maximally isotropic subspaces of \(A[2]\) (with respect to the Weil pairing) that contain \(P\); each of them can be used as the kernel of the first Richelot isogeny. Since none of these Richelot isogenies is defined over \(k\), the absolute Galois group of \(k\) has to permute these three spaces transitively. We now assume \(\hbox{char}(k) \neq 2\) and that \(A = J\) is the Jacobian of a curve of the form \(y^2 = f(x,z)\). Then
with a quadratic \(q\) and a quartic \(h\) whose Galois group is not contained in \(D_4\) (which is equivalent to saying that the three factorizations of \(h\) into two quadratics are all Galois-conjugate. These factorizations correspond to the maximal isotropic subspaces containing the point \(P\) given by \(q\)).
We note that if \(A'\) is doubly Richelot isogenous to \(J\), then the intermediate abelian surface \(A''\) must also be a Jacobian. Since the three possible Richelot isogenies \(J \to A''\) are Galois-conjugate, the same is true for the three possible \(A''\), so they are either all Jacobians or all products of two elliptic curves (at least geometrically). Working over a suitable field extension, we can assume that \(h(x,z) = x z (x - z) (x - az)\). Recall that the condition for \(A''\) to be split is that the three quadratic factors corresponding to the kernel of \(J \to A''\) are linearly dependent. If all versions of \(A''\) are split, this means that there is some quadratic \(q(x,z) = r x^2 + s xz + t z^2\) such that the triples of vectors
\((r, s, t)\), \((0, 1, 0)\), \((1, -1-a, a)\)
\((r, s, t)\), \((1, -1, 0)\), \((0, 1, -a)\)
\((r, s, t)\), \((1, -a, 0)\), \((0, 1, -1)\)
are linearly dependent. Equivalently, the cross products of the last two vectors in each triple all lie in the plane orthogonal to \((r, s, t)\); in particular, these cross products are linearly dependent. But
and
So we obtain all surfaces doubly Richelot isogenous to \(J\) by taking a Richelot isogeny to another Jacobian over a cubic field and composing it with a further Richelot isogeny (over the same field) from that Jaocbian.
The DoubleRichelot.. intrinsics below compute a sequence containing all double Richelot isogenous Jacobians (resp., the corresponding curves) for a given genus 2 Jacobian over the rational numbers such that the intermediate Richelot isogenous abelian surface (which will be defined over a cubic number field) is a Jacobian.
Let \(\varphi \colon J \to J'\) be an isogeny between two Jacobians of genus 2 curves whose degree is a power of 2 (and such that the canonical principal polarizations are compatible via the isogeny). Then \(\varphi\) factors as a product of Richelot isogenies and double Richelot isogenies defined over the base field. So we can find all \(J'\) that are 2-power isogenous to a given \(J\) by constructing the connected component of the graph of Richelot and double Richelot isogenies that contains \(J\). The TwoPowerIsogenies intrinsic is provided for this purpose.
Caveat: right now, we cannot guarantee that we really find all isogenous Jacobians (say), since the graph traversal stops at points where we reach a non-Jacobian abelian surface. It might be possible that a 2-power isogenous Jacobian can only be reached via an isogeny to a non-Jacobian.
- RichelotIsogenousSurfaces(J): JacHyp -> List, List#
- RichelotIsogenousSurfaces(C): CrvHyp -> List, List#
Kernels: BoolElt Default: true
Computes the richelot isogenies defined over the basefield of the given 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
Kernelsis 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 \(C:y^2=f(x)\) and the kernel must be a quadratic polynomial \(Q(x)\) over a cubic algebra \(L\) such that \({\rm Norm}_{L[x]/k[x]} Q(x)= cf(x)\). The elements of the second list returned by
RichelotIsogenousSurfaceswhen givenKernels:=trueare valid kernel descriptions. The codomain is returned using the same conventions as forRichelotIsogenousSurfaces.
- Example: Richelot Isogeny (ex-194252)#
We will determine the Richelot isogenies on the Jacobian of \(y^2=x^5+x\). 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
Qgives us \(x^5+x\) again and that callingRichelotIsogenousSurfaceallows 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 \({\mathbb{Q}}\) 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 : LocalData:="Ogg"); > myL:=func< A| > case<Type(A) | SetCart : LSeries(A[1])*LSeries(A[2]), > JacHyp : LSeries(Curve(A) : LocalData:="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 ]
- DoubleRichelotIsogenies(J): JacHyp -> SeqEnum#
Given a genus 2 Jacobian \(J\), this finds Jacobians \(J'\) related to \(J\) by a “double Richelot isogeny”, i.e., an isogeny with kernel isomorphic to \({\mathbb{Z}}/2 \times {\mathbb{Z}}/2 \times {\mathbb{Z}}/4\), but which does not factor into two Richelot isogenies over \({\mathbb{Q}}\). A sequence containing these Jacobians \(J'\) is returned.
