Siegel families with application to class fields

0.2 MB·

by  Ja Kyung Koo

Siegel families with application to class fields
Ja Kyung Koo, Dong Hwa Shin and Dong Sung Yoon∗
6
1
0
2
Abstract

r
p
We investigate certain families of meromorphic Siegel modular functions on which Galois
A
groups act in a natural way. By using Shimura’s reciprocity law we construct some algebraic

8
numbersinthe rayclassfieldsofCM-fieldsintermsofspecialvaluesoffunctions intheseSiegel

] families.
T
N
. 1 Introduction
h
t
a
m For a positive integer N let FN be the field of meromorphic modular functions of level N (defined
[ on H = τ C Im(τ) > 0 ) whose Fourier coefficients belong to the Nth cyclotomic field. As is
{ ∈ | }

2 well known, FN is a Galois extension of F1 whose Galois group is isomorphic to GL2(Z/NZ)/ I2
{± }
v
([8, 6.1–6.2]). Now, let N 2 and consider a set
4 § ≥
0
4 V = v Q2 N is the smallest positive integer for which Nv Z2
N
{ ∈ | ∈ }
0
.0 as the index set. We call a family {fv(τ)}v∈VN of functions in FN a Fricke family of level N if each
1 fv(τ) depends only on v (mod Z2) and satisfies
0 ±
6
1 fv(τ)α = fαTv(τ) (α GL2(Z/NZ)/ I2 ),
∈ {± }
:
v
where αT means the transpose of α. For example, Siegel functions of one-variable form such a
i
X
Fricke family of level N ([5, Proposition 1.3 in Chapter 2]). See also [2] or [4].
r
a Let K be an imaginary quadratic field with the ring of integers , and let f be a proper non-
K
O
trivial ideal of . We denote by Cl(f) and K the ray class group modulo f and its corresponding
K f
O
ray class field modulo f, respectively. If fv(τ) v is a Fricke family of level N in which every fv(τ)
{ }
is holomorphic on H, then we can assign to each ray class Cl(f) an algebraic number f ( ) as a
f
C ∈ C
special value of a function in fv(τ) v. Furthermore, we attain by Shimura’s reciprocity law that
{ }
f ( ) belongs to K and satisfies
f f
C
f ( )σf(D) = f ( ) ( Cl(f)),
f f
C CD D ∈
2010 Mathematics Subject Classification. Primary 11F46, Secondary 11G15.
Keywords andphrases. abelianvarieties,classfieldtheory,CM-fields,Shimura’sreciprocitylaw,Siegelmodular
functions.
∗Corresponding author.
The second named author was supported by HankukUniversity of Foreign StudiesResearch Fundof 2016.
1
where σ is the Artin reciprocity map for f ([5, Theorem 1.1 in Chapter 11]).
f
In this paper, we shall define a Siegel family h (Z) of level N consisting of meromorphic
M M
{ }
Siegel modular functions of (higher) genus g and level N, which would be a generalization of a
Fricke family of level N in case g = 1 (Definition 3.1). It turns out that every Siegel family of level
N is induced from a meromorphic Siegel modular function for the congruence subgroup Γ1(N)
(Theorem 3.5).
Let K be a CM-field and let f = N . Given a Siegel family h (Z) of level N, we
