Siegel Moduli Space of Principally Polarized Abelian Manifolds
Andrea Anselli
8 february 2017
1
Recall
Definition 1.1. A complex torus T of dimension d is given by V /Λ, where V is a C-vector space of dimension
d and Λ is a lattice in V with rank 2d.
Let T = V /Λ be a complex torus of dimension d.
Definition 1.2. A hermitian form on V is given by H : V × V → C such that H(z, ·) is C-linear for every
z ∈ V and H(w, z) = H(z, w) for every z, w ∈ V .
Remark 1.3. If H is a hermitian form on V then, for every z, w ∈ V :
H(z, w) = E(iz, w) + iE(z, w),
where E : V × V → R is R-bilinear, antisymmetric (E(w, z) = −E(z, w) for every z, w ∈ V ) and E(iz, iw) =
E(z, w) for every z, w ∈ V . Moreover H is determined by E.
Definition 1.4. A Riemann form on T is given by a hermitian form H on V such that E(Λ, Λ) ⊆ Z, where
E := ImH. It is called positive if it is positive definite as bilinear form, that is H(z, z) ≥ 0 for every z ∈ V
and the equality holds if and only if z = 0.
Definition 1.5. An abelian manifold (a.m.) A is a complex torus which posses a positive Riemann form.
We have seen the following theorem.
Theorem 1.6. The torus T is the manifold of complex points of an abelian variety if and only if it is an
abelian manifold.
2
Polarization
Let A = V /Λ be an abelian manifold of dimension d.
Definition 2.1. Let H1 , H2 be two Riemann forms on A. They are equivalent if there exists n1 , n2 ∈ N
e the equivalence class of a Riemann form H on A.
such that n1 H1 = n2 H2 . We denote by H
e is given by an abelian manifold A together
Definition 2.2. A polarized abelian manifold (p.a.m.) (A,H)
e that contains a positive Riemann form (we can suppose that
with an equivalence class of Riemann forms H
e
H is positive definite). The class H is called (homogeneous) polarization of A.
e be a polarized abelian manifold, we have seen that every D ∈ Div(A) whose
Remark 2.3. Let (A,H)
associated Riemann form is H is an ample divisor. Thus for every such D the following map is a polarization
(as defined in the notes):
∗
ϕD : A → Ǎ := P ic0 (A), a 7→ [τ−a
D − D].
(1)
1
We have seen that ϕD is an isogeny of degree det E, where E := ImH. Moreover ϕD depends only on
[D] ∈ Div(A)/Div 0 (A). In facts D and D0 defines the same element in Div(A)/Div 0 (A) if and only if they
have the same Riemann form associated and for every a ∈ A the divisor τa∗ D − D correspond to the character
λ 7→ e2πiE(w,λ) , where w ∈ V represents a and E = ImH where H is the Riemann form associated to D,
thus depends only on the class of D.
f1 ) → (A2 H
f2 ) is given by a morphism
Definition 2.4. A morphism of polarized abelian manifold φ : (A1 H
of complex abelian variety φ : A1 = V1 /Λ1 → A2 = V2 /Λ2 (a holomorphic map which is a homomorphism
of groups) such that φe∗ H2 is equivalent to H1 , where φe : V1 → V2 is the C-linear map that lifts such that
e 1 ) ⊆ Λ2 .
φ(Λ
One can prove the following theorem.
e the automorphism group Aut(A, H)
e is a finite
Theorem 2.5. For every polarized abelian manifold (A, H)
group.
e is called principally polarized abelian manifold (p.p.a.m.)
Definition 2.6. A polarized abelian manifold (A, H)
if there is an element in the homogeneous polarization class of A (we can suppose that this element is H)
such that P f (E) = 1, where E := ImH. It means that with respect to a symplectic basis {λ1 , . . . , λ2d } of
Λ the bilinear form E is given by the following matrix:
0
Id
J :=
.
−Id 0
Proposition 2.7. Up to isogenies every polarized abelian manifold is a principally polarized abelian manifold.
e is a polarized abelian manifold of dimension d, with E := ImH
Sketch of proof. Suppose that (A = V /Λ, H)
and {λ1 , . . . , λ2d } a symplectic basis of Λ. Thus E is given by the following matrix, where E = diag(e1 , . . . , ed )
and ej := E(λj , λj+d ) ∈ N for every j = 1, . . . , d:
0
E
.
