Computing modular correspondences for abelian varieties

37 Pages·2011·0.51 MB·English

Computing modular correspondences for abelian
varieties
Jean-Charles Faug`ere1, David Lubicz2,3, Damien Robert4
1 INRIA, Centre Paris-Rocquencourt, SALSA Project
UPMC, Univ Paris 06, LIP6
CNRS, UMR 7606, LIP6
UFR Ing´enierie 919, LIP6 Passy Kennedy, Boite courrier 169,
4, place Jussieu, F-75252 Paris Cedex 05
2 CE´LAR, BP 7419, F-35174 Bruz
3 IRMAR, Universit´e de Rennes 1, Campus de Beaulieu, F-35042 Rennes
4 LORIA, CACAO Project
Campus Scientifique
BP 239
54506 Vandoeuvre-l`es-Nancy Cedex
Abstract. In this paper, we describe an algorithm to compute modular
correspondences in the coordinate system provided by the theta null
points of abelian varieties together with a theta structure. As an applica-
tion, this algorithm can be used to speed up the initialisation phase of a
point counting algorithm [CL09]. The main part of the algorithm is the
resolutionofanalgebraicsystemforwhichwehavedesignedaspecialized
Gro¨bner basis algorithm. Our algorithm takes advantage of the structure
of the algebraic system in order to speed up the resolution. We remark
thatthisspecialstructurecomesfromtheactionoftheautomorphismsof
the theta group on the solutions of the system which has a nice geomet-
ric interpretation. In particular we were able count the solutions of the
system and to identify which one correspond to valid theta null points.
Keywords: Abelian varieties, Theta functions, Isogenies, Modular cor-
respondences.
1 Introduction
The aim of this paper is to compute a higher-dimensional analog of the classical
modular polynomials Φ (X,Y). We recall that Φ (X,Y) is a polynomial with
(cid:96) (cid:96)
integercoefficients.Moreover,ifj isthej-invariantassociatedtoanellipticcurve
E over a field k then the roots of Φ (j,X) correspond to the j-invariants of
k (cid:96)
elliptic curves that are (cid:96)-isogeneous to E . These modular polynomials have
k
important algorithmic applications. For instance, Atkin and Elkies (see [Elk98])
take advantage of the modular parametrisation of (cid:96)-torsion subgroups of an
elliptic curve to improve the original point counting algorithm of Schoof [Sch95].
In [Sat00], Satoh introduced an algorithm to count the number of rational
points of an elliptic curve E defined over a finite field k of small characteristic
k
p that relies on the computation of the canonical lift of the j-invariant of E .
k
Here again it is possible to improve the original lifting algorithm of Satoh
[VPV01,LL06] by solving over the p-adics the equation given by the modular
polynomial Φ (X,Y).
p
This last algorithm has been improved by Kohel in [Koh03] using the notion
of modular correspondence. For N a strictly positive integer, the modular curve
X (N) parametrizes the set of isomorphism classes of elliptic curves together
0
with an N-torsion subgroup. For instance, the curve X (1) is just the line of j-
0
invariants.LetpbeprimetoN.AmapX (pN)→X (N)×X (N)isamodular
0 0 0
correspondence if the image of each point represented by a pair (E,G), where
G is a subgroup of order pN of E, is a couple ((E ,G ),(E ,G )) with E =E
1 1 2 2 1
and G is the unique subgroup of index p of G, and E =E/H where H is the
1 2
unique subgroup of order p of G. In the case that the curve X (N) has genus
0
zero, the correspondence can be expressed as a binary equation Φ(X,Y)=0 in
X (N)×X (N) cutting out a curve isomorphic to X (pN) inside the product.
0 0 0
For instance, if one considers the correspondence X ((cid:96))→X (1)×X (1) for (cid:96)
0 0 0
a prime number then the polynomial defining its image in the product is the
modular polynomial Φ (X,Y).
