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Computing low-degree isogenies in genus 2 with the Dolgachev-Lehavi method
Benjamin Smith
To cite this version:
Benjamin Smith. Computing low-degree isogenies in genus 2 with the Dolgachev-Lehavi method. Con-
temporary mathematics, American Mathematical Society, 2012, Arithmetic, Geometry, Cryptography
and Coding Theory, 574, pp.159-170. �inria-00632118v2�
WITH THE DOLGACHEV–LEHAVI METHOD
BENJAMIN SMITH
ABSTRACT. Letℓbe a prime, andHa curve of genus 2 over a fieldkof characteristic not 2 orℓ. IfSis a maximal Weil-isotropic subgroup ofJH[ℓ], thenJH/Sis isomorphic to the JacobianJXof some (possibly reducible) curveX. We investigate the Dolgachev–
Lehavi method for constructing the curveX, simplifying their approach and making it more explicit. The result, at least forℓ=3, is an efficient and easily programmable algorithm suitable for number-theoretic calculations.
1. INTRODUCTION
Letℓ≥3 be prime, and letH be a curve of genus 2 over a perfect fieldkof character- istic not 2 orℓ. LetJH be the Jacobian ofH, and letSbe a maximalℓ-Weil isotropic subgroup ofJH[ℓ]; sinceℓis prime,S∼=(Z/ℓZ)2. The quotientJH/S is isomorphic (as a principally polarized abelian variety) to a JacobianJX, whereXis some curve of genus 2 (see [14]); hence, there exists an isogeny
φ:JH→JX
with kernelS(note thatX may be reducible, in which caseJX is a product of elliptic curves). Our aim is to compute an explicit form forXgivenH andS.
In the caseℓ=2, the problem is resolved by the well-known Richelot construction (see [3] and [5, Chapter 9]). More generally, ifkis finite, then we can apply the explicit theta function-based algorithms of Lubicz and Robert [10], implemented in the freely- availableavIsogeniespackage [1].
Alternatively, there is the algebraic-geometric approach described by Dolgachev and Lehavi [7], which computes the image of the theta divisor onJH in the Kummer surface ofJX. As presented in [7], this approach has two drawbacks:
(1) it is not effective forℓ6=3, and
(2) forℓ=3, where theta structures are involved, it assumesk⊂C.
In this work we render the kernel of the Dolgachev–Lehavi method completely ex- plicit, with a view to computations in number theory. Our intention is to provide a sort of “user’s guide” to the algorithm and its concrete implementation. Forℓ=3 we obtain a simple, efficient, and easily-programmable algorithm (that does not requirek⊂C).
Our algorithm retains the pleasing geometric flavour of the original, but is better-suited to everyday calculations.
2. AN OVERVIEW OF THEDOLGACHEV–LEHAVI CONSTRUCTION
We begin by briefly recalling the Dolgachev–Lehavi construction, before treating it in detail in the following sections. SupposeH/k,S,φ, andXare as above; we assume we are given an explicit form forH andS, and we want to compute an explicit form forX. Dolgachev and Lehavi observe that ifΘH andΘX are theta divisors onJH andJX, respectively, thenφ(ΘH) is in|ℓΘX|(see [7, Proposition 2.4]); and as such, the image of φ(ΘH) in the Kummer surfaceKX=JX/〈±1〉is a degree-2ℓrational curve1inP3of
2010Mathematics Subject Classification. 11Y99;14Q05,14H45.
1By “rational curve” we mean a curve of genus 0. In all other contexts, “rational” means “defined overk”.
1
arithmetic genus (ℓ2−1)/2 and with (ℓ2−1)/2 ordinary double points corresponding to the nonzero elements ofS, up to sign [7, Proposition 3.1]. We can compute this curve withoutknowingφby expressing the mapΦ:H∼=ΘH ⊂JH →JX→KX⊂P3as the composition of a double coverρ2ℓof a rational normal curve inP2ℓwith a projection π:P2ℓ→P3whose centre depends on certain secants corresponding to the nonzero elements ofS, up to sign. The images underΦof the Weierstrass points ofH lie on a conicQcontained in a hyperplane ofP3; that is, a trope ofKX. The double cover ofQ ramified over the Weierstrass point images is then (a quadratic twist of)X.