−E 0
We define the following lattice in V :
Λ0 :=
λd
λ1
Z + ... +
Z + λd+1 Z + . . . + λ2d Z.
e1
ed
It is a lattice of rank 2d contained in Λ, the form E is alternating on Λ0 , by definition E(Λ0 , Λ0 ) ⊆ Z and
det E = 1, thus A0 := V /Λ0 is a principally polarized abelian manifold. The map A = V /Λ → V /Λ0 = A0
defines an isogeny.
e is a principally polarized abelian manifold then A is autodual. It sufficies to consider an
Remark 2.8. If (A, H)
ample divisor D ∈ Div(A) whose associated Riemann form is H and the associated polarization φD : A → Ǎ.
The map φD is an isogeny of degree det E = 1, hence it is an isomorphism of complex abelian variety.
Example 2.9. The major example of principally polarized abelian manifold is the Jacobian of a compact
Riemann surface C of genus g > 0 (even of non singular algebraic curve). In fact one can show there is a
positive Riemann form H on Cg with respect to the lattice Λ ⊆ Cg , where Λ is the additive subgroup of Cg
generated by:
Z
Z
Z
Z
ω1 , . . . ,
ai
ωg ,
ai
ω1 , . . . ,
bi
ωg
: i = 1, . . . , g ,
bi
where H1 (C, Z) = π1 (C) = ha1 , . . . , ag , b1 , . . . , bg i and {ω1 , . . . , ωg } is a C-basis of the g-dimensional C-vector
space of holomorphic differential forms on C. The Riemann form H is induced by the intersection pairing on
H1 (C, Z) ' Λ, the determinant of the intersection pairing is 1 hence H determines a principal polarization
on Jac(C) := Cg /Λ, the Jacobian of C.
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The following theorem is a deep result and encourages us to proceed in the study of the moduli space of
principally polarized abelian manifolds.
Theorem 2.10 (Torelli). Let C1 and C2 be two compact Riemann surfaces and let H1 and H2 be the
e 1 ) ' (Jac(C2 ), H
e 2 ) as (principally)
Riemann forms induced by the intersection pairings. If (Jac(C1 ), H
polarized abelian manifolds, then C1 ' C2 as complex manifolds.
3
Siegel Moduli Space
For every d ≥ 1 we define:
Ad := {isomorphism classes of principally polarized abelian manifold of dimension d},
we want to give to Ad the structure of a complex analytic space.
Example 3.1. Case d = 1. We have seen that every abelian manifold of dimension 1 (i.e. every elliptic curve)
is principally polarized by the Riemann form associated to the divisor (0) and all the Riemann forms are
equivalent (they are all integers multiple of a Riemann form, it’s Neron-Severi group is isomorphic to Z).
Let H := {τ ∈ C : Imτ > 0} be the Poincaré upper half plane and let:
a b
SL2 (Z) :=
: a, b, c, d ∈ Z, ad − bc = 1 .
c d
The group SL2 (Z) acts on H in the following way:
aτ + b
a b
.
, τ 7→
c d
cτ + d
One has:
A1 ' SL2 (Z)\H,
this bijection is defined as follows. Any element in A1 can be represented by the elliptic curve C/(Zω1 +Zω2 ),
where {ω1 , ω2 } is a R-basis of C such that Im(ω1 /ω2 ) > 0. To this element of A1 correspond the class of
τ := ω1 /ω2 . Moreover, by the j-invariant, A1 inherits from C the structure of a complex manifold. This
material will be seen as a special case of our further considerations.
e a principal polarized abelian manifold. By choosing a C-basis {v1 , . . . , vd } for V and
Let (A = V /Λ, H)
a symplectic basis {ω1 , . . . , ω2d } of Λ one can represent the polarized abelian manifold in the following way:
(Cd , Ω, J),
where Ω ∈ M atd×2d (C) is called the period matrix and has as columns the components of the elements of
the symplectic basis {ω1 , . . . , ω2d } in the basis {v1 , . . . , vd } of V and J is the matrix that represents the
imaginary part E of the Riemann form H in the symplectic basis. The following two lemmas prove that the
period matrices are characterize by the following two conditions, called Riemann’s relations:
(RI) ΩJΩT = 0,
(RII) 2i(ΩJ −1 ΩT )−1 > 0.
Remark 3.2. Every element of Cd can be written in the form Ωx, for some x ∈ R2d . We have for every
x, y ∈ R2d :
E(Ωx, Ωy) = xT Jy.
Let C ∈ M at2d (R) be the matrix such that iΩ = ΩC (iωj = c1,j ω1 + . . . c2d,j ωd for every j = 1, . . . , 2d), as
E(i·, ·) is a symmetric bilinear form and E(i·, i·) = E(·, ·) the following equation hold:
C T J = −JC.
