Extended Gauss AGM and corresponding Picard modular forms

Transcription

1 Extended Gauss AGM and corresponding Picard modular forms Kenji Koike and Hironori Shiga August,006 0 Introduction For two positive numers a,, set ψ g (a, ) = ( a+, a) and inductively set ψg n (a, ) = ψ g (ψg n (a, )) = (a n, n ) Then we have a common limit M g (a, ) = lim n a n = lim n n This is called the Gauss Arithmetic Geometric Mean (we use the areviation Gauss AGM) Concerning this Gauss AGM we have two classical identities oth we can find in the literature of Gauss himself Theorem 0 (Gauss [GaT] 799) For x (0, ) we have AGM(, x) = dz = F ( π z(z )(z λ),, ; λ), (λ = x ) (0) The right hand side F means the Gauss hypergeometric function Setting the Jacoi theta function and theta constant for p, q {0, } we have [ p ϑ q [ p ϑ q ] (z, τ) = n Z exp[πi(n + p ) τ + πi(n + p )(z + q )], ] (τ) = n Z exp[πi(n + p ) τ + πi(n + p )q], Theorem 0 (Gauss [GaH] 88) ( ϑ (τ) = 0 ϑ (τ) + ϑ (τ)) ϑ (τ) = ϑ (τ)ϑ (τ) 0 (0) The latter theorem shows the relation of the (coefficients of the realized) elliptic curves corresponding to two isogenous torus C/Z + τz and C/Z + τz So in general this theorem is referred as the isogeny formula for the Jacoi theta constants Any way these two theorems are telling us a very interesting story concerned with AGM, periods of algeraic varieties, hypergeometric functions and modular forms at a same time But still now there is no sufficiently nice generalization for it In this article we show an extended story of Gauss AGM to two variales case y using Picard modular forms for Q( ) studied y Matsumoto [Mat] which is coming from one of the Terada and Deligne-Mostow list

2 Extended Gauss AGM and the Appell F (First Main Theorem) Extended AGM system At first let us consider a trivially extended Gauss AGM system (a,, ( a + a), ) a + a, a We generalize it to the system of three terms Let a, and c e positive real numers Set ( ) a + ψ(a,, c) =, c, 4 4 (c + a)(c + a + ), () and set (a n, n, c n ) = ψ(a n, n, c n ) Theorem Suppose 0 < a,, c, then the sequences {a n }, { n } and {c n } have a common limit Definition Let us denote the aove limit y M(a,, c), and call it Extended Gauss AGM Lemma We have min{a,, c} min{a,, c } Max{a,, c } Max{a,, c} [proof] Set min = min{a,, c}, Max = Max{a,, c} Oviously we have a, [min, Max] Set ( ) c c + a =, c = c + a + We see easily c, c [min, Max] Because c + a c + a +, we have c c c So we have c [min, Max] qed Remark It happens that For example (a,, c) = (,, ) min{a,, c} = min{a,, c } < Max{a,, c } = Max{a,, c} The following Proposition assures Theorem Proposition Suppose 0 < µ a,, c ν for fixed positive numers µ, ν Then Max{a,, c } min{a,, c } ρ(max min) for some positive numer ρ(µ, ν) <

3 [proof] Assume a c Then and a = a + c c c = + a = (a + )/ + c a + c, (c ) + a = a + c (c + a)(c + ) We have = c c c + a c + a = By cosidering ovious inequality a,, c c, we have ac Max{a,, c } min{a,, c } c a Assume a c, then we have c ac c a = c c + a ν µ + ν < and a = a + c c c = + a = (a + )/ + c a + a, (c ) + a = a + (c + a)(c + ) a We have So we have By putting /a = t, we have = c c c + a = Max{a,, c } min{a,, c } a a + a a+ a a We have for t > So Max{a,, c } min{a,, c } a ( d dt t ) ( + t)/ = t Max t (,ν/µ] t ( + t)/ t This argument works for the case c a also So we otain the required inequality t ( + t)/ t (t ) 3 + t t > 0 = ν µ(µ + ν)/ ν µ < qed Functional equations Set x y = (x + y) ( + x + y xy) = ( x + y) ( + x)( + y) ( + xy) 3