3. THE DOMAIN CURVE
We suppose thatH/kis presented as a nonsingular projective model
(1) H :Y2=F(X,Z)=
X6 i=0
FiXiZ6−i⊂P(1,3,1),
whereFis a squarefree homogeneous sextic overk(such a model always exists whenk is perfect and has characteristic not 2: see [5, §1.3]). The hyperelliptic involution ofHis
ιH : (X:Y :Z)7−→(X:−Y :Z).
The divisor at infinity onH is D∞=¡
1 :p F6: 0¢
+¡ 1 :−p
F6: 0¢
;
we observe thatD∞is defined overk, fixed byιH, and equal to 2(1 : 0 : 0) ifF6=0.
The six Weierstrass points ofH are the fixed points ofιH; they correspond to the projective roots of the sexticF. The Weierstrass divisorWH ofH is the effective divisor cut out byY =0; ifF(X,Z)=Q6
i=1(ziX−xiZ) overk, then WH=(x1: 0 :z1)+ ··· +(x6: 0 :z6).
Note thatWH is defined overk. Finally, we fix a canonical divisor onH, defining KH=WH−2D∞.
4. THE KERNEL OF THE ISOGENY
When defining their method forℓ=3, Dolgachev and Lehavi state “unfortunately, we do not know how to input explicitly the pair (H,S). Instead we considerH with an odd theta structure.” We will take a rather more middlebrow approach to the problem: we suppose thatH is presented in the form (1), and thatSis given as a collection of divisor classes onH expressed using an extended Mumford representation (detailed below).
Our motivation for this choice is simple: this is precisely how one computes with hyperelliptic Jacobians in computational algebra systems such as Magma [11, 2] and Sage [13]. This choice also radically simplifies the algorithm: we can omit the theta structure calculations and pass directly to the secant computations (short-circuiting the first four steps of the algorithm in [7, §5.1]).
Points onJH correspond to divisor classes of degree zero onH. The Riemann–
Roch theorem tells us that every nontrivial degree-0 class has a unique representative in the formP+Q−D∞(this representation fails to be unique for the trivial class, because [P+ιH(P)−D∞]=0 for everyPinH(k)).
Let ebe a point ofJH, corresponding to a divisor class [P+Q−D∞]. The effec- tive divisorP+Q is cut out by an ideal in the form (A(X,Z),Y−B(X,Z)), whereBis a homogeneous cubic andAa homogeneous polynomial of degreed≤2. The triple
〈a(x),b(x),d〉:= 〈A(x,1),B(x,1),d〉
then encodes the pointe(with the convention thatAis chosen such thatais monic).
Note that ifk′is an extension ofk, thene= 〈a,b,d〉is inJH(k′) if and only ifaandb have coefficients ink′.
Conversely, given a triple 〈a,b,d〉, we recover the corresponding point ofJH by computing the effective divisor cut out by (A(X,Z),Y−B(X,Z)), whereBis the degree-3 homogenization ofbandAis the degree-dhomogenization ofa, and then subtracting (d/2)D∞. IfH has two points at infinity (that is, ifF66=0) thendmust be either 2 or 0.
In the case whereH has a single point at infinity (that is, whenF6=0) we always have d=dega, and the pair〈a,bmoda〉is the standard Mumford representation. The ad- vantage of the extended representation above is that it gracefully handles the general case where there are two points at infinity.
Example4.1. Consider the following points on the Jacobian ofH :Y2=X6−Z6.
• 0 is represented by〈1,0,0〉.
• [(1 : 0 : 1)+(−1 : 0 : 1)−D∞] is represented by〈x2−1,0,2〉.
• [(1 : 1 : 0)−(1 :−1 : 0)]=[2(1 : 1 : 0)−D∞] is represented by〈1,x3,2〉.
In this article, we will assume that the points ofSare allk-rational. This simplifies the exposition and the computations; however, all of our calculations are symmetric in the elements ofS. The algorithm should therefore be easily adapted to the case whereS is rational but its elements are not.