(cid:96)
In this paper, we are interested in the computation of an analog of modular
correspondencesforhigherdimensionalabelianvarietiesoverafieldk.Wesuppose
that the characteristic of k is different from 2 and that it is possible to represent
the elements of k and compute efficiently the addition and multiplication laws
of k: this is the case for instance for finite fields of characteristic different from
2. We use a model of moduli space which is amenable to computations. We fix
an integer g > 0 for the rest of the paper. In the following if n is a positive
integer, n denotes the element (n,...,n) ∈ Ng. We consider the set of triples
of the form (A ,L,ΘB) where A is a g dimensional abelian variety equipped
k n k
with a symmetric ample line bundle L and a symmetric theta structure ΘB of
n
type n. Such a triple is called an abelian variety with an n-marking. To a triple
(A ,L,ΘB),onecanassociatefollowing[Mum66]itsthetanullpoint(seeSection
k n
2). The locus of theta null points corresponding to the set of abelian varieties
with an n-marking is a quasi-projective variety M . Moreover, it is proved in
n
[Mum67a] that if 8|n then M is a classifying space for abelian varieties with an
n
n-marking. We would like to compute an analog of modular correspondences in
M .
n
For this, let (A ,L,ΘA) be an abelian variety with a ((cid:96)n)-marking. We
k
(cid:96)n
suppose that (cid:96) and n are relatively prime. From the theta structure ΘA, we
(cid:96)n
deduceadecompositionofthekernelofthepolarizationK(L)=K (L)×K (L)
1 2
intomaximalisotropicsubspacesforthecommutatorpairingassociatedtoL.Let
K(L)[(cid:96)]=K (L)[(cid:96)]×K (L)[(cid:96)] be the induced decomposition of the (cid:96)-torsion
1 2
partofK(L).LetB bethequotientofA byK (L)[(cid:96)]andC bethequotient
k k 2 k
of A by K (L)[(cid:96)]. In this paper, we show that the theta structure of type (cid:96)n of
k 1
A induces in a natural manner theta structures of type n on B and C . As a
k k k
consequence, we obtain a modular correspondence, Φ :M →M ×M . In
(cid:96) (cid:96)n n n
the projective coordinate system provided by theta constants, we give a system
of equations for the image of M in the product M ×M as well as an
(cid:96)n n n
efficient algorithm to compute, given the theta null point (b ) of B with
u u∈Z(n) k
a theta structure of type n, all the theta null points (c ) of C with a
u u∈Z(n) k
theta structure of type n such that ((b ) ,(c ) ) is in the image of
u u∈Z(n) u u∈Z(n)
Φ . It should be remarked that in genus 1 our notion of modular correspondence
(cid:96)
does not coincide with the definition of [Koh03] which gives a parametrisation of
(cid:96)-isogenies while with our definition with obtain a parametrisation of (cid:96)2-isogenies.
Still in the case of genus 1, our modular correspondence can be used in the
aforementioned applications.
Thispaperisorganizedasfollows.InSection2werecallsomebasicdefinitions
and properties relating to algebraic theta functions. In Section 3, we define
formally the modular correspondence, and then in Section 4 we give explicit
equations for the computation of this correspondence. In particular, given the
theta null an abelian variety B with an n-marking, we define a polynomial
k
system (the equations of the image of M ), of which the solutions give theta
(cid:96)n
null points of varities isogenous to B . In Section 5, we describe the geometry of
k
these solutions. The last section is devoted to the description of a fast algorithm
compute the solutions.
2 Some notations and basic facts
In this section, we fix some notations for the rest of the paper and recall well
known results on abelian varieties and theta structures.
Let A be a g-dimensional abelian variety over a field k. Let L be a degree-d
k
ample symmetric line bundle on A . From here, we suppose that d is prime to
k
the characteristic of k. Denote by K(L) the group given by the geometric points
in the kernel of the polarization corresponding to L and by G(L) the theta
group (see [Mum66, p. 289]) associated to L. For x a geometric point of A ,
k
we denote by τ the translation by x map on A . The theta group G(L) (see
x k
the definition of [Mum66, p. 289]) is by definition the group (representable by
a group scheme by [Mum84, prop. 1]) given by the set of pairs (x,ψ), where
x is a point of K(L) and ψ is an isomorphism of line bundles ψ : L → τ∗L,
x
together with the composition law (x,ψ)◦(y,ϕ) = (x+y,τ∗ψ◦ϕ). Let δ =
y
(δ ,...,δ ) be a finite sequence of integers such that δ |δ . We consider the
1 g i i+1
finite group Z(δ) = (Z/δ Z)×...×(Z/δ Z) with elementary divisors given
1 g
by δ. For a well chosen unique δ, the finite group K(δ) = Z(δ)×Zˆ(δ) (where
Zˆ(δ) is the Cartier dual of Z(δ)) is isomorphic to K(L) (see [Mum70a, p.
