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
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)) 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) f(Z)= S?/(SK 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). 112 Shigeaki Tsuyumine The proof is given as a slight modification of that of Lemma 3in Andrianov and Maloletkin [1]. So we skip the proof. Proof of Proposition 1. dtf-J:>" ](Z)is equalto LZ^(^M(Z, F^X)) | ^ SubstitutingZby F1/2Xinthe formula in Lemma 2 and applying LIJr]at X=0 we get, by Lemma 3, d^''n＼U＼MZ) =:^(M)^((")(M)|CZ+JDr/2sgn(/)sgn(/) xSn ＼r]i(CZ+DY(G+uM)F1"V＼ U ＼Vj(CZ+DY(G+um)F1'27]＼ Xz(tr(^ZF＼_G + UM] + ＼G + uM)vM)). Using the Laplace expansion IVi(CZ + DY(G + uM)F^V |= S |Vi(CZ + D)% ||v.KG + UmW1"] I S=l the equality becomes 0#'J>[*](MZ) =XF(M)EF((U), M)|CZ+Dr/2sgn(/)sgn(/)S(SS) ＼＼y / / gst XII |i7i(CZ+Z>)^,| II ＼Vj＼CZ+ Dyr]t＼-＼Vs＼G+ uM)F^7]＼ Xl^t'CG+M^F^iyle^l-ZFCG + Ujfl+ 'CG + M^i;^)), S, T running through all the collectionsof r integers in {1, ･･･,n} (admitting a repeated choice), =1F{M)EF((U), M)＼CZ+D＼ml*Z IL(-l)i+s＼Vi(CZ+DyVs＼ ＼＼V' ' S.Tiels (-l)i+s＼rji(CZ+Dyr}s＼ is an (s, z)-entry of the cofactor matrix (CZ+D)* of CZ +D, and so ＼mZ)=If(M)Ef((U), ≫≫< M)＼CZ+D＼mlzYl{t(CZ+D)*Y1-^ x6^s-T>＼UM＼z)((cz+D)*yT-J＼ Multi-tensors of differential forms 113 Thus we h ave ](MZ) Iv =XAM)EF((U),M)＼CZ + D＼m'＼＼CZ+ D)*)RrWF r＼Um](Z)((CZ+D)*)Rt =xF(M)EF(( u),m)＼cz+d＼ ww(＼cz+Dyvf q. e. d. 3. Further properties of ＼F,r＼U . The following formula is easy to see. Lemma 4. Let WF ＼(Z) be as in the preceding section. Let P be any symmetric matrix of size n with rational coefficients,and let k^O be an integer such that k2PF[G] is even for any GeMm Then B(Z). ]<*+/･)=*-<-ws<-i*wc≪-+(≫/≪])rw,[ "^l ]<*) Lv where w runs over the representativesof MmiJl(-j―Zj modZ. Now we show that for any F and for any characteristic of the form (o) ＼F,r 5]is non-trivial for infinitely many r. Let lij = (i, ･■■,i)&MUZr(Z) Then the Fourier coefficienta(S) in the expansion of 0jFi:D)[o]is given by a(S) = 2l^£(G + u)F1/2>?|2r G where G runs over the finite set {G^Mm,tn(Z)＼F[G + u~]= S}. There is an 5 such that {G＼F＼_g+ii＼=S}±e for r=l, 2, ･･･, #{G|,F[G+w] = S}, then every ＼r)it{G+ u)FllH＼z must be zero). Lemma 5. Suppose that WF,r＼U ＼{Z)is non-trivial. Let us take any complex symmetric matrix W of size n. Then tr(WF,r＼u ＼(Z)WRrJ is identicallyzero in Z if and only if W―0. Proof. Wf r ."](Z) satisfiesa transformation formula in Corollary to Pro- position 1. It follows that ##C1]])m (Z) is not identically zero. Let us suppose 114 Shigeaki Tsuyumine W±?0. If we put W=(Wij), then it is impossible that only wn is non-zero. So some Wij is not zero with (z,j)*?(l,1). Again by the transformation formula, it follows from our assumption, that tr(wFJ Uyl(Z)(£LW£/)Rr)=0 for any £/e GLn{Z){l)-{UE:GLn{Z)＼U=ln modi), I being as in Corollary to Proposition 1, and ln denoting the identity matrix of size n. Then we may assume Wu^O for some i with 2 t 1 " where b is an (1, f)-entry,then km:] (zwuwu^^ep ･m wti-b2r + (lower degree terms of b) which cannot be zero, a contradiction. q. e. d [;] 4. Construction of ＼(Z). Let WF~ (Z), rn(l) be as in Corollary to Pro- )Rr' position 1. Then for some positive integer r (''･'["I satisfies / r u i ＼Rr' = |CZ+JD|(Cm/2)+2r)r'(t(CZ+D)-1)Rrr x(^>r[^](Z))0r' ((cz+Drrrr' for M=(^j^J^rn(l). Let {Mj} be any system of representativesof Fn mod rn(l). Let us put where ^=(^0* Then ^(Z) satisfies (*) W(MZ)= CZ+D＼am/2>+2r>r' (HCZ+D)-Trr'W(Z)UCZ+D)-1frr' for M _(AB＼ r Proposition 2. Let Zo be any point of Hn, and let W be any non-zero com- plex symmetric matrix of size n. Let m be an integer with m^2(n ―1). Then for infinitelymany r and for infinitelymany r', thereis a symmetric matrix W{Z) Multi-tensors of differentialforms 115 of size nrr' satisfying the above transformation formula (*) for Fn such that tr(W(ZoWRrr')*O. Proof. Let F' be a positive symmetric matrix of size m with rational co- efficients.By