5. THE RATIONAL NORMAL CURVE
The Riemann–Roch spaceL(2ℓKH) is a direct sum of subspaces L(2ℓKH)=L(2ℓKH)+⊕L(2ℓKH)−,
whereιHacts as+1 on the elements ofL(2ℓKH)+and−1 on the elements ofL(2ℓKH)−. Writingx=X/Zandy=Y/Z3, we have
L(2ℓKH)+=
xi/y2ℓ®2ℓ
i=0 and L(2ℓKH)−=
xi/y2ℓ−1®2ℓ−3 i=0 .
The spaceL(2ℓKH)+corresponds to the linear system|2ℓKH|〈ιH〉; we see immedi- ately that it is (2ℓ+1)-dimensional, and therefore defines a map
ρ2ℓ:H −→R2ℓ⊂P2ℓ
onto a curveR2ℓinP2ℓ. Fixing coordinates onP2ℓ, we takeρ2ℓto be defined by ρ2ℓ: (X:Y :Z)7−→(U0:···:U2ℓ)=(X0Z2ℓ:X Z2ℓ−1:···:X2ℓ−1Z:X2ℓ).
We see thatR2ℓis a rational normal curve of degree 2ℓinP2ℓ, andρ2ℓis a double cover:
(2) ρ2ℓ(P)=ρ2ℓ(Q)⇐⇒¡
P=Q or P=ιH(Q)¢ .
(Essentially,ρ2ℓis a composition of the canonical map ofH and anℓ-uple embedding.) 6. THE SECANT LINES
We adopt the following convention: ifSis a set of points in some projective spacePn, then〈S〉denotes the linear subspace ofPngenerated byS.
For any pair of pointsPandQonH, we defineLP,Qto be the line inP2ℓintersecting R2ℓinρ2ℓ(P)+ρ2ℓ(Q); that is,
LP,Q:=
½ ρ2ℓ(P),ρ2ℓ(Q)®
ifP∉{Q,ιH(Q)}
Tρ2ℓ(P)(R2ℓ) otherwise.
We can also define secant lines corresponding to nonzero Jacobian elements: ifeis a nonzero point onJH, then we define
Le:=LP,Q wheree=[P+Q−D∞] .
Observe thatLP,Q=LP,ιH(Q)=LιH(P),Q=LιH(P),ιH(Q)for allPandQonH, so Le=L−e
for alleinJH\ {0}.
Remark6.1. Dolgachev and Lehavi define secantsle=
ρ2ℓ(P1),ρ2ℓ(P2)®
for each non- trivial pointe=[P1−P2] inJH (see [7, Theorem 1.1]). OurLeis equal tole, because [P1−P2]=[P1+ιH(P2)−D∞] andLP1,P2=LP1,ιH(P2).
The following lemma gives explicit and rational formulæ for the secantsLeand their intersections with arbitrary hyperplanes inP2. These formulæ are central to the explicit Dolgachev–Lehavi method.
Lemma 6.2. Let e= 〈a,b,d〉be a nonzero point ofJH. Let H:P2ℓ
i=0HiUi=0be a hyper- plane inP2ℓ, and write h(x) :=P2ℓ
i=0Hixi. (1) If a=1and d=2, then
Le=
(0 :···: 0 : 1 : 0),(0 :···: 0 : 0 : 1)® . (a) If H2ℓ=H2ℓ−1=0, thenLe⊂H .
(b) Otherwise,Le∩H=(0 :···: 0 :H2ℓ:−H2ℓ−1).
(2) If a(x)=x−α, then Le=
(0 :···: 0 : 1),(1 :···:α2ℓ)® . (a) If h(α)=0and H2ℓ=0, thenLe⊂H .
(b) Otherwise,Le∩H=¡
H2ℓ:H2ℓα:···:H2ℓα2ℓ−1:H2ℓα2ℓ−h(α)¢ . (3) If a(x)=x2+a1x+a0with a126=4a0, then
Le=
(π0:···:π2ℓ),(−a1:a2π0:a2π1:···:a2π2ℓ−1)® whereπ0=2,π1= −a1, andπi= −a1πi−1−a2πi−2for i>1.
(a) If a(x)divides h(x), thenLe⊂H . (b) Otherwise,Le∩H=(γ0:···:γ2ℓ)where
γi= X
0≤j≤2ℓ
Hj(a2jσi−j−ai2σj−i)
withσk=0for k<1,σ1=1, andσk= −a1σk−1−a2σk−2for k>1.