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Lemma 3.3. The matrix C T J is symmetric if and only if (RI) holds.
Proof. Clearly J T = −J, thus:
(C T J)T = J T C = −JC.
If C T J is symmetric then C T J = −JC, thus −CJ −1 = J −1 C T , thus −ΩCJ −1 ΩT = ΩJ −1 C T ΩT . By
definition of C we obtain:
−iΩJ −1 ΩT = −ΩCJ −1 ΩT = ΩJ −1 C T ΩT = iΩJ −1 ΩT ,
therefore 2iΩJ −1 ΩT = 0. By applying the same argument we can obtain the converse.
Lemma 3.4. The matrix associated to H with respect the canonical basis of Cd is M := 2i(ΩJ −1 ΩT )−1 .
Proof. The matrix M is hermitian, in fact:
M
T
= −2i(Ω(J T )−1 ΩT )−1 = −2i(−ΩJ −1 ΩT )−1 = M,
thus it sufficies to show that H(u, u) = uT M u for every u ∈ Cd (and then we conclude by the polarization
identity). We denote by id := iId , clearly we have:
Ω
Ω
id
0
=
C.
0 −id
Ω
Ω
Recalling that H(u, u) = E(iu, u) for every u ∈ Cd we obtain, for x ∈ R2d such that u = Ωx:
H(u, u) = − xT JCx =
−1 Ω
id
= − xT J
0
Ω
Ω
0
x=
−id
Ω
−1 T
T −1
Ω
Ω
id
0
T
T
T
= − x [Ω , Ω ][Ω , Ω ] J
x.
0 −id
Ω
Ω
Clearly:
T
xT [ΩT , Ω ] = [uT , uT ],
Ω
Ω
x=
u
u
.
Moreover, by using and Lemma 3.3 and the following relation:
−1 0 X
0
Y −1
=
,
Y
0
X −1
0
we obtain:
T
[ΩT , Ω ]−1 J
Ω
Ω
−1 id
0
0
−id
=
"
=
Ω
Ω
=
"
=
−1 T
ΩJ −1 Ω
T
ΩJ −1 Ω
0
ΩJ −1 Ω
0
ΩJ −1 ΩT
"
T
J −1 [ΩT , Ω ]
ΩJ −1 ΩT
ΩJ −1 ΩT
"
=
0
−iΩJ −1 ΩT
0
−1 T −1
i(ΩJ Ω )
4
T
id
0
#−1 #−1 iΩJ −1 Ω
0
T
0
−id
=
id
0
0
−id
id
0
0
−id
−1
#−1
=
−i(ΩJ −1 ΩT )−1
0
#
.
=
=
Thus:
"
T
T
H(u, u) = −[u , u ]
0
−1 T −1
i(ΩJ Ω )
−i(ΩJ −1 ΩT )−1
0
#
u
u
=
T
=iuT (ΩJ −1 ΩT )−1 u − iuT (ΩJ −1 Ω )−1 u = 2iuT M u.
If we write Ω = [Ω1 , Ω2 ], for suitable Ω1 , Ω2 ∈ M atd (C), the Riemann’s conditions can be rewritten in the
following way (one has J −1 = −J):
(RI’) Ω2 ΩT1 − Ω1 ΩT2 = 0,
T
T
(RII’) 2i(Ω2 Ω1 − Ω1 Ω2 ) > 0.
We define the following:
R := {Ω = [Ω1 , Ω2 ] : Ω1 , Ω2 ∈ M atd (C) and the conditions (RI 0 ), (RII 0 ) hold}.
and recall the definition of the symplectic group:
SP2d (R) := {M ∈ M at2d (R) : M JM T = J}.
The property of the symplectic group are collected in the following proposition.
Proposition 3.5. The symplectic group SP2d (R) is a subgroup of GL2d (R) and it is closed by transposition
(if M ∈ SP2d (R), then M T ∈ SP2d (R)).
Proof. Clearly det J 6= 0, thus if M ∈ SP2d (R) then (det M )2 = 1, thus M ∈ GL2d (R). Moreover from
M JM T = J we obtain J = M −1 J(M −1 )T , thus M −1 ∈ SP2d (R). For the product and for the identity
element analogous arguments hols, therefore SP2d (R) is a subgroup of GL2d (R). Moreover let M ∈ SP2d (R),
taking the inverse of the relation J = M −1 J(M −1 )T , we obtain:
−J = J −1 = (M −1 J(M −1 )T )−1 = M T J −1 M = −M T J(M T )T ,
thus M T ∈ SP2d (R).