K M M
O { }
shall introduce a number h ( ) by a special value of a function in h (Z) for each ray class
f M M
C { }
Cl(f) (Definition 4.4). Under certain assumptions on K (Assumption 4.1) we shall prove that
C ∈
if h ( ) is finite, then it lies in the ray class field K whose Galois conjugates are of the same form
f f
C
(Theorem 6.2 and Corollary 6.3). To this end, we assign a principally polarized abelian variety to
each nontrivial ideal of , and apply Shimura’s reciprocity law to h ( ).
K f
O C
2 Actions on Siegel modular functions
First, we shall describe the Galois group between fields of meromorphic Siegel modular functions
in a concrete way.
O I
g g
Let g be a positive integer, and let η = − . For every commutative ring R with unity
g
"Ig Og #
we denote by
GSp (R) = α GL (R) αTη α = ν(α)η with ν(α) R× ,
2g ∈ 2g | g g ∈
Sp (R) = (cid:8)α GSp (R) ν(α) = 1 . (cid:9)
2g { ∈ 2g | }
Let
G = GSp (Q),
2g
and let G be the adelization of G, G its non-archimedean part and G its archimedean part.
A 0 ∞
One can extend the multiplier map ν : G Q× continuously to the map ν :G Q×, and set
→ A → A
G = α G ν(α) > 0 , G = G G , G = G G .
∞+ ∞ A+ 0 ∞+ + A+
{ ∈ | } ∩
Furthermore, let
I O
∆ = g g s Z× ,
("Og sIg# | ∈ p p)
Y
U = GSp (Z ) G ,
1 2g p × ∞+
p
Y
U = x U x I (mod N M (Z )) for all rational primes p
N 1 p 2g 2g p
{ ∈ | ≡ · }
for every positive integer N. Then we have
U EU G and G = U ∆G
N 1 A+ A+ N +
≤
2
([10, Lemma 8.3 (1)]).
Note that the group G acts on the Siegel upper half-space H = Z M (C) ZT =
∞+ g g
{ ∈ |
Z, Im(Z) is positive definite by
}
α(Z) = (AZ +B)(CZ +D)−1 (α G , Z H ),
∞+ g
∈ ∈
where A,B,C,D are g g block matrices of α. Let be the field of meromorphic Siegel modular
N
× F
functions of genus g for the congruence subgroup
Γ(N) = γ Sp (Z) γ I (mod N M (Z))
∈ 2g | ≡ 2g · 2g
(cid:8) (cid:9)
ofthesymplecticgroupSp (Z)whoseFouriercoefficients belongtotheNthcyclotomic fieldQ(ζ )
2g N
with ζ = e2πi/N. That is, if f , then
N N
∈ F
c(h)e(tr(hZ)/N)
f(Z)= h for some c(h),d(h) Q(ζ ),
N
d(h)e(tr(hZ)/N) ∈
Ph
where the denominator andPnumerator of f are Siegel modular forms of the same weight, h runs
over all g g positive semi-definite symmetric matrices over half integers with integral diagonal
×
entries, and e(w) = e2πiw for w C ([3, Theorem 1 in 4]). Let
∈ §
∞
= .
N
F F
N=1
[
Proposition 2.1. There exists a homomorphism τ : G Aut( ) satisfying the following
A+
→ F
c(h)e(tr(hZ)/N)
properties: Let f(Z)= h .
Phd(h)e(tr(hZ)/N) ∈ FN
P
(i) If α G = α G ν(α) > 0 , then
+
∈ { ∈ | }
fτ(α) = f α.
◦
I O
g g
(ii) If β = ∆ and t is a positive integer such that t s (mod NZ ) for all rational
p p
"Og sIg# ∈ ≡
primes p, then
c(h)σe(tr(hZ)/N)
fτ(β) = h ,
d(h)σe(tr(hZ)/N)
Ph
where σ is the automorphism of Q(ζN)Pgiven by ζNσ = ζNt .
(iii) For every positive integer N we have