Lemma 5, and by the argument just before it, tr＼WF',rL/K.ZW2"') is not identically zero for infinitelymany r if we take a suitable theta charac- teristic (",). Since the analytic closure of {Zo+P＼ rational symmetric matrices P of size n}(ZHn equals Hn, there is P such that tr(WF>,rK~＼(Z0+P)WRr)*0. [VI """ J By Lemma 4, WF> r , (Z+P) is written as a linear combination of the similar u, matrices as WF> whose entries are theta series in Z. It follows that there T ["](Z0Wr)^a is F of size m, and ("j such that tr＼WF,r We make ＼(Z) [-1 ' from WF,r " KZ) as in the above manner Let Mj=(j＼ijjty be as above. We may assume that M1=l2n. Then tr(W(Z0)W9rr') =tr((WF,r[Uv](Zo)yr'w^) + I^r(|QZo+£i|-<(m/2>+2^'J(QZo+^^^ X(CjZ0+Dj)Rrr'WRrr') =(tr(＼F,r[Uv](Z0W^)J' ](M,-Zo)(QZo+DJ.)0WRr)r' LV which is not zero for infinitelymany r', since the firstterm is not zero. q. e.d. Remark. Our argument actually shows that we can take infinitely many r, r' from the set of multiples of any fixed integer, which makes tr(W(Z0)WRrr>) non-zero in the proposition. Let us put Xm,r,r'=HW{Z)O)Rrr'). Bv Lemma 1. we have the following: 116 Shigeaki Tsuyumine Proposition 3. Suppose r(n―l)^m/2. Then for any modular form f for Fn of weight (r(n―1)―m/2)r', /Am,r,r 5. Extensibility. Let AnA be a toroidalcompactification of Hn/rn{l), which is taken to be smooth and projective for /^3 ([2]). We may assume that ATl is a quotient of An,i by Fn/rn(l). In this section we assume nS^3. Then the singularities of An are just the ramification locus of An,i-^An. Let A°nbe the smooth locus of An. Let flm.r.r1be as in Proposition 3. flm,r,r'is holomorphic multi-tensor oi differentials on the smooth locus of An. However we need to find a condition that it extends holomorphically to a projective non-singular model. By the similar argument as in the case of pluri-canonical differentialforms ([2], Tai [10]), we have the following; Lemma 6. // / vanishes at the cusp of order ^.rr'',then fAm,r>r>extends holomorphically to A°n. Now we discuss the extensibility of fkm,r,T'over the quotient singularities. As in the similar way as in Tai [10], Theorem 3.3, we have the following, whose proof is easy to recover, so that we skip it. Lemma 7. Let X be a quotient S)/G of an open domain 3) in CN by a finite group G acting on S>> Let X-*X be a desingularization. Suppose that g^G acts on the tangent space of a fixed point as a multiplicationby e(Si),z= l, ･･･,N with s^Q, O^st SSj^l+max{Si} i is satisfiedfor each g^l and for each of its fixed point, then a G-invariantform in (Q%~1)Rr extends holomorphically to X. Tai [10] showed that every fixed point in An satisfies Ss^l for n^5. i Following his method, we can find when the condition in the lemma is satisfied. Indeed it is sufficientfor the condition, that Sss^2. It is shown that every i fixed point of order k satisfiesit unless k equals 2,3,4,5,6,8,10,12 orl4. Then we check the condition in the lemma for the individual case of these k, and get our bound for it to hold; n>7. Multi-tensors of differentialforms 117 Now we have the following by Proposition 3, and by Lemmas 6 and 7. Proposition 4. Let n^7. Let Xm>rir>be as in the preceding section. Then if f is a modular form of weight (r{n―l)―m/2)r' with ord(f)^rr', then a multi- tensor fAm,r,r' of differentialsextends holomorphically to a protective non-singular model of An. 6. Let n^7. Suppose that there is a non-trivialmodular form / such that (7i-l)ord(/) >1 weight(/) Then for suitableintegers a, /3,every modular form g in fakA{rn)pk, k=l, 2,･･･, has enough vanishing order at the cusp, i.e., (n―1) ord(g)/weight (g) 2^1/ (l―(m/2r(n―1))) for a fixed m, A(rn)pk denoting the vector space of modular form of weight fik. If weight(g)=(r(n―l)―m/2)r'! then gAm,r.r' extends to a section of (J2f-1)R"" (Proposition 4), and it is taken to be non-trivial (cf. the remark in §4). So, in this case there are many global sections of (Q%~1)Rr for some r. Freitag [6], [7] introduced the desired modular forms