We define the Siegel upper half space:
Hd := {τ ∈ M atd (C) : τ is symmetric and Imτ > 0}.
It is an open of the complex manifold of the symmetric d × d matrices in complex coefficients, that has
dimension d(d + 1)/2.
Proposition 3.6. Let Ω = [Ω1 , Ω2 ] ∈ R, the following hold:
(i) If g ∈ GLd (C), then gΩ := [gΩ1 , gΩ2 ] ∈ R.
A B
(ii) If M =
∈ SP2d (R), then ΩM := [Ω1 A + Ω2 C, Ω1 B + Ω2 D] ∈ R.
C D
(iii) Ω1 , Ω2 ∈ GLd (C).
(iv) Ω−1
2 Ω1 ∈ Hd .
Proof.
(i) We have to prove that the conditions (RI 0 ), (RII 0 ) hold for gΩ = [gΩ1 , gΩ2 ].
5
(RI’) gΩ2 (gΩ1 )T − gΩ1 (gΩ2 )T = g(Ω2 ΩT1 − Ω1 ΩT2 )g T = 0,
T
T
T
T
(RII’) 2i(gΩ2 gΩ1 − gΩ1 gΩ2 ) = g2i(Ω2 Ω1 − Ω1 Ω2 )g T > 0.
(ii) As M ∈ SP2d (R) we have:
BAT − AB T = 0,
BC T − ADT = Id ,
DAT − CB T = −Id ,
DC T − CDT = 0.
To prove (RI 0 ) we observe that:
(Ω1 B + Ω2 D)(AT ΩT1 + C T ΩT2 ) − (Ω1 A + Ω2 C)(B T ΩT1 + DT ΩT2 ) = −Ω1 ΩT2 + Ω2 ΩT1 .
An analogous argument hold for proving (RII 0 ).
(iii) Suppose that there exists v ∈ Cd such that v T Ω1 = 0. Thus:
T
T
v T (Ω1 Ω2 − Ω2 Ω1 )v = 0.
T
T
For (RII 0 ) the matrix 2i(Ω1 Ω2 − Ω2 Ω1 ) is positive definite, therefore v = 0. Thus Ω1 ∈ GLd (C). The
matrix J ∈ SP2d (R), thus [−Ω2 , Ω1 ] = ΩJ ∈ R for (ii). Hence also Ω2 ∈ GLd (C).
−1
−1
0
(iv) For (iii) and (i) the element [Ω−1
2 Ω1 , Id ] = Ω2 Ω ∈ R. The condition (RI ) means that Ω2 Ω1 is
−1
0
symmetric, the condition (RII ) means that Im(Ω2 Ω1 ) > 0.
Lemma 3.7. Every element of Ad contains a representative of the form:
(Cd , [τ, Id ], J),
for some τ ∈ Hd .
Proof. We have alredy seen that each element can be represented by a triple (Cd , Ω, J), where Ω = [Ω1 , Ω2 ] ∈
−1
−1
R. For Proposition 3.6 (i) we have Ω−1
2 Ω = [Ω2 Ω1 , Id ] ∈ R and for (iv) the matrix τ := Ω2 Ω1 ∈ Hd . The
−1
d
multiplication by Ω2 corresponds to a change of basis of C , thus it doesn’t change the isomorphism class
and the matrix that represent the imaginary part of the positive Riemann form (in fact the symplectic basis
remains unchanged, it only change the period matrix, that is the description of the symplectic basis in the
basis of Cd ).
To get uniqueness we have to factor out by an appropriate group of automorphisms.
Remark 3.8. The group SP2d (R) acts on Hd in the following way:
A B
M=
, τ 7→ M (τ ) := (Aτ + B)(Cτ + D)−1 .
C D
To prove that the action is well defined note that Ω := [τ, Id ] ∈ R, thus by Proposition 3.6 (ii):
[τ A + C, τ B + D] = ΩM ∈ R.
Thus, by Proposition 3.6 (iv), (τ B + D)−1 (τ A + C) ∈ Hd . Clearly Hd is closed by transposition and τ T = τ ,
thus:
M T (τ ) = (AT τ + C T )(B T τ + DT )−1 = [(τ B + D)−1 (τ A + C)]T ∈ Hd .
To conclude it sufficies to consider M T instead of M (SP2d (R) is closed under transposition).