4 for real numers 0 < x, y <, namely here we choose real positive square roots We have x = ( x + y) ( + x)( + y) + i( x y) ( x)( y) ( + xy) y = ( x + y) ( + x)( + y) i( x y) ( x)( y) ( + xy), Proposition M (, xy, ) x + y = + ( xy M, x y, ) x + y [proof] It holds ( M, ) ( x + y + xy x + y 4 xy, = M,, 4 (x + y + xy)( x + y ( + xy x + y 4 = M,, ( ) x + y) ( + x)( + y) = + ( xy (x + y) M, ( + xy), ( x + ) y) ( + x)( + y) 4 ( + xy) 4 = + xy (x + y) M, ( + xy), ( x + y) ( + x)( + y) ( + xy) = + ( xy M, ) x x y + y, ) + + xy ) Let F (a,,, c; x, y) e the Appell hypergeometric function F (a,,, c; x, y) = m,n 0 (a, m + n)(, m)(, n) x m y n, ( x, y < ) (c, m + n)m!n! qed of complex variales x, y with the Pochmammer notation { a(a + ) (a + k ), (k ) (a, k) = 0, (k = 0) This is a solution of the differential equation x( x)f xx + ( x)yf xy + (c (a + + )x)f x yf y af = 0, E (a,,, c) : y( y)f yy + ( y)xf xy + (c (a + + )y)f y xf x a f = 0, (x y)f xy f x + f y = 0, and it is characterized as the solution f(x, y) which is holomorphic at (x, y) = (0, 0) and f(0, 0) = Theorem We have for real variales 0 < x, y < F (, 4, 4, ; x, y ) = + xy F (, 4, 4, ; (x ), (y ) ) 4

5 ( [proof] First let us note the system E, 4, 4, ) : P = P (x, y, x, y ) = x( x) x + y( x) x y + ( 7 4 x) x 4 y y 8 P = P (x, y, x, y ) = y( y) y + x( y) x y + ( 7 4 y) y 4 x x 8 P 3 = ( x) x P ( y) y P + 4 (P P ) = (x y) x y 8 ( x y ) () gives the differential equation for F (, 4, 4, ; x, y) From the definition of x and y we have Putting x = X, y = Y we are requested to show (x ) = ( ( x )( y ) i(x y)) ( + xy) 4 (y ) = ( ( x )( y ) + i(x y)) ( + xy) 4 with + XY F ( X 4, Y 4 ) = F (Φ (X, Y ), Φ (X, Y )) (3) ( ) ( ) ( X4 )( Y Φ (X, Y ) = 4 ) i(x Y ) ( X4 )( Y ( + XY ), Φ (X, Y ) = 4 ) + i(x Y ) ( + XY ) Now we are going to descrie the differential operators which annihilate F ( X 4, Y 4 ) and F (Φ (X, Y ), Φ (X, Y )), respectively Lemma The system Q = X4 6X X + X( Y 4 ) 6Y 3 X Y 4 X X + Y 4 8Y 3 Y 8 Q = Y 4 6Y Y + Y ( X4 ) 6X 3 X Y 4 Y Y + X4 6X 3 X 8 Q 3 = X4 Y 4 3X 3 Y 3 X Y + 3X 3 X 3Y 3 Y (4) annihilates F ( X 4, Y 4 ) [proof] We get the aove system y just putting in the system () x = X 4, y = Y 4, x = 4X 3 X, y = 4Y 3 Y Lemma 3 Putting R i = ( + XY ) Q i ( + XY ) the system R = X4 6X X + X( Y 4 ) 6Y 3 X Y X4 +Y 4 +4X 3 Y 3 +X 4 Y 4 6X Y 3 (+XY ) X + ( Y 4 ) 6Y 3 (+XY ) Y Y 3 (X Y ) X 3 ( Y 4 ) 8X Y 3 (+XY ) R = Y 4 6Y Y + Y ( X4 ) 6X 3 X Y Y 4 +X 4 +4Y 3 X 3 +Y 4 X 4 6Y X 3 (+Y X) Y + ( X4 ) 6X 3 (+Y X) X X3 (Y X ) Y 3 ( X 4 ) 8Y X 3 (+Y X) ( ) R 3 = 3X 3 Y (X 4 Y 4 ) 3 X Y X5 +Y 3 +XY X + X +Y 5 +XY Y (X4 Y 4 ) (+XY ) annihilates + XY F ( X 4, Y 4 ) 5

