Background on Weil Descent
7.4 Zariski closed subsets inside of the Weil descent
Theorem 7.8 The varietyW is equal to the Weil restrictionWL/K(V).
Proof. To prove this theorem we check the properties characterizing the Weil descent.
FirstW is a variety defined overL. As we have seen its set of points is invariant under the action ofGK. SoW is a variety defined overK.
A pointP ∈W isK-rational if and only if it isL-rational and for allτ ∈Gwe have τ(Pτ−1◦σ) =Pσ.
Takingτ = σ−1this means that Pσ = σ−1PId for allσ ∈ GwithP ∈ V(L). It follows that W(K) =V(L).
Next we extend the ground fieldKtoLand look atWL. On the Galois theoretic side this means that we restrict the Galois action ofGK onW to an action ofGL. But this group leaves eachVσ invariant and soWLis isomorphic to
σ∈GV =Vd.
We shall be interested in the special case thatK=FqandL=Fqd.
Corollary 7.9 LetV be a (projective or affine) variety defined overFqd of dimensiont. Fori = 0, . . . , d−1letVibe the image ofV with respect toφiq(cf. Proposition 5.67).
Then
W(V) :=
d−1
i=0
Vi
is a variety defined overFqof dimensiontdwhich isK-isomorphic toWF
qd/Fq(V).
IfV is affine (respectively projective) thenW(V)is an affine (respectively projective) variety defined overK.
IfV is an abelian variety overFqdthenW(V)is an abelian variety overFq.
The action ofφonW(V)is given as follows: LetP = (. . . , Pi, . . .)be a point inW(V)(K).
Thenφq(P) = (. . . , Qi, . . .)withQi=φq(P)(i−1 modd).
Remark 7.10 In general the Weil restriction of a Jacobian variety is not a Jacobian variety.
7.4 Zariski closed subsets inside of the Weil descent
As mentioned already, one main application of the Weil descent method is that inWL/K there are Zariski closed subsets which cannot be defined inV.
In the following we shall describe strategies to find such subsets.
7.4.1 Hyperplane sections
To simplify the discussion we assume thatV is affine with coordinate functionsx1, . . . , xnand we take the description ofWL/K(V)given in Proposition 7.1. There we have introducedndcoordi- nates functionsyi,jforWL/K(V)by
xi=u1y1,i+· · ·+udyd,i, fori= 1, . . . , n,
where{u1, . . . , ud}is a basis ofL/K. TakeJ ⊂ {1, . . . , d} × {1, . . . , n}and adjoin the equations yi,j= 0 for(i, j)∈J to the equations definingWL/K(V).
The resulting Zariski closed set inside ofWL/K(V)is denoted byWJ. It is the intersection of the Weil restriction ofV with the affine hyperplanes defined byyi,j= 0; (i, j)∈J.
“In general” we can expect thatWJis again a variety overKof dimensiontd− |J|.
130 Ch. 7 Background on Weil Descent
Example 7.11 LetEbe an elliptic curve defined overLgiven by a Weierstraß equation E:x21+a1x1x2+a3x1=f(x2),
wheref is monic of degree3. Take1md−1andJ ={1, . . . , m} × {2}.
ThenWJ(K)consists of all points inE(L)whosex2-coordinate is aK-linear combination of the elementsu1, . . . , um.
Remark 7.12 This example is the mathematical background of a subexponential attack to the dis- crete logarithm in elliptic curves over nonprime fields found recently by Gaudry and Diem (cf.
Section 22.3.5).
7.4.2 Trace zero varieties
We assume for simplicity thatL=FqdandK=Fqand we use the Galois theoretic description of the Weil descent.
LetV be a variety defined overK. So we getVφq =V. Note that neverthelessWF
qd/Fq(V) isnot Fq-isomorphic to Vd because of the twisted Galois operation. But we can embedV into WF
qd/Fq(V)as diagonal:
Map the pointP ∈V(K)to the point(. . . , φiq(P), . . .)∈d−1
i=0 V. By this map we can identify V with a subvariety ofWF
qd/Fq(V).
Now assume in addition thatV =Ais an abelian variety. Then we find a complementary abelian subvariety toAinside ofWF
qd/Fq(A).
We use the existence of an automorphismπof orderdofWF
qd/Fq(A)defined by P = (. . . , Pi, . . .)→π(P) = (. . . , Qi, . . .) with Qi=Pi−1 modd.