(4) If a(x)=x2+a1x+a0with a12=4a0, then writingαfor−a1/2, we have Le=
(1 :α:···:α2ℓ),(0 : 1 : 2α:···: 2ℓα2ℓ−1)® . (a) If a(x)divides h(x), thenLe⊂H .
(b) Otherwise,Le∩H=(γ0:···:γ2ℓ)whereγi=iαi−1h(α)−αih′(α).
Proof. In general, given pointsα=(α0:···:α2ℓ) andβ=(β0:···:β2ℓ) inP2ℓ, we have H∩Lα,β=(Aβ0−Bα0:···:Aβ2ℓ−Bα2ℓ)
where A=P2ℓ
i=0Hiαi andB=P2ℓ
i=0Hiβi; if A=B=0, thenLα,β⊂H(and the point above is not defined). In the following, we supposee=[P+Q−D∞]; we havee6=0, so we can supposeP6=ιH(Q).
In case (1) bothP andQare at infinity, soρ2ℓ(P)=ρ2ℓ(Q)=(0 :···: 0 : 1); our ex- pression forLegives generators for the tangent toR2ℓat (0 :···: 0 : 1). The intersection formula follows immediately.
In case (2), we haveP=(1 : 0 : 0) andQ=(α:±p
F(α,1) : 1), soρ2ℓ(P)=(0 :···: 0 : 1) andρ2ℓ(Q)=(1 :α:···:α2ℓ). The intersection formula follows immediately.
In case (3), we haveP =(α:±p
F(α,1) : 1) andQ=(β:±p
F(β,1) : 1) withα6=β, α+β= −a1, andαβ=a2; soρ2ℓ(P)=(1 :α:···:α2ℓ) andρ2ℓ(Q)=(1 :β:···:β2ℓ). If we take
T=(2 :α+β:···:α2ℓ+β2ℓ) and S=(α+β: 2βα:···:αβ2ℓ+βα2ℓ),
then we easily verify thatLP,Q=LT,S; it is a straightforward exercise with symmetric polynomials to show thatαi+βi =πi for 0≤i≤2ℓandαβi+βαi =a2πi−1fori>0, whence our formula forLe. The intersectionH∩Leis
H∩LP,Q=(h(α)−h(β) :h(α)β−h(β)α:···:h(α)β2ℓ−h(β)α2ℓ);
it is another straightforward exercise to show that
αjβi−βjαi=(β−α)(a2jσi−j−ai2σj−i), soh(α)βi−h(β)αi=P2ℓ
j=0Hj(β−α)(a2jσi−j−a2iσj−i)=(β−α)γifor 0≤i≤2ℓ, and thus H∩Le=(γ0:···:γ2ℓ).
In case (4), we haveP=Q=(α:±p
F(α,1) : 1); our expression forLegives generators for the tangent toR2ℓatρ2ℓ(P)=(1 :α:···:α2ℓ). The intersection formula follows.
7. THEWEIERSTRASS SUBSPACE
SinceR2ℓis a rational normal curve of degree 2ℓ, any 2ℓ+1 distinct points onR2ℓ are linearly independent. In particular, the images of the six Weierstrass points ofH underρ2ℓare linearly independent becauseℓ≥3. In view of (2) the images are distinct, so the subspace
W:=
ρ2ℓ(WH)®
⊂P2ℓ is five-dimensional.
For each 0≤i≤2ℓ−6, we define a linear form Wi:=X6
j=0
FjUi+j. Lemma 7.1. The space W is
W=
2ℓ\−6 i=0
V(Wi)=V({Wi: 0≤i≤2ℓ−6}).
Proof. Each hyperplaneV(Wi) contains W, sinceWi◦ρ2ℓ =XiZ2ℓ−6−iF(X,Z). But theWiare linearly independent, so the intersectionT2ℓ−6
i=0 V(Wi) is 5-dimensional, and
hence equal toW.
8. THE THEOREM OFDOLGACHEV ANDLEHAVI
We are now ready to state the main theorem behind the Dolgachev–Lehavi method.
Theorem 8.1([7, Theorem 1.1]). There exists a unique hyperplane H⊂P2ℓsuch that (1) H contains W , and
(2) the intersection points of H with the secantsLefor each nonzero e in S are con- tained in a subspace N of codimension3in H .
