On elliptic modular surfaces - Project Euclid

J. Math. Soc. Japan Vol. 24, No. 1, 1972

On elliptic modular surfaces By Tetsuji SHIODA* (Received March 1, 1971)

Introduction

The purpose of this paper is to study a certain class of algebraic elliptic surfaces called elliptic modular surfaces from both analytic and arithmetic point of view. Our results are based on the general theory of elliptic surfaces due to Kodaira [11]. Let $B$ denote an (algebraic) elliptic surface having a global section over and $G$ the functional and homological its base curve . We denote by $B$ the family of (not necessarily algeinvariants of over , and by braic) elliptic surfaces over with the functional and homological invariants is non-constant and that and . We assume throughout the paper that contain no exceptional curves of the first kind. The the fibres of $B$ over part I is devoted to the generalities on such an elliptic surface $B$ . In \S 1, we give an explicit description of the structure of the N\’eron-Severi group of $B$ ; for the sake of later use, the results are formulated over an arbitrary algebraically closed ground field. In \S 2, we compute the cohomology groups with coefficients in the of the base curve (or Riemann surface) following Kodaira. This gives an analytic proof of the so-called sheaf of formula. In \S 3, it is shown that, in the family $J$

$\Delta$

$\mathcal{F}(J, G)$

$\Delta$

$\Delta$

$J$

$G$

$J$

$\Delta$

$\Delta$

$H^{i}(\Delta, G)$

$G$

$Ogg-\check{S}afarevi\check{c}’ s$

$\mathcal{F}(J, G)$

analytic elliptic surfaces, algebraic surfaces are ”dense” (Theorem 3.2); this answers a question raised by Kodaira. The results in \S 1 or 2 must be wellknown, but they are included here because we could not find a suitable

reference. In the part II, we develop the analytic theory of elliptic modular surfaces. for each subgroup First in \S 4 we define the elliptic modular surface of finite index of $SL(2, Z)$ such that $\Gamma*-1,$ and examine its singular fibres of and numerical characters. The base is the compact Riemann surface associated with . In \S 5, we show that the group of global sections over of an elliptic modular surface is a finite group (Theorem 5.1). In other words, the generic fibre of is an elliptic curve deflned over the field of modular functions with respect to , and it has only a finite $B_{\Gamma}$

$\Delta_{\Gamma}$

$B_{\Gamma}$

$\Gamma$

$B_{\Gamma}$

$\Delta_{\Gamma}$

$B_{\Gamma}$

$K_{\Gamma}$

*Partially supported by Fujukai.

$\Gamma$

$\Gamma$

Elliptic modular

21

surfaces

-rational points. A few examples are given. In \S 6, we prove number of of -cusp forms of weight 3 is canonically isomorphic that the space (Theorem 6.1). In \S 7, we give to the space of holomorphic 2-forms on ; a geometric interpretation of Shimura’s complex torus attached to containing namely it is essentially the parameter space of the family (Theorem 7.3 and Remark 7.5). Moreover, the group of division points of this complex torus has an algebro-geometric (or rather arithmetic) meaning as essentially the group of locally trivial algebraic principal homogeneous spaces for . over In the appendix, we shall consider arithmetic questions concerning elliptic modular surfaces. As is well-known, the fibre systems of (self-products of) being a elliptic curves or abelian varieties parametrized by a curve certain arithmetic subgroup of $SL(2, R))$ have been considered by several people–notably by Sato, Kuga, Shimura, Ihara, Morita and Deligne ([14], [8], [16], [2]). Their main result was to establish the relation between the local zeta functions of the fibre varieties and the Hecke polynomials, and thereby to reduce the Ramanujan-Petersson conjecture on the eigenvalues of Hecke of -cusp forms to the Weil conjecture operators acting on the space poles of the zeta function of a (nonand zeroes on the absolute values of the singular complete) variety defined over a finite field. In these treatments the fibre varieties seem to play a somewhat auxiliary role (except for [14]). Now we think it worthwhile to study a suitable compactification of such a fibre variety as an example of a non-singular complete variety. In particular, ) will provide (attached to certain groups the elliptic modular surfaces important examples for the arithmetic theory of non-singular complete surfaces. From such a viewpoint, we recall the algebraic formulation of the elliptic modular surface of level $n(n\geqq 3)$ in section $A$ , and the arithmetic theory of surfaces in section B. Then we determine the zeta function of the in characteristic , using the results of elliptic modular surface of level Deligne [2] or Ihara [9]. We discuss the validity of various conjectures for such a surface. The main results of this paper were announced in two short notes [24]. We wish to take this opportunity to thank Professor Kodaira for his interest in this work and for showing us his notes on the results of \S 2. We also wish to thank Professor Shimura for several valuable remarks. $K_{\Gamma}$

$S_{3}(\Gamma)$

$\Gamma$

$B_{\Gamma}$

$S_{3}(\Gamma)$

$\mathcal{F}(J, G)$

$B_{\Gamma}$

$B_{\Gamma}$

$\Delta_{\Gamma}$

$\Gamma\backslash \mathfrak{H}(\Gamma$

$S_{w}(\Gamma)$

$\Gamma$

$\Gamma$

$B_{\Gamma}$

$n$

$p$

T.

22

$s_{HlOD_{-\{}}$

Part I. Generalities.

\S 1. N\’eron-Severi group of an elliptic surface. denote a non-singular proWe fix an algebraically closed field . Let and let $B$ denote a non-singular projective surface havjective curve over with the canonical projection ing a structure of an elliptic surface over $B$ . We assume that admits a section over and that the fibres contain no exceptional curves of the first kind. In the following, we of shall describe the structure of the N\’eron-Severi group $NS(B)$ of the surface $B,$ . . the group of algebraic equivalence classes of divisors on $B$ . over the For that purpose we first consider the fibre $E=\Phi^{-1}(u)$ of of . $E$ is an elliptic curve defined over the function field generic point $K=k(u)$ of , given with a K-rational point $0=o(u)$ . Let $E(K)$ denote the group of K-rational points of $E$. Then, by the Mordell-Weil theorem ([15] p. 71), $E(K)$ is a finitely generated abelian group provided that the absolute invariant of $E$ is transcendental over the constant field ; we always assume that this condition is satisfied in what follows. Let be the rank of of $E(K)$ modulo the torsion subgroup generators $E(K)$ and take $E(K)_{tor}$ of order $E(K)_{tor}$ . is generated by at most two elements ; $|E(K)_{tor}|=e_{1}e_{2}$ . Now the group $E(K)$ of K-rational points with $E$ is canonically identified with the group of sections of $B$ over . For of the image (curve) in $B$ of the section correeach $s\in E(K)$ , we denote by sponding to . We put $k$

$\Delta$

$k$

$\Delta$

$\Phi:B\rightarrow\Delta$

$\Delta$

$0$

$B$

$i$

$e$

$B$

$\Delta$

$u$

$\Delta$

$k$

$J$

$r$

$r$

$s_{1},$

$\cdots$

$s_{r}$

$t_{1},$

$1\leqq e_{2},$

$t_{2}$

$e_{1},$

$e_{2}$

$e_{2}|e_{1}$

$\Delta$

$(s)$

$s$

(1.1)

$D_{\alpha}=(s_{a})-(0)$

$D_{\beta}^{\prime}=(t_{\beta})-(0)$

, ,

$1\leqq\alpha\leqq r$

$\beta=1,2$

,

.

Next we consider the singular fibres of $B$ over . The classification of singular fibres are given in Kodaira [11] (cited as [K] in the following) or denote the finite in N\’eron [17]. We shall follow Kodaira’s notation. Let , for which $C_{v}=\Phi^{-1}(v)$ is a singular fibre. For each of set of points $\Theta_{v,i}(0\leqq i\leqq m_{v}-1)$ components of the divisor the irreducible we denote by being the number of irreducible components. We take to be the containing the identity $o(v)$ . Then we have unique components of $\Delta$

$\Sigma$

$v$

$C_{v},$

$ v\in\Sigma$

$\Delta$

$\Theta_{v,0}$

$m_{v}$

$C_{v}$

(1.2)

$C_{v}=\Theta_{v,0}+\sum_{i\geqq 1}\mu_{v,i}\Theta_{v,i}$

,

$\mu_{v,i}\geqq 1$

.

denote the square matrix of size $(m_{v}-1)$ whose $(i, j)$ -coefficient is , where denotes the intersection number of the divisors $B$ $D$ . on . Finally we take and fix a non-singular fibre and With these notations, we can state

Let

$A_{v}$

$(\Theta_{v,i}\Theta_{v,j})(i, j\geqq 1)$

$D^{\prime}$

$(DD^{\prime})$

$C_{u_{0}}(u_{0}\not\in\Sigma)$

Elliptic modular

THEOREM 1.1. The N\’eron-Severi generated by the following divisors: (1.3)

$C_{u_{0}}$

(0),

The

fundamental

$(\beta=1,2)$

,

group

23

surfaces

of

$NS(B)$

$\Theta_{v,i}(1\leqq i\leqq m_{v}-1, v\in\Sigma)$

and

$D_{\alpha}(1\leqq\alpha\leqq r)$

the elliptic

surface

$B$

is

,

$D_{\beta^{\prime}}(\beta=1,2)$

.

relations among them are given by (at most) two relations

:

(1.4)

$e_{\beta}D_{\beta^{\prime}}\approx e_{\beta}(D_{\beta^{\prime}}(0))\cdot C_{u_{0}}+\sum(\Theta_{v,1}, \Theta_{v,m_{v}- 1})e_{\beta}A_{v}^{-1}(((D^{\beta^{\prime}}\dot{\Theta}_{v,m}))^{*)}D_{\beta}\Theta_{v,1})_{O-1}$

.

and LEMMA 1.2. Any two fibres valent to each other. In particular, for $C_{u_{1}}$

$C_{u_{2}}(u_{1}, u_{2}\in\Delta)$

$ v\in\Sigma$

(1.5)

are algebraically equi-

,

$C_{u_{0}}\approx\Theta_{v,0}+\sum_{\ell\geq 1}\mu_{v,i}\Theta_{v,i}$

.

PROOF. This is clear from the definition of algebraic equivalence. (1.5) follows from (1.2). LEMMA 1.3. The matrix $A_{v}=((\Theta_{v,t}\Theta_{v,j}))_{1\leqq i,j\leqq m_{\emptyset}-1}$

is negative definite and the absolute value of $\det(A_{v})$ is equal to the number is isoof simple components of . Moreover, the group morphic to the finite abelian group attached to the singular fibre [K] 604). (see below and p. PROOF. This can be checked case by case for each type of singular fibre $-A_{v}$ gives a ([K], \S 6). For example, for the singular fibre of type positive definite even integral quadratic form of discriminant 1 in 8 variables. LEMMA 1.4. Suppose that a divisor $D$ on $B$ does not meet the generic fibre E. Then the algebraic equivalence class of $D$ is uniquely expressed as a linear combination of and $\Theta_{v,i}(v\in\Sigma, i\geqq 1)$ : $A_{v}^{-1}Z^{m_{v^{-1}}}/Z^{m_{v}-1}$

$C_{v}$

$m_{v}^{(1)}$

$C_{v}^{*}/\Theta_{v^{\#},0}$

$C_{v}$

$I1^{*},$

$C_{v}$

$C_{uo}$

$D\approx(D(0))C_{uo}+\sum_{v}(\Theta_{v,1}, \Theta_{v,m_{v}-1})A_{v}^{-1}(((DD\dot{\Theta}_{v,m-1}^{v,1}\Theta)_{v}))$

.

PROOF. By assumption, each component of $D$ is contained in a fibre. Hence the assertion follows immediately from Lemmas 1.2 and 1.3. PROOF OF THEOREM 1.1. First we show that an arbitrary divisor $D$ on $B$ is algebraically equivalent to a linear combination of divisors in (1.3). By Lemma 1.2 we may assume that no component of $D$ is contained in a fibre. If we put $d=(DC_{u})$ , the divisor $D-d(0)$ cuts out on the generic fibre $E$ a gives a K-rational of points in divisor of degree zero. The sum $\mathfrak{d}$

$\mathfrak{d}$

$*)$

The symbol

$S(\mathfrak{d})$

$\approx$

indicates algebraic equivalence.

