A character of Siegel modular group of level 2 from theta constants

0.11 MB·

by  Xinhua Xiong

A CHARACTER OF THE SIEGEL MODULAR GROUP
OF LEVEL 2 FROM THETA CONSTANTS
7
1
0 XINHUA XIONG
2

r
Abstract. Given a characteristic, we define a character of the
a
M Siegel modular group of level 2, the computations of their values
are also obtained. By using our theorems, some key theorems of

3 Igusa [1] can be recovered.
2

]
T
N
1. Introduction.
.
h
t The theta function of characteristic m of degree g is the series
a
m
m′ m′
[ θ (τ,z) := expπi p+ ,τ(p+ )
m
(cid:20)(cid:18) 2 2 (cid:19)
2 pX∈Zg
v
m′ m′′
1
+2 p+ ,z + ,
5 (cid:18) 2 2 (cid:19)(cid:21)
5
2
m′
0 where τ ∈ H , z ∈ Cg, m = ∈ Z2g, H is the Siegel upper
. g (cid:18)m′′(cid:19) g
1
0 half-plane, m′ and m′′ denote vectors in Zg determined by the first and
7 last g coefficients of m. If we put z = 0, we get the theta constant
1
θ (τ) = θ (τ,0). The study of theta functions and theta constants
: m m
v
has a long history, and they are very important objects in arithmetic
i
X
and geometry. They can be used to construct modular forms and to
r study geometric properties of abelian varieties. Farkas and Kra’s book
a
[1] contains very detailed descriptions for the case of degree one. In
[2], [3], and [4], Matsuda gives new formulas and applications. It is
Igusa in [5] who began to study the cases of higher degrees. He used
θ (τ)θ (τ) to determine the structure of the graded rings of modular
m n
forms belonging to the group Γ (4,8).
g
In this note, we will define a character of the group Γ (2), the prin-
g
cipal congruence group of degree g and of level 2. We obtained its
computation formula. Using our results, Igusa’s key Theorem 3 in [5]
can be recovered.
1991 Mathematics Subject Classification. 11F46, 11F27.
1
2 XINHUAXIONG
2. The Siegel modular group of level 2.
The Siegel modular group Sp(g,Z) of degree g is the group of 2g×2g
integral matrices M satisfying
0 I 0 I
M g tM = g ,
(cid:18)−Ig 0(cid:19) (cid:18)−Ig 0(cid:19)
in which tM is the transposition of M, I is the identity of degree g. If
g
a b
we put M = , the condition for M in Sp(g,Z) is atd−btc = I ,
(cid:18)c d(cid:19) g
atbandctdaresymmetric matrices. Infact, ifM isinSp(g,Z), thentbd
andtacarealsosymmetric, see[6, p. 437]. Inthispaper, wediscuss two
special subgroupsoftheSiegel modulargroup. Thefirst istheprincipal
congruence subgroup Γ (2) of degree g and of level 2 which is defined
g
by M ≡ I (mod 2). The second is the Igusa modular group Γ (4,8),
2g g
which is defined by M ≡ I (mod 4) and (atb) ≡ (ctd) ≡ 0 (mod 8).
2g 0 0
Where if s is a square matrix, we arrange its diagonal coefficients in a
natural order to form a vector (s) . The Siegel modular group Sp(g,Z)
0
acts on H by the formula
g
Mτ = (aτ +b)(cτ +d)−1,
a b
where τ ∈ H ,M = ∈ Sp(g,Z). An element m ∈ Z2g is called
g (cid:18)c d(cid:19)
a theta characteristic of degree g. If n is another characteristic, then
we have
θ (τ,z) = (−1)tm′n′′θ (τ,z).
m+2n m
Since
θ (τ,−z) = (−1)tm′m′′θ (τ,z),
m m
m is called even or odd according as tm′m′′ is even or odd. Only for
even m, theta constants are none zero. Given a characteristic m and
an element M in the Siegel modular group, we define
d −c m′ (ctd)
M ◦m = + 0 .