The image of the Weierstrass divisor under the projectionP2ℓ→P3with centre N lies on a conicQ(which may be reducible), and the double cover ofQramified over this divisor is a stable curveX of arithmetic genus2such thatJX∼=JH/S.
It is crucial to note that Theorem 8.1 is not constructive: it does not in itself yield the hyperplaneH, nor the centreNof the projection toP3. It is noted in [7, §3.4] thatHis defined byφ∗(ΘX), but in our situation we do not yet have an expression forφorΘX.
In the caseℓ=3, we are saved by a happy coincidence: 2ℓ−1=5, soH=W (we return to this case in §11 below). For ℓ>3, we must computeHin some other way;
Lemma 8.2, an easy corollary of Lemma 7.1, characterizes the possible hyperplanes.
Lemma 8.2. The linear system of all hyperplanes inP2ℓcontaining W is generated by the 2ℓ−5hyperplanes V(Wi)for0≤i≤2ℓ−6. That is, if H⊃W is a hyperplane inP2ℓ, then
H=V(α0W0+ ··· +α2ℓ−6W2ℓ−6) for some(α0:···:α2ℓ−6)inP2ℓ−6(k).
In view of Lemma 8.2, one naïve approach to computingHforℓ>3 would be to take a genericH=V¡P2ℓ−6
i=0 αiWi
¢and compute its intersection with the secantsLe. This yields (ℓ2−1)/2 points whose coordinates are linear expressions in theαi. We could then solve for the values ofαiby computing the zero locus of the (2ℓ−2)×(2ℓ−2) minors of the matrix formed by the intersectionsH∩Le; but each minor is still a degree-(2ℓ−2) polynomial in 2ℓ−5 variables, and the number of minors is exponential inℓ.
Alternatively, we could take a generic set of linear equations determiningNinside the genericH; requiring that this centre intersects any one of the (ℓ2−1)/2 secants imposes O(ℓ4) quartic polynomial conditions on theO(ℓ) unknowns.
In each approach the system is highly overdetermined, and with a clever choice of minors we might hope to get lucky and find solutions for toy examples. However, both approaches already represent a significant undertaking forℓ=5, even over finite fields;
they are totally impractical for largerℓand for infinite fields.
We continue the treatment for generalℓin §9 and §10, supposing that an equation for Hhas been found; without such an equation, theavIsogeniespackage [1] represents a much more sensible approach forℓ≥5 (ifkis finite). Forℓ=3, the Dolgachev–Lehavi method is as practical as it is interesting; we specialize to this case in §11 and §12.
9. FROM THEORY TO PRACTICE
To computeXvia Theorem 8.1, we must compute the map Φ:=π◦ρ2ℓ:H →P3,
whereπ:P2ℓ→P3is the projection with centreN. Suppose that we have an equation H:X
i
αiWi=0
forH. We can then apply Lemma 6.2 to compute the centreN= 〈Le∩H:e∈S\ {0}〉.
SinceN⊂H, we may computeν0,0,... ,ν0,2ℓ,ν1,0,... ,ν1,2ℓ,ν2,0,... ,ν2,2ℓinksuch that N=V
Ã2ℓ X
i=0
ν0,iUi,X2ℓ
i=0
ν1,iUi,X2ℓ
i=0
ν2,iUi,2ℓX−6
i=0
αiWi
! .
(This amounts to computing the kernel of the matrix whose rows are formed by the coordinates of theLe∩H; the choice ofP6
i=0αiWi for the fourth defining equation will be convenient later in the procedure.)
Fixing coordinates onP3, the projectionπwith centreNis defined by π: (U0:···:U2ℓ)7−→(V0:V1:V2:V3)=
Ã2ℓ X
i=0
ν0,iUi,X2ℓ
i=0
ν1,iUi, X2ℓ
i=0
ν2,iUi,2ℓX−6
i=0
αiWi
!
