# Richelot Isogenies Let $k$ be a field of characteristic different from $2$. We consider a curve of genus $2$, given by an equation $$ C:y^2=f(x) $$ 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 $\Phi: J \rightarrow A$ 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 $$ f(x)=cQ_1(x)Q_2(x)Q_3(x), $$ 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 $$ \{0,[Q_1(x)=0]-[Q_2(x)=0],[Q_1(x)=0]-[Q_3(x)=0], [Q_2(x)=0]-[Q_3(x)=0] \} $$ 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 $$ f(x)=c{\rm Norm}_{L[x]/k[x]} Q(x) $$ 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](../../references.md#cite-smith-phd)], Chapter $8$ for an exposition that is relatively close to the description given here. See [[Bruin and Doerksen, 2009](../../references.md#cite-brudoe-44split)] 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 $$ f(x,z) = q(x,z) h(x,z) \qquad \text{in }k[x] $$ 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 $$ \begin{aligned}(0, 1, 0) \times (1, -1-a, a) &= (a, 0, -1) \\ (1, -1, 0) \times (0, 1, -a) &= (a, a, 1) \\ (1, -a, 0) \times (0, 1, -1) &= (a, 1, 1)\end{aligned} $$ and $$ \left|\begin{matrix}a & 0 & -1 \\ a & a & 1 \\ a & 1 & 1\end{matrix}\right| = 2 a (a-1) \neq 0 \,. $$ 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` ```magma 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 `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 $C:y^2=f(x)$ 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 [`RichelotIsogenousSurfaces`](#function-richelotisogenoussurfaces) when given `Kernels:=true` are valid kernel descriptions. The codomain is returned using the same conventions as for [`RichelotIsogenousSurfaces`](#function-richelotisogenoussurfaces). ## `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. ```magma > R:=PolynomialRing(Rationals()); > C:=HyperellipticCurve(x^5+x); > J:=Jacobian(C); > RichelotIsogenousSurfaces(J); [* Cartesian Product, 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. ```magma > codomains,kernels:=RichelotIsogenousSurfaces(J:Kernels); > Q:=kernels[1]; > LX:=Parent(Q); > L:=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 $x^5+x$ again and that calling [`RichelotIsogenousSurface`](#function-richelotisogenoussurface) allows us to recreate the corresponding codomain. ```magma > _,swp:=SwapExtension(LX); > Norm(swp(Q)); x^5 + x ``` We can use Q to recreate the corresponding codomain. ```magma > 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. ```magma > LC:=LSeries(C : LocalData:="Ogg"); > myL:=func< A| > case 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. ```magma > R:=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. ```magma > R:=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 ] ```