132]). We note G the group k∗. The Heisenberg group of type δ is the group
m,k
H(δ) = G ×Z(δ)×Zˆ(δ) together with the group law defined on points by
m,k
(α,x ,x ).(β,y ,y )=(α.β.y (x ),x +y ,x +y ). We recall (see [Mum66, cor.
1 2 1 2 2 1 1 1 2 2
of Th. 1, p. 294]) that a theta structure Θ of type δ is an isomorphism of central
δ
extensions from H(δ) to G(L) fitting in the following diagram:
0 (cid:47)(cid:47)G (cid:47)(cid:47)H(δ) (cid:47)(cid:47)K(δ) (cid:47)(cid:47)0, (1)
m,k
(cid:15)(cid:15)Θδ (cid:15)(cid:15)Θδ
0 (cid:47)(cid:47)G (cid:47)(cid:47)G(L) κ (cid:47)(cid:47)K(L) (cid:47)(cid:47)0
m,k
where κ is the natural projection.
We note that Θ induces an isomorphism, denoted Θ in the preceding
δ δ
diagram, from K(δ) into K(L) and as a consequence a decomposition K(L)=
K (L)×K (L) where K (L) is the Cartier dual of K (L). As it will be
1 2 2 1
explained shortly, the data of a triple (A ,L,Θ ) defines a basis of global
k δ
sections of L that we denote (ϑ ) and as a consequence an morphism of
i i∈Z(δ)
A into Pd−1 where d=(cid:81)g δ is the degree of L. We recall the construction
k k i=1 i
of this basis. We recall ([Mum66, p. 291]) that a level subgroup K˜ of G(L) is a
subgroupsuchthatK˜ isisomorphictoitsimagebyκinK(L)whereκisdefined
in (1). We define the maximal level subgroups K˜ over K (L) and K˜ over
1 1 2
K (L) as the image by Θ of the subgroups (1,x,0) and (1,0,y) of
2 δ x∈Z(δ) y∈Zˆ(δ)
H(δ). Let A0 be the quotient of A by K (L) and π :A →A0 be the natural
k k 2 k k
projection. By the descent theory of Grothendieck (see [Mum66, p. 290]), the
data of K˜ is equivalent to the data of a couple (L ,λ) where L is a degree-one
2 0 0
ample line bundle on A0 and λ is an isomorphism λ:π∗(L )→L. Let s be
k 0 0
theuniqueglobalsectionofL uptoaconstantfactorandlets=λ(π∗(s )).We
0 0
have the following proposition which is an immediate consequence of [Mum66,
th. 2 p. 297] and Step I of [Mum66]:
Proposition 1: For all i ∈ Z(δ), let (x ,ψ ) = Θ ((1,i,0)). We set ϑΘδ =
i i δ i
(τ∗ ψ (s)). The elements (ϑΘδ) form a basis of the global sections of L
−xi i i i∈Z(δ)
which is uniquely determined, up to a multiplication by a factor independent of i,
by the data of Θ .
δ
If no ambiguity is possible, we let ϑΘδ =ϑ for i∈Z(δ).
i i
The image of the zero point 0 of A by the projective embeding defined by
k
Θ , which has homogeneous coordinates (ϑ (0)) , is by definition the theta
δ i i∈Z(δ)
null point associated to (A ,L,Θ ). If Θ is symmetric [Mum66, p. 308 and
k δ δ
p. 317], we say that (A ,L,Θ ) is an abelian variety with a δ-marking. The
k δ
locus of the theta null points associated to abelian varieties with a δ-marking is
a quasi-projective variety denoted M .