; the composed mapΦ=π◦ρ2ℓis then
Φ: (X:Y :Z)7−→(V0:V1:V2:V3)=¡
Φ0(X,Z) :Φ1(X,Z) :Φ2(X,Z) :Φ3(X,Z)¢ , where
Φ0:= X2ℓ i=0
ν0,iXiZ2ℓ−i, Φ2:= X2ℓ i=0
ν1,iXiZ2ℓ−i, Φ2:= X2ℓ i=0
ν2,iXiZ2ℓ−i, and
Φ3:=
2ℓX−6 i=0
αiXiZ2ℓ−6−iF(X,Z).
The image ofΦis a rational curve of degree 2ℓinP3. It lies on the Kummer sur- faceKXof the unknown codomain JacobianJX, and is therefore the intersection of a quadric and a cubic hypersurface inP3(see [9, Chapter XIII]):
Φ(P1)=Qe∩Ce where Qe=V¡Q˜(V0,V1,V2,V3)¢
and Ce=V¡C˜(V0,V1,V2,V3)¢
for some forms ˜Qand ˜Cof degree 2 and 3, respectively. The forms ˜Qand ˜Cgenerate the elimination ideal
¡Q˜, ˜C¢
=(V0−Φ0,V1−Φ1,V2−Φ2,V3−Φ3)∩k[V0,V1,V2,V3];
note that ˜Qis uniquely determined, and ˜Cis determined modulo (V0Q,V1Q,V2Q,V3Q).
The Weierstrass points of H map into the hyperplaneV3=0, which we identify withP2. (This simplification motivates our choice ofΦ3.) Theorem 8.1 asserts that a conicQpasses through the six images, and indeed
Q=V(Q(V0,V1,V2))⊂P2, where Q(V0,V1,V2)=Q(V˜ 0,V1,V2,0).
The image of the Weierstrass divisor underΦis thereforeQ∩C, where C =V(C(V0,V1,V2))⊂P2 with C(V0,V1,V2)=C(V˜ 0,V1,V2,0).
We are more interested in the formsQandCthan in ˜Qand ˜C, and it is a simple matter to interpolate them. ForQ, we compute the six quintic polynomialsΦiΦj(x,1) modF(x,1) for 0≤i ≤j≤2; the unique linear relation between them (and between theνi,0νj,0if F6=0) yields the coefficients ofQ. Similarly, to findC we compute the ten quintics ΦiΦjΦk(x,1) modF(x,1) for 0≤i≤j≤k≤2; any one of the linear relations between them (and theνi,0νj,0νk,0ifF6=0) gives an equation for a valid cubicC.
10. THE CODOMAIN CURVE
The data (Q,Q∩C) specifies a genus 2 curveX(up to a quadratic twist) as a double cover ofQramified over the six points ofQ∩C. This is the output of the Dolgachev–
Lehavi algorithm and of Theorem 8.1, and it is sufficient for computing isomorphism invariants ofX (see, for example, [4] and [12]).
In some situations, however, we would like to derive a defining equation forXitself.
WhenQis nonsingular, we recover a hyperelliptic curve; in the degenerate case where Qis singular, we recover a union of two elliptic curvesX+andX−, which are generally defined over a quadratic extension ofk(in which case they are Galois conjugates). The procedure is essentially standard (cf. [4, §2]), but we recall it here for completeness.
Algorithm 10.1. Computes a (possibly reducible) genus 2 curve representing a double cover of a given plane conic ramified over the intersection with a plane cubic.
Input: A plane conicQ:Q(V0,V1,V2)=0 and cubicC:C(V0,V1,V2)=0.
Output: A genus 2 curveX forming a double cover ofQramified overQ∩C. If Qis singular, thenX will be a one-point union of elliptic curvesX+andX−, withX±ramified overP0andC∩L±whereQ=L++L−andP0=L+∩L−. 1: LetMbe the matrix defined by
M:=
2q0,0 q0,1 q0,2
q0,1 2q1,1 q1,2
q0,2 q1,2 2q2,2
, where X
0≤i≤j≤2
qi,jViVj=Q(V0,V1,V2).
2: If det(M)=0, thenQis singular.
2a: Compute a diagonal matrixD =diag(a,b,0) and an invertible matrixT such thatM=T DT−1.
2b: Setδ=p
−a/b, and define homogeneous cubicsC+(X,Z) andC−(X,Z) byC±:=C¡
(t00±δt01)Z+t02X,(t10±δt11)Z+t12X,(t20±δt21)Z+t22X¢ where
t00 t01 t02
t10 t11 t12
t20 t21 t22
=T.
