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1 1.1 SL (R 1.1.1 SL (R H SL (R SL (R H H H = {z = x + iy C; x, y R, y > 0}, SL (R = {g M (R; dt(g = 1}, gτ = aτ + b a b g = SL (R cτ + d c d 1.1. Γ H H SL (R f(τ f(gτ G SL (R G H J(g, τ τ g G Hol f(τ f(gτj(g, τ 1 Hol 1

G H J(g, τ G g G J(g, τ τ J(g 1 g, τ = J(g 1, g τj(g, τ J J(1, τ = 1 k g = ( a b c d J(g, τ = (cτ + dk G = SL (R SL (R G G 1.1.3 α, β C α = α iθ (θ R α β = α β iθβ θ π < θ π θ α β 1.1.4 Γ multiplir systm J(g, τ G r J(g, τ = cτ + d r g = ( a c d b G J(g, τ r g H J(g, τ/(cτ + d r 1 g G v(g = J(g, τ (cτ + d r τ J(g, τ multiplir

1.1.5 Γ G r J 1, J multiplir v 1, v J 1 (g, τ J (g, τ = v 1(g v (g τ v 1 (g/v (g G G G/[G.G] [G, G] G 1. SL (R 1..1 n SL (R n G n (.g. Yoshida [9] G n = {(g, ϕ(g, τ; ϕ(g, τ n = (cτ + d} µ(g, τ cτ + d n g G n n G n (g 1, ϕ(g 1, τ(g, ψ(g, τ = (g 1 g, ϕ(g 1, g τψ(g, τ G n G n SL (R SL (R Z n G n 1.. SL (R Γ G n SL (R Γ SL (R 1/n J(M, τ Γ = {(M, J(M, τ; M Γ} G n G n Γ G n 3

SL (R G n Γ split n G n (g, ϕ(g, τf = f(gτϕ(g, τ 1 Hol.1 Γ SL (R ±1 γ Γ 1 1 γ γ(x = x R x Γ Γ SL (Z Q SL (Z (commnsurabl Γ SL (Z Γ Γ\SL (Q/(P SL (Q. P SL (R SL (Z Borl {( a Γ = Γ 0 (4 = c } b SL (Z; c 0 mod 4 d i, 0, 1/. Γ r multiplir systm v(m J(M, τ 4

M Γ v(m = 1 M Γ (f J [M](τ = f(gτj(m, τ 1 Dfinition.1 H f M Γ f J [M] = f Γ r multiplir v(m r T SL (R µ r (g, τ = (cτ + d r S, T σ(s, T = µ r(s, T τµ r (T, τ µ r (ST, τ µ r (T, τ Γ r J(M, τ multiplir v Γ L 1 ΓL r J L (L 1 T L, τ, multiplir systm v L T Γ J L (L 1 T L, τ = J(T, Lτµ r(l, τ µ r (L, L 1 T Lτ, v L (L 1 T L = v(t σ(t, L σ(l, L 1 T L Lmma. (1 J L (L 1 T L, τ L 1 ΓL L 5

( v L multiplir systm (3 L, T Γ J L (L 1 T L, τ = v L (L 1 T Lµ r (L 1 T L, τ J L (L 1 T L, τ = J(L 1 T L, τ J L (L 1 T 1 T L, τ = J(T 1T, Lτµ r (L, τ µ r (L, L 1 T 1 T Lτ = J(T 1, T LτJ(T, Lτµ r (L, τ, µ r (L, L 1 T 1 T Lτ J L (L 1 T 1 L, L 1 T Lτ = J(T 1, T Lτµ r (L, L 1 T Lτ, µ r (L, L 1 T 1 LL 1 T Lτ J L (L 1 T L, τ = J(T, Lτµ r (L, τ µ r (L, L 1 T Lτ. (1 σ(t, Lµ r (T L, τ = µ r (T, Lτµ r (L, τ, σ(l, L 1 T Lµ r (T L, τ = σ(l, L 1 T Lµ r (LL 1 T L, τ = µ r (L, L 1 T Lτµ r (L 1 T L, τ. v L (L 1 T Lµ r (L 1 T L, τ = J(T, Lτµ r(l, τ µ r (L, L 1 T Lτ v(t σ(t, L = σ(l, L 1 T L. ( L Γ J(L, τ = v(lµ r (L, τ, J(L, L 1 T Lτ = v(lµ r (L, L 1 T Lτ J(T L, τ = J(LL 1 T L, τ = J(L, L 1 T LτJ(L 1 T L, τ = J(T, LτJ(L, τ 6