δ
Let (A ,L,Θ ) be an abelian variety with a δ-marking. We recall that the
k δ
naturalactionofG(L)ontheglobalsectionsofL isgivenby(x,ψ).f =τ∗ ψ(f)
−x
forf ∈Γ(L)and(x,ψ)∈G(L).ThereisanactionofH(δ)ontheglobalsections
of L. After an immediate computation using the group law of H(δ) and the
definitionof(ϑ ) givenbyProposition1,onobtainsthefollowingexpression
i i∈Z(δ)
for this action:
(α,i,j).ϑ =αe (m+i,−j)ϑ , (2)
m δ m+i
for (α,i,j) ∈ H(δ) and e the commutator pairing on K(δ). By construction,
δ
this action is compatible via Θ with the natural action of G(L) on (ϑ ) .
δ i i∈Z(δ)
Using (2), one can compute the coordinates in the projective system given
by the (ϑ ) of any point of K(L) from the theta null point associated
i i∈Z(δ)
to (A ,L,Θ ). Indeed, let (x,ψ) ∈ G(L) be any lift of x ∈ K(L) and let
k δ
(α,i,j)=Θ−1((x,ψ)) then the coordinates of x in the projective system given
δ
by the (ϑ ) are ((α,i,j).ϑ )(0)) .
i i∈Z(δ) m m∈Z(δ)
For δ = (δ ,...,δ ) ∈ Ng and δ(cid:48) = (δ(cid:48),...,δ(cid:48)) ∈ Ng, we write δ|δ(cid:48) if for
1 g 1 g
i = 1,...,g, we have δ |δ(cid:48). If n ∈ N, then n|δ means that n|δ for all i. If δ|δ(cid:48),
i i i
then we have the usual embedding
i:Z(δ)→Z(δ(cid:48)),(x ) (cid:55)→(δ(cid:48)/δ .x ). (3)
i i∈{1,...,g} i i i
A basic ingredient of our algorithm is given by the Riemann relations which
are algebraic relations satisfied by the theta null values if 2|δ.
Theorem 2 (Mumdord [Mum66] p. 333): Denote by Zˆ(2) the dual group
of Z(2). Let (a ) be the theta null point associated to an abelian variety with
i i∈Z(δ)
a δ-marking (A ,L,Θ ) where 2|δ and δ is not divisible by the characteristic of
k δ
k. For all x,y,u,v ∈Z(2δ) that are congruent modulo 2Z(2δ), and all χ∈Zˆ(2),
we have
(cid:0) (cid:88) (cid:1)(cid:0) (cid:88) (cid:1)
χ(t)ϑ ϑ . χ(t)a a =
x+y+t x−y+t u+v+t u−v+t
t∈Z(2) t∈Z(2)
(cid:0) (cid:88) (cid:1)(cid:0) (cid:88) (cid:1)
= χ(t)ϑ ϑ . χ(t)a a .
x+u+t x−u+t y+v+t y−v+t
t∈Z(2) t∈Z(2)
Here we embed Z(2) into Z(δ) and Z(δ) into Z(2δ) using (3).
Remark 3: It is moreover proved in [Mum66, Cor. p. 349] that if 4|δ the image
of A by the projective morphism defined by Θ is the closed subvariety of Pd−1
k δ k
defined by the homogeneous ideal generated by the relations of Theorem 2. (This
result can be sharpened, see [Kem89, Section 8]).
A consequence of Theorem 2 is the fact that if 4|δ, from the knowledge of a
validthetanullpoint(a ) ,onecanrecoveracouple(A ,L)whichitcomes
i i∈Z(δ) k
from. In fact, the abelian variety A is defined by the homogeneous equations
k
of Theorem 2. Moreover, from the knowledge of the projective embedding of
A , one recover immediately L by pulling back the sheaf O(1) of the projective
k
space.