6 Let S (S ) e the operator otained from P (P ), respectively, y sustituting [ x Φ (X, Y ) x Φ / X Φ =, = / X X y Φ (X, Y ) Φ / Y Φ / Y They take some complicated shapes, ut y T = 8(X + Y )( XY ) 3 (S + S ), T = 8(X Y )( XY ) 3 i ( X 4 )( Y 4 ) (S S ) can e represented in an explicit way: Lemma 4 Y ] T = T = Y ( X 4 ) (X Y 3 ) X + ( X4 )( Y 4 )(X + Y )( XY ) X Y +X( Y 4 ) (Y X 3 ) Y ( X4 )(X 3 Y + X 4 Y + 4XY 4X Y 3 X 3 Y 4 Y 5 ) X ( Y 4 )(Y 3 X + Y 4 X + 4Y X 4Y X 3 Y 3 X 4 X 5 ) Y (X + Y )( XY ) 3 Y ( X 4 )(Y X 3 ) X (X4 Y 4 )( XY ) X Y X( Y 4 )(X Y 3 ) Y +(X 5 X Y + X 6 Y X 3 Y Y 3 ) X (Y 5 Y X + Y 6 X Y 3 X X 3 ) Y By comparing the results in Lemma 3 and Lemma 4 we otain: Lemma 5 6XY T = X( X 4 )(X Y 3 )R +Y ( Y 4 )(Y X 3 )R + ( X 4 )( Y 4 )(X Y )R 3 6XY T = X(Y X 3 )R Y (X Y 3 )R +(XY + X Y X 3 Y 3 + X 4 Y 4 X 4 Y 4 )R 3 So T and T annihilate +XY F ( X 4, Y 4 ) Hence S and S annihilate +XY F ( X 4, Y 4 ) also Namely +XY F ( X 4, Y 4 ) satisfies the same differential equation S + S with three dimensional solution space as F (Φ (X, Y ), Φ (X, Y )) This system has unique holomorphic solution at (X, Y ) = (, ) up to a constant factor So we have the required equality qed 3 First Main theorem Theorem 3 Assume 0 < x, y < and set λ = x, µ = y Then we have the following trinity theorem for Extended Gauss AGM: [proof] M(, xy, (x + y)/) = π According to Proposition and Theorem, we know that F (, 4, 4, ; λ, µ) satisfy the same functional equation Set du 4 u (u ) (u λ)(u µ) = F (, 4,, ; λ, µ) (5) 4 g(x, y) = F (, 4, 4, ; λ, µ)m(, xy, (x + y)/) M(, xy, (x + y)/) and Then we have Note that we have g(x, y) = g(x, y ) n lim = n c n 6