= f fτ(x) = f for all x U .
N N
F { ∈F | ∈ }
(iv) ker(τ) = Q×G .
∞+
Proof. See [10, Theorem 8.10].
3
Since
U / G if N = 1,
U (Q×G )/Q×G U /(U Q×G ) 1 ± ∞+
N ∞+ ∞+ N N ∞+
≃ ∩ ≃ ( UN/G∞+ if N > 1,
we see by Proposition 2.1 (iii) and (iv) that is a Galois extension of with
N 1
F F
Gal( / ) U / U . (1)
N 1 1 N
F F ≃ ±
Proposition 2.2. We have
Gal( / ) GSp (Z/NZ)/ I .
FN F1 ≃ 2g {± 2g}
Proof. Let α U . Take a matrix A in M (Z) for which A α (mod N M (Z )) for
1 2g p 2g p
∈ ≡ ·
all rational primes p. Define a matrix ψ(α) M (Z/NZ) by the image of A under the natural
2g
∈
reduction M (Z) M (Z/NZ). Then by the Chinese remainder theorem ψ(α) is well defined
2g 2g
→
and independent of the choice of A. Furthermore, let t be an integer relatively prime to N such
that t ν(α ) (mod NZ ) for all rational primes p. We then derive that
p p
≡
tη ν(α )η αTη α ATη A ψ(α)Tη ψ(α) (mod N M (Z ))
g ≡ p g ≡ p g p ≡ g ≡ g · 2g p
for all rational primes p, and hence ψ(α) GSp (Z/NZ). Thus we obtain a group homomorphism
∈ 2g
ψ : U GSp (Z/NZ).
1 → 2g
Let β GSp (Z/NZ), and take a preimage B of β under the natural reduction M (Z)
∈ 2g 2g →
M (Z/NZ). Since ν(β) (Z/NZ)× and
2g
∈
BTη B βTη β ν(β)η (mod N M (Z)),
g g g 2g
≡ ≡ ·
B belongs to GSp (Z ) for every rational prime p dividing N. Let α = (α ) be the element of
2g p p p
GSp (Z ) given by
p 2g p
B if p N,
Q
α = |
p
( I2g otherwise.
We then see that α U and ψ(α) = β. Thus ψ is surjective.
1
∈
Clearly, U is contained in ker(ψ). Let γ ker(ψ). Since γ I (mod N M (Z )) for all
N p 2g 2g p
∈ ≡ ·
rational primes p, we get γ U , and hence ker(ψ) = U . Therefore ψ induces an isomorphism
N N
∈
U /U GSp (Z/NZ), from which we achieve by (1)
1 N 2g
≃
Gal( / ) U / U GSp (Z/NZ)/ I .
N 1 1 N 2g 2g
F F ≃ ± ≃ {± }
Remark 2.3. We have the decomposition
Gal( / ) GSp (Z/NZ)/ I G Sp (Z/NZ)/ I ,
FN F1 ≃ 2g {± 2g} ≃ N · 2g {± 2g}
4
where
I O
G = g g ν (Z/NZ)× .
N
("Og νIg# | ∈ )
By Proposition 2.1 one can describe the action of GSp (Z/NZ)/ I on as follows:
2g 2g N
{± } F
c(h)e(tr(hZ)/N)
Let f(Z)= h .
Phd(h)e(tr(hZ)/N) ∈ FN
P
I O
g g
(i) An element β = of G acts on f by
N
"Og νIg#
c(h)σe(tr(hZ)/N)
fβ = h ,
d(h)σe(tr(hZ)/N)
Ph
where σ is the automorphism of Q(ζ P) satisfying ζσ = ζν .
N N N
(ii) An element γ of Sp (Z/NZ)/ I acts on f by
2g 2g
{± }
fγ = f γ′,
◦
where γ′ is any preimage of γ under the natural reduction Sp (Z) Sp (Z/NZ)/ I .
2g → 2g {± 2g}
3 Siegel families of level N
By making use of the description of Gal( / ) in 2 we shall introduce a generalization of a
N 1
F F §
Fricke family in higher dimensional cases.
Let N 2. For α M (Z) we denote by α its reduction modulo N. Define a set
2g
≥ ∈
AT A B
e
= (1/N) α= M (Z) such that α GSp (Z/NZ) .
VN ( "BT# | "C D#∈ 2g ∈ 2g )
e
Let M be an element of stemmed from α M (Z) such that α GSp (Z/NZ), and let β be