An immediate consequence of the preceding theorem is the
Theorem 4: Let(a ) bethethetanullpointassociatedtoanabelianvariety
i i∈Z(δ)
with a δ-marking (A ,L,Θ ) where 2|δ. For all x,y,u,v ∈ Z(2δ) that are
k δ
congruent modulo 2Z(2δ), and all χ∈Zˆ(2), we have
(cid:0) (cid:88) (cid:1)(cid:0) (cid:88) (cid:1)
χ(t)a a . χ(t)a a =
x+y+t x−y+t u+v+t u−v+t
t∈Z(2) t∈Z(2)
(cid:0) (cid:88) (cid:1)(cid:0) (cid:88) (cid:1)
= χ(t)a a . χ(t)a a .
x+u+t x−u+t y+v+t y−v+t
t∈Z(2) t∈Z(2)
As Θ is symmetric, the theta constants also satisfy the additional symmetry
δ
relations a =a , i∈Z(δ).
i −i
Theorem4givesequationssatisfiedbythethetanullpointsofabelianvarieties
together with a δ-marking. Let M be the projective variety over k defined by
δ
the relations from Theorem 4. Mumford proved in [Mum67a, p. 83] the following
Theorem 5: Suppose that 8|δ. Then
1. M is a classifying space for abelian varieties with a δ-marking: to a theta
δ
null point corresponds a unique triple (A ,L,Θ ).
k δ
2. M is an open subset of M .
δ δ
A geometric point P of M is called a theta constant. If a theta constant P
δ
is in M we say that P is a valid theta null point, otherwise we say that P is a
δ
degenerate theta null point.
Remark 6: As the results of Section 5 show, M may not be a projective closure
δ
of M . Nonetheless, every degenerate theta null point can be obtained from a
δ
valid theta null point by a “degenerate” group action (see the discussion after
Proposition 18), hence the terminology.
3 Theta null points and isogenies
Let k be a field. Let (cid:96) and n be relatively prime integers and suppose that n is
divisible by 2 and that n(cid:96) is prime to the characteristic of k. Let (A ,L,ΘA) be
k
(cid:96)n
a g-dimensional abelian variety together with an ((cid:96)n)-marking. We recall from
just above Proposition 1 that the theta structure ΘA induces a decomposition
(cid:96)n
of the kernel of the polarization
K(L)=K (L)×K (L) (4)
1 2
into maximal isotropic subgroups for the commutator pairing associated to L.
Let K be either K (L)[(cid:96)] or K (L)[(cid:96)]. There are two possible choices for K,
1 2
one contained in K (L), the other one in K (L). In the next subsection, we
1 2
explain that a choice of K determines a certain abelian variety together with an
n-marking. The main results of this section are Corollary 8 and Proposition 9,
which explain how its theta null point is related to A.
3.1 The isogenies defined by K
LetX bethequotientofA byK andletπ :A →X bethenaturalprojection.
k k k k
Let κ:G(L)→K(L) be the natural projection. As K is a subgroup of K(L),
we can consider the subgroup G of G(L) defined as G = κ−1(K). Let K˜ be
the level subgroup of G(L) defined as the intersection of G with the image of
(1,x,y) ⊂H((cid:96)n) by ΘA. By the descent theory of Grothendieck,
(x,y)∈Z((cid:96)n)×Zˆ((cid:96)n) (cid:96)n
we know that the data of K˜ is equivalent to the data of a line bundle X on X
k
and an isomorphism λ:π∗(X)→L.
Now, we explain that the ((cid:96)n)-marking on A induces an n-marking on X .
k k
Let G∗(L) be the centralizer of K˜ in G(L). Applying [Mum66, Proposition 2 p.
291], we obtain an isomorphism
G∗(L)/K˜ (cid:39)G(X) (5)
and as a consequence a natural projection q :G∗(L)→G(X).
As H(n) is generated by the subgroups Gm×0×0, 1Gm ×Z(n)×0Zˆ(n) and
1Gm×0Z(n)×Zˆ(n),inordertodefineathetastructureΘnB :H(n)→G(X),itis
enoughtogivemorphisms1Gm×Z(n)×0Zˆ(n) →G(X)and1Gm×0Z(n)×Zˆ(n)→
G(X) such that the resulting ΘB is an isomorphism. Let Z∗((cid:96)n), Zˆ∗((cid:96)n), K∗
n 1
and K∗ be such that
2
1Gm ×Z∗((cid:96)n)×0Zˆ((cid:96)n) =Θ(cid:96)An−1(G∗(L))∩(1Gm ×Z((cid:96)n)×0Zˆ((cid:96)n)),
1Gm ×0Z((cid:96)n)×Zˆ∗((cid:96)n)=Θ(cid:96)An−1(G∗(L))∩(1Gm ×0Z((cid:96)n)×Zˆ((cid:96)n)),
1Gm ×K1∗×0Zˆ((cid:96)n) =Θ(cid:96)An−1(K˜)∩(1Gm ×Z((cid:96)n)×0Zˆ((cid:96)n)),
1Gm ×0Z((cid:96)n)×K2∗ =Θ(cid:96)An−1(K˜)∩(1Gm ×0Z((cid:96)n)×Zˆ((cid:96)n)).