7 for the Extended AGM system (a n, n, c n ) So, y the iteration of the procedure (x, y) (x, y ), we have g(x, y) = Hence the left hand side and the right hand side in (5) coincide The second equality in the statement of the Theorem is nothing ut the integral representation of the Appell hypergeometric function Namely we otained the required equalities qed Modular interpretation and isogeny formula Matsumoto hyperelliptic modular function for C(x, y) (sum up with corrections) Here we review the result of K Matsumoto [Mat] with some modifications and small corrections Let us start from the family of algeraic curves C(x, y) : w 4 = z (z ) (z x)(z y) ((x, y) C {xy(x )(y )(x y) = 0})() C(x, y) has a model of compact nonsinguler hyperelliptic curve of genus 3 We denote it y the same notation The differentials dz/w, z(z )dz/w 3, z (z )dz/w 3 form a asis of holomorphic -forms on C(x, y) For the moment suppose x and y are real and satisfy < x < y Let α e the automorphism (z, w) (z, w) of C(x, y) Let us make three -cycles Choose ranches w (0), w (), w (), w (3) of w over {Im z 0} {0,, x, y} so as to e α(z, w (k) ) = (z, w (k+) ) with w (4) = w (0) For real numers z, z let [z, z > e an oriented line segment connecting z and z in this order Let [z, z > (k) e an arc on C(x, y) over [z, z > (k) with the ranch w (k) Set A = [0, > (0) +[, 0 > (), B = [, 0 > (0) +[0, > (), A 3 = [y, x > (0) +[x, y > () They are considered as -cycles on C(x, y) By making their analytic continuations we can define multivalued analytic functions dz a (x, y) = A w, a dz (x, y) = B w, a dz 5(x, y) = A 3 w on C {xy(x )(y )(x y) = 0} These setting and notation are the same in [Mat] [Fact ] a (x, y), a (x, y), a 5 (x, y) satisfy the differential equation E (, 4, 4 ) of the Appell hypergeometric function F (, 4, 4 ; x, y) and they form a asis of the space of solutions We define the Schwarz map for E (, 4, 4 ) y Φ : (x, y) (u, v) = (a 5 /a, a /a ) C [Fact ] The image of Φ is contained in the hyperall B = {(u, v) C : Im v u > 0} and it is open dense in B The map Φ has a continuation on P P {(0, 0), (, ), (, )} and the image is equal to B [Fact 3] The fundamental group π (C {xy(x )(y )(x y) = 0}, ) induces the monodromy group G of Φ acting on B The following five transformations give a generator system of G: g = i i, i 0 i + i i i g = + i i i, i + i i 7

8 g 3 = , 0 0 g 4 = i i i 0 i 0, 0 i g 5 = + i i i + i i i, i + i i where g i (i =,, 3, 4, 5) acts on t (, v, u) from left Their orders are 4, 4,, 4, 4 and eigen values are {,, i}, {,, i}, {,, }, {, i, i}, {,, i}, respectively (the original article contains a printing error for g 5 ) The Schawarz map Φ induces a iholomorphic correspondence P P = B /G, and three points (0, 0), (, ), (, ) correspond to the oundaries B /G B /G [Fact 5] We can realize B in the form The map D = {ξ = [ξ 0, ξ, ξ ] P : ξh t ξ < 0}, H = 0 i 0 i Ω(u, v) = v + iu / u / iu u / v iu / u iu u i gives an modular emedding of B into the Siegel upper half space of degree 3 Set G 0 = U(,, Z[ ]) = {g GL(3, Z[ ]) : gh t g = H} G 0 ecomes a restriction of Sp(6, Z) on Ω(B ) Note that the projective group G 0 = G 0 / is isomorphic to SU(,, Z[ ]) = {g SL(3, Z[ ]) : gh t g = H} [Fact 6] Set g 6 = i Then the group generated y G and g 6 together with ecomes the congruent sugroup G( + ) = {g G 0 : g id (mod ( + )} Fact 7] For p = (p, p, p 3 ), q = (q, q, q 3 ) Z 3, let p p p 3 p p (u, v) = ϑ p 3 (Ω(u, v)) q q q 3 q q q 3 = exp[πi(n + p )Ω(u, v)t (n + p ) + πi(n + p )t q] n=(n,n,n 3) Z 3 e the Riemann theta constant with half integral characteristics defined on B via the emedding Ω 8

9 Theorem (Matsumoto hyperelliptic theta map theorem) (u, v) (x, y) = (u, v) 0 0, (u, v) 0 4 (u, v) 0 Remark The denominators and the numerators on the right hand side have common zero on some divisors of B But according to the argument on the period map, we have extensions x = x(u, v), y = y(u, v) as holomorphic maps on the whole domain B [Fact 8] Lemma (Fourier expansion) p p p 3 (u, v) q q q 3 = exp[ π {(n + p ) + i(n + p )} u ] exp[πi{(n + p )q + (n + p )q }] n,n ( p3 ϑ {(n + p ) i(n + p ) )}u, i exp[πi{(n + p ) + (n + p ) }v] q 3 Lemma (Matsumoto exchange formula The multiplication factor is misplaced in the original paper)) We have exp[πip q + πi p p p 3q 3 ] q 3 p p (u, v) = p 3 (u, v) q q p 3 q q q 3 9 theta constants with monodromy invariant characteristics Recall the transformation formula for theta constants: ( ) Ag B Proposition (Transformation formula (see for example [I])) Let g G 0 and N g = g e its symplectic representation Then we have [ [ a N g Where ]] (u, v)) = ε, a, ) (det (C g Ω(u, v) + D g )) / N g a Dg a C = g + (/)dv(c t g D g ) B g a + A g + (/)dv(a t, g B g ) ( dv indicates the diagonal vector) and ε, a, ) is a certain 8-th root of unity C g a (u, v) By use of the transformation formula we otain the following two propositions Because the calculation is straight forward we omit the detail a Proposition () There are 36 even theta constants ϑ (Ω) with a t 0 (mod ) Among them a we have 0 different (u, v): D g 9