when n^lO. Let ■9[w~＼(Z)be a theta constant: #M(Z)= S e(＼Kg+u)Z{g+u)+＼g+u)v) with a theta characteristic w=( " Z2n (modZ), e(2{My)=l. Then the func- tions n^M(Z) and ft = 2 S #[>](Z)8* (/: a natural number) IVn W*Wn have the desired property when n^lO. Moreover in Freitag [7], it is shown that on any subvariety in An of codimension one, all of /{, 1=1, 2, ･･･, do not vanish identically. Theorem 1. Let An be a non-singular model of the moduli An of principally polarized abelian varietiesover C of dimension n. Suppose n^:10. Then for any elements alf ･･･,at of the modular function field, there exists r such that there are t+l elements Ao,Xlf ■■■,Xt^{Q^~1)9r with ai=h/X0, N being n(n+l)/2. 7. Proof of Theorem 2. Let D be any subvariety in An of codimension one. When n^3, there is a modular form h(Z) of some weight p whose divisor 118 Shigeaki Tsuyumine equals D (Freitag [4], [5], or Tsuyumine [13]). Let us put which is a symmetric matrix of size n. Let %: Hn-*An be the canonical projection. Let 7i~＼D)° be the smooth locus of 7:~1(D). Then by using =0, we get Lemma 8. Let n^3. Let D be any subvariety in An of codimension one. Then for infinitelymany r and for infinitelymany r' there are Xm,r,r Proof. Let h be as above. If Zo is a non-singular point of D, then is non-zero. Since Xm,r,r'＼n-icD)°:=tr(W(Z) For D, we can take a modular form /with (n―l)ord(/)/weight(/)>l, not vanishing identicallyon D, provided that n2slO. By Lemma 8, there are many multi-tensors of the form g2.m,r,r',g being a modular form, extending holomor- phically to a non-singular model of An whose restrictionsto D do not vanish identically. This proves Theorem 2. Proof of Corollary. We denote by An,t*,the Satake compactificationof AnA. Let 0 be any birationalautomorphism of Anil*. We can eliminatepoints of indeterminancy for 6, by taking a commutative diagram /v *l/ X An,* where induces an automorphism of Hn/Fn(l). Since Fn is the maximal discrete sub- group of Sp2n(R) having Fn(l) as its normal subgroup, Remark. By our argument, the assertion in the corollary holds for n^2 if / is sufficientlylarere. References [1] Andrianov, A. V. and Maloletkin, G. N., Behavior of theta series of degree N under modular substitutions, Math. USSR Izvest. 9 (1975), 227-241. [2] Ash, A., Mumford, D., Rapoport, M. and Tai, Y., Smooth compactification of locally symmetric varieties, Math. Sci. Press, Brookline, Massachusetts, 1975. [3] Barnes, E. S. and Cohn, M. J., On the inner product of positive quadratic forms, 1 London Math. Soc. (2) 12 (1975). 32-36. [4] Freitag, E., Stabile Modulformen, Math. Ann. 230 (1977), 197-211. [5] Die Irreducibilitat der Schottky relation (Bemerkungen zu einem Satz von J. Igusa), Archiv der Math. 40 (1983), 255-259. [6] Siegelsche Modulfunktionen, Grundlehren 254, Springer-Verlag, Berlin Heidelberg, New York, 1983. [7] Holomorphic tensors on subvarieties of the Siegel modular variety (Automorphic forms of several variables, Taniguchi symposium, Katata, 1983), Birkhaser, Progress in Math. 46 (1984), 93-113. [8] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero: I. II, Ann. of Math. 79 (1964), 109-326. [9] Mumford, D., On the Kodaira dimension of the Siegel modular variety (Algebraic Geometry―Open problems), Lect. Notes in Math. 997 (1982), 348-375, Springer- Verlag, Berlin, Heidelberg, New York, Tokyo. [10] Tai, Y., On the Kodaira dimension of the moduli space of abelian varieties,Invent. Math. 68 (1982), 425-439. [11] Tsuyumine, S., Constructions of modular forms by means of transformation formulas for theta series. Tsukuba I. Math. 3 (1979). 59-80. [12] Theta series of a real algebraic number field, Manuscripta Math. 52 (1985), 131-149. [13] Factorial property of a ring of automorphic forms, Trans. Amer. Math. Soc, 296 (1986), 111-123. Sigeaki Tsuyumine SFB 170, "Geometrie und Anolysis" Mathematisches Institut, Bunsenstr. 3-5, 3400 Gottingen WEST GERMANY current address: Department of Mathematics Mie University Tsu, 514 JAPAN