There are natural isomorphisms Z∗((cid:96)n)/K∗ (cid:39)Z(n) and Zˆ∗((cid:96)n)/K∗ (cid:39)Zˆ(n)
1 2
from which we deduce projections p :Z∗((cid:96)n)→Z(n) and p :Zˆ∗((cid:96)n)→Zˆ(n)
1 2
(compare with the diagram of [Mum67a, p. 303]).
We define ΘB as the unique theta structure for X such that the following
n
diagrams are commutative
(1,x,0) Θ(cid:96)An (cid:47)(cid:47)G∗(L), (6)
x∈Z∗((cid:96)n)
p˜1 q
(cid:15)(cid:15) (cid:15)(cid:15)
(1,x,0) ΘnB (cid:47)(cid:47)G(X)
x∈Z(n)
(1,0,y) Θ(cid:96)An (cid:47)(cid:47)G∗(L), (7)
y∈Zˆ∗((cid:96)n)
p˜2 q
(cid:15)(cid:15) (cid:15)(cid:15)
(1,0,y) ΘnB (cid:47)(cid:47)G(X)
y∈Zˆ(n)
where p˜ is induced by p and p˜ is induced by p . Using the fact that ΘA is
1 1 2 2
(cid:96)n
symmetric, it is easy to see that ΘB is also symmetric.
n
WesaythatthethetastructuresΘA andΘB areπ-compatible(orcompatible)
(cid:96)n n
if the diagrams (6) and (7) commute.
Let K and K be the maximal (cid:96)-torsion subgroups of respectively K (L)
1 2 1
and K (L). By taking K =K and K =K in the preceding construction, we
2 2 1
obtain respectively (B ,L ,ΘB) and (C ,L ,ΘC) two abelian varieties with an
k 0 n k 1 n
n-marking. As a consequence, we have a well defined modular correspondence
Φ :M →M ×M . (8)
(cid:96) (cid:96)n n n
Letπ :A →B andπ(cid:48) :A →C betheisogeniesfromtheconstruction.Let
k k k k
[(cid:96)]betheisogenyofmultiplicationby(cid:96)onB andletπˆ :B →A betheisogeny
k k k
such that [(cid:96)]=π◦πˆ. From the symmetry of L we deduce that L is symmetric
0
and by applying the formula of [Mum66, p. 289], we have [(cid:96)]∗L =L(cid:96)2. Denote
0 0
byK thekernelofπ(cid:48).Thenπ(K )=π(A [(cid:96)])sothissubgroupofB isexactly
π(cid:48) π(cid:48) k k
the kernel of πˆ:
B (9)
k
πˆ
(cid:33)(cid:33)
[(cid:96)] Ak
π(cid:48)
(cid:15)(cid:15) (cid:125)(cid:125) π (cid:32)(cid:32)
B C
k k
3.2 The theta null points defined by K
The following two propositions explain the relation between the theta null point
of (A ,L,ΘA) and the theta null points of (B ,L ,ΘB) and (C ,L ,ΘC).
k (cid:96)n k 0 n k 1 n
Keeping the notations of the previous paragraph, we have
Proposition 7: Let (A ,L,ΘA), (B ,L ,ΘB) and π :A →B be defined as
k (cid:96)n k 0 n k k
above. There exists a constant factor ω ∈k such that for all i∈Z(n), we have
π∗(ϑΘnB)=ωϑΘ(cid:96)An. (10)
i i
In this identity, Z(n) is identified with a subgroup of Z((cid:96)n) via the map x(cid:55)→(cid:96)x.
Proof: The theta structure ΘA (resp. ΘB) induces a decomposition of the kernel
(cid:96)n n
ofthepolarizationK(L)=K (L)×K (L)(resp.K(L )=K (L )×K (L )).