- DoubleRichelotIsogenies(C): CrvHyp -> SeqEnum#
This finds curves \(C'\) such that the Jacobians of (genus \(2\)) \(C\) and \(C'\) have a “double Richelot isogeny” between them, i.e., an isogeny with kernel isomorphic to \({\mathbb{Z}}/2 \times {\mathbb{Z}}/2 \times {\mathbb{Z}}/4\), but which does not factor into two Richelot isogenies over \({\mathbb{Q}}\). A sequence containing these curves \(C'\) is returned.
- TwoPowerIsogenies(J): JacHyp -> SeqEnum, SeqEnum, SeqEnum#
For a genus 2 Jacobian \(J\) over \({\mathbb{Q}}\), determine (hopefully) all principally polarized Jacobians and some other principally polyarized abelian surfaces over \({\mathbb{Q}}\) up to isomorphism that are isogenous to \(J\) (with compatible polarization) by an isogeny of degree a power of 2. This returns a sequence whose elements are isogenous Jacobians (other than \(J\)), a sequence whose elements are products of two elliptic curves over \({\mathbb{Q}}\) and a sequence whose elements are elliptic curves over quadratic fields (whose restriction of scalars down to \({\mathbb{Q}}\) are isogenous to \(J\)).
- Example: Richelot Double Isogeny (ex-bdb4ad)#
We consider the Jacobian of the curve \(y^2 + (x^3 + 1) y = x^2 + x\) (this is curve 249.a.249.1 in the LMFDB). There are no Richelot isogenous abelian surfaces, but there are some double Richelot isogenous Jacobians.
> R<x>:=PolynomialRing(Rationals()); > C:=HyperellipticCurve(x^2+x, x^3+1); > J:=Jacobian(C); > RichelotIsogenousSurfaces(J); [* *] > DoubleRichelotIsogenies(J); [ Jacobian of Hyperelliptic Curve defined by y^2 = -15*x^6 - 220*x^5 - 960*x^4 - 770*x^3 + 1780*x^2 - 780*x + 105 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = x^6 - 4*x^5 + 6*x^3 + 4*x^2 + 12*x + 9 over Rational Field ]
Now we look at the curve \(y^2 = (x-2)(x-1)x(x+1)(x+2)\) and its Jacobian. The Jacobian has full rational 2-torsion, so there are many Richelot isogenies. There are no double Richelot isogenies, but there are some additional 2-power isogenous abelian surfaces.
> R<x>:=PolynomialRing(Rationals()); > C:=HyperellipticCurve((x-2)*(x-1)*x*(x+1)*(x+2)); > J:=Jacobian(C); > RichelotIsogenousSurfaces(J); [* Jacobian of Hyperelliptic Curve defined by y^2 = -40*x^6 + 280*x^5 - 60*x^4 - 1520*x^3 - 120*x^2 + 1120*x - 320 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -10*x^6 + 70*x^5 - 60*x^4 - 170*x^3 - 210*x^2 - 80*x + 40 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -8*x^5 + 60*x^4 - 80*x^3 - 120*x^2 - 32*x over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -35*x^6 + 112*x^5 + 105*x^4 - 350*x^3 + 210*x^2 + 448*x - 280 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -15*x^6 + 40*x^5 - 15*x^4 + 10*x^3 + 30*x^2 + 160*x + 120 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -5*x^6 - 20*x^5 + 105*x^4 - 170*x^3 + 120*x^2 + 280*x + 80 over Rational Field, Elliptic Curve defined by y^2 = x^3 - 35/256*$.1*x^2 + 35/32768*x - 1/8388608*$.1 over Number Field with defining polynomial x^2 - 2 over the Rational Field, Elliptic Curve defined by y^2 = x^3 + 5/256*$.1*x^2 + 45/32768*x + 729/8388608*$.1 over Number Field with defining polynomial x^2 + 2 over the Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -18*x^5 - 90*x^3 - 72*x over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 35*x^6 + 112*x^5 - 105*x^4 - 350*x^3 - 210*x^2 + 448*x + 280 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 15*x^6 + 40*x^5 + 15*x^4 + 10*x^3 - 30*x^2 + 160*x - 120 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 5*x^6 - 20*x^5 - 105*x^4 - 170*x^3 - 120*x^2 + 280*x - 80 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 40*x^6 + 280*x^5 + 60*x^4 - 1520*x^3 + 120*x^2 + 1120*x + 320 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 10*x^6 + 70*x^5 + 60*x^4 - 170*x^3 + 210*x^2 - 80*x - 40 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -8*x^5 - 60*x^4 - 80*x^3 + 120*x^2 - 32*x over Rational Field *] > DoubleRichelotIsogenies(J); [] > TwoPowerIsogenies(J); [ Jacobian of Hyperelliptic Curve defined by y^2 = -10*x^6 + 70*x^5 - 15*x^4 - 380*x^3 - 30*x^2 + 280*x - 80 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -10*x^6 + 70*x^5 - 60*x^4 - 170*x^3 - 210*x^2 - 80*x + 40 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -2*x^5 + 15*x^4 - 20*x^3 - 30*x^2 - 8*x over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -35*x^6 + 112*x^5 + 105*x^4 - 350*x^3 + 210*x^2 + 448*x - 280 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -15*x^6 + 40*x^5 - 15*x^4 + 10*x^3 + 30*x^2 + 160*x + 120 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 5*x^6 - 35*x^5 + 30*x^4 + 85*x^3 + 105*x^2 + 40*x - 20 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 5*x^3 + 4*x over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 35*x^6 + 112*x^5 - 105*x^4 - 350*x^3 - 210*x^2 + 448*x + 280 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 15*x^6 + 40*x^5 + 15*x^4 + 10*x^3 - 30*x^2 + 160*x - 120 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -5*x^6 - 35*x^5 - 30*x^4 + 85*x^3 - 105*x^2 + 40*x + 20 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 10*x^6 + 70*x^5 + 15*x^4 - 380*x^3 + 30*x^2 + 280*x + 80 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 10*x^6 + 70*x^5 + 60*x^4 - 170*x^3 + 210*x^2 - 80*x - 40 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -2*x^5 - 15*x^4 - 20*x^3 + 30*x^2 - 8*x over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 105*x^6 + 20440*x^5 - 6930*x^4 - 81760*x^3 - 13860*x^2 + 81760*x + 840 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = 45*x^6 - 680*x^5 + 2970*x^4 - 2720*x^3 - 5940*x^2 - 2720*x - 360 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -105*x^6 + 20440*x^5 + 6930*x^4 - 81760*x^3 + 13860*x^2 + 81760*x - 840 over Rational Field, Jacobian of Hyperelliptic Curve defined by y^2 = -45*x^6 - 680*x^5 - 2970*x^4 - 2720*x^3 + 5940*x^2 - 2720*x + 360 over Rational Field ] [] [ Elliptic Curve defined by y^2 = x^3 - 35/256*$.1*x^2 + 35/32768*x - 1/8388608*$.1 over Number Field with defining polynomial x^2 - 2 over the Rational Field, Elliptic Curve defined by y^2 = x^3 + 5/256*$.1*x^2 + 45/32768*x + 729/8388608*$.1 over Number Field with defining polynomial x^2 + 2 over the Rational Field, Elliptic Curve defined by y^2 = x^3 + 1/64*(-635*$.1 - 285)*x^2 + 1/4096*(-29350*$.1 + 41725)*x + 1/262144*(-625*$.1 + 875) over Number Field with defining polynomial x^2 - 2 over the Rational Field, Elliptic Curve defined by y^2 = x^3 + 1/128*(415*$.1 + 930)*x^2 + 1/8192*(28650*$.1 + 45075)*x + 1/1048576*(455625*$.1 + 182250) over Number Field with defining polynomial x^2 + 2 over the Rational Field, Elliptic Curve defined by y^2 = x^3 + 1/128*(-1645*$.1 + 2310)*x^2 + 1/8192*(-114170*$.1 - 161455)*x + 1/1048576*(-2401*$.1 - 3430) over Number Field with defining polynomial x^2 - 2 over the Rational Field, Elliptic Curve defined by y^2 = x^3 - 5/256*$.1*x^2 - 45/32768*x + 729/8388608*$.1 over Number Field with defining polynomial x^2 - 2 over the Rational Field, Elliptic Curve defined by y^2 = x^3 + 35/256*$.1*x^2 - 35/32768*x - 1/8388608*$.1 over Number Field with defining polynomial x^2 + 2 over the Rational Field, Elliptic Curve defined by y^2 = x^3 + 1/128*(415*$.1 - 930)*x^2 + 1/8192*(-28650*$.1 + 45075)*x + 1/1048576*(455625*$.1 - 182250) over Number Field with defining polynomial x^2 + 2 over the Rational Field ]