T. SHIODA

24

point of

$E$

, say

$s$

.

Since

$E(K)$

is generated by

$s_{1}$

,

$\cdot$

.. ,

$s_{r}$

and

$t_{1},$

$t_{2}$

, we can

write ,

$s=\sum_{\alpha=1}a_{a}s_{a}+\sum_{\beta=1}^{2}b_{\beta}t_{\beta}$

where

$a_{\alpha},$

are integers. Putting

$b_{\beta}$

$D^{\prime}=\sum_{\alpha}a_{a}D_{\alpha}+\sum_{\beta}b_{\beta}D_{\beta}^{\prime}$

sor

.

By Abel’s theorem on an elliptic curve, the divion $E$. Therefore the divisor $D-d(0)-D^{J}$ is linearly equivalent to

we see that $\mathfrak{d}$

,

$S(\mathfrak{d})=S(D^{\prime}\cdot E)$

$D^{\prime}\cdot E$

does not meet the generic fibre. Applying Lemma 1.4, we conclude that $D$ is algebraically equivalent to a linear combination of divisors in (1.3). To prove the second part of the theorem, suppose that there is a relation: (1.6)

$\sum_{\alpha}a_{\alpha}D_{\alpha}+\sum_{\rho}b_{\beta}D_{\beta}^{\prime}+cC_{u_{0}}+\sum_{v}\sum_{\iota\geq 1}d_{v,i}\Theta_{v,i}+e(0)\approx 0$

,

.

By considering the intersection number with , $B$ $e=0$ variety canonically of isomorphic Picard the is . Since get to the we (1.6) linearly , variety side of equivalent left is of the to a divisor Jacobian , where is a divisor of degree zero on . Restricting it of the form $E$ , get we fibre generic to the

with integers

$a_{\alpha},$

$e$

$b_{\beta},$

$C_{u_{0}}$

$\Delta$

$\Phi^{-1}(c)$

$\Delta$

$c$

$\sum_{\alpha}a_{\alpha}D_{\alpha}\cdot E+\sum_{\beta}b_{\beta}D_{\beta}^{\prime}\cdot E\sim o.**)$

Again by Abel’s theorem, this is equivalent to $\sum_{\alpha}a_{a}s_{\alpha}+\sum_{\beta}b_{\beta}f_{\beta}=0$

.

and $b_{\beta}\equiv 0mod e_{\beta}(\beta=1,2)$ . On the other Hence we get and hand Lemma 1.4 implies the relations (1.4) for $\beta=1,2$ and also that algebraic independent equivalence. Therefore modulo are (1.4) consequence (1.6) of the relations a for $\beta=1$ and 2. This is relation the completes the proof of Theorem 1.1. The rank of N\’eron-Severi group $NS(B)$ is called the Picard number of $B$ . $a_{\alpha}=0(1\leqq\alpha\leqq r)$

$C_{u_{0}}$

$\Theta_{v,i}(v\in\Sigma, i\geqq 1)$

Obviously

we

get (cf. [27] p. 15)

COROLLARY 1.5. The Picard number

$\rho$

of

the

$\rho=r+2+\sum_{v\in\Sigma}(m_{v}-1)$

surface

$B$

is given by

.

denote the set of points of multiplicity one on the divisor Let . III, [17] Ch. is a commutative algebraic group As is shown in [K] \S 9 or $o(v)$ , identity is the connected component of in which with is a multiis a singular fibre of type $I_{b}(b\geqq 1)$ , then the identity. If $C_{v}^{*}$

$C_{v}$

$C_{v}^{*}$

$\Theta_{v.0}^{*}=\Theta_{v,0}\cap C_{V}^{*}$

$\Theta_{v^{*}.0}$

$C_{v}$

$**)$

The symbol

$\sim$

indicates linear equivalence.

Elliptic modular

25

surfaces

plicative group and the quotient group is a cyclic group of order . If is a singular fibre of other type, then is an additive group and is a group of order at most 4. PROPOSITION 1.6. Let $E(K)_{0}$ be the subgroup of $E(K)$ consisting of such that . Then $E(K)_{0}$ is a torsion-free subgroup of finite for all $b$

$C_{v}^{\#}/\Theta_{v^{\#},0}$

$C_{v}$

$\Theta_{v^{\#}.0}$

$C_{v}^{\#}/\Theta_{v^{\#}.0}$

$s$

$ v\in\Sigma$

$s(v)\in\Theta_{v^{\#}.0}$

index in $E(K)$ . PROOF. Suppose that is an element of $E(K)_{0}$ of finite order Applying Lemma 1.4 to the divisor $D=n[(s)-(0)]$ , we get $s$

$n[(s)-(0)]\approx n([(s)-(0)](0))C_{u_{0}}$

$n>1$

.

,

since $D$ does not meet . By taking the intersection number of for both side with the divisor , we have $i\geqq 1$

$\Theta_{v,i}$

$(s)$

$((s)(s))+((0)(0))=2((s)(0))\geqq 0$

.

This contradicts to the fact that $((s)(s))=((0)(0))=-(p_{a}+1)0$

$b_{J)}1$

.

This contradicts the relation (cf. (2.1)) $A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}\cdots C_{1}\cdots C_{t}=1$

.

PROPOSITION 2.2. $H^{2}(\Delta, G)$ is a finite group. For the proof, see [K] Theorem 11.7. (Note that the sheaf $G$ is nonis non-constant.) We remark that trivial since the functional invariant $H^{2}(M)=G_{0}/{\rm Im}(\partial_{2})$ $H^{2}(\Delta, G)$ is isomorphic to where is the subgroup of $J$

${\rm Im}(\partial_{2})$

$G_{0}=Z\oplus Z$

generated by $G_{0}({}^{t}\varphi(\gamma)-1)$

PROPOSITION 2.3.

$H^{1}(\Delta, G)$

(2.6)

,

$\gamma\in\pi_{1}(\Delta^{\prime})$

.

is a finitely generated group of rank $4g-4+2t-t_{1}$

,

is the number of singular fibres of types $I_{b}(b\geqq 1)$ . where . By Proposition 2.1 the map PROOF. We consider $2+t_{1}$ , since $G_{j}(1\leqq j\leqq t)$ is of is is injective. The rank of according to whether the singular fibre is of types $1_{b}(b\geqq 1)$ rank 1 or or not. Obviously is of rank $2(2g+t)$ , while the rank of is 2 by $H^{1}(M)\cong H^{1}(\Delta, G)$ is Proposition 2.2. Hence the rank of the remark after $t_{1}$

$H^{1}(M)=Ker(\partial_{2})/{\rm Im}(\partial_{1})$ $M_{0}\cong 1m(\partial_{1})$

$\partial_{1}$

$0$

$C_{v_{j}}$

$M_{1}$

${\rm Im}(\partial_{1})$

equal to $2(2g+t)-2-(2+t_{1})=4g-4+2t-t_{1}$

.

REMARK 2.4. Actually is a free module. This follows from the exact sequence below (2.8) and Proposition 1.6. Let denote the group scheme over associated with the elliptic surface $B$ and let be the connected component of the identity section in we have $H^{1}(\Delta, G)$

$B^{\#}$

$\Delta$

$B_{0}^{*}$

$B^{*};$

$0$

Elliptic modular

29

surfaces

$B_{0}^{*}=\bigcup_{u\subseteq\Delta^{J}}C_{u}\cup(\bigcup_{v\in\Sigma}\Theta_{v^{\#}.0})$

$B^{\#}=\bigcup_{v\in\Delta}C_{v\prime}^{\#}$

defined in the notation of \S 1 (cf. [K] \S 9). Let be the line bundle over of the section in [K] \S 11; is the normal bundle of the image (curve) $B$ the sheaves of germs of holoin . We denote by $o(t),$ or By morphic sections of Theorem 11.2 of [K], we have over . or then the exact sequence: $\Delta$

$\mathfrak{f}$

$o(\Delta)$

$f$

$\Omega(B^{*})$

$o$

$f,$

$B^{\#}$

$\Omega(B_{0}^{\#})$

$\Delta$

$B_{0}^{\#}$

$h$

$i$

(2.7)

$0\rightarrow G\rightarrow \mathcal{O}(f)\rightarrow\Omega(B_{0}^{\#})\rightarrow 0$

.

It induces the exact sequence of cohomology groups: $0\rightarrow H^{0}(\Delta, \Omega(B_{0}^{*}))\rightarrow H^{1}(\Delta, G)\rightarrow^{i^{*}}H^{1}(\Delta, \mathcal{O}(f))$

(2.8)

$h^{*}$

$\rightarrow H^{1}(\Delta, \Omega(B_{0}\#))\rightarrow H^{2}(\Delta, G)\rightarrow 0$

.

) may (respectively Note that the group of sections be identified with the group $E(K)$ of K-rational points of the generic fibre $E$ (respectively, with the subgroup $E(K)_{0}$ (cf. \S 1)). (Incidentally (2.7) gives another proof of Proposition 2.1 since the degree of is negative and hence $H^{0}(\Delta, \Omega(B^{\#}))$

$H^{0}(\Delta,$

$\Omega(B_{0}^{*}))$

$f$

Let

be the rank of the and let image group can be defined as the maximum number such that the group contains a subgroup isomorphic to $(Q/Z)^{\nu}$ , product of copies of the group of rational numbers modulc integers. As an immediate consequence of Proposition 2.3 and (2.8), we get $H^{0}(\Delta, \mathcal{O}(f))=0.)$

$r$

be the rank of ; equivalently

$i^{*}H^{1}(\Delta, G)$

$E(K)$

$r^{\prime}$

$r^{\prime}$

$H^{1}(\Delta, \Omega(B_{0}^{*}))$

$\nu$

$\nu$

THEOREM 2.5. $r+r^{\prime}=4g-4+2t-t_{1}$

.

formula, which has been proved This is a special case of for abelian varieties over function fields of arbitrary characteristic (cf. [18], $ogg-\check{S}afarevi\check{c}’ s$

[19], [20]).

Let

$b_{i}$

(2.9)

denote the i-th Betti number of $b_{1}=2g$

where

$c_{2}$

(2.10)

,

is the Euler number of

$B$

.

Then we have

$b_{2}=c_{2}+2b_{1}-2$ $B$

.

,

By [K] Theorem 12.2,

$c_{2}=12(p_{a}+1)=\sum_{v\in\Sigma}\epsilon_{v}$

$=\mu+6\sum_{b\geqq 0}\nu(I_{b}^{*})+2\nu(II)+10\nu(II^{*})$

$+3\nu(III)+9\nu(III^{*})+4\nu(IV)+8\nu(IV^{*})$

where fibre

is the arithmetic genus of is the Euler number of singular ; $\nu(T)$ is the number of singular fibres of type $T$ : and is the total

$p_{a}$

$C_{v}$

,

$B;\epsilon_{v}$

$\mu$

30

T. SHIODA

. On the other hand, the degree of the map $B$ the Picard number of is given by Corollary 1.5. By a direct computation, one can check that $\epsilon_{v}-m_{v}=0$ or 1 according to whether is of type $1_{b}(b\geqq 1)$ or not. Hence one gets (cf. [19] VI) COROLLARY 2.6. multiplicity of

$J,$

$i$

. .

$J:\Delta\rightarrow P^{1}$

$e$

$\rho$

$C_{v}$

$b_{2}-\rho=r^{\prime}$

.

Further we know from the Lefschetz-Hodge theory that (2.11)

$(h^{pq}=\dim H^{q}(B, \Omega^{p}))$

$b_{2}=2p_{g}+h^{11}$

$\rho\leqq h^{11}$

where

$p_{g}=h^{02}=h^{20}$

,

,

is the geometric genus of

$B$

. Hence we

get

COROLLARY 2.7. $r^{\prime}\geqq 2p_{g}$

or equivalently,

;

$r\leqq 4g-4+2t-t_{1}-2p_{g}$

.