1 2 0 1 0 2 0
DenotebyK thekernelofπ.WehavethatK isasubgroupofK(L)contained
2 2
in K (L).
2
The hypotheses of [Mum66, Th. 4] are verified by construction of ΘB and
n
Equation (10) is an immediate application of this theorem. (cid:4)
As an immediate consequence of the preceding proposition, we have
Corollary 8: Let(A ,L,ΘA)and(B ,L ,ΘB)bedefinedasabove.Let(a )
k (cid:96)n k 0 n u u∈Z((cid:96)n)
and (b ) be theta null points respectively associated to (A ,L,ΘA) and
u u∈Z(n) k (cid:96)n
(B ,L ,ΘB). Considering Z(n) as a subgroup of Z((cid:96)n) via the map x (cid:55)→ (cid:96)x,
k 0 n
there exists a constant factor ω ∈k such that for all u∈Z(n), b =ωa .
u u
Proposition 9: Let (A ,L,ΘA) and (C ,L ,ΘB) be defined as above. Let
k (cid:96)n k 0 n
(a ) and (c ) be the theta null points respectively associated to
u u∈Z((cid:96)n) u u∈Z(n)
(A ,L,ΘA) and (C ,L ,ΘC). We have for all u∈Z(n),
k (cid:96)n k 1 n
(cid:88)
c = a , (11)
u u+t
t∈Z((cid:96))
where Z(n) and Z((cid:96)) are considered as subgroups of Z((cid:96)n) via the maps j (cid:55)→(cid:96)j
and j (cid:55)→nj.
Proof: The proof follows the same line as that of Proposition 7. The theta
structureΘA (resp.ΘC)inducesadecompositionofthekernelofthepolarization
(cid:96)n n
K(L) = K (L)×K (L) (resp. K(L ) = K (L )×K (L )). Denote by K
1 2 1 1 1 2 1 1
the kernel of π(cid:48). We have that K is a subgroup of K (L) and we have an
1 1
isomorphism:
σ(cid:48) :K (L)/K →K (L ),
1 1 1 1
which translate via ΘA and ΘC into the natural isomorphism
(cid:96)n n
σ(cid:48) :Z((cid:96)n)/Z((cid:96))→Z(n).
0
The hypotheses of [Mum66, Th. 4] are then verified and Equation (11) is an
immediate application of this theorem. (cid:4)
4 The image of the modular correspondence
Inthissection,weusetheresultsoftheprevioussectioninordertogiveequations
for the image of the modular correspondence Φ given by (8). That is, for a given
(cid:96)
point x of M , we give equations for the set of points in M that correspond to
n n
x via the map defined by Φ (M ).
(cid:96) (cid:96)n
In order to make this precise, we let (B ,L ,ΘB) be an abelian variety
k 0 n
together with an n-marking and denote by (b ) its associated theta null
u u∈Z(n)
point. Unless specified, we shall assume that 4|n.
Denotebyp (resp.p )thefirst(resp.second)projectionfromM ×M into
1 2 n n
M , and let π =p ◦Φ , π =p ◦Φ . We would like to compute the algebraic
n 1 1 (cid:96) 2 2 (cid:96)
set π (π−1((b ) )) which we call the image of the modular correspondence.
2 1 u u∈Z(n)
We remark that this question is the analog in our situation of the computation
of the solutions of the equation Φ (j,X) obtained by substituting in the modular
(cid:96)
polynomial Φ a certain j-invariant j ∈ k. The only difference is that our
(cid:96)
modular correspondence parametrizes (cid:96)2-isogenies while the usual one deals with
(cid:96)-isogenies.
Let PZ((cid:96)n) =Proj(k[x |u∈Z((cid:96)n)]) be the ambient projective space of M ,
k u (cid:96)n
and let I be the homogeneous ideal defining M , which is spanned by the
(cid:96)n
relationsofTheorem4,togetherwiththesymmetryrelations.LetJ betheimage
of I under the specialization map
(cid:26)
b , if u∈Z(n)
k[x |u∈Z((cid:96)n)]→k[x |u∈Z((cid:96)n),nu(cid:54)=0], x (cid:55)→ u .
u u u x , else
u
and let V be the affine variety defined by J.
