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: [email protected]