2c: Define elliptic curvesX+andX−overk(δ) inP(2,3,2) by X+:Y2=C+(X,Z) and X−:Y2=C−(X,Z),
and return the union ofX+andX−identifying the points at infinity.
3: Otherwise,Qis nonsingular.
3a: Compute a rational pointP=(α0:α1:α2) inQ(k) (see Remark 10.2).
3b: Letπ:P1→Qbe the corresponding parametrization, defined by π: (X:Z)7−→(V0:V1:V2)=(P0(X,Z) :P1(X,Z) :P2(X,Z)) (thePiare quadratic forms).
3c: ReturnX:Y2=C(P0(X,Z),P1(X,Z),P2(X,Z)).
Remark10.2. Step 3a of Algorithm 10.1 requires us to compute ak-rational pointPon the conicQ. IfH has a rational Weierstrass pointW0, then we may takeP =Ψ(W0).
Generically, however,H has no rational Weierstrass points, and then we are obliged to search for a rational point onQ. We are guaranteed that such a rational point exists (cf. [12, Lemme 1]). Over a finite field, finding a rational point is straightforward; over the rationals, we can apply (for example) the Cremona–Rusin algorithm [6].
11. THE ALGORITHM FORℓ=3
Consider now the special caseℓ=3. The mapρ6:H →R6⊂P6is defined by ρ6: (X:Y :Z)7−→(U0:U1:···:U5:U6)=¡
Z6:X Z5:···:X5Z:X6¢ . The hyperplaneHof Theorem 8.1 containsW =
ρ6(WH)®
by definition; but dimH= dimW =5, soH=W. Applying Lemma 7.1, we find
(3) H=V(W0)=V
à 6 X
i=0
FiUi
!
⊂P6; this allows us to simplify Lemma 6.2 for the caseℓ=3.
Proposition 11.1. If e= 〈a,b,d〉is a nonzero 3-torsion point ofJH, then H∩Le=¡
γ0(e) :···:γ6(e)¢ , where theγiare defined as follows:
(1) If a=1, thenγi(e)=0for0≤i<5, withγ5(e)=F6andγ6(e)= −F5. (2) If a is linear, thenγi(e)=0for0≤i<6, andγ6(e)=1.
(3) If a(x)=x2+a1x+a0with a126=4a0, then γi(e)=
X6 j=0
Fj(a2jσi−j+ai2σj−i) for 0≤i≤6
withσk=0for k<1,σ1=1, andσk= −a1σk−1−a2σk−2for k>1.
(4) If a(x)=x2+a1x+a0with a12=4a0, then γi(e)=
X6 j=0
(i−j)Fj(−a1/2)i+j−1 for 0≤i≤6.
Proof. This follows immediately from Lemma 6.2 on settingH=V¡P6 i=0FiUi¢
and not- ing thata(x) cannot divideh(x)=P6
i=0Fixi(since otherwiseewould have order 2).
We now present a version of the Dolgachev–Lehavi algorithm forℓ=3 based on the extended Mumford representation. The algorithm requires only elementary matrix al- gebra and polynomial arithmetic, and should be easily implemented in most computa- tional algebra systems.
Algorithm 11.2. A streamlined Dolgachev–Lehavi-style algorithm forℓ=3.
Input: A genus 2 curveH :Y2=F(X,Z)=P6
i=0FiXiZ6−i overkand a maximal Weil-isotropic subgroupSofJH[3], its elements defined overkand presented as in §4.
Output: A genus 2 curveX/ksuch that there exists an isogenyφ:JH→JXwith kernelS(the curveX is computed up to a quadratic twist, so the isogeny may only be defined over a quadratic extension ofk).
1: Compute a minimal subsetS±ofSsuch thatS={e:e∈S±}∪{−e:e∈S±}∪{0}
(then {Le:e∈S±}={Le:e∈S\ {0}}; this avoids redundancy in Steps 2 and 3).
2: For eacheinS±, compute the vectorve=(γ0(e),... ,γ6(e)) using the formulæ in Proposition 11.1.
3: Compute vectorsni =(νi,0,... ,νi,6) such that {n0,n1,n2,(Fj : 0≤ j ≤6)} is a basis for the (left) kernel of the 7×4 matrix (vet:e∈S±). Set
Φi= X6 j=0
νi,jXjZ6−j for 0≤i≤2.