10 numer name characteristic {a, } n h {{0, 0, 0}, {0, 0, 0}} n {{, 0, 0}, {0, 0, 0}} n3 {{0, 0, 0}, {, 0, 0}} n4 {{0, 0, }, {0, 0, 0}} n5 h {{,, 0}, {0, 0, 0}} n6 yn {{, 0, }, {0, 0, 0}} n7 {{, 0, 0}, {0,, 0}} n8 xn {{, 0, 0}, {0, 0, }} n9 h3 {{0, 0, 0}, {,, 0}} n0 xz {{0, 0, }, {, 0, 0}} n yz {{0, 0, }, {0,, 0}} n {{,, }, {0, 0, 0}} n3 yd {{, 0, }, {0,, 0}} n4 xd {{, 0, 0}, {0,, }} n5 {{0, 0, }, {,, 0}} n6 {{, 0, }, {, 0, }} n7 {{,, }, {,, 0}} n8 {{,, }, {, 0, }} n9 {{, 0, }, {,, }} n0 {{,, 0}, {,, 0}} Tale 3 () We have 9 even theta s with N g with proper names a a (mod ) for g G, those are indicated in Tale 3 We call the nine theta constants in Proposition () G-invariant thetas We note here that the rational representation of the transformation a a a 3 3 G 0 c c c 3 is given y Re Im Im 3 Re Im Re 3 Im Re Re 3 Im Re Im 3 Im c Re c Re c 3 Im c Re c Im c 3 Re a Im a Im a 3 Re a Im a Re a 3 Im a Re a Re a 3 Im a Re a Im a 3 Re c Im c Im c 3 Re c Im c Re c 3 and that we have det g det (C g Ω(u, v) + D g )) = (a + a v + a 3 u), (u, v)) det g = (u, v) (a + a v + a 3 u) 3 for g = a a a 3 3 generator systems of the structure ring G 0 A holomorphic function f(u, v) on B is said to e a modular form of weight d with respect to G 0 provided f(u, v)) = ((a + a v + a 3 u)) d f(u, v), g = a a a 3 G 0 () 0

11 We define the modular form for other discrete groups in the same way We use the following notation: M d (G) : the vector space of modular forms of weight d with respect to G, M d (G( + i)) : the vector space of modular forms of weight d with respect to G( + i), M(G)(M(G( + i))) : the graded ring of modular forms with respect to G(G( + i)), respectively We note that the definition () has a meaning only for d of divisile y 4 Proposition 3 Every fourth power of the G-invariant theta constant is a modular form of weight 4 with respect to G, namely it satisfies 4 a (u, v)) = (a + a v + a 3 u) 4 4 a (u, v), g = a a a 3 G [proof] According to Proposition, for any G-invariant theta it holds 4 [ a ] (u, v)) = ε, a, ) 4 (a + a v + a 3 u) 4 4 a (u, v), g = a a a 3 G Because ε, a, ) 4 (a + a v + a 3 u) 4 [{, ] } we may calculate the approximate value of the ratio of a a 4 (u, v))/(a +a v +a 3 u) 4 4 (u, v) just to know the signature As a consequence we otain the required equalities qed Note that we have G( + i) = g,, g 5, g 6, and that we have the index etween two projective groups [G( + i), G] = And the generator g 6 of G( + i)/g induces the involution ι : (x, y) (y, x) So we otain the diagram with the Segre emedding map (u, v) [X 0, X, X, X 3 ] P 3 : P P (x, y) Φ B /G = {X 0 X 3 = X X } P 3 (u, v) π π Ψ P = (P P )/ι B /G( + i) = P Proposition 4 ) We have ) M 4 (G) is generated y 4 xn, 4 yn, 4 h, 4 h Diagram : Period diagram dimm 4 (G) = 4 dimm 8 (G) = 9 [proof] Note that B is the uniformization of the orifold P P with the arrangement of weighted divisors: name divisor weight D 0x {(x, y) : x = 0} 4 D 0y {(x, y) : y = 0} 4 D x {(x, y) : x = } 4 D y {(x, y) : y = 0} 4 D x {(x, y) : x = } 4 D y {(x, y) : y = } 4 D xy {(x, y) : x = y}