J L (L 1 T L, τ = (3 L SL (R J(T, LτJ(L, τ J(L, L 1 T Lτ = J(L 1 T L, τ (f r [L](τ = f(lτµ r (L, τ 1 = f(lτ(cτ + d r T Γ f(t τ = f(τj(t, τ (f r [L](L 1 T Lτ = (f r [L](τJ L (L 1 T L, τ f L (L 1 T Lτ = f(t Lτµ r (L, L 1 T Lτ = f(lτj(t, Lτµ r (L, L 1 T Lτ 1 = f(lτj L (L 1 T L, τµ r (L, τ = f L (τj L (L 1 T L, τ. Γ κ R κ L(i = κ L SL (R i L Γ J(L, τ J µ r f κ = f r [L] = f(lτµ r (L, τ 1 Γ κ κ Γ h 0 R ϵ = ±1 m 1 h0 {±1} { ; m Z}, ±1 Γ, L 1 Γ κ L = L 1 0 1 ΓL P = ( m 1 h0 { ϵ ; m Z} ±1 Γ. 0 1 7

{ h0 ±1 Γ ϵ = 1 h = h 0 ±1 Γ ϵ = 1 f Γ f κ L 1 ΓL f κ (L 1 T Lτ = J L (L 1 T L, τf κ (τ. f κ L 1 Γ k L f κ F F (τ + n = F (τ n R P R L SL (R σ(l 1 RL, L = σ(l, R = 1 (cf. [1] h U = ( 1 h 0 1 L 1 T L = U T Γ J L (U, τ = v L (U = v(t σ(lul 1, L σ(l, U = v(t = v(lul 1. multiplir systm 1 v(t = v L (U = πin κ (0 n κ < 1 f L (τ + h = πin κ f L (τ F (τ = πin κτ f L (hτ F (τ + 1 = F (τ F (τ = n= a(n πinτ 8

f L (τ = πinκτ/h n= a(n πinτ/h n < 0 a(n = 0 κ Γ L f κ a(n 0 n N κ n κ + N κ > 0.3 Rimann Roch rgular cusp irrgular cusp multiplir systm [] 9

3 3.1 Ddkind ta Ramanujan Dlta q = πiτ η(τ = q 1/4 (1 q n, n=1 (τ = η(τ 4. (τ SL (Z 1 η(τ SL (Z 1/ multiplir η(τ multiplir systm ( a b b c, d Z c 0 d ( c d = ( c d = c, d c ( 1 (sgn(c 1(sgn(d 1/4. d (/ 0 0 = = 1 ±1 ±1 Ptrsson c d d d ( c = = ( 1 c c d (c 1(d 1/4. a b M = SL (Z c d 10

Proposition 3.1 (Mrtns,Hrmit,Ptrsson,Radmachr tc whr ( d c v(m = ( c d = ( c d η(mτ = v(m(cτ + d 1/ η(τ, xp( πi((a + d bdc 3c + bd if c is odd 1 πi xp( ((a + d bdc 3dc + bd + 3d 3 1 ((a d bdcc + bd + 3d 3 if c is vn xp( πi 1 [] Knopp, Radmachr Ptrsson [6] T. Asai, H. Saito log(η(τ Ddkind η 1/ η(τ = πiτ/1 p Z ( 1 p πip(3p+1τ = πi/1 θ 1/6,1/ (3τ [8] p. 145 SL (Z Proposition 3.1 11

3. α (α = πiα m = (m, m Q τ H, z C θ m (τ, z = 1 (p + m τ + (p + m (z + m p Z charactristic ( m a b M = SL (Z M m c d d c M m = m + 1 cd b a ab Proposition 3. ( aτ + b θ M m cτ + d, z cz = κ(m(ϕ m (M(cτ+d 1/ θ m (τ, z. cτ + d (cz + d (cτ + d 1/ ϕ m (M = (bdm + acm bcm m ab(dm cm /, κ(m τ m κ(m 8 = 1 Igusa [4] Sigl Igusa [4] Igusa I II 3.3 κ(m M SL (Z SL (Z 1