VN ∈ 2g ∈ 2g
an element of M (Z) satisfying β GSp (Z/NZ). Then it is straightforward that βTM is also
2g ∈ 2g
an element of given by the product αβ. e
N
V e
Definition 3.1. We call a family h (Z) a Siegel family of level N if it satisfies the
{ M }M∈VN
following properties:
(S1) Each h (Z) belongs to .
M N
F
(S2) h (Z) depends only on M (mod M (Z)).
M 2g×g
±
(S3) h (Z)σ = h (Z) for all σ GSp (Z/NZ)/ I Gal( / ).
M σTM ∈ 2g {± 2g} ≃ FN F1
By we mean the set of such Siegel families of level N.
N
S
Remark 3.2. Let h (Z) .
M M N
{ } ∈ S
(i) The property (S3) yields a right action of the group GSp (Z/NZ)/ I on h (Z) .
2g {± 2g} { M }M
5
AT A B
(ii) Let M = (1/N) , and so there is a matrix α = M (Z) such that
N 2g
"BT# ∈ V "C D# ∈
α GSp (Z/NZ). Considering α as an element of GSp (Z/NZ)/ I we obtain
∈ 2g 2g {± 2g}
h (Z)α = h (Z) = h (Z).
e (1/N) Ige (1/N)αT Ig M
(cid:20)Og(cid:21) e (cid:20)Og(cid:21)
Thus the action of GSp (Z/NZ)/ I on h (Z) is transitive.
2g {± 2g} { M }M
Let
I O
Γ1(N) = γ Sp (Z) γ g g (mod N M (Z)) ,
2g 2g
( ∈ | ≡ " Ig# · )
∗
andlet 1(Q)bethefieldofmeromorphicSiegelmodularfunctionsforΓ1(N)withrationalFourier
FN
coefficients.
Lemma 3.3. If h (Z) , then h (Z) 1(Q).
{ M }M ∈ SN (1/N)Ig ∈ FN
(cid:20) Og (cid:21)
A B
Proof. For any γ = Γ1(N) we deduce by (S2) and (S3) that
"C D#∈
h (γ(Z)) = h (Z)γ = h (Z) = h (Z) = h (Z)
(cid:20)(1/ONg)Ig(cid:21) (cid:20)(1/ONg)Ig(cid:21) e γT(cid:20)(1/ONg)Ig(cid:21) (1/N)hBATTi (cid:20)(1/ONg)Ig(cid:21)
because A I , B O (mod N M (Z)). Thus h (Z) is modular for Γ1(N).
≡ g ≡ g · g (1/N)Ig
(cid:20) Og (cid:21)
For every ν (Z/NZ)× we see by (S2) and (S3) that
∈
Ig Og
h (Z)(cid:20)Og νIg(cid:21) = h (Z)= h (Z),
(1/N)Ig Ig Og (1/N)Ig (1/N)Ig
(cid:20) Og (cid:21) (cid:20)Og νIg(cid:21)(cid:20) Og (cid:21) (cid:20) Og (cid:21)
which implies that h (Z) has rational Fourier coefficients. This proves the lemma.
(1/N)Ig
(cid:20) Og (cid:21)
One can consider as a field under the binary operations
N
S
h (Z) + k (Z) = (h +k )(Z) ,
M M M M M M M
{ } { } { }
h (Z) k (Z) = (h k )(Z) .
M M M M M M M
{ } ·{ } { }
By Lemma 3.3 we get the ring homomorphism
φ : 1(Q)
N SN → FN
h (Z) h (Z).
{ M }M 7→ (1/N)Ig
(cid:20) Og (cid:21)
A B
Lemma 3.4. If M , then there is γ = M (Z) such that γ Sp (Z/NZ) and
∈ VN "C D# ∈ 2g ∈ 2g
AT
e
M = (1/N) .
"BT#
6
A B AT
Proof. Let α = M (Z) such that α GSp (Z/NZ) and M = (1/N) . In
"U V# ∈ 2g ∈ 2g "BT#
M (Z/NZ), decompose α as
2g e
I O A B
α= eg g with ν = ν(α) (Z/NZ)×
"Og νIg#"ν−1U ν−1V# ∈
e e
A B
so that belongs to Sp (Z/NZ). Since the reduction Sp (Z) Sp (Z/NZ) is
"ν−1U ν−1V# 2g 2g → 2g
A B
surjective ([7]), we can take γ M (Z) satisfying γ = .
∈ 2g "ν−1U ν−1V#
Theorem 3.5. and 1(Q) are isomorphic evia φ .
SN FN N
Proof. Since and 1(Q) are fields, it suffices to show that φ is surjective.