The following result is due to Kodaira. PROPOSITION 2.8. Let $h^{1}=the$ rank of

$H^{1}(\Delta, G)$

.

Then

$h^{1}-2p_{g}\geqq\nu(1_{0}^{*})+\nu(II)+\nu(III)+\nu(IV)$

.

PROOF. We consider the meromorphic differential $\omega=dJ/J$ on . If we denote respectively by the number of zeroes of , of zeroes of and $J-1$ , and the number of poles of , then the divisor of poles of has degree , while the divisor of zeroes of we have Hence . has degree $\Delta$

$\nu^{0},$

$\nu^{1}$

$J$

$\nu^{\infty}$

$J$

$\omega$

$\nu^{0}+\nu^{\infty}$

(2.12)

$2g-2\geqq\mu-\nu^{1}-(\nu^{0}+\nu^{\infty})$

For $i=1,2,3$ , let

(2.13)

.

denote the number of zeroes of whose order is conSimilarly $\nu^{1}(i)(i=1,2)$ denotes the number of zeroes of $J$

$\nu^{0}(\iota)$

gruent to $imod 3$ . $J-1$

$\geqq\mu-\nu^{1}$

$\omega$

whose order is congruent to $imod 2$ . $\mu\geqq\nu^{0}(1)+2\nu^{0}(2)+3\nu^{0}(3)$

,

Obviously we have $\mu\geqq\nu^{1}(1)+2\nu^{1}(2)$

.

From (2.12) and (2.13), we get (2.14)

$61\mu\leqq 2g-2+-2^{-\nu^{1}(1)+-}3^{-\nu^{9}(1)+}123^{-\nu^{0}(2)+\nu^{\infty}}1$

.

On the other hand, we know from [K] \S 8 that $\nu^{\infty}=\sum_{b\geqq 1}(\nu(I_{b})+\nu(I_{b}^{*}))$

(2.15)

$\nu^{0}(1)=\nu(II)+\nu(IV^{*})$

,

$\nu^{1}(1)=\nu(III)+\nu(11I^{*})$

Then, computing

$h^{1}-2p_{g}$

, $\nu^{0}(2)=\nu(II^{*})+\nu(IV)$

,

.

by (2.6), (2.10) and comparing it with (2.14), (2.15),

Elliptic modular

we

31

surfaces

get the assertion.

\S 3. A density theorem. denote the family of all (not necessarily algebraic) elliptic $G$ having the same functional and homological invariants as the elliptic surface $B$ considered in \S 2. We refer to [K] \S 9, 10, 11 for over what follows. Let $A$ be one of the sheaves of groups or . For each cohomology class $\eta\in H^{1}(\Delta, A)$ , let denote the elliptic surface modulo Ain obtained by twisting $B$ with . The family equivalence is parametrized by the cohomology group . Moreover is an algebraic surface if and only if is an element of finite order of ( $[K]$ Theorem 11.5). Now from the exact sequence (2.8) we see that

Let surfaces over

$\mathcal{F}(J, G)$

$\Delta$

$J,$

$\Omega(B^{*})$

$\Delta$

$\Omega(B_{0^{\#}})$

$B^{\eta}$

$\mathcal{F}(J, G)$

$\mathcal{F}(J, G)$

$\eta$

$H^{1}(\Delta, A)$

$B^{\eta}$

$\eta$

$H^{1}(\Delta, A)$

(3.1)

$H^{1}(\Delta, \Omega(B_{0}^{*}))\cong h^{*}H^{1}(\Delta, \mathcal{O}(f))\times H^{2}(\Delta, G)$

,

is a divisible group. For a fixed , we denote by the element of corresponding to $(h^{*}(t), c)$ under the isomorphism (3.1). Then the collection ( $[K]$ Theorem 11.3). forms a complex analytic family We shall be concerned with the following question: “Are algebraic suris faces dense in the family Since or in the family a finite group by Proposition 2.2, is an algebraic surface if and only if is a rational linear combination of elements of . Thus the above question is equivalent to whether or not the vector space is spanned over $R$ (the real numbers) by . Note that the rank of by Corollary 2.7, while is is a complex vector space of dimension ([K] p. 15). Let us consider the exponential exact sequence of sheaves on $B$

since

$h^{*}H^{1}(\Delta, \mathcal{O}(f))=H^{1}(\Delta, o(f))/i^{*}H^{1}(\Delta, G)$

$c\in H^{2}(\Delta, G)$

$H^{1}(\Delta, \Omega(B_{0}^{\#}))$

$\eta(t)(t\in H^{1}(\Delta, \mathcal{O}(\mathfrak{f})))$

$\{B^{r(t)}|t\in H^{1}(\Delta, \mathcal{O}(f))\}$

$\mathcal{V}^{(c)}$

$\mathcal{V}^{(c)}$

$H^{2}(\Delta, G)$

$\mathcal{F}(J, G)^{p}$

$B^{\eta(t)}$

$t$

$i^{*}H^{1}(\Delta, G)$

$H^{1}(\Delta, \mathcal{O}(f))$

$i^{*}H^{1}(\Delta, G)$

$i^{*}H^{1}(\Delta, G)$

$H^{1}(\Delta, \mathcal{O}(f))$

$r^{\prime}\geqq 2p_{g}$

$p_{g}$

$j$

$0\rightarrow Z\rightarrow \mathcal{O}\rightarrow \mathcal{O}^{x}\rightarrow 0$

.

In the corresponding exact sequence of cohomology groups, the image of in $H^{2}(B, Z)$ can be identified with the N\’eron-Severi group of $B$ . Thus we have the exact sequence: $H^{1}(B, \mathcal{O}^{\times})$

$j^{*}$

(3.2)

$ 0\rightarrow NS(B)\rightarrow H^{2}(B, Z)\rightarrow H^{2}(B, \mathcal{O})\rightarrow\ldots$

We shall compare this with the exact sequence (2.8)

$0\rightarrow H^{0}(\Delta, \Omega(B_{0}^{\#}))\rightarrow H^{1}(\Delta, G)\rightarrow^{i^{*}}H^{1}(\Delta, \mathcal{O}(f))$

.

THEOREM 3.1. There exists an isomorphism of $H^{1}(\Delta, o(f))$ onto such that is commensurable with $j^{*}H^{2}(B, Z)$ in . $\psi$

$\psi(i^{*}H^{1}(\Delta, G))$

$H^{2}(B, \mathcal{O})$

$H^{2}(B, \mathcal{O})$

32

T. SHIODA

PROOF. Let

denote the canonical projection of the Leray spectral sequence (cf. [2] p. 30-31): $\Phi$

$B$

$E\oint^{?}=H^{p}(\Delta, R^{q}\Phi_{*}(\mathcal{O}))\Rightarrow H^{p\cdot\vdash q}(B, \mathcal{O})$

onto

$\Delta$

. We

consider

.

For a positive integer $m>1$ , we denote by the rational map of $B$ to itself over , which induces multiplication by $m$ on the generic fibre $E$ . It can be checked that, for suitably chosen $m,$ is a holomorphic map of $B$ onto $B$ . (For instance, we can take $m$ such that $m\equiv 1mod m_{0}$ , where is the least $B$ , supposing that has singular fibres common multiple of 12 and b\’is .. $b=b_{1}$ ) The map , , of of types for induces an endomorphism by multiplication spectral sequence, the acts on as and : comsince it acts on as such. The map mutes with are modules over and this implies that $d_{r}=0(r\geqq 2)$ since a field of characteristic zero. Therefore is isomorphic . to the submodule of on which acts as multiplication by In particular, we get an isomorphism $\varphi_{m}$

$\Delta$

$\varphi_{m}$

$m_{0}$

$(1\leqq i\leqq t_{1})$

$I_{b}$

$\varphi_{m}^{*}$

$b_{t_{1}}.$

$\cdot$

$\varphi_{m}$

$m^{q}$

$E_{r}^{p\cdot q}(r\geqq 2)$

$\varphi_{m}^{*}$

$R^{q}\Phi_{*}(o)$

$E_{\gamma}^{p,q}\rightarrow E_{r}^{p+r.q-r+1}(r\geqq 2)$

$d_{r}$

$E_{r}^{pq}$

$\varphi_{m}^{*}$

$E_{2}^{p.q}=H^{p}(\Delta, R^{q}\Phi_{*}(\mathcal{O}))$

$H^{p^{4}q}(B, \mathcal{O})$

(3.3)

$\psi:$

If

$m^{q}$

$\varphi_{m}^{*}$

$(\Delta, \mathcal{O}(t))\rightarrow^{\sim}H^{2}(B, \mathcal{O})$

,

using the fact and $\dim H^{1}(\Delta, \mathcal{O}(f))=p_{g}=\dim H^{2}(B, \mathcal{O})$ . Applying the above argument to the constant sheaf $Q$ on $B$ instead of the sheaf and $R^{1}\Phi_{*}(Q)\cong G\otimes Q$ noting that , we get a homomorphism $R^{1}\Phi_{*}(o)\cong \mathcal{O}(\overline{|})$

$\mathcal{O}$

(3.4)

$\psi^{\prime}:H^{1}(\Delta, G\otimes Q)\rightarrow H^{2}(B, Q)$

.

maps the subgroup of It follows from (3.3) and (3.4) that $j^{*}H^{2}(B, Z)\otimes Q$ . Since the rank of into the subgroup $j^{*}H^{2}(B, Z)$ by Corollary 2.6, we of equal of to the rank is is commensurable with $j^{*}H^{2}(B, Z)$ , which completes conclude that the proof. is non-constant on THEOREM 3.2. Assume that the functional invariant . Then algebraic surfaces are dense in the family of elliptic surfaces with the functional and homological invariants and . over PROOF. By Theorem 3.1 and the argument preceding it, it is sufficient is spanned over $R$ by the image to show that the vector space $j^{*}H^{2}(B, Z)$ of $H^{2}(B, Z)$ . This follows from the following general fact. $X$ LEMMA 3.3. Let denote a compact Kahler manifold and let be the natural injection of sheaves on X. Then, for each $n\geqq 1$ , the vector space is spanned over $R$ by the image $j^{*}H^{n}(X, Z)$ . PROOF. It is enough to show that the map $i^{*}H^{1}(\Delta, G)\otimes Q$

$\psi$

$H^{2}(B, \mathcal{O})$

$H^{1}(\Delta, \mathcal{O}(f))$

$i^{*}H^{1}(\Delta, G)$

$r^{\prime}$

$ b_{2}-\rho$

$\psi(i^{*}H^{1}(\Delta, G))$

$J$

$\mathcal{F}(J, G)$

$\Delta$

$G$

$J$

$\Delta$

$H^{2}(B, \mathcal{O})$

$j:Z\rightarrow \mathcal{O}$

$H^{n}(X, \mathcal{O})$

$j_{1}^{*}$

:

$H^{n}(X, R)\rightarrow H^{n}(X, \mathcal{O})$

induced by the canonical map

$j_{1}$

:

$R\rightarrow \mathcal{O}$

is surjective.

