TSUKUBA J. MATH. Vol. 11 No. 1 (1987). 107-119 MULTI-TENSORS OF DIFFERENTIAL FORMS ON THE SIEGEL MODULAR VARIETY AND ON ITS SUBVARIETIES By Shigeaki Tsuyumine Let An=Hn/rn> where Hn is the Siegel space and rn―Sp2n{Z). An is called a Siegel modular variety, which is the coarse moduli variety of n-dimen- sional principallypolarized abelian varieties over C. Let An be a projective non-singular model of An. An is shown to be of general type for n2i9 by Tai [10] (n=8 by Freitag [6], n―1 by Mumford [9]). Subvarieties of An are ex- pected to have the same property if they are not too special. Freitag [7] showed that An carries many global sections of Symmd(i2f~1), JV=n(n+l)/2, if n is bounded from below by some n0, and that any subvariety in An (n^n0) of codimension one is of type 'G' which is some weakened notion of "general type". He conjectured that n0 would be taken to be ten,in connec- tion with the argument of extensibility of holomorphic differentials to a non- singular model which is similar to Tai's one [9]. In this paper, we show that for n^lO, An carries many global sections of (Q*~1)Rr for some r (Theorem 1), and show the following; Theorem 2. Let n^lO. Then any subvariety in An of codimension one it nf aanam I tvfia. We have the following corollary to this theorem (cf. Freitag [7]). We denote by /＼≫(/),the principal congruence subgroup {Me/7n|M=l2n mod/}, l2n being the identity matrix of size 2n, and by AnA, the quotient space Hn/Fn{l). Corollary. // K(rn{l)) denotes the function fieldof AUi t{namely, the Siegel modular function fieldfor rn(l)), then the automorphism group Autc(K(rn(l))) over C is isomorphic to rn/±rn(l) for n^lO, i.e., the birationalautomorphism group of AnA equals A＼itc(Anil)s.rn/±rn(l). In particular, An (mS^IO) has no non-trivialhirational automorthism. This result is shown to be true under the condition that n is sufficiently laro^e.in Freitaer T71. Received March 1, 1986. 108 Shigeaki Tsuyumine Let co be a matrix as in §1, whose entries are differentialsin Qh'1. Then ofr satisfiesthe formula M-ft>Rr=|CZ+D|-rCn+1)(CZ+I>)Rr6>0r＼CZ+D)Rr, M= (^^)^Sp2n(R) (Lemma 1). So if we construct a square matrix A=A(Z) of size nr whose entriesare holomorphic functions on Hn such that A(MZ)= ICZ+D＼rin+i:> X(KCZ+D)-TrA(Z)((CZ+D)-Tr, M=(4^Vrn, then tr{Av*r)is /^-invariant and is regarded as a section of (QN~1)Rr (n^3), A°ndenoting the smooth locus An of An. The multi-tensor we consider is of this kind. Extensibility of tr(Ao)Rr) to An is proven under some conditions on A and on the degree n, which are similar to the case of pluri-canonical differentialforms (Tai [10]). A restriction of such a multi-tensor of differentialsto a subvariety D of codimension one gives a pluri-canonical differentialform on D. Construction of a desired A is done by using transformation formulas for theta series of quadratic forms with spherical functions. Using this we show that for any D, there are such multi- tensors whose restriction to D gives enough non-trivial pluricanonical differen- tials. 1. Mm,n(*) denotes the set of mXn matrices with entriesin *, and Mm― Mm,m. Let Hn be the Siegelspace of degree n ; Hn={Z<=Mn(C)＼tZ=Z, ImZ>0}. The symplecticgroup Spzn(R) acts on Hn by the usual symplecticsubstitution ) Z-^MZ^iAZ+B^CZ+D)'1, M= Let Z―{Zij), and let <oy=(―l)l+:'eijdznAdz12A ･･■AdztjA ■･･Adznn (l^z^/^n) where the symbol e^ denotes 1 if i^j, 2 if /=y, and dzij means that dztJ is omitted. 