The mapπis obviously an automorphism overFqd. To prove thatπis defined overFq we have to show thatπcommutes with the action ofφq. But
π(φq(P)) = (. . . , Qi, . . .) with Qi =φq(Qi−2 modd) and this is equal toφq
π(P) .
Denote byA0the kernel of the endomorphismd−1
i=0 πi. It is an abelian subvariety ofAand it is called thetrace zero subvarietyofA. Note that the intersection set ofA— embedded as diagonal into WF
qd/Fq(A)— withA0 consists of the points ofAof order dividingd, and theFq-rational points ofA0are the pointsPinA(Fqd)withTr(φq)(P) = 0.
To see thatAandA0generateWF
qd/Fq(A)we use thatAis the kernel ofπ−Idand thatA0
contains(π−Id) WF
qd/Fq(A) . We summarize:
Proposition 7.13 LetAbe an abelian variety defined overFq. We use the product representation ofWF
qd/Fq(A)and defineπas automorphism induced by a cyclic permutation of the factors. Then we have the following results:
1. Acan be embedded (as diagonal) intoWF
qd/Fq(A). Its image under this embedding is the kernel ofπ−Id.
2. The image ofπ−Idis the trace zero subvarietyA0.
3. TheFq-rational points ofA0are the images of pointsP∈ A(Fqd)withTr(π)(P) = 0.
4. Inside ofWF
qd/Fq(A)the subvarietiesAandA0intersect in the group of points ofAof order dividingd.
§ 7.4 Zariski closed subsets inside of the Weil descent 131
For an example withA=Ean elliptic curve andd= 3we refer to [FRE 2001]; forA=JCbeing the Jacobian of a hyperelliptic curveC, see [LAN2004c]. We further investigate these constructive applications of Weil descent in Section 15.3.
7.4.3 Covers of curves
LetCbe a curve defined overFqdwith Jacobian varietyJC. We want to apply Weil descent to get information aboutPic0CfromWF
qd/Fq(JC)(Fq).
Here we investigate the idea of looking for curvesC defined overFq that are embedded into WF
qd/Fq(JC). Then the Jacobian ofC hasWF
qd/Fq(JC)as a factor and we can use information aboutPic0Cto studyPic0C. Of course this is only a promising approach if the genus ofCis not too large.
One can try to constructC directly, for instance, by using hyperplane sections. But it is very improbable that this will work if we are not in very special situations. Hence, it is not clear whether this variant can be used in practice. But this approach leads to interesting mathematical questions:
• Which abelian varieties have curves of small genus as sub-schemes?
• Which curves can be embedded into Jacobian varieties of modular curves?
• Which curves have the scalar restriction of an abelian variety (e.g., an elliptic curve) as Jacobian?
In [BODI+ 2004] one finds families of curves for which the last question is answered positively.
7.4.4 The GHS approach
In practice another approach is surprisingly successful. A prioriit has nothing to do with Weil descent, but as a background and in order to prove results the Weil descent method is useful.
LetL be a Galois extension of the fieldK. In our applications we shall takeL = Fqd and K =Fq. Assume thatC is a projective irreducible nonsingular curvedefined overL, andDis a projective irreducible nonsingular curvedefined overK.
Let
ϕ:DL→C
be a nonconstant morphism defined overL. As usual we denote byϕ∗the induced map fromPic0C toPic0DL. It corresponds to the conorm map of divisors in the function fieldsϕ∗
L(C)
⊂L(DL).
Next we use the inclusionK(D)⊂L(DL)to define a correspondence map on divisor classes ψ: Pic0(C)→Pic0(D)
given by
ψ:= NL/K◦ϕ∗, whereNL/K is the norm ofL/K.
Assume that we are interested in a subgroupG(for instance, of large prime order) inPic0C and assume that we can prove thatG
ker(ψ) ={0}. Then we have transferred the study ofG as subgroup of a Jacobian variety overLto the study of a subgroup of a Jacobian variety overK which may be easier.
The relation with the Weil descent method is that by the Weil descent of the cover mapϕwe get an embedding ofD into WL/K(JC). This method is the background of the so-called GHS algorithm. We shall come to this in more detail in 22.3.2.
132 Ch. 7 Background on Weil Descent
The mathematically interesting aspect of this method is that it relates the study of Picard groups of curves to the highly interesting theory of fundamental groups of curves over non-algebraically closed ground fields.