(cid:18)−b a (cid:19)(cid:18)m′′(cid:19) (cid:18)(atb) (cid:19)
0
This operation modulo 2 is a group action, i.e. M ◦ (M ◦ m) ≡
1 2
(M M ) ◦ m (mod 2). Next we explain the transformation formulas
1 2
for theta constants: for any m ∈ Z2g and M ∈ Sp(g,Z), we put
1
Φ (M) := − tm′tbdm′ +tm′′tacm′′
m
8
(cid:0)
−2tm′tbcm′′ −2t(atb) (dm′ −cm′′) ,
0
(cid:1)
A CHARACTER OF THE SIEGEL MODULAR GROUP OF LEVEL 2 FROM THETA CONSTANTS3
then we have
1
θM◦m(Mτ) = a(M)e(Φm(M))det(cτ +d)2θm(τ),
in which a(M) is an eighth root of unity depending only on M and the
choice of square root sign for det(cτ+d)21, and e(Φm(M)) = e2πiΦm(M).
From now on, we always discuss the group Γ (2), unless specified.
g
Hence, we can write
1
Φ (M) = − tm′tbdm′ +tm′′tacm′′ −2t(atb) dm′ .
m 0
8
(cid:0) (cid:1)
3. Main theorems and proofs.
The character mentioned in the abstract is as follows:
Definition. Let m ∈ Z2g,M ∈ Γ (2), we define χ (M) by
g m
1
θm(Mτ) = a(M)χm(M)det(cτ +d)2θm(τ),
wherea(M)comesfromthetransformationformulasofthetaconstants.
Theorem 1. For a fixed m, χ (M) is a character of Γ (2).
m g
The proof of Theorem 3.1 needs two lemmas, in which Γ (2) is es-
g
sential.
Lemma 2. If m,n ∈ Z2g and m ≡ n (mod 2), then Φ (M) ≡ Φ (M)
m n
(mod 1).
Proof. Let m = n+2∆, then
tm′tbdm′
= t(n′ +2∆′)tbd(n′ +2∆′)
= tn′tbdn′ +4t∆′tbd∆′ +2tn′tbd∆′ +2t∆′tbdn′
≡ tn′tbdn′ (mod 8),
since b ≡ 0 (mod 2) and tbd is symmetric, the last two terms are equal.
Similarly, tac is symmetric which implies that tm′′tacm′′ ≡ tn′′tacn′′
(mod 8). Moreover, 2t(atb) dm′ ≡ 2t(atb) dn′ (mod 8) is trivial. By
0 0
the definition of Φ (M), Lemma 3.2 is true.
m
Lemma 3. ForM,M′ ∈Γg(2), we haveΦM′◦m(M) ≡ Φm(M) (mod 1).
Proof. This lemma can be proved from the definition of M′ ◦m, M′ is
in Γ (2) and Lemma 3.2.
g
Proof of Theorem 3.1.
We firstly give a formula for χ (M). By the definition of the opera-
m
tion ◦, we can find a unique n in Z2g with M ◦n = m. Define ∆ ∈ Z2g
by m+2∆ = n, then by Lemma 3.2 and Lemma 3.3, we have
θ (Mτ)
m
4 XINHUAXIONG
= θ (Mτ)
M◦n
1
= a(M)e(Φn(M))det(cτ +d)2θn(τ)
1
= a(M)e(Φm(M))det(cτ +d)2θn(τ)
1
= a(M)e(Φm(M))det(cτ +d)2θm+2∆(τ)
= a(M)e(Φm(M))det(cτ +d)21(−1)tm′∆′′θm(τ).
Hence,
(1) χ (M) = e(Φ (M))(−1)tm′∆′′.
m m
To prove Theorem 3.1 is equivalent to prove χ (M )χ (M ) =
m 1 m 2
χ (M M ) for any M ,M ∈ Γ (2). Now fix M ,M and m, define
m 1 2 1 2 g 1 2
∆ ,∆ by m+2∆ = n ,M ◦n = m; m+2∆ = n ,M ◦n = m.
1 2 1 1 1 1 2 2 2 2
Write
a b a b
M = 1 1 ,M = 2 2 ,
1 (cid:18)c d (cid:19) 2 (cid:18)c d (cid:19)
1 1 2 2
by (1), we have
χm(M1) = e(Φm(M1))(−1)tm′∆′1′
and
χm(M2) = e(Φm(M2))(−1)tm′∆′2′.
In order to compute χ (M M ), we write τ′ = M τ, M = M M =
m 1 2 2 1 2
a b
, and define ∆ by m+2∆ = n¯ with M ◦n = m,M ◦n¯ = n .