12 So the divisor of du dv corresponds to 3 4 D 0x 3 4 D 0y 3 4 D x 3 4 D y D xy 3 4 D x 3 4 D y via the period map Φ Suppose f(u, v) elongs to M 4 (G) Then the divisor (f(u, v)(du dv) 4/3 ) should satisfy (f(u, v)(du dv) 4/3 ) D 0x D 0y D x D y 3 D xy D x D y That is equivalent to give an element of H 0 (P P, O( 3 D 0x + 3 D 0y + 4 3K)) Where K denotes the canonical divisor Because we have K = D 0x D 0y, we know the isomorphism M 4 (G) = H 0 (P P, O(D 0x + D 0y )) The vector space H 0 (P P, O(D 0x + D 0y ) is generated y the system {, x, y, xy}, and it is four dimensional By the same way we have So we otain dimm 8 (G) = 9 M 8 (G) = H 0 (P P, O(D 0x + D 0y ) Oserving the definition of Ω(u, v) we know that [ p p p 3 p (0, v) = ϑ q q q 3 q ] (v) ϑ [ p q ] (v) ϑ [ p3 q 3 ] (v) So our modular forms 4 xn, 4 xd, 4 yd, 4 h are linearly independent, and they give a asis of M 4(G) qed Proposition 5 The space M 4 (G) is generated y fourth powers of G-invariant thetas, and all algeraic relations are induced from 4 xn 4 xd = 4 yn 4 yd 4 h = ( 4 xn 4 xd) 4 h 4 h3 = ( 4 xd + 4 yn ) 4 xz 4 yz = 4 xn 4 yn 4 h = ( 4 xz + 4 yz) 4 xd 4 yn = 4 xz( 4 xn ( 4 xd) 4 xn 4 yd = 4 yz( 4 yn 4 yd) [proof] Let us consider the truncation of the Fourier expansions of the fourth powers of our G invariant Theta constants up to some fixed hight l Set V l e the vector space generated y these polynomials There is the canonical surjective homomorphism ω : M 4 (G) V l We can easily find that V l is four dimensional So ω should e an isomorphism Then we can otain the aove relations y explicit calculations of their coefficients The situation is the same for the quadratic relations qed Remark By oserving the last identity in the aove Proposition we have a system of generators 4 xn, 4 yz, 4 yn 4 yd, 4 yd of the structure ring C[X 0, X, X, X 3 ]/(X 0 X 3 X X ) of B /G = P P 4 CM-isogeny on Jac(C(x, y)) Let us consider an isogeny map of B : σ(u, v) = (( + i)u, v)