$j_{1}^{*}$

factors as

Elliptic modular

33

surfaces $j_{2}^{*}$

$H^{n}(X, R)\rightarrow H^{n}(X, C)\rightarrow H^{n}(X, \mathcal{O})$

,

. The Hodge decomposition $H^{n}(X, C)$ is the canonical map are related so that and the Dolbeault isomorphism . Hence corresponds to the projection the surjectivity of is clear. be COROLLARY 3.4. Let $X$ be a non-singular projective variety and let the rank of the N\’eron-Severi group of X. If $\rho=h^{11}$ , then $H^{2}(X, \mathcal{O})/j^{*}H^{2}(XZ)$ has a structure of a complex torus. For the elliptic surface $B$ , we get , then COROLLARY 3.5. If $r^{\prime}=ranki^{*}H^{1}(\Delta, G)$ is equal to (or group cohomology is a complex torus. Hence the is a product of a complex torus and a finite group. REMARK 3.6. Theorem 3.2 and Corollary 3.5 were observed in [24] in the special case where $B$ is an elliptic modular surface (cf. \S 7 below). REMARK 3.7. It is unknown whether or not every elliptic surface in the family is a K\"ahler surface. Using Theorem 3.2 this question can be reduced to a local one around the singular fibres as follows. For each point , let $(\omega(u), 1)$ be a normalized period of the elliptic curve $C_{u}=\Phi^{-1}(u)$ . is a multivalued holomorphic function on with $1m(\omega(u))>0$ . Let be the universal covering of ; we consider as a single-valued function by a certain on . Now is a quotient of the product group of holomorphic automorphisms (cf. [K] p. 580 (8.5)). For each point , we put of where

$C\rightarrow \mathcal{O}$

$j_{2}$

$H^{n}(X, \mathcal{O})\cong H^{0,n}$

$=\bigoplus_{p+q=n}H^{pq}$

$j_{2}^{*}:$

$\oplus H^{p,q}\rightarrow H^{0n}$

$H^{n}(X, C)\rightarrow H^{n}(X, \mathcal{O})$

$j_{1}^{*}$

$\rho$

$H^{1}(\Delta, \mathcal{O}(f))/$

$2p_{g}$

$H^{1}(\Delta, \Omega(BO))$

$i^{*}H^{1}(\Delta, G)$

$H^{1}(\Delta, \Omega(B\#)))$

$\mathcal{F}(J, G)$

$ u\in\Delta^{\prime}\subset\Delta$

$\omega(u)$

$U^{\prime}$

$\Delta^{\prime}$

$\Delta^{\prime}$

$U^{\prime}$

$\omega$

$U^{\prime}\times C$

$B^{\gamma}=\Phi^{-1}(\Delta^{\prime})$

$\mathcal{G}$

$(\tilde{u}, \zeta)$

$U^{\prime}\times C$

$\zeta=\xi_{1}\cdot\omega(\tilde{u})+\xi_{2}$

Consider the metric on (3.7)

$U^{\prime}\times C$

,

$\xi_{1},$

$\xi_{2}\in R$

.

:

$\Phi^{*}(ds_{0}^{2})+\frac{\omega}{m}\frac{\tilde{u}}{\omega}-\underline{|d\xi}_{1}\cdot()+\underline{d\xi_{2}}|^{2}I((\tilde{u}))$

is a K\"ahler metric on . Then it is easy to see that (3.7) is a K\"ahler metric on , invariant under the group . Thus it defines a K\"ahler metric ”constant” translaunder on . Moreover it is invariant tions: where

$\Delta$

$ds_{0}^{2}$

$U^{\prime}\times C$

$ds^{2}$

$\mathcal{G}$

$B^{\prime}$

$(\tilde{u}, \zeta)\leftrightarrow(\tilde{u}, \zeta+a_{1}\omega(\tilde{u})+a_{2})$

are real constants

.

,

is spanned By Theorem 3.2, with over $R$ by . Hence we see that the elliptic surface $B$ $t\in H^{1}(\Delta, o(t))$ is obtained by twisting with local ”constant” translations. on Therefore will be a K\"ahler surface provided that the metric can be extended to a K\"ahler metric on $B$ by modifying at singular fibres. where

$a_{1},$

$a_{2}$

$i^{*}H^{1}(\Delta, G)$

$B^{\eta(t)}$

$mod 1$

$H^{1}(\Delta, \mathcal{O}(f))$

$B^{\eta^{(t)}}$

$ds^{2}$

$B^{f}$

34

T. SHIODA

For instance, for the singular fibre of type verified.

$I_{0}^{*}$

, this last extendability

can be

Part II. Analytic theory of elliptic modular surfaces.

\S 4. Elliptic modular surfaces. In this section we shall introduce a certain class of elliptic surfaces, called elliptic modular surfaces, connected with the theory of automorphic functions of one variable. For the theory of automorphic functions, see for example [10] or [29]. Let denote a subgroup of finite index of the homogeneous modular group $SL(2, Z)$ . The group acts on the upper half plane , we put in the usual manner: for and $\Gamma$

$\Gamma$

$\mathfrak{H}$

$\gamma\in\Gamma$

(4.1)

$z\in \mathfrak{H}$

$\gamma\cdot z=\frac{az+b}{cz+d}$

,

$\gamma=\left(\begin{array}{ll}a & b\\c & d\end{array}\right)$

.

by , together with a finite number of cusps, forms The quotient of , then the canonical map of a compact Riemann surface, say . If holomorphic map onto of extends to a . In particular, onto we have a holomorphic map of : onto the projective line $\Gamma\backslash \mathfrak{H}$

$\Gamma$

$\mathfrak{H}$

$\Gamma\subset\Gamma_{1}$

$\Delta_{\Gamma}$

$\Gamma\backslash \mathfrak{H}$

$\Gamma_{1}\backslash \mathfrak{H}$

$\Delta_{\Gamma}$

$P^{1}$

$\Delta_{\Gamma}$

$J_{\Gamma}$

(4.2)

$J_{\Gamma}:\Delta_{\Gamma}\rightarrow P^{1}$

,

by taking $\Gamma_{1}=SL(2, Z)$ and identifying with elliptic modular function Now we make the following assumption on $\Delta_{r_{1}}$

$j$

$(^{*})$

$\Gamma$

$\Delta_{\Gamma_{1}}$

means of the ordinary

by

$P^{1}$

.

acts effectively

:

$\Gamma$

on the upper half plane

$\mathfrak{H}$

(i.

e.

$\Gamma$

D-l).

We put in

$\mu=the$

index of

$t^{\prime}=the$

number of cusps in

$\Gamma\cdot\{\pm 1\}$

$SL(2, Z)$

,

$\Delta_{\Gamma}(t^{\prime}\geqq 1)$

,

\langle 4.3) $s=the$ number of elliptic points in $t=t^{\prime}+s$

$\Delta_{\Gamma}(s\geqq 0)$

,

.

, take a point representing . Then the For an elliptic point in generator of the stabilizer of in and is of order 3 by the assumption is conjugate in $SL(2, Z)$ to either $v\in\Delta_{\Gamma}$

$z$

$z$

(4.4)

$(^{-1}1$

$-10)$

$v$

$\mathfrak{H}$

$\Gamma$

$(^{*})$

or

$(_{-1}0$

$-11)$

.

, the stabilizer of a representative in For a cusp generator which is conjugate in $SL(2, Z)$ to either $v\in\Delta_{\Gamma}$

$ Q\cup t\infty$

} of

$v$

has a

Elliptic modular

or

\langle 4.5) accordingly, the cusp of is given by

$v$

35

surfaces ,

$b>0$

;

is called of the first or second kind. The genus

$g$

$\Delta_{\Gamma}$

\langle 4.6)

$ 2g-2+t^{\prime}+(1_{36}^{11}---)s=--\mu$

.

be the set of cusps and elliptic points in and put If we denote by with the projection the universal covering of , there is a holomorphic map into of such that

Let

$\Sigma$

$\Delta^{\prime}=\Delta_{\Gamma}-\sum\subset\Gamma\backslash \mathfrak{H}$

$\Delta_{\Gamma}$

$U^{\prime}$

$\Delta^{\gamma}$

$U^{\prime}\rightarrow\Delta^{\prime}$

$U^{\prime}$

$\omega$

\langle 4.7)

,

(cf. Remark 3.7). being the elliptic modular function on of the fundamental group is a unique representation

$j$

;

$\mathfrak{H}$

$(\tilde{u}\in U^{\prime})$

$J_{\Gamma}(\pi(\tilde{u}))=j(\omega(\tilde{u}))$

$\pi$

.

$\mathfrak{H}$

$\varphi$

Moreover there of into

$\pi_{1}(\Delta^{\prime})$

$\Delta^{\prime}$

$SL(2, Z)$

(4.8)

$\varphi;\pi_{1}(\Delta^{\prime})\rightarrow\Gamma\subset SL(2, Z)$

,

such that (4.9)

$\omega(\gamma\cdot\tilde{u})=\varphi(\gamma)\cdot\omega(\tilde{u})$

,

$\tilde{u}\in U^{\prime}$

,

$\gamma\in\pi_{1}(\Delta^{\prime})$

,

where the right side is defined as in (4.1). The representation determines a sheaf over , locally constant over with the general stalk $Z\oplus Z$. We can apply to this situation Kodaira’s construction of elliptic surfaces , \langle $[K]$ \S 8). Namely there exists a non-singular algebraic elliptic surface, with a global section having and over as its functional and homo$G)$ in [K]); it is unique up logical invariants ( . . the basic member of to a biregular fibre-preserving map over . DEFINITION 4.1. The elliptic surface over will be called the elliptic modular surface attached to the group . ; obviously they lie We shall examine the singular fibres of over consisting of the elliptic points and the cusps. The of over the subset , where singular type of the fibre $C_{v}(v\in\Sigma)$ is determined by is (cf. [K] p. 604). If is an elliptic a positively oriented loop around on point, is conjugate to either one of the normal form (4.4). Hence the singular fibre is of type or IV. Let (or ) denote the number of (or IV); $s=s_{1}+s_{2}$ . (We shall see below that singular fibres of type $s_{2}=0.)$ If is a cusp, is conjugate to one of the normal forms (4.5). or , according to whether the Hence the singular fibre is of type (or ) denote the number of cusp is of the first or second kind. Let cusps of the first (or second) kind, and put . The numerical characters of are computed as follows. First the irre$\varphi$

$G_{\Gamma}$

$\Delta^{\prime}$

$\Delta_{\Gamma}$

$B_{\Gamma}$

$\Delta_{\Gamma}$

$G_{\Gamma}$

$J_{\Gamma}$

$i$

$\mathcal{F}(J,$

$e$

$\Delta$

$B_{\Gamma}$

$\Delta_{\Gamma}$

$\Gamma$

$B_{\Gamma}$

$\Delta_{\Gamma}$

$\Sigma$

$\Delta_{\Gamma}$

$\varphi(\gamma_{v})\in\Gamma$

$v$

$v$

$\Delta_{\Gamma}$

$\varphi(\gamma_{v})$

$IV^{*}$

$C_{v}$

$s_{2}$

$s_{1}$

$IV^{*}$

$v$

$\varphi(\gamma_{v})$

$C_{v}$

$I_{b}^{*}$

$I_{b}$

$v$

$l_{1}$

$t_{2}$

$t^{\prime}=t_{1}+t_{2}$

$B_{\Gamma}$

$\gamma_{v}$

T. SHIODA

36

of is equal to the genus is, by (2.10), arithmetic genus

gularity

$q$

$B_{\Gamma}$

$g$

of

$\Delta_{\Gamma}$

, which is given by (4.6).

The

$p_{\alpha}$

(4.10)

$12(p_{a}+1)=\mu+6t_{2}+4s_{2}+8s_{1}$

Hence the geometric genus (4.11)

$p_{g}=p_{a}+q$

.

is given by

.

$p_{g}=2g-2+t-t_{1}/2-s_{2}/3$

Comparing (4.11) with Proposition 2.3,

we

get

$h^{1}-2p_{g}=32s_{2}$

$(h^{1}=rankH^{1}(\Delta, G))$

.

On the other hand, Proposition 2.8 implies $h^{1}-2p_{g}\geqq\nu(IV)=s_{2}$

Hence we get (4.12)

$s_{2}=0$

and

.

$h^{1}=2p_{g}$

.

Thus we have proved the following singular fibres of has PROPOSITION 4.2. The elliptic modular surface singular fibres of types $I_{b}(b\geqq 1),$ singular fibres of types 1* $(b\geqq 1)$ and type , where and are respectively the number of cusps of the first kind, the number of cusps of the second kind and the number of elliptic points $t_{1}$

$B_{\Gamma}$

$s$

$t_{2}$

$1V^{*}$

for

$\Gamma$

$t_{1},$

$s$

$t_{2}$

.