(cid:18)c d(cid:19) 3 3 1 1 2 1
Then by Lemma 3.2 and 3.3, we have
θ (M M τ)
m 1 2
= θ (M τ′)
M1◦n1 1
= a(M1)e(Φn1(M1))det(c1τ′ +d1)21θn1(τ′)
= a(M1)e(Φm(M1))det(c1τ′ +d1)21θM2◦n¯(M2τ)
= a(M1)e(Φm(M1))det(c1τ′ +d1)21a(M2)e(Φn¯(M2))
1
×det(c2τ +d2)2θn¯(τ)
= a(M1)a(M2)e(Φm(M1))e(Φn1(M2))det(c1τ′ +d1)21
1
×det(c2τ +d2)2θn¯(τ)
= a(M )a(M )e(Φ (M ))e(Φ (M ))
1 2 m 1 m 2
1
×det(cτ +d)2θm+2∆3(τ)
= a(M M )e(Φ (M ))e(Φ (M ))
1 2 m 1 m 2
×det(cτ +d)21(−1)tm′∆′3′θm(τ),
A CHARACTER OF THE SIEGEL MODULAR GROUP OF LEVEL 2 FROM THETA CONSTANTS5
inwhich, weuse cτ+d = (c τ′+d )(c τ+d ) anda(M) = a(M )a(M ),
1 1 2 2 1 2
the later is implied by Igusa’s Theorem 2 in [5]. Hence,
χm(M1M2) = e(Φm(M1))e(Φm(M2))(−1)tm′∆′3′.
Now we compute
(−1)tm′∆′1′,(−1)tm′∆′2′ and (−1)tm′∆′3′.
From M ◦n = m, we get
1 1
ta m′ +tc m′′ −ta (c td ) −tc (atb )
(2) n = 1 1 1 1 1 0 1 1 1 0 .
1 (cid:18)tb m′ +td m′′ −tb (c td ) −td (atb ) (cid:19)
1 1 1 1 1 0 1 1 1 0
Therefore, from m+2∆ = n , we have
1 1
n′′ −m′′
∆′′ = 1
1 2
tb m′ (td −I )m′′ tb (c td ) td (atb )
= 1 + 1 g − 1 1 1 0 − 1 1 1 0
2 2 2 2
tb m′ (td −I )m′′ (tb )
≡ 1 + 1 g − 1 0 (mod 2),
2 2 2
by noting that b ≡ c ≡ 0 (mod 2) and a ≡ d ≡ I (mod 2). So
1 1 1 1 g
tm′∆′′
1
tm′tb m′ tm′(td −I )m′′ tm′(tb )
≡ 1 + 1 g − 1 0 (mod 2)
2 2 2
tm′(td −I )m′′
≡ 1 g (mod 2),
2
because tb1 (mod 2) is symmetric, which follows from the fact atb is
2 1
symmetric. Similarly,
tm′(td −I )m′′
tm′∆′′ ≡ 2 g (mod 2).
2 2
The computation for ∆′′ is more complicated. Recall that M ◦ n =
3 1 1
m, M ◦n¯ = n ,m+2∆ = n¯ and (2), we have
2 1 3
ta m′ +tc m′′ −ta (c td )
n ≡ 1 1 1 1 1 0 (mod 4).
1 (cid:18)tb m′ +td m′′ −td (atb ) (cid:19)
1 1 1 1 1 0
From M ◦n¯ = n , we get
2 1
ta n′ +tc n′′ −ta (c td )
n¯ ≡ 2 1 2 1 2 2 2 0 (mod 4).
(cid:18)tb n′ +td n′′ −td (atb ) (cid:19)
2 1 2 1 2 2 2 0
Therefore,
n¯′′ ≡ tb (ta m′ +tc m′′ −ta (c td ) )+td (tb m′
2 1 1 1 1 1 0 2 1
+td m′′ −td (atb ) )−td (atb ) (mod 4)
1 1 1 1 0 2 2 2 0
6 XINHUAXIONG
≡ tb ta m′ +td tb m′ +td td m′′
2 1 2 1 2 1
−td td (atb ) −td (a tb ) (mod 4).