SN FN N
A B
Let h(Z) 1(Q). For each M , take any γ = M (Z) such that γ
∈ FN ∈ VN "C D# ∈ 2g ∈
AT
e
Sp (Z/NZ) and M = (1/N) by using Lemma 3.4. And, set
2g "BT#
h (Z)= h(Z)γ.
M
e
A B
We claim that h (Z) is independent of the choice of γ. Indeed, if γ′ = M (Z) such
M "C′ D′# ∈ 2g
that γ′ Sp (Z/NZ), then we attain in M (Z/NZ) that
∈ 2g 2g
e A B DT BT I O
γ′γ−1 = − = g g
"C′ D′#" CT AT # " Ig#
− ∗
ee
by the fact γ,γ′ Sp (Z/NZ). Let δ be an element of Sp (Z) such that δ = γ′γ−1. We then
2g 2g
∈
achieve
e e h(Z)γ′ = (h(Z)γ′γ−1)γ = h(δ(Z))γ = h(Z)γ e ee
e ee e e e
because h(Z) is modular for Γ1(N) and δ Γ1(N).
∈
P Q
Now, for any σ = GSp (Z/NZ)/ I with ν = ν(σ) we derive that
"R S# ∈ 2g {± 2g}
h (Z)σ = h(Z)γσ
M
eA B P Q
= h(Z)hC DihR Si
Ig Og AP+BR AQ+BS
= h(Z)(cid:20)Og νIg(cid:21)(cid:20)ν−1(CP+DR) ν−1(CQ+DS)(cid:21)
AP+BR AQ+BS
= h(Z)(cid:20)ν−1(CP+DR) ν−1(CQ+DS)(cid:21) since h(Z) has rational Fourier coefficients
= h (Z)
(AP+BR)T
(cid:20)(AQ+BS)T (cid:21)
7
= h (Z)
PT RT AT
(cid:20)QT ST (cid:21)hBTi
= h (Z).
σTM
This shows that the family h (Z) belongs to . Furthermore, since
M M N
{ } S
Ig Og
φ ( h (Z) )= h (Z) = h(Z)(cid:20)Og Ig (cid:21) = h(Z),
N { M }M (1/N)Ig
(cid:20) Og (cid:21)
φ is surjective as desired.
Remark 3.6. (i) By Proposition 2.2 and Remark 2.3 we obtain
I O
Gal( / 1(Q)) G γ Sp (Z/NZ)/ I γ = g g .
FN FN ≃ N ·( ∈ 2g {± 2g} | ±" Ig#)
∗
(ii) Let (Q) be the field of meromorphic Siegel modular functions for
1,N
F
I
g
Γ (N) = γ Sp (Z) γ ∗ (mod N M (Z))
1 2g 2g
( ∈ | ≡ "Og Ig# · )
with rational Fourier coefficients. If we set
(1/√N)I O
g g
ω = ,
" Og √NIg#
then we know that ω Sp (R) and
∈ 2g
A B A (1/N)B A B
ω ω−1 = for all Sp (R).
"C D# "NC D # "C D#∈ 2g
This implies
ωΓ1(N)ω−1 = Γ (N),
1
and so (Q) and 1(Q) are isomorphic via
F1,N FN
(Q) 1(Q)
F1,N → FN
h(Z) (h ω)(Z) = h((1/N)Z).
7→ ◦
4 Special values associated with a Siegel family
As an application of a Siegel family of level N we shall construct a number associated with each
ray class modulo N of a CM-field.
Let n be a positive integer, K be a CM-field with [K : Q] = 2n and ϕ ,...,ϕ be a set of
1 n
{ }
embeddings of K into C such that (K, ϕ n ) is a CM-type. We fix a finite Galois extension L of
{ i}i=1
Q containing K, and set
S = σ Gal(L/Q) σ = ϕ for some i 1,2,...,n ,
K i
{ ∈ | | ∈ { }}
8
S∗ = σ−1 σ S ,
{ | ∈ }
H∗ = γ Gal(L/Q) γS∗ = S∗ .
{ ∈ | }
Let K∗ be the subfield of L corresponding to the subgroup H∗ of Gal(L/Q), and let ψ ,...,ψ
1 g
{ }
be the set of all embeddings of K∗ into C arising from the elements of S∗. Then we know that
(K∗, ψ g ) is a primitive CM-type and
{ j}j=1
n
K∗ = Q aϕi a K
| ∈
!
i=1
X
([9, Proposition 28 in 8.3]). We call this CM-type (K∗, ψ g ) the reflex of (K, ϕ n ). Using
§ { j}j=1 { i}i=1
this CM-type we define an embedding
Ψ :K∗ Cg
→
aψ1
.
a  .. .
7→
aψg
 