Recall the following table from [K] \S 6 and \S 9; as in \S 1, denotes the number of components (resp. simple components) of

$m_{v}$

(resp.

$C_{v}$

$m_{v}^{(1)}$

)

.

.

$Type_{b_{*}}1^{b}V^{o_{*}f}IIC_{v}$

$|^{1}|$

$b^{m}7b+^{v}5$

$-\left|\begin{array}{lll} & m_{v}^{(1)} & \\ & b & \\- & 43 & ---\end{array}\right|-\frac{b)}{Z/(}Z/_{--}(4)\times\overline{Z/(2)}$

denotes a cyclic group of order ) For the sake of later reference, we rewrite (4.11), (4.12): PROPOSITION 4.3. The geometric genus of the elliptic modular is given by the formula:

(Here

$Z/(n)$

$n.$

$p_{g}$

surface

$B_{\Gamma}$

(4.13)

$p_{g}=2g-2+t-f_{1}/2$ ,

$(t=t_{1}+t_{2}+s)$

.

only for In [24], we defined elliptic modular surfaces $SL(2, Z)$ subgroups of present finite index of . Thus the torsion-free definition 4.1 is slightly more general than [24] \S 2. This generalization allows us to consider some examples of elliptic modular surfaces which may be of some arithmetic interest (\S 5. Example 5.8).

REMARK 4.4.

$B_{\Gamma}$

$\Gamma$

Elliptic modular

37

surfaces

\S 5. The group of sections. Examples. THEOREM 5.1. An elliptic modular surface has only a finite number of global holomorphic sections over its base curve. PROOF. From the second relation of (4.12) and Corollary 2.7, we get (5.1)

(and

$r=0$

),

$r^{\prime}=2p_{g}$

where is the rank of the group of global sections. Hence the assertion follows. We can give a more precise result. For brevity, we denote by $S(B)$ the group of global holomorphic sections of $B$ over its base curve $\Delta:S(B)$ $r$

$=H^{0}(\Delta, \Omega(B^{\#}))$

.

THEOREM 5.2. (i) If has trivial or (ii) If has $\Gamma$

$\Gamma$

sections

be the elliptic modular surface attached to . is either torsion $(i. e. s>0)$ , then the group of sections a cyclic group of order 3. a cusp of the second kind (i. e. $t_{2}>0$), then the group of is either trivial or isomorphic to one of the groups Let

$\Gamma$

$B_{\Gamma}$

$S(B_{\Gamma})$

$S(B_{\Gamma})$

or

$Z/(2),$ $Z/(4)$

(iii)

$Z/(2)\times Z/(2)$

.

is torsion-free and all cusps are of the first If is isomorphic to a subgroup of of sections denotes the least common multiple of b\’is pose that the singular fibres of are of types $\Gamma$

$S(B_{\Gamma})$

kind, then the

group

$Z/(m)\times Z/(m)$ ,

where

$(1\leqq i\leqq t_{1})$

$m$

$B_{\Gamma}$

.

Here we

$I_{b_{i}}(1\leqq i\leqq t_{1})$

$\sup-$

.

PROOF. If satisfies the condition of (i) (or (ii)), then contains a (or ) by Proposition 4.2. Since the of type singular fibre isomorphic subgroup to of is torsion $\Gamma$

$B_{\Gamma}$

$IV^{*}$

$C_{v}$

$I_{b}^{*}(b\geqq 1)$

$C_{v}^{*}$

$Z/(3)$

(or

$Z/(4),$ $Z/(2)\times Z/(2)$ ),

the assertion (i) (or (ii)) follows from Remark 1.10 (and Theorem 5.1). The assertion (iii) is an immediate consequence of Proposition 1.6, . . the injectivity of the homomorphism $i$

$S(B_{\Gamma})\rightarrow\prod_{v}C_{v}^{\#}/\Theta_{v0}\#,\cong\prod_{0)$

$\epsilon\left(\begin{array}{ll}1 & b\\0 & 1\end{array}\right)$

,

$\epsilon=\pm 1$

where the sign depends on whether the cusp kind (cf. (4.5)). If we put

$v$

$\epsilon$

(6.3)

$g(z)=(c^{\prime}z+d^{\prime})^{-w}f(\delta^{-1}\cdot z)$

,

,

is of the first or second

$\delta^{-1}=(_{c},$

$d)$

,

we have (6.4)

,

$g(z+b)=\epsilon^{w}\cdot g(z)$

) or $g(z)^{2}$ (if Thus $g(z)$ (if for of function dition can be stated as follows: $\epsilon^{w}=1$

$\tau=e^{2\pi iz/b}$

$h(\tau)$

(6.5)

$h(\tau)$

$\epsilon^{w}=-1$

$|\tau|\neq 0$

.

$\epsilon^{w}=\pm 1$

)

can be considered as a holomorphic

. With these

notations, the second

is holomorphic and vanishes at

$\tau=0$

con-

.

-cusp forms of weight $w$ will be denoted by The vector space of . is isomorphic to the space of holomorphic As is well-known, the space under the correspondence . In particular, l-forms on the curve $\dim S_{2}(\Gamma)$ is equal to the genus . For $w\geqq 3$ , the dimension of of can be calculated, for instance, with the aid of the Riemann-Roch theorem : on the curve $\Gamma$

$S_{w}(\Gamma)$

$S_{2}(\Gamma)$

$f\leftrightarrow f(z)dz$

$\Delta_{\Gamma}$

$g$

$S_{w}(\Gamma)$

$\Delta_{\Gamma}$

$\Delta_{\Gamma}$

(6.6)

$\dim S_{w}(\Gamma)=(w-1)(g-1)+s[w/3]+(w/2-1)t^{\prime}+\delta(w)t_{2}/2$ ,

have the same meaning as in \S 4 (4.3); $[w/3]$ denotes the ; and $\delta(w)=0$ or 1 according to whether the weight $w$ largest integer is even or odd. Note that, for $w=3$ , we have

where

$s,$

$t^{\prime},$

$f_{\underline{o}}$

$\leqq w/3$

(6.7) $p_{g}$

$\Gamma$

$\dim S_{3}(\Gamma)=p_{g}$

,

being the geometric genus of the elliptic modular surface , cf. (4.13).

The space $C$ isomorphic (over ) to the space THEOREM 6.1.

-cusp forms of weight 3 is canonically . . of holomorphic 2-forms on

$S_{s}(\Gamma)$

of

$\Gamma$

$B_{\Gamma},$

$S_{\theta}(\Gamma)\cong H^{0}(B_{\Gamma}, \Omega^{2})$

PROOF. We put $B=B_{\Gamma},$ elliptic points and cusps in under the canonical map

$\Delta$

$\mathfrak{H}\rightarrow\Gamma\backslash \mathfrak{H}$

antomorphisms:

$i$

$e$

.

being the set of and If we denote by the inverse image of , then is the quotient of by

$\Delta=\Delta_{\Gamma}$

.

attached to

$B_{\Gamma}$

$\Delta^{\prime}=\Delta-\Sigma(\subset\Gamma\backslash \mathfrak{H}),$

$\Sigma$

$\Delta^{r}$

$\mathfrak{H}^{\prime}$

$B^{\prime}=B|_{A^{\prime}}$

$\mathfrak{H}^{\prime}\times C$

Elliptic modular

(6.8)

43

surfaces

$(z, \zeta)-(\gamma\cdot z, (cz+d)^{-1}(\zeta+n_{1}z+n_{2}))$

where

$\gamma=\left(\begin{array}{ll}a & b\\c & d\end{array}\right)\in\Gamma$

and

$n_{1},$

.

Now, taking a

$n_{2}\in Z$

3, we consider the holomorphic 2-form on (6.9)

$\mathfrak{H}\times C$

$\Gamma$

,

-cusp

formf of

weight

:

.

$\omega=\omega_{f}=f(z)dz\wedge d\zeta$

is invariant under the automorphisms (6.8). Hence It is easily seen that . We shall , which we also denote by defines a holomorphic 2-form on see in the following that extends to a holomorphic form in a neighborhood of $B(v\in\Sigma)$ . of each singular fibre is a singular fibre of is an elliptic point and Case i) The point be the stabilizer of of and let . Take a representative type is a cyclic group of order 3. We may assume without loss of in $\omega$

$\omega$

$B^{\prime}$

$\omega$

$\omega$

$C_{v}$

$C_{v}$

$v$

$IV^{*}$

$z_{0}\in \mathfrak{H}$

$\Gamma_{z_{0}}$

$v$

$\Gamma;\Gamma_{z_{0}}$

$z_{0}$

generality that

transformed to

and

$z_{0}=e^{2\pi i/3}$

$e^{2\pi i/3}$

(6.10)

$\Gamma_{z_{0}}$

is generated by

an element of

by

$SL(2, Z)$

$\sigma=(z-z_{0})/(z-z_{0}^{2})$

,

.

$\gamma_{0}=\left(\begin{array}{ll}-1 & -1\\1 & 0\end{array}\right)$

, since

$z_{0}$

is

We put

$\tau=\sigma^{3}$

defined by $|\sigma|0)$ , and We denote by $D$ a small neighborhood of . by $F$ the quotient of $D\times C$ by the group of automorphisms (6.8) with Then we are exactly in the situation of [K] p. 591-592, Case (2). A cyclic acts on $F$ and the quotient $F/C$ has group of order 3 corresponding to three singular points $p_{\nu}(\nu=0,1,2)$ . Moreover the non-singular model of (de$F/C$ obtained by a reduction of singularities gives a neighborhood under consideration. Now it is in [K]) of the singular fibre noted by of (6.9) is holomorphic on $F/C-\{p_{0}, p_{1}, p_{2}\}$ , since obvious that the form in (6.10), we have is invariant under all automorphisms (6.8). Using $z_{0}$

$\gamma=1$

$C$

$\Gamma_{z_{0}}$

$B_{v}$

$C_{v}$

$B_{\rho}$

$\omega$

$\omega$

$\sigma,$

$\omega\sim h(\tau)d\sigma\wedge d\zeta$

$\tau$

,

and the symbol\sim denotes the equality up to a (locally) non-vanishing holomorphic function. Then, using the local coordigiven by [K] p. 592 (8.24), we see immediately that our form nates on . of is holomorphic on the whole neighborhood Case ii) The point is a cusp of the first kind and the singular fibre ), we have is of type . With the notations (6.2), .. , (6.5) (noting where

$h(\tau)$

is holomorphic in

$\tau$

$\omega$

$B_{v}$

$B_{v}$

$C_{v}$

$v$

$I_{b}$

$C_{v}$

$\epsilon=1$

$\cdot$

(6.11)

$\delta^{-1}\cdot\omega=h(\tau)2\pi ib-d_{\tau}\tau_{-}\wedge d\zeta$

.

vanishes at $\tau=0$ by (6.5), the right side of (6.11) is holomorphic Since (cf. [K] . in and . Therefore, by the structure of a neighborhood of 599-600, Case (1)), our form is holomorphic at every point of $C_{v}^{\#}=C_{v}-$ in $B$ . must be holomorphic in a neighborhood of { points}. Hence $h(\tau)$

$\tau$

$C_{v}$

$\zeta$

$p$

$\omega$

$b$

$\omega$

$C_{v}$

T. SHIODA

44

Case iii) The point is a cusp of the second kind and the singular fibre is of type . With the notations (6.2), .. , (6.5) (noting $\epsilon=-1$ ), we have $v$

$C_{v}$

$I_{b}^{*}$

$\cdot$

(6.12)

$\delta^{-1}\cdot\omega=\sqrt{h(\tau)}2\pi i^{-}b$

$d_{\frac{\tau}{\tau}\wedge d\zeta}$

.