2 1 1 1 0 2 2 2 0
By the definition of ∆ , we have
3
n¯′′ −m′′
∆′′ =
3 2
tb ta m′ td tb m′ (td td −I )m′′
≡ 2 1 + 2 1 + 2 1 g
2 2 2
td td (atb ) td (a tb )
− 2 1 1 1 0 − 2 2 2 0 (mod 2)
2 2
tb m′ tb m′ (td td −I )m′′
≡ 2 + 1 + 2 1 g
2 2 2
(tb ) (tb )
− 1 0 − 2 0 (mod 2),
2 2
and
tm′∆′′
3
tm′tb m′ tm′tb m′ tm′(td td −I )m′′
≡ 2 + 1 + 2 1 g
2 2 2
tm′(tb ) tm′(tb )
− 1 0 − 2 0 (mod 2)
2 2
tm′(td td −I )m′′
≡ 2 1 g (mod 2),
2
by using the expansions of quadratic forms and the fact that tb2, tb1
2 2
are symmetric modulo 2. Finally, the verification of
tm′∆′′ ≡ tm′∆′′ +tm′∆′′ (mod 2)
1 2
is easy, which comes from the simple fact
(td −I )(td −I ) ≡ 0 (mod 4).
2 g 1 g
This completes the proof of Theorem 3.1.
In [5], Igusa gave the generators A ,B ,C of Γ (2), where
ij ij ij g
a 0
(1) 1 ≤ i 6= j ≤ g, A = ,d = ta−1,a is obtained by
ij (cid:18)0 d(cid:19)
replacing (i,j)-coefficient in I by 2;
g
a 0
(2) 1 ≤ i ≤ g, A = ,d = ta−1,a is obtained by replacing
ii (cid:18)0 d(cid:19)
(i,i)-coefficient in I by −1;
g
I b
(3) 1 ≤ i < j ≤ g, B = g ,b is obtained by replacing (i,j)-
ij (cid:18)0 Ig(cid:19)
and (j,i)-coefficients in 0 by 2;
A CHARACTER OF THE SIEGEL MODULAR GROUP OF LEVEL 2 FROM THETA CONSTANTS7
I b
(4) 1 ≤ i ≤ g, B = g ,b is obtained by replacing (i,i)-
ii (cid:18)0 Ig(cid:19)
coefficient in 0 by 2;
(5) 1 ≤ i ≤ j ≤ g, C = tB .
ij ij
By noting the computation of ∆, which depends on m and M, we find
(−1)tm′∆′′ = 1forM = B orC , becauseinthesecases, d = I , hence
ij ij g
tm′∆′′ = 0. IfM = A , itiseasytofindthattm′∆′′ = −m′m′′ ≡ m′m′′
ij i j i j
(mod 2). We can easily compute Φ (M) for M = A ,B and C .
m ij ij ij
Now the values of χ (M) for the generators are
m
χm(Aij) = (−1)m′im′j′,χm(Bij) = (−1)m′im′j,
χm(Bii) = (−1)m′ie −(m4′i)2 ,
(cid:16) (cid:17)
χm(Cii) = e −(m4′i′)2 ,χm(Cij) = (−1)m′i′m′j′.
(cid:16) (cid:17)
Usingthedefinitions ofΦ (M) andtm′∆′′, itiseasytoprove χ (M) =
m m
1 for M in the Igusa modular group Γ (4,8). Hence, by the computa-
g
tions above, we get
Theorem 4. Write M in the form
M = Apij Bqij CrijM′
ij ij ij
1≤Yi,j≤g 1≤Yi≤j≤g 1≤Yi≤j≤g
with p ,q ,r ∈ Z and M′ is in the commutator subgroup of Γ (2),
ij ij ij g
which is in Γ (4,8), then
g
B
χ (M) = (−1)Ae(− )
m
4
with
A = p m′m′′ + q m′m′
ij i j ij i j
1≤Xi,j≤g 1≤Xi≤j≤g
+ r m′′m′′,
ij i j
1≤Xi<j≤g
B = q (m′)2 + r (m′′)2.
ii i ii i
1X≤i≤g 1X≤i≤g
4. Applications.
If we define ψ(τ) = θ (τ)θ (τ), then for M ∈ Γ (2),
m n g
ψ(Mτ) = a2(M)χ (M)χ (M)det(cτ +d)θ (τ)θ (τ),
m n m n
we find χ (M)χ (M) is exactly the character defined by Igusa in [5],
m n
hence ourtheorems canrecover Igusa’sTheorem 3in[5]. Ourcharacter
8 XINHUAXIONG
χ (M)ismorefundamental, moreoverwecanseetherelationsbetween
m
χ (M) and Φ (M).