13 It is induced from the Q-symplectic transformation: and we have Set A 0 = ( ), ( A O N σ = O t A ), A = N σ τ = Aτ t A, τ H 3 Λ = {(n + n, n n ) Z : n, n Z} ϑ Λ a, (τ ) := exp[πi(m + a )τ t (m + a a t ) + πi(m + ) ], τ H, a, Z m Λ ϑ a, (τ) := ϑ a,(aτ t A), τ H 3, a, Z 3 ϑ a, (τ ) := ϑ a, (A 0τ t A 0 ), τ H, a, Z, where H g denotes the Siegel upper half space of degree g By elementary caluculations we have Lemma 3 ϑ a, (τ ) = ϑ Λ (τ ) a A 0, t A 0 According to Matsumoto, Minowa and Nishimura [Mat-Min-Nish] we have Theorem ( Λ-theta formula) ( ϑ Λ a, (τ ) ) ϑ Λ a +{,0}, (τ = ( exp[ πi (a ) ( + a )] ϑ a, (τ) ) ) exp[ πi (a + a )] ϑ a, +{,}(τ ) By virtue of Theirem together with Lemma 3 we otain: Proposition 6 ϑ 0 0 (τ) = ϑ (τ) ϑ (τ) ϑ 0 0 (τ) = ϑ (τ) ϑ (τ) ϑ (τ) = ϑ (τ) ϑ 0 (τ) = + i ϑ (τ) 0 ϑ (τ) = ϑ (τ) + ϑ (τ) 0 ϑ 0 (τ) = ϑ (τ) ϑ (τ) 0 5 modular interpretation of the extended Gauss AGM Set Now we have H (u, v) = h (u, v) = ϑ (Ω(u, v)), H 3 (u, v) = h3 (u, v) = ϑ (Ω(u, v)), 0 Z(u, v) = Z (u, v) = ϑ (Ω(u, v)) 0 0 3

14 Theorem 3 (Second Main Theorem = CM-isogeny formula) It holds H (( + i)u, v)) = (H (u, v) + H 3 (u, v)) H 3 (( + i)u, v)) = Z(u, v) Z(( + i)u, v)) = 4 4 ( Z + H H 3 )(Z + H + H 3 [proof] The first and the second equalities are already [ otained ] in Proposition [ 6 ] So we concentrate to show the last equality The theta constants (u, v) and (u, v) are not G- invariant thetas Setting g i = a i a i a i3 (i =,, 5), according to Proposition we have 0 0 i (u, v)) = (a i + a iv + a i3u) 0 0 i (u, v)) (i =, 3), 0 i (u, v)) = (a i + a iv + a i3u) 0 i (u, v)) (i =, 3), 0 0 i (u, v)) = ( ) i+ (a i + a iv + a i3u) 0 i (u, v)) (i =, 4, 5) 0 i (u, v)) = ( ) i+ (a i + a iv + a i3u) 0 0 i (u, v)) (i =, 4, 5) 4 4 Then (u, v) + (u, v) and M 4 (G) By oserving the truncations we otain (u, v) 0 0 ) (u, v) elong to (u, v) + (u, v) = ( h (u, v) 4 h(u, v) + 4 h3(u, v) ) (u, v) (u, v) = ( xn (u, v) 4 yn (u, v) ) By elliminating from these two equalities we otain the description of Z(u, v) 0 4 in terms of h, h, h3, xn and yn Proposition 6 gives the description of 4 h (( + i)u, v) 4 h (( + i)u, v) + 4 h3 (( + i)u, v) and 4 xn (( + i)u, v) 4 yn (( + i)u, v) in terms of h(u, v), h3 (u, v) and Z(u, v) As a consequence we otain the required relation qed References [GaT] C,F, Gauss, Mathematisches Tageuch , Ostwalds Kalassiker der exakte Wissenschaften 56,Geest und Portig, Leizig,976 [GaH] C,F, Gauss, Hundert Theoreme üer die neuen Transzendenten, 88 [I] J Igusa, Theta functions, Springer-Verlag, Heidelerg, New-York(97) [Ko-Shi] K Koike and H Shiga, Isogeny formulas for the Picard modular form and a three terms arithmetic geometric mean, J Numer Theory, to appear [Bi-La] Ch Birkenhake and H Lange, Complex Aelian Varieties, Springer-Verlag (99) [Mat] K Matsumoto, On Modular Functions in Variales Attached to a Family of Hyperelliptic Curves of Genus 3, Ann SNS Pisa, Ser IV-XVI(989),

15 [Mat-Min-Nish] K Matsumoto, T Minowa and R Nishimura, Automorphic forms on the 5-dimensional complex all with respect to the Picard modular group over Z[i], to appear in Hokkaido Math J [Mum] D Mumford, Tata Lectures on Theta I, Birkhäuser, Boston-Basel-Stuttgard(983) [Re-Ta] Resnikoff and Tai, On the Structure of a Graded Ring of Automorphic Forms on the -dimensional Complex Ball, Math Ann 38 (978),