Again, by [K] p. 600-602, Case (2), we know that a neighborhood of a singular fibre of type is obtained by a reduction of singularities from the by a group of quotient of a neighborhood of a singular fibre of type order 2. Using the result of case ii), we can argue as in case i). defined by Thus we have seen that, for each $fES_{3}(\Gamma)$ , the 2-form $B$ (6.9) is holomorphic on the whole surface . Obviously the map is injective. In view of (6.7), this completes the proof of Theorem 6.1. Let be the line bundle over be the canonical as in (2.7), and let $B$ bundle of . Then the canonical bundle of is induced from the line bundle $f-f$ over ([K] Theorem 12.1). Therefore by the canonical projection we have a canonical isomorphism: $C_{v}$

$I_{b}^{*}$

$1_{2b}$

$\omega=\omega_{f}$

$f\leftrightarrow\omega_{f}$

$\Delta$

$f$

$f$

$\Delta$

$ B\rightarrow\Delta$

$\Delta$

(6.13)

$H^{0}(B, \Omega^{2})\cong H^{0}(\Delta, \mathcal{O}(f-f))$

.

Hence we get COROLLARY 6.2. There is a canonical isomorphism (over $H^{0}(\Delta, \mathcal{O}(f-\mathfrak{f}))\cong S_{s}(\Gamma)$

$C$

):

.

We identify the two spaces by the canonical isomorphism. By the duality theorem on a curve, we have a natural (C-bilinear) non-degenerate pairing: (6.14)

$H^{0}(\Delta, \mathcal{O}(f-\uparrow))\times H^{1}(\Delta, \mathcal{O}(\mathfrak{f}))\rightarrow C$

$(f, \xi)\leftrightarrow\langle f, \xi\rangle$

The value

$\langle f, \xi\rangle$

.

is given as follows. We take a sufficiently fine finite open

covering of and represent the cohomology class , where is a holomorphic section of over determines a cohomology class in sheaf of germs of holomorphic l-forms on . We have $\mathfrak{U}=\{U_{i}\}$

$(\xi_{ij})$

$\Delta$

$f$

$\xi_{ij}$

$\xi$

$ U_{i}\cap U_{j}\neq\emptyset$

by

.

$H^{1}(\Delta, \mathcal{O}(f))=H^{1}(\Delta, \Omega^{1}),$

$(f\xi_{ij})$

a l-cocycle

The l-cocycle being the $\Omega^{1}$

$\Delta$

(6.15)

$H^{1}(\Delta, \Omega^{1})\cong H^{2}(\Delta, C)\cong C$

,

$(f\xi_{ij})\leftrightarrow\langle f, \xi\rangle$

,

, where the first isomorphism comes from the exact sequence and the second comes from the evaluation of a 2-cocycle on the fundamental is the complex number corresponding to the class . Then the value cohomology class of under (6.15). In the next section, we shall explicitly compute when is an element of the image . hand, On the other the space of -cusp forms of weight 3 is selfdual (over $R$ ) with respect to the Petersson metric. Recall that, for any $0\rightarrow C\rightarrow \mathcal{O}\rightarrow\Omega^{1}\rightarrow 0$

$\Delta$

$\langle f, \xi\rangle$

$(f\xi_{ij})$

$\langle f, \xi\rangle$

$i^{*}H^{1}(\Delta, G)\subset H^{1}(\Delta, \mathcal{O}(\uparrow))$

$\xi$

$S_{3}(\Gamma)$

$\Gamma$

Elliptic modular

of -cusp forms of weight is a positive definite Hermitian scalar product defined by

weight $w$

45

surfaces

$w$

on the space

, the Petersson metric

(6.16)

$(f, g)=\int_{\Gamma \mathfrak{H}}f(z)\overline{g(z)}y^{w- 2}dxdy$

$S_{w}(\Gamma)$

,

$\Gamma$

$z=x+iy\in \mathfrak{H}$

,

.

Comparing (6.14) and (6.16), we get the following result. denote the unique ele, let PROPOSITION 6.3. For each ment of satisfying

for

$f,$

$g\in S_{w}(\Gamma)$

$\psi(\xi)$

$\xi\in H^{1}(\Delta_{\Gamma}, \mathcal{O}(t))$

$S_{3}(\Gamma)$

(6.17)

Then

for all

$\langle f, \xi\rangle=4(f, \psi(\xi))$

$\psi$

$f\in S_{3}(\Gamma)$

.

is a C-antilinear isomorphism: $\psi:H^{1}(\Delta_{\Gamma}, \mathcal{O}(f))\rightarrow\sim S_{3}(\Gamma)$

.

is dual Similarly the space of holomorphic 2-forms on (over $C$ ) to the space by Serre duality. Hence Theorem 6.1 implies together , the space that is canonically isomorphic to with the complex structure which is conjugate to the usual one. In view of (5.1) and (2.11), we get PROPOSITION 6.4. The Hodge decomposition of the two dimensional cohomology of the surface is given as follows: $B_{\Gamma}$

$H^{0}(B_{\Gamma}, \Omega^{2})$

$H^{2}(B_{\Gamma}, \mathcal{O})$

$S_{3}(\Gamma)$

$\overline{S_{3}(\Gamma)}$

$H^{2}(B_{\Gamma}, \mathcal{O})$

$B_{\Gamma}$

(6.18)

$H^{2}(B_{\Gamma}, C)\cong S_{\theta}(\Gamma)\oplus\overline{S_{3}(\Gamma)}\oplus(NS(B_{\Gamma})\otimes C)$

\S 7.

.

Shimura’s complex torus for weight 3.

As we have noted in Remark 5.3, for the elliptic modular surface (over ), the quotients

$B=B_{\Gamma}$

$\Delta=\Delta_{\Gamma}$

$H^{1}(\Delta, \mathcal{O}(f))/i^{*}H^{1}(\Delta, G)$

are complex tori of dimension

and

$H^{2}(B, \mathcal{O})/j^{*}H^{2}(B, Z)$

.

In this section we shall show that these complex tori have another interpretation as an analogue for weight 3 of Shimura’s abelian varieties attached to cusp forms of even weights (cf. Shimura [22], cited as [S] in the following). Incidentally this will give another proof that is a complex torus, independent of the results in \S 3 (cf. [24]). We shall begin with the definition of the parabolic cohomology groups of following [S], restricting our attention to the weight 3 case. Let $R$ denote one of the rings $Z,$ $R$ or $C$, and let denote the module of column vectors $R$ with coefficients in . By an R-valued parabolic cocycle of , we mean a map $p_{g}$

$H^{1}(\Delta, \mathcal{O}(f))/i^{*}H^{1}(\Delta, G)$

$\Gamma$

$R^{2}$

$\Gamma$

$\mathfrak{x}:\Gamma\rightarrow R^{2}$

satisfying the two conditions:

46

T. SHIODA )

(7.1)

$\mathfrak{x}(\sigma\sigma^{\prime})=\mathfrak{x}\langle\sigma$

(7.2)

$x(\gamma)E(\gamma-1)R^{2}$

where

$\Gamma\subset SL(2, Z)$

for

$+\sigma \mathfrak{x}(\sigma^{\prime})$

naturally acts

$\sigma^{\prime}\in\Gamma$ $\sigma,$

for parabolic

on

$R^{2}$

;

$\gamma\in\Gamma$

,

from the left. A coboundary is a

cocycle X of the form

for all

$\mathfrak{x}(\sigma)=(\sigma-1)\mathfrak{x}_{0}$

$\sigma\in\Gamma$

,

. The parabolic cohomology is an arbitrary (fixed) element of where , is defined as the quotient of the group group of , denoted by of all R-valued parabolic cocycles modulo the subgroup of coboundaries. The natural injection induces a canonical homomorphism: $R^{2}$

$\mathfrak{x}_{0}$

$\Gamma$

$H_{par}^{1}(\Gamma, R^{2})$

$Z\rightarrow R$

$c:H_{par}^{1}(\Gamma, Z^{2})\rightarrow H_{par}^{1}(\Gamma, R^{2})$

.

The following is a special case of Proposition 1 of [S] \S 3. under is a lattice in the real PROPOSITION 7.1. The image of vector space . We shall next consider the relation of cusp forms (of weight 3) to para, we put bolic cohomology. For each $H_{p\backslash r}^{1}(\Gamma, Z^{2})$

$c$

$H_{par}^{1}(\Gamma, R^{2})$

$f\in S_{3}(\Gamma)$

(7.3)

$F_{f}(z)=\int_{z_{0}^{z}}(z1)f(z)dz$

where

$z_{0}$

is a fixed base point in

and $\mathfrak{H}$

$r_{f}(\sigma)=F_{f}(\sigma\cdot z_{0})$

. Since we

$(\sigma\in\Gamma)$

,

have

$\left(\begin{array}{l}\sigma\cdot z\\1\end{array}\right)f(\sigma\cdot z)d(\sigma\cdot z)=\sigma(_{1}^{z})f(z)dz$

,

we get \langle 7.4)

$F_{f}(\sigma\cdot z)=\sigma F_{f}(z)+\mathfrak{x}_{f}(\sigma)$

,

and is a C-valued parabolic cocycle; note that the cohomology class of . If we is uniquely determined by and independent of the choice of containing the real cocycle the cohomology class in denote by , we get an R-linear homomorphism:

$\mathfrak{x}_{f}$

$\mathfrak{x}_{f}$

$z_{0}\in \mathfrak{H}$

$f$

$H_{par}^{1}(\Gamma, R^{2})$

$\varphi(f)$

${\rm Re}(\mathfrak{x}_{f})$

$\varphi:S_{3}(\Gamma)\rightarrow H_{par}^{1}(\Gamma, R^{2})$

.

. onto is an isomorphism of PROPOSITION 7.2. We omit the proof, because this is a special case of a general result of Shimura ([29] Chapter 8). The purpose of this section is to prove: , THEOREM 7.3. There is an isomorphism of $H^{1}(\Delta, G)$ onto which makes the following diagram commute: $S_{3}(\Gamma)$

$H_{par}^{1}(\Gamma, R^{2})$

$\varphi$

$H_{par}^{1}(\Gamma, Z^{2})$

$\eta$

Elliptic modular

47

surfaces

$i^{*}$

$(Prop(Prop..72)6..3)$

$H_{par}^{1}(\Gamma,Z)H_{p^{3}ar}^{S_{1}^{1}(\Gamma)}H^{1}(\Delta,G_{2}).*H(\Delta_{(\Gamma,R)}\eta\downarrow\underline{\overline{(2_{c|\downarrow\varphi}8)}}|\downarrow\psi^{\mathcal{O}(f))_{2}}$

.