m m
We can use our results to give a more transparent proof of the key
part of Theorem 5 in Igusa’s paper [5]. The key part of Theorem 5 in
that paper is from the invariant condition that θm(Mτ) = θm(τ) holds
θn(Mτ) θn(τ)
for all even m,n to infer M is in Γ (4,8), here M is in Γ (2). By the
g g
definition of χ (M), this is equivalent to the congruence χ (M) =
m m
χ (M) holds for all even m,n, i.e.
n
tm′(td−Ig)m′′
e(Φm(M))(−1) 2
is equal to
tn′(td−Ig)n′′
e(Φn(M))(−1) 2 .
Let n′ = n′′ = 0, we get for any even m,
tm′(td −I )m′′
g
≡ 0 (mod 2)
2
and
tm′tbdm′ +tm′′tacm′′ −2t(atb) dm′ ≡ 0 (mod 8).
0
The first congruence implies d ≡ I (mod 4), from atd−btc = I , we
g g
get a ≡ I (mod 4). In the second congruence, let m′ = 0, we get
g
tm′′tacm′′ ≡ 0 (mod 8), this implies tac ≡ 0 (mod 4), hence c ≡ 0
(mod 4), and (tac) ≡ 0 (mod 8). If we write ta = I + 4a¯, then we
0 g
have
(tac) = ((I +4a¯)c) ≡ (c) ≡ 0 (mod 8).
0 g 0 0
Let m′′ = 0, we get the congruence
tm′tbdm′ −2t(atb) dm′ ≡ 0 (mod 8),
0
which is equivalent to the congruence
(3) tm′tbdm′ −2t(b) m′ ≡ 0 (mod 8).
0
Write tb = (2b ) and d = I +4d¯, then
ij g
tm′tbdm′ −2t(b) m′
0
= tm′2b (I +4d)m′ −2(2b ) m′
ij g ij 0
≡ 2tm′b m′ −4(b ) m′ (mod 8)
ij ij 0
≡ 2b m′2 −4b m′ (mod 8)
11 1 11 1
by taking tm′ = (m′,0,0, ··· ,0). Hence (3) implies b ≡ 0 (mod 4).
1 11
Similarly, we can prove b ≡ 0 (mod 4) holds for each 1 ≤ i ≤ g.
ii
Therefore, we have (b) ≡ 0 (mod 8). Combing it with (3), we find the
0
congruence tm′tbdm′ ≡ 0 (mod 8) holds for any even m′, which implies
A CHARACTER OF THE SIEGEL MODULAR GROUP OF LEVEL 2 FROM THETA CONSTANTS9
tbd ≡ 0 (mod 4), hence b ≡ 0 (mod 4). The analysis above shows that
M is in Γ (4) and (b) ≡ (c) ≡ 0 (mod 8). The observation of Igusa
g 0 0
in [5, p. 222, line 4-line 6] shows M is in Γ (4,8).
g
Acknowledgments. The author would like to thank the referee for
his/her helpful corrections and suggestions.
References
[1] Farkas, H.M. and Kra, I.Theta constants, Riemann surfaces and the Modular
group, AMS Grad. Studies in Math., vol 37, (2001)
[2] Matsuda,K.Generalizationsofthe Farkasidentity formodulus 4and7.Proc.
Japan Acad. Ser. A 89 (2013), 129-132.
[3] Matsuda, K. The determinant expressions of some theta constant identities.
Ramanujan J. 34 (2014), 449-456.
[4] Matsuda, K. Analogues of Jacobi’s derivative formula. Ramanujan J. 39
(2016), 31-47.
[5] Igusa, J. -I, On the graded ring of theta constants. Am. J. Math. 86 (1964),
no. 1, 219-246.
[6] Salvati Manni, S., Thetanullwerte and stable modular forms. Am. J. Math.
111 (1980), no. 3, 435-455.
DepartmentofMathematics, ChinaThreeGorgesUniversity,Yichang,
Hubei Province, 443002, P.R. China,
E-mail address: [e-mail]