6
0
Remark 3.9. Let Ω0 = [Ω01 , Ω02 ] be another period matrix, defined by a different symplectic basis {ω10 , . . . , ω2d
}
0
T
for Λ. Clearly there exists M ∈ GL2d (Z) such that Ω = ΩM . The form E, with matrix J with respect
0
to the basis {ω1 , . . . , ω2d }, has the matrix M JM T with respect to the basis {ω10 , . . . , ω2d
}. Therefore the
description of the form E (and thus H) remains unchanged if and only if M ∈ SP2d (R). We define:
Γ := Aut(J) = GL2d (Z) ∩ SP2d (R) = SP2d (Z).
Any two representative of an element in Ad of the form (Cd , [τ, Id ], J) and (Cd , [τ 0 , Id ], J) respectively, with
τ, τ 0 ∈ Hd , differ by a change of symplectic basis preserving J (as above) followed by a change of basis in Cd
transforming the second half of the lattice vectors into the unit vectors.
By this two remarks we obtain the following Theorem.
Theorem 3.10. The following is a bijection:
Ad ' Hd /Γ = SP2d (Z)\Hd .
It is defined as follows. Every element of Ad admits a representant of the form (Cd , [τ, Id ], J), where τ ∈ Hd .
To this element correspond the class of τ .
We recall the following facts and an important theorem. Let X be a complex analytic space and let G
be a subgroup of Aut(X) acting on X. The quotient X/G, endowed with the quotient topology, admits the
structure of a ringed space. For every U open of X/G:
OX/G (U) := {f : U → C : f ◦ π ∈ OX (π −1 U)}.
The group G acts properly and discontinuously on X if for any K1 , K2 compact subsets of X the following
set is finite:
{g ∈ G : gK1 ∩ K2 6= ∅}.
Theorem 3.11. Let X be a complex analytic space and let G be a group acting properly and discontinuously
on X. The quotient X/G is a complex analityc space. Moreover, if X is normal (in particular if X is a
complex manifold), so is X/G.
One can prove the following proposition.
Proposition 3.12. Any discrete subgroup of SP2d (R) (in particular Γ) acts properly and discontinuously
on Hd .
Thus we can apply the Theorem 3.11 obtaining the following.
Theorem 3.13. The quotient Hd /Γ is a normal complex analytic space of dimension d(d + 1/2).
Therefore we have reach the goal of this section: give to Ad the structure of a complex analytic space.
Moreover we have recover the familiar situation when d = 1.
4
Application
1. There are complex tori which are not the manifold of complex points of an abelian variety.
Consider the following example for d = 2. In C2 consider the following period matrix, for α, β, γ, δ ∈ R
algebraically indipendent over Q:
α+i
β
1 0
Ω :=
.
γ
δ+i 0 1
Let T := C2 /Λ, where Λ is the lattice generated by the columns of the period matrix. One can prove
that M(T ) = C. The following theorem due to Siegel:
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Theorem 4.1. Let M be a compact, connected, complex manifold of dimension d. Then M(M ) has
trascendence degree over C at most d. If d is atteined then M(M ) is a finitely generated field over C.
If M = X(C), the complex points on a nonsingular algebraic variety X, then M(M ) ' C(X), the
field of rational functions on X. Thus, in this case, M(M ) is a finitely generated field of trascendence
degree d.
2. The generic abelian variety has Z as endomorphism ring.
Let A be the representant of an element in Ad and let τ ∈ Hd the corresponding matrix. As we have
seen before the multiplication by g ∈ Md (C) gives an element of End(A) if and only if there exists
M ∈ SP2d (Z) such that g[τ, Id ] = [τ, Id ]M . Therefore:
gτ = τ B + D,
gτ = τ A + C,
thus τ Bτ + Dτ − τ A − C = 0. If B = C = 0 and A = D = nId for some n ∈ Z there are no conditions
imposed on τ . Otherwise the coefficients of τ must satisfy certain non trivial quadratic polynomials
with coefficients in Z, but in general this cannot happen.
References
[1] G. Cornell, J.H. Silverman Arithmetic Geometry, Springer-Verlag New York Inc., 1986.
[2] C. Birkenhake, H. Lange, Complex Abelian Varieties, Springer-Verlag Berlin Heidelberg, 2004.
[3] P. Griffiths, J. Harris, Principle of Algebraic Geometry, John Wiley & Sons, Inc., 1978.
[4] S. Lang, Introduction to Algebraic and Abelian Functions, Springer-Verlag New York, 1983.
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