(Prop. 7.1)

In order to apply the results of \S 2, we need to make explicit the relation of and $H^{1}(\Delta, G)$ . We consider a sufficiently fine simplicial decomposition , which is a subdivision of the of the Riemann surface decomposition, say , of considered in \S 2 to compute $H^{i}(\Delta, G)$ . We denote by , and the vertices of , by the l-simplex connecting by and the 2-simplex with the vertices . Let be the dual cell decomposition of . We denote by the dual cells in or (of dimension 2, 1, respectively) corresponding to the simplices or . For an alternating l-chain with respect to with coefficients in the sheaf $G$ : $H_{1}(\Delta, G)$

$\mathcal{T}$

$\Delta=\Delta_{\Gamma}$

$\Delta$

$\mathcal{T}_{0}$

$(\lambda)(\lambda\in\Lambda)$

$\mathcal{T}$

$(\lambda\mu\nu)$

$(\lambda),$

$\mathcal{T}$

$\mathcal{T}^{*}$

$[\lambda],$

$(\mu),$

$\mathcal{T}^{*}$

$(\nu)$

$[\lambda\mu\nu]$

$[\lambda\mu]$

$0$

$(\lambda),$

$(\lambda\mu\nu)$

$\mathcal{T}$

,

$c_{1}=\sum g_{\lambda\mu}(\lambda\mu)$

we can define a l-cochain

$c^{1}$

with respect to

(7.6)

$c^{1}$

as is

$(\lambda\mu)$

$c_{1}$

(7.5)

and,

$(\mu)$

$(\lambda)$

$(\lambda\mu)$

:

easily seen, the map

(7.7)

$\mathcal{T}^{*};$

$[\lambda\mu]\leftrightarrow g_{\lambda\mu}$

$c_{1}\mapsto c^{1}$

,

induces an isomorphism:

$H_{1}(\Delta, G)\rightarrow\sim H^{1}(\Delta, G)$

Moreover, if the vertex

we denote

$(\lambda)$

.

the interior of the union of 2-simplices having in common, then by

$U_{\lambda}$

(7.8)

$u=\{U_{\lambda}\}_{\lambda\in\Lambda}$

forms a (sufficiently fine) open covering of and the l-cochain c’ (7.6) may be considered as a l-cochain on the nerve of the covering U. Hence the right side of (7.7) can be considered as the cohomology group in Cech’s sense. We take a suitable fundamental domain in the upper half plane of corresponding to on . and consider the simplicial decomposition of .. , paths , corresponding to the If we denote by the sides of conin (2.1), then the boundary of of the fundamental group of sists of $4g+2t$ sides $\Delta$

$\Gamma$

$\mathcal{F}$

$\tilde{\alpha}_{i},\tilde{\beta}_{i}$

$\mathcal{F}$

$\cdot$

$\pi_{1}(\Delta^{\prime})$

(7.9)

$\tilde{\alpha}_{i},\tilde{\beta}_{i},$

$\tilde{\gamma}_{j},$

where $\varphi(\alpha_{i}),$

$\varphi$

$-\varphi(\alpha_{i})(\tilde{a}_{i}),$

is the representation

$\varphi(\beta_{i}),$

$\cdots$

$\alpha_{i},$

$-\varphi(\beta_{i}^{-1})(\tilde{\beta}_{i})$

$(1\leqq i\leqq g)$

,

$(1\leqq j\leqq t=s+f^{\prime})$

we

.

,

Writing for have obtained standard gen-

$\pi_{1}(\Delta^{\prime})\rightarrow\Gamma\subset SL(2, Z)$

to simplify the notation,

$\beta_{i}$

$\overline{\mathcal{F}}$

$\Delta^{\prime}$

$-\varphi(\gamma_{j})(\tilde{\gamma}_{j})$

$\Delta$

$\mathcal{T}$

$\overline{\mathcal{F}}$

$\mathfrak{H}$

$\alpha_{i},$

$\beta_{i},$

$\cdots$

T. SHIODA

48

erators of

$\Gamma$

with the relations (cf. (2.1)): $\gamma_{t}\cdots\gamma_{1}\cdots\beta_{1}^{-1}a_{1}^{-1}\beta_{1}a_{1}=1$

(7.10)

,

$\gamma_{1}^{\epsilon}=\ldots=\gamma_{s}^{e}=1$

assuming that

points

$s$

$v_{i}\in\Sigma(1\leqq i\leqq s)$

$(e=3)$

are the

,

elliptic points of

$\Delta_{\Gamma}$

.

Put

(cf. (2.4)) $\kappa_{i}=\beta_{t}^{-1}a_{i}^{-1}\beta_{i}a_{i}$

$\gamma^{(j)}=\gamma_{j}\cdots\gamma_{1}$

,

,

$\kappa^{(j)}=\kappa_{i}\cdots\kappa_{1}$

,

$\kappa=\kappa^{(g)}$

;

.

$\gamma^{(0)}=\kappa^{(0)}=1$

PROPOSITION 7.4. The Petersson metric $(f, g)(f, g\in S_{3}(\Gamma))$ can be expressed (complex conjugate) as follows. Pu in terms of the parabolic cocycles and and . Then $t$

$\overline{\mathfrak{x}_{g}}$

$\mathfrak{x}_{f}$

$\mathfrak{y}=\overline{\mathfrak{x}_{g}}$

$\mathfrak{x}=\mathfrak{x}_{f}$

$4(f, g)=\sum_{i=1}^{g}t\mathfrak{x}(a_{i}^{-1})P[\mathfrak{y}(a_{i}^{-1}\beta_{i}\alpha_{i}\kappa_{i-1})-t)(\kappa_{i-1})]$

(7.11)

$+\sum_{:=\iota}^{g}t\mathfrak{x}(\beta_{i})P[\mathfrak{y}(\beta_{i}\alpha_{i}\kappa_{i- 1})-\mathfrak{y}(a_{i}^{-1}\beta_{i}\alpha_{i}\kappa_{i- 1})]$

$-\sum_{j=\iota}^{t}{}^{t}r,(\gamma_{j}^{-1})P[\mathfrak{y}(\gamma^{(j- 1)}\kappa)-\mathfrak{y}_{j}]$

where

of

$v_{j}$

and

$P=$

in

$\overline{\mathcal{F}}(1\leqq j\leqq t)$

$\mathfrak{y}_{j}$

of

is the value

,

function

the

at the representative

$\overline{F}_{g}$

.

The proof can be found in [S] \S 4 (especially the formula (19)) by suitably changing the notations. The idea of the proof is similar to that of the well-known Riemann bilinear relation on a compact Riemann surface. Note (cf. [S] (14)). that the above satisfies the relation: Now we shall compute the value for $f\in S_{3}(\Gamma)$ and $\xi=i^{*}(g)\in$ (cf. (6.14)). We take a representative cocycle of $G$ , where (or and put (or f) over the open ) is a section of set open . We lift an set in the upper half plane to so that either is contained in or or meets one of the sides . Then can be identified with a holomorphic function on of the form $\mathfrak{y}(\gamma_{j})=(1-\gamma_{j})\mathfrak{y}_{j}$

$\mathfrak{y}_{j}$

$\langle f, \xi\rangle$

$i^{*}H^{1}(\Delta, G)\subset H^{1}(\Delta, \mathcal{O}(t))$

$g$

$(g_{\lambda\gamma\iota})$

$\xi_{\lambda/\ell}=i^{*}g_{\lambda\mu}$

$\xi_{\lambda/\iota}$

$g_{\lambda/l}$

$ U_{\lambda\mu}=U_{\lambda}\cap U_{\mu}\neq\emptyset$

$\tilde{\gamma}_{j}$

$\tilde{U}_{\lambda/\iota}$

$U_{\lambda\mu}$

$0_{\lambda\mu}$

$O_{\lambda_{l^{p}}}$

$\mathcal{F}$

$\tilde{\alpha}_{i},\tilde{\beta}_{i}$

$O_{\lambda/p}$

$\xi_{\lambda_{f^{p}}}$

$\xi_{\lambda\mu}=n_{1}z+n_{2}$

,

$(n_{1}, n_{2})=g_{\lambda\mu}\in Z\oplus Z$

.

If we put $Y_{\lambda\mu}(u)=\int_{z_{0}^{z}}f(z)\xi_{\lambda\mu}dz=g_{\lambda/4}F_{f}(z)$

,

$z\in O_{\lambda_{l^{p}}}$

(7.12) $c_{\lambda\mu\nu}=Y_{\lambda\mu}(u)+Y_{\mu\nu}(u)+Y_{\nu\lambda}(u)$

$u$

being the image in

(7.13)

$\Delta_{\Gamma}$

of the point

$z\in \mathfrak{H}$

,

, then

$\langle f, \xi\rangle=\sum c_{\lambda\mu\nu}$

,

$u\in U_{\lambda\mu\nu}$

,

we have

by (6.15)

where the summation is Obviously unless Suppose that lies on as a side; we take $c_{\lambda\mu\nu}=0$

$(\lambda\mu)$

$(\lambda\mu)$

$\nu$

49

surfaces

Elliptic modular

of . 2-simplices paths or has a side lying on the be the 2-simplices having and Let

over all positively oriented $(\lambda\mu\nu)$

. so that

$\alpha_{i}$

$a_{i},$

$(\lambda\mu\nu)$

$\tilde{U}_{\lambda\mu}$

$\mathcal{T}$

$(\lambda\mu\nu)$

$\beta_{i}$

$\gamma_{j}$

$(\lambda_{\sim}\mu\nu^{\prime})$

meets

$U_{\lambda\nu}$

. Then we have

,

$c_{\lambda\mu\nu}=0$

and $c_{\lambda\mu\nu^{\prime}}=g_{\lambda\mu}\mathfrak{x}_{J}(\alpha_{i}^{-1})$

since, for

$u\in U_{\lambda\mu\nu^{\prime}}$

and

$z\in\alpha_{i}(0_{\lambda\mu})\cap O_{\lambda\nu^{\ovalbox{\tt\small REJECT}}}$

,

,

we have

by (7.4)

$Y_{\lambda\mu}(u)=g_{\lambda\mu}F(\alpha_{i}^{-1}\cdot z)=g_{\lambda_{l^{\ell}}}(\alpha_{i}^{-1}F_{f}(z)+\mathfrak{x}_{f}(a_{i}^{1}))$

and

$(g_{\lambda\mu})$

is a cocycle with coefficients in

$G$

.

we put

Therefore, if

$a_{i}=\Sigma g_{\lambda\mu}(\in Z\oplus Z)$

with define

$(\lambda\mu)$

$b_{i},$

running

over all positive

$c_{j}\in Z\oplus Z$

(7.14)

similarly for

l-simplices contained in the path , then we get

$\alpha_{i}$

, and

$\beta_{i},$

$\gamma_{j}$

$-\langle f, \xi\rangle=\sum_{t=\iota}^{g}[a_{i}\mathfrak{x}_{f}(\alpha_{i}^{-1})+b_{i}\mathfrak{x}_{f}(\beta_{i})]+\sum_{J=1}^{t}c_{j}\mathfrak{x}_{f}(\gamma_{j}^{-1})$

,

with (7.15)

$\sum_{i=1}^{g}[a_{i}(1-\alpha_{i}^{-1})+b_{i}(1-\beta_{i})]+\sum_{j=1}^{t}c_{j}(1-\gamma_{j}^{-1})=0$

.

By comparing (7.11) and (7.14), we want to define an integral parabolic cocycle for $g=(g_{\lambda\mu})\in H^{1}(\Delta, G)$ by the conditions: $\mathfrak{y}=\eta(g)$

$-{}^{t}a_{i}=P[\mathfrak{y}(a_{i}^{-1}\beta_{i}\alpha_{i}\kappa_{i- 1})-\mathfrak{h}(\kappa_{i- 1})]$

(7.16)

,

$-{}^{t}b_{i}=P[\mathfrak{y}(\beta_{i}\alpha_{i}\kappa_{i- 1})-\mathfrak{y}(a_{i}^{-1}\beta_{i}a_{i}\kappa_{i-1})]$

${}^{t}c_{j}=P[\mathfrak{y}(\gamma^{(j- 1)}\kappa)-\mathfrak{h}_{j}]$

$\mathfrak{y}(\gamma_{j})=(1-\gamma_{j})\mathfrak{y}_{j}$

,

,

,

$(1\leqq i\leqq g, 1\leqq j\leqq t)$

.

By the cocycle condition (7.1), we can express the values and as integral linear combinations of they hence are integral. and Moreover it follows from (7.15) that the map thus defined is compatible with the relations (7.10). Hence is really an integral parabolic cocycle of and we get a homomorphism: $\mathfrak{y}(\alpha_{i}),$

${}^{t}a_{i},{}^{t}b_{i}$

$\mathfrak{y}_{j}$

$\mathfrak{y}(\beta_{i}),$

$\mathfrak{y}(\gamma_{j})$

$t_{C_{j};}$

$\mathfrak{y}:\Gamma\rightarrow Z^{2}$

$\mathfrak{y}=\eta(g)$

$\Gamma$

$\eta:H^{1}(\Delta, G)\rightarrow H_{par}^{1}(\Gamma, Z^{2})$

.