4: For each 0≤i≤j≤2, compute the vectorri,j of length 6 whosenthentry is the coefficient ofxn−1in (ΦiΦj)(x,1) modF(x,1). IfF6=0, then take the 6thentry ofri,j to beνi,0νj,0: this allows us to correctly interpolate through the image of the Weierstrass point at infinity.
5: Compute a generator (qi,j: 0≤i≤j≤2) for the (left) kernel of the 6×6 matrix whose rows are theri,jfor 0≤i≤j≤2. Set
Q(V0,V1,V2) :=q0,0V02+q0,1V0V1+q0,2V0V2+q1,1V12+q1,2V1V2+q2,2V22. 6: For each 0≤i≤j≤k≤2, compute the vectorsi,j,kof length 6 whosenthentry
is the coefficient ofxn−1in (ΦiΦjΦk)(x,1) modF(x,1). IfF6=0, then take the 6thentry ofsi,j,kto beνi,0νj,0νk,0.
7: Compute any nontrivial element (ci,j,k: 0≤i≤j≤k≤2) of the (left) kernel of the 10×6 matrix whose rows are thesi,j,kfor 0≤i≤j≤k≤2, and set
C(V0,V1,V2) := X
0≤i≤j≤k≤2
ci,j,kViVjVk.
8: Return the resultX of Algorithm 10.1 applied toQ=V(Q) andC=V(C).
12. THE ALGORITHM IN PRACTICE
We conclude with an example forℓ=3. To avoid a visually overwhelming mass of coefficients, we will work over a small finite field; the curve was chosen at random.
Consider the genus 2 curve overF997defined by
H:Y2=X6+113X5Z+99X4Z2+363X3Z3+64X2Z4+503X Z5+630Z6. Computing the zeta function ofH (using Magma), we see that its Weil polynomial is
P(T)=T4−31T3+54T2−30907T+994009,
soJH is absolutely simple by the Howe–Zhu criterion [8, Theorem 6]. The elements D1= 〈x2+392x+208,579x+603,2〉andD2= 〈x2+48x+527,918x+832,2〉ofJHhave order 3, andS= 〈D1,D2〉is a maximal 3-Weil isotropic subgroup ofJH[3]. Applying Algorithm 11.2, we may take
S±=
½〈x2+392x+208,579x+603,2〉,〈x2+48x+527,918x+832,2〉,
〈x2+428x+880,252x+901,2〉,〈x2+348x+292,596x+269,2〉
¾
in Step 1. Equation (3) shows that the hyperplaneH⊂P6is defined by H: 630U0+503U1+64U2+363U3+99U4+113U5+U6=0,
so the matrix in Step 3 is
234 319 906 896
780 16 29 754
500 565 703 398 680 329 823 248 324 68 779 868 742 416 468 392 664 395 698 952
;
computing kernel vectors, we take
Φ0=121X6+742X5Z+549X4Z2+X Z5, Φ1=285X6+642X5Z+332X4Z2+X2Z4, Φ2=889X6+701X5Z+454X4Z2+X3Z3. The quadratic form of Step 5 is then
Q(V0,V1,V2)=V02+52V0V1+361V12+548V0V2+715V1V2+296V22, and we may take the cubic form in Step 7 to be
C(V0,V1,V2)=V03+167V13+149V0V1V2+836V12V2+885V0V22+538V1V22+294V23. We now apply Algorithm 10.1 to Q:Q(V0,V1,V2)=0 andC :C(V0,V1,V2)=0. The conicQis nonsingular, andH has a rational Weierstrass point (−76 : 0 : 1) mapping to the point (−36 :−80 : 1) onQ. The associated parametrizationP1→Qis defined by
(X:Z)7−→(36X2+781X Z+109Z2: 80X2+865X Z+17Z2: 996X2+945X Z+636Z2);
substituting its defining polynomials intoC, we find thatX has a model X:Y2=118X5Z+183X4Z2+613X3Z3+35X2Z4+174X Z5+474Z6.
In fact, this is the quadratic twist of the trueX: explicit calculation shows that its Weil polynomial isP(−T).
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