 
For each purely imaginary element c of K∗ we associate an R-bilinear form
E : Cg Cg R
c
× →
u v
g 1 1
(u,v) cψj(ujvj ujvj) (u =  ... ,v =  ... ).
7→ −
j=1
X ug vg
   
   
Then, one can readily check that
Ec(Ψ(a),Ψ(b)) = TrK∗/Q(cab) for all a,b K∗ (2)
∈
by utilizing the fact aψj = aψj for all a K∗ (1 j g).
∈ ≤ ≤
Assumption 4.1. In what follows we assume the following conditions:
(i) (K∗)∗ = K.
(ii) Thereis apurelyimaginary element ξ ofK∗ andaZ-basis a1,...,a2g ofthelattice Ψ( K∗)
{ } O
in Cg for which
O I
E (a ,a ) = g − g .
ξ i j
1≤i,j≤2g "Ig Og #
h i
Inthiscase, wesay thatthecomplex torus(Cg/Ψ( K∗),Eξ)is aprincipallypolarized abelian
O
variety with a symplectic basis a ,...,a . See [9, 6.2].
1 2g
{ } §
(iii) f = N for an integer N 2.
K
O ≥
Remark 4.2. The Assumption 4.1 (i) is equivalent to saying that (K, ϕ n ) is a primitive
{ i}i=1
CM-type, namely, the abelian varieties of this CM-type are simple ([9, Proposition 26 in 8.2]).
§
9
By Assumption 4.1 (i) one can define a group homomorphism
g : K× (K∗)×
→
n
d dϕi,
7→
i=1
Y
and extend it continuously to the homomorphismg : K× (K∗)× of idele groups. Itis also known
A → A
that for each fractional ideal a of K there is a fractional ideal (a) of K∗ such that
G
n
(a) = (a )ϕi
L L
G O O
i=1
Y
([9, 8.3]). Let be a given ray class in Cl(f). Take any integral ideal c in , and let
§ C C
(c) = (c) = /c.
K/Q K
N N |O |
Lemma 4.3. (Cg/Ψ( (c)−1),E ) is also a principally polarized abelian variety.
ξN(c)
G
Proof. It follows from (2) that
EξN(c)(Ψ( (c)−1),Ψ( (c)−1)) = TrK∗/Q(ξ (c) (c)−1 (c)−1)
G G N G G
= TrK∗/Q(ξ K∗)
O
= Eξ(Ψ( K∗),Ψ( K∗))
O O
Z
⊆
becauseEξ isaRiemannformonCg/Ψ( K∗). ThusEξN(c)definesaRiemannformonCg/Ψ( (c)−1).
O G
Now, let b ,...,b be a symplectic basis of the abelian variety (Cg/Ψ( (c)−1),E ) so
1 2g ξN(c)
{ } G
that
2g
O
Ψ( (c)−1)= Zb and E (b ,b ) = g −E ,
G Xj=1 j h ξN(c) i j i1≤i,j≤2g " E Og#
ε 0
1
···
where =  ... ... ...  is a g g diagonal matrix for some positive integers ε1,...,εg satisfying
E ×
0 εg
 ··· 
ε ε .Furthermore, let b ...,b be elements of (c)−1 such that b = Ψ(b ) (1 j 2g).
1 g 1 2g j j
| ··· | G ≤ ≤
Since K∗ (c)−1, we have
O ⊆ G
a a = b b α for some α M (Z) GL (Q), (3)
1 2g 1 2g 2g 2g
··· ··· ∈ ∩
h i h i
and hence
aψ1 aψ1 bψ1 bψ1
1 ··· 2g 1 ··· 2g
. . . .
. . . .
 . .   . . 
aψg aψg bψg bψg
 1 ··· 2g =  1 ··· 2gα.
aψ1 aψ1 bψ1 bψ1
 1 ··· 2g  1 ··· 2g
 . .   . . 
 .. ..   .. .. 
   
   
aψg aψg bψg bψg
 1 ··· 2g  1 ··· 2g
   
10