Proposition 3.3 M = ( a c d b SL (Z c 0 κ(m = 1 c 1/ πi abcd 4 sgn(c + acd a a(b + d 4 8 c x + x x mod c abcd = + acd a(1 + bc + cd x 4 8 c x mod c Thorm 3.4 M SL (Z κ(m ( abcd + acd c a c 4 8 8 c κ(m = ( c d ( 1 (d 1 = ( c 8 d ϵ 1 d c. d 1 mod 4 d 3 mod 4 ϵ d = 1 i M SL (Z κ(m M SL (Z θ 00 θ(τ = θ 00 (τ = p Z (p τ Γ 0 (4 Proposition 3.5 M = ( a c d b Γ 0(4 aτ + b ( c θ = ϵ 1 d (cτ + d cτ + d d 1/ θ(τ. 13

a b 1 c d k 1/ Γ 0 (4 j(m, τ = (θ(mτ/θ(τ k 1 4 4.1 θ 00 θ m (τ, z m mod 1 n = t (n, n Z (n i Z θ m+n (τ, z = (m n θ m (τ, z θ m (τ, z m mod 1 θ m (τ, z m mod 1 charactristic m Q 0 = t (0, 0 1 θ m (τ, z = m τ + m (τ + m θ 0 (τ, z + m τ + m. θ 0 (τ, z θ m (τ, z m = 0 ϕ m (M = 0 (M SL (Z 4. θ 00 (τ, z τ H, a, z C (x = πix Poisson formula n Z 1 (n + a τ + z(n + a = ( iτ ( 1/ 1 (z n τ 1 + na. n Z 14

( θ 00 1 τ, z = (τ/i 1/ τ ( z τ θ 00 (τ, z. U = ( 1 1 0 1 U 0 = (0, 1/. 01 θ 01 (τ + 1, z = θ 00 (τ, z SL (Z 0 1 1 1 1 0 0 1 up to constant κ(m 4.3 κ(m κ(m θ 00 (τ τ τ 1 4.3.1 Sigl [7] M0 = 1 cd ab M SL (Z θ M 0 (Mτ/θ(τ c = 0 a = d = ±1 θ M 0 (Mτ = θ M 0 (τ + d 1 b = ( 1 p (τ + d 1 b + p( ab. p Z d 1 bp / abp/ mod 1 θ 0 (τ d = 1 d = 1 cτ + d = d = 1 i 15

c = 0 d = 1 d = 1 κ(m = 1 i 1 c 0 c 0 1/ ( cτ + d = c 1/ πi ci 4 sgn(c (cτ + d 1/. M z = τ + d c, z 1 = 1 z Mτ = a c + z 1 c. ( a θ M 0 c + z 1 = ( 1 c (p + cd ( a c + z 1 c + (p + cd ab p Z p = p 1 + cp 0 (p 0 p 1 c ( a p 1 + cp 0 + cd a ( p 1 + cd mod 1 c c θ M 0 (Mτ = p 1 mod c ( a c (p 1 + cd ( 1 (p 0 + p 1 c + d 4 (4z 1 + (p 0 + p 1 c + d 4 abc. p 0 Z p 0 1/ 4z1 i p 0 Z ( z 1/ = 4i p 0 Z ( 1 (abc p 0 (4z 1 1 + p 0 ( p 1 c + d 4 ( p 0 8 z + (p 0 + abc( p 1 c + d 4. p 1 ( a c (p 1 + cd + (p 0 + abcp 1 c p 1 mod c 16

p 0 p 1 p 1 + c ( a p 1 + c + cd + (p 1 + c(p 0 + abc c c a c ( p 1 + cd + p 1(p 0 + abc + c ca(1 + d + b + p 0 mod 1. ac ad bc = 1 b, d 1 + d + b 0 mod. ac(1 + b + d 0 mod. (p 0 / p 0 0 mod p 0 = p 1/ cτ + d θ M 0 (Mτ = 4ci f(p 1, p = a c p 1 mod c p Z 1 p τ (f(p 1, p. ( p 1 + cd ( p1 + (p + abc c + d + d 4 c p p p p 1 p 1 dp bd(1 a c 0 mod f(p 1 dp, p a c p 1 + a(b + d p 1 + abcd + acd 4 8 a(b + d p a(b + d 1 p 1 mod 1 c 0 κ(m = 1 ( c 1/ πi abcd 4 sgn(c θ M 0 (Mτ = κ(m(cτ + d 1/ θ(τ = 1 ( c 1/ πi abcd 4 sgn(c + acd 4 8 4 + acd 8 17 x mod c x mod c a a(b + d c x + x a(1 + bc + cd x c