0)^ is a section of i2$~＼ iV=n(n+l)/2, where i2^~J is a sheaf of holomorphic (iV―l)-form. Let (i>={o)il).Then we have a transformation M-a>=|CZ+£>|-B-1(CZ+Z))<ut(CZ+D). Let A, B=(bij) be square matrices of size n, m respectively. A tensor pro- duct ARB is defined to be Ablt ― Abx m gMm aL m Then we have (i)(ARB)(A'RB')=AA'RBB', A', B' beingmatricesof the same sizeas A, B respectively,(ii)KAQB^'Agi'B, (iii)c(ARB)=(cA)RB = AR(cB) Multi-tensors of differentialforms 109 for a scalar c, (iv) tr{A(^B)―tr{A)-tr{B). The following is a direct consequence of the above: Lemma 1. For a non-negative integer r, we have M-a)Rr=＼CZ+D＼-nn+1＼CZ+D)Rr(DRrt(CZ + D)R＼ Let A=(ay)eMB and let us fix a positiveinteger r. Let /, / be ordered collections of r integers in {1, ･･･, n} where a repeated choice is allowed. We define Aa-J:> by where I={ilf ･･■, iT}, J={ji, ･･･, iA- Then a (k, /)-entry of a matrix ARr is equal to Au-J*> if fe= l+2(*Y sgn (/) s=l is defined by sgn(/)= II (―1)* i'El A holomorphic function/ on Hn satisfying (AB)<zr f(MZ)=＼CZ+D＼kf(Z) for M= ＼cd) n> is called a (Siegel) rnodular form of weight k (when n = l, we need an additional condition that / is holomorphic also at the cusp). / admits the Fourier expan- sion f(Z)= S?/(SK<YZS)) where e(*) stands for exp(27iV―1*), and 5 runs over the set of semi-positive symmetric even matrices of size n. f is said to vanish to order a at the cusp if a is the minimum integer such that a(S)=0 for S with min iTrS[j?]r<<2, S[g~] denoting lgSg. The minimum is equal to min＼― tr(ST)＼,T running over the set of non-zero symmetric positive semi-definiteintegral matrices of size n (cf. Barnes and Cohn [3]). We denote by ord(/), the order a of vanishing at the cusp. 2. Theta series and a matrix WF>T＼_w~].Let m be an integer with m2s2(n ―1), and let rj be a complex mX(n―1) matrix satisfying both ^rj―O rank -q―n―1 (such exists), w (l^z'^n) denotes an (n-l)Xn matrix given by 110 Shigeaki Tsuyumine 1 ) 10 Vi= 1 i We fix a positive symmetric matrix F of size m with rational coefficients.Let r be a positive integer, and let /, / be as in the preceding section. We define a theta series associated with F by setting 0y-° r ](Z)=sgn(/)sgn(y)S.II lyAG+uW^yl U ＼r]/(G+u)F1/2V＼ LV Xe(trO=rZPlG + u-]+ ＼G+u)vj) where G runs through allmXn integralmatrices,and u, v are mXn matrices with rational coefficients. (*M is called a theta characteristic. If {/',/'} equals {/,/} up to orders, then obviously ^'■JI)["](Z)=^/-J)[j5](Z). Then we de- fine WF,r＼ U ＼(Z) to be a square matrix of size nr whose {k, /)-entry equals Otf-^l^iZ) where £=l+2(is-l)ns-＼ ^l+SO'.-D"'"1 with I={iu-i,}, J={jx, ･■■, jr}- y?F,r＼ is a symmetric matrix. Our purpose of this section is y to prove the following; Proposition 1. Let M=(^^)^Sp2n(Q) such that A, D, F05, F-10C are integral. Put Um^uA+F-'vC+^HF-^aCAC),. Vu=FuB+vD+-^t(F/1)(tBD)/,. £f((W), M)=e(^tr(-t(uA+F-1vC)(FuB+vD+t(FA)(tBD)j) + tuv))> where for a square matrix P, Pa denotes the vector composed of diagonal elements of P. Then we have a transformation formula m: ](MZ)=fc.(M)£*.(( Uv),M)＼ CZ+D＼'m'^r x(t(cz+Dyirr＼F,r＼UM](Z)((cz+DyTr ＼-VmJ where XF(M) is an eighth root of unity depending only on F and M. Since F, u, v are of rational coefficients,we have the following corollary. Multi-tensors of differentialforms Ill Corollary. There is an integer I such that ＼mZ)=1{M) ICZ+L>|cm/2)+2r Vf.t Iv X{l{CZ + D)-l)RrWF,r＼ U ](Z){(CZ+DYTr v for any M=(CDJ^rn(l) where 1 is a map of Fn{l) to the set of rootsof unity 1 is killed by some power. Let X=(Xij) be an mXn variable matrix. We define another theta series associated with F by setting ](Z, X)=^e(tr(^ZFlG + u^+ t(G+ u)(X+v)) LV where G runs through all mXn integral matrices. Lemma 2. Let the notations be as in Proposition 1. Then ＼MZ, X)=1f{M)Ef{(kUv), M)e(^~tr(Ct(CZ + D)tXF~1X)) vF＼Iv X ＼CZ+ D＼ml26f＼Um](Z,X(CZ+D)). The proof of the above lemma was given in Andrianov and Maloletkin[1] under some condition on yt＼ M, and in Tsuyumine [11], [12], the general case (not exactly this form). Let 3=f-~ j be an mXn matrix of differentialoperators. Let r,and/,/ be as above. We define a differentialoperator LIJv by sgn(/)sgn(/) //)? (2^V~l)2rCn"1) Lemma 3. Let P be a complex symmetric matrix of degree n, and Q, a com- plex nXm matrix and c, a constant. Then we have the following identity: LIJjl(e(tr(jPtXX+QX) + c)) =sgn{I)sgn(J)II＼yi(PtX+Q)7]＼ U ＼f]j{PtX+Q)r]＼ Xe^jP'XX+QX^c).