J
Let π˜0 : PZ((cid:96)n) → PZ(n) and π˜0 : PZ((cid:96)n) → PZ(n) be the rational maps
1 k k 2 k k
of the ambient projective spaces respectively defined on geometric points by
(cid:80)
(a ) (cid:55)→ (a ) and (a ) (cid:55)→ ( a ) . Clearly, π
u u∈Z((cid:96)n) u u∈Z(n) u u∈Z((cid:96)n) t∈Z((cid:96)) u+t u∈Z(n) 1
and π are the restrictions of π˜0 and π˜0 to M . The rational map π˜0 (resp
2 1 2 (cid:96)n 1
π˜0) restricts to a rational map π˜ : M → M (resp π˜ : M → M ). By
2 1 (cid:96)n n 2 (cid:96)n n
definition of J, we have V =π˜−1((b ) ).
J 1 u u∈Z(n)
Let S =k[y ,x |u∈Z(n),v ∈Z((cid:96)n)], we can consider J as a subset of S via
u v
thenaturalinclusionofk[x |u∈Z((cid:96)n)]intoS.LetL(cid:48) betheidealofS generated
u
byJ togetherwiththeelementsy −(cid:80) x andletL=L(cid:48)∩k[y |u∈Z(n)].
u t∈Z((cid:96)) u+t u
Let V be the subvariety of AZ(n) defined by the ideal L. By the definition of L,
L
V is the image by π˜ of the fiber V , so that V =π˜ (π˜−1((b ) )).
L 2 J L 2 1 u u∈Z(n)
Proposition 10: Keeping the notations from above, we suppose that 4|n and
let (b ) be the geometric point of M corresponding to (B ,L ,ΘB). The
u u∈Z(n) n k 0 n
algebraic variety V0 =π (π−1(b ) ) has dimension 0 and is isomorphic to
L 2 1 u u∈Z(n)
a subvariety of V .
L
Proof: From the preceding discussion the only thing left to prove is that V0 has
L
dimension 0. But this follows from the fact that the algebraic variety V has
J
dimension 0 [CL09, Th. 2.7] which generalize easily to the case where n is not a
power of 2. (cid:4)
From an algorithmic point of view, with our method the hard part of
the computation of the modular correspondence is the computation of V0 =
J
π−1((b ) ), the set of points in V that are valid theta null points. From
1 u u∈Z(n) J
nowon,weconsideronlyV0,sincecomputingV0 fromitistrivialbyProposition
J L
9.
We proceed in two steps. First we compute the solutions in V using a
J
specialized Gr¨obner basis algorithm (Section 6.3) and then we detect the valid
theta null points using the results of Section 5 (see Theorem 23). But first we
recall the moduli interpretation of V0 given by Section 3:
J
Proposition 11: We suppose that 4|n, then V0 is the locus of theta null points
J
(a ) in M such that if (A ,L,Θ ) is the corresponding variety with
u u∈Z((cid:96)n) (cid:96)n k (cid:96)n
an ((cid:96)n)-marking then ΘA is compatible with the theta structure ΘB of B .
(cid:96)n n k
Proof: Let (a ) be a geometric point of V0. Let (A ,L,Θ ) be a corre-
u u∈Z((cid:96)n) J k (cid:96)n
sponding variety with ((cid:96)n)-marking. If we apply the construction of Section 3,
we get an abelian variety (B(cid:48),L(cid:48),Θ(cid:48)) with an n-marking and an isogeny
k 0 n
π :A →B(cid:48) suchthatΘA iscompatiblewithΘ(cid:48).BydefinitionofJ,Corollary8
k k (cid:96)n n
shows that the theta null point of B(cid:48) is (b ) . As 4 | n, the paragraph
u u∈Z(n)
directly below Theorem 2 shows that (B(cid:48),L(cid:48))(cid:39)(B ,L ). By [Mum67b, p. 82]
k 0 k 0
we then have that the triples (B(cid:48),L(cid:48),Θ(cid:48)) and (B ,L ,Θ ) are isomorphic, so
k 0 n k 0 n
that ΘA is compatible with ΘB. (cid:4)
(cid:96)n n