κ(m 1 1 8 cd 0 mod x x cd/ κ(m = 1 c 1/ ( πi 4 sgn(c x mod c ( a c x + ab x abx/ abx / mod 1 κ(m = 1 ( c 1/ πi a(1 + bc 4 sgn(c x c x mod c = 1 ( c 1/ πi a 4 sgn(c d c x. x mod c (a, c = 1 c a c cd 0 mod d x modc x ax κ(m = 1 ( c 1/ πi ( = c 1/ πi 4 sgn(c 4 sgn(c x mod c x mod c ( d c x ( d c x. 4.3. M SL (Z κ(m ( κ (M = a a(b + dx c x + x mod c 18

1. c c a κ (M = a/ x mod c = x mod c c x ( a/ c x a sgn(c = c ϵ c. c a b + d κ (M = a(1 + c x c x mod c = a(1 + c/ x c x mod c a sgn(c = c 1/ ϵ c. c c ( sgn(c sgn(c ( ϵ c = c 8 c 8 c κ(m = πi abcd 4 sgn(c + acd asgn(c ϵ c 4 8 c ( abcd = + acd c a 4 8 8 c ( ad bc = 1 a c ( = d c. c c a cd 0 mod κ(m κ (M = d c x x mod c 19

κ(m = 1 c 1/ ( 18 sgn(c κ (M c = s c 0 (s 1, c 0 s+1 x 0 + c 0 y 0 = 1 x 1 = xx 0, x = xy 0 x = s+1 x 1 + c 0 x ( ( s+1 d κ (M = x dc0 c 0 x 1 mod c 0 x 1 mod s+1 = c 0 1/ s+1 d sgn(c 0 ϵ c0 (s+/ ( dc 0 c 0 8 { 1 s ( 1 (d c 0 1/8 s x s+1 ( κ(m = πi 4 sgn(c c0 d 8 = d c 0 ( c 0(d 1 8 ( d sgn(c0 ϵ c0 ( c 0 1 s s d c 0 c 0 ( d s s Ptrsson d ( c d = c ( 1 (sgn(c 1(sgn(d 1/4 d c 0, d d c0 = ( 1 ( c 0 1( d 1/4 c 0 d d c0 = c 0 d d 0 ( 1 (c 0 1(d 1/4

( c κ(m = d = = = ( c d c d c d ( 1 8 c 0(d 1 1 8 (c 0 1(d 1 1 (d 1 8 ( 1 16 (d 1 + 1 (d 1 8 ϵ 1 d. 3.4 5 [1], [] [3] [1] R. A. Rankin, Modular forms and functions, Cambridg Univ. Prss, 1977. [] T. Ibukiyama, Modular forms of rational wights and modular varitis, to appar in Abhand. Math. Smi. Univ. Hamburg 70 (000. [3] 003 [4] J. Igusa, On th gradd ring of thta constants I, II, Amr. J. Math. 86(1964, 19 45; ibid. 88(1966, 1-36. [5] J. Igusa, Thta functions, Springr Vrlag 197. [6] H. Ptrsson, Übr di arithmtischn Eignschaftn ins Systms multiplikativr Modulfunktionn von Primzahlstuf, Acta Math.,95(1956, 57 110. 1

[7] C. L. Sigl, Indfinit quadratisch Formn und Funktionn thori I, Math. Ann. 14, (1951. [8] [9] H. Yoshida, Rmarks on mtaplctic rprsntations of SL(, J. Math. Soc. Japan, Vol. 44 No. 3 (199, 351 373. 560-0043 ibukiyam@math.wani.osaka-u.ac.jp

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