It is clear by the above definition of that satisfies the conditions of Theorem 7.3. This completes the proof. REMARK 7.5. Let consisting of cusp denote the subgroup of $w$ forms of weight whose ”period” has integral real part. Then is a lattice of the complex vector space by Propositions 7.1 and 7.2, $\eta$

$\eta$

$D_{w}(\Gamma)$

$f$

$S_{w}(\Gamma)$

$D_{w}(\Gamma)$

$\mathfrak{x}_{f}$

$S_{w}(\Gamma)$

50

T. SHIODA

and the quotient -cusp forms of weight

is called Shimura’s complex torus attached to For $w=3$ , Theorem 7.3 implies that

$S_{w}(\Gamma)/D_{w}(\Gamma)$

$\Gamma$

$w$

.

,

$S_{3}(\Gamma)/D_{3}(\Gamma)\cong H^{1}(\Delta, \mathcal{O}(t))/i^{*}H^{1}(\Delta, G)$

if we take the complex structure on that is conjugate to the usual one. Therefore Shimura’s complex torus for weight 3 is essentially the same as (cf. (3.1)), which has the geometric significance as the the group parameter space of the family of elliptic surfaces. In particular, the subgroup of division points of has an algebro-geometric (or arithmetic) interpretation as essentially the group of locally trivial principal $B$ homogeneous spaces for $B$ over being the elliptic modular surface attached to . REMARK 7.6. As to the question of whether or not the complex torus for an odd weight $w$ has a structure of abelian variety (as in even weight case), the following has been remarked by Prof. Shimura. In general, a complex torus of dimension has a structure of abelian variety if its endomorphism algebra (tensored by $Q$) contains a totally real field of (degree . By this fact and the theory of Hecke operators, it can be shown that has a structure of abelian variety for a certain class of , for instance for the groups of Example 5.8. $\Gamma=\Gamma(4)$ , the congruence subgroup of level 4 of EXAMPLE 7.7. For $SL(2, Z)$ , the elliptic modular surface $B(4)$ for level 4 is a $K3$ surface (cf. [12]), since we have $S_{3}(\Gamma)$

$H^{1}(\Delta, \Omega(B^{*}))$

$\mathcal{F}(J, G)$

$S_{3}(\Gamma)/D_{3}(\Gamma)$

$\Delta,$

$\Gamma$

$S_{w}(\Gamma)/D_{w}(\Gamma)$

$n$

$n$

$\Gamma$

$S_{3}(\Gamma)/D_{3}(\Gamma)$

$\Gamma_{0}^{\prime}(q)$

$g=0$

by (5.3), (5.5).

The complex torus

and

$p_{g}=1$

$H^{1}(\Delta, \mathcal{O}(f))/i^{*}H^{1}(\Delta, G)$

dimension 1, . . an elliptic curve. The space weight 3 is spanned by one element : $i$

$e$

or

$S_{3}(\Gamma)$

is of -cusp forms of

$S_{3}(\Gamma)/D_{3}(\Gamma)$

of

$\Gamma$

$f$

$f(z)=\Delta(z)^{1/4}$

,

is the well-known cusp form of weight 12 for $SL(2, Z)$ . By an argument similar to [S] p. 309, we see that is an elliptic curve with complex multiplication by $Q(\sqrt{-}1)$ . REMARK 7.8. Kuga-Satake [13] has attached to a polarized $K3$ surface , and has shown among others that, , of dimension an abelian variety, is 20 $(=b_{2}-2p_{g})$ , if is ”singular” in the sense that the Picard number of copies of is isogenous to the self-product of then the abelian variety $K3$ surface $B(4)$ is an elliptic curve with complex multiplication. Now the $(5.4)$ singular by –in fact, every elliptic modular surface is ”singular”, . . $\rho=b_{2}-2p_{g}$ by (5.1) and Corollary 2.6, and the elliptic curve $S_{3}(\Gamma(4))/D_{3}(\Gamma(4))$ . is presumably isogenous to the simple component of the abelian variety where

$\Delta(z)$

$S_{3}(\Gamma)/D_{3}(\Gamma)$

$S$

$2^{19}$

$A_{s}$

$S$

$S$

$A_{s}$

$2^{19}$

$i$

$e$

$A_{B(4)}$

Elliptic modular

Appendix.

51

surfaces

Arithmetic applications.

A. Algebraic reformulations. We first recall Igusa’s theory of elliptic modular functions in arbitrary characteristic not dividing the level, as reformulated by Deligne (see [5], [6], [2]). Fix a positive integer . Let denote the moduli scheme for the elliptic curves with level structure; exists and is an affine curve over $Spec(Z[1/n])$ . We recall the following facts: i) The scheme , projective and is compactified to a curve scheme smooth over $Spec(Z[1/n])$ , and $M_{n}^{*}-M_{n}$ is an \’etale covering of $Spec(Z[1/n])$ $n\geqq 3$

$M_{n}$

$M_{n}$

$n$

$M_{n}^{*}$

$M_{n}$

\langle $[2]$ Theorem 4.1).

curve

over

is analytically isomorphic (or its compactification ), cf. Example 5.4. the Riemann surface is iii) The algebraic closure of $Q$ in the function field of , being a primitive n-th root of unity (cf. [23]), and hence . All can be considered as a non-singular projective curve defined over points of $\Sigma_{0}=M_{n}^{*}\otimes Q-M_{n}\otimes Q$ are rational over . iv) Let be be an arbitrary prime number not dividing , and let $mod n$ . Then can be conthe smallest -power such that . All points of sidered as a non-singular projective curve defined over (cf. [5]). are rational over Now, let denote the universal family of elliptic curves with level structure. Then v) $E$ admits . sections of order over vi) Let . Then denote the N\’eron model of over (non-singular projective) a having is sections elliptic surface over of order , all defined over . It can be verified with the aid of Theorem 5.5 that is analytically isomorphic to the elliptic modular surface (of The

ii)

$M_{n}\otimes C$

(or

$M_{n}^{*}\otimes C$

)

$t0$

$C$

$\Delta(n)$

$\Gamma(n)\backslash \mathfrak{H}$

$M_{n}^{*}\otimes Q$

$K_{n}$

$Q(\zeta_{n})$

$M_{n}^{*}\otimes Q$

$\zeta_{n}$

$Q(\zeta_{n})$

$Q(\zeta_{n})$

$p$

$p^{f}$

$n$

$p^{f}\equiv 1$

$p$

$M_{n}^{*}\otimes F_{p}$

$F_{p^{f}}$

$\Sigma_{p}=M_{n}^{*}\otimes F_{p}-M_{n}\otimes F_{p}$

$F_{p^{f}}$

$E\rightarrow M_{n}$

$n$

$n^{2}$

$M_{n}$

$n$

$E_{0}^{*}$

$E_{0}^{*}$

$M_{n}^{*}\otimes Q$

$E\otimes Q$

$n^{2}$

$M_{n}^{*}\otimes Q$

$Q(\zeta_{n})$

$n$

$E_{0}^{*}\otimes C$

level n)

$B(n)$

over

$\Delta(n)$

.

. Then denote the N\’eron model of over , having is a (non-singular projective) elliptic surface over sections of order being as in iv). defined over will be called the elliptic modular surface of level in characteristic . viii) The singular fibres (or ) and they (or ) lie over of (or of are of type . Since each point of ) is rational over (or ), the divisor is rational over the same field. By the construction of N\’eron model ([17]), we see that the components are rational of (or over an at most quadratic extension of ). Finally we note the following. vii)

Let

$M_{n}^{*}\otimes F_{p}$

$E\otimes F_{p}$

$E_{p}^{*}$

$M_{n}^{*}\otimes F_{p}$

$E_{p}^{*}$

$n$

$F_{p^{f}},$

$f$

$E_{p}^{*}$

$p$

$n$

$E_{0}^{*}$

$C_{v}$

$I_{n}$

$F_{p^{f}}$

$n^{2}$

$v$

$\Sigma_{0}$

$E_{p}^{*}$

$\Sigma_{0}$

$\Sigma_{p}$

$Q(\zeta_{n})$

$\Sigma_{p}$

$C_{v}$

$\Theta_{v,i}$

$Q(\zeta_{n})$

$F_{p^{f}}$

$C_{v}$

T. SHIODA

52 ix)

The Betti number

$b_{2}(n)$

of

$E_{0}^{*}$

(

$i$

. .

of

$e$

$b_{2}(n)=2p_{g}(n)+\rho(n)$ $p_{g}(n)=\dim S_{3}(\Gamma(n))$

satisfies the relation:

$B(n)$)

,

:

cf. (5.4), (5.6) and (6.7). B.

Arithmetic theory of surfaces over a finite field.

In order to consider arithmetic questions on the elliptic modular surfaces of level , we recall the known facts and conjectures for an algebraic surface over a finite field (cf. [26], [27]). Let $X$ denote a non-singular projective is connected such that surface defined over a finite field ). Then the zeta function of $X$ is of the form: is an algebraic closure of $n$

$(\overline{F}_{q}$

$\overline{X}=X\otimes\overline{F_{q}}$

$F_{q}$

$F_{q}$

$P_{1}(X, T)P_{1}(X, qT)$

$\zeta(X, T)=(1-T)P_{2}(X, T)(1-q^{2}T)^{-}$

,

$T=q^{-s}$

,

is the characteristic polynomial of the endogroup cohomology morphism of the l-adic induced by the Froof $X$ , 1 being a prime number different from the benius endomorphism (and hence also ) has integral coefficients characteristic. It is known that and is independent of . The degree is equal to of the polynomial . In particular, if $X$ is obtained as a reduction of a non-singular in characteristic zero, surface is equal to the i-th Betti number of . We put

where

$P_{i}(X, T)=\det(1-\varphi_{i,\iota}T)$

$H^{i}(\overline{X}, Q_{\iota})$

$\varphi_{i,l}$

$\varphi$

$P_{2}$

$P_{1}$

$l$

$P_{i}$

$b_{i}$

$\dim_{Q}{}_{\iota}H^{i}(\overline{X}, Q_{\iota})$

$\tilde{X}$

$\tilde{X}$

$b_{i}$

$P_{2}(X, T)=\prod_{j=1}^{b_{2}}(1-\alpha_{j}T)$

CONJECTURE 1

,

$a_{j}\in C$

The algebraic integers The corresponding fact for is known (Weil). , and we write such that (Weil).

$j’ s$

have absolute value . Let denote the number $q$

$\alpha_{j}$

$P_{1}$

of

.

$\rho^{\prime}$

$\alpha_{j}=q$

$P_{2}(X, T)=(1-qT)^{\rho^{\prime}}R(T)$

CONJECTURE 2 of $X$.

(Tate).

$\rho^{\prime}$

,

$R(q^{-1})\neq 0$

is equal to the rank

.

of N\’eron-Severi group

$\rho$

$NS(X)$

is known; see [26] \S 3. CONJECTURE 3 (Artin-Tate). The Brauer group The inequality

$\rho\leqq\rho^{\prime}$

$Br(X)$

$R(q^{-1})=\frac{|Br(X)||\det((D_{i}D_{j}))|}{q^{a(X)}|NS(X)_{tor}|^{2}}$

of

$X$

is finite, and

,

where is a basis of $NS(X)$ mod torsion and $a(X)$ is a suitably defined integer with $0\leqq\alpha(X)\leqq p_{g}(X)$ . This is the conjecture $(C)$ of [27] \S 4 and its non-p part is known to be $D_{\iota}(1\leqq i\leqq\rho)$

Elliptic modular

53

surfaces

([27] Theorem 5.2). The order of Brauer group true if conjectured to be a square or twice a square. $\rho^{\prime}=\rho$

$|Br(X)|$

is

C. Main results. and a prime number We fix a positive integer $H_{w,p}(u)$ Let denote the Hecke polynomial: $n\geqq 3$

$H_{w,p}(u)=\det(1-T_{p}u+p^{w-1}R_{p}u^{2})$

$p$

not dividing

$\Gamma(n)$

$H_{w,p^{f}}(u)=\prod_{j}(1-\beta_{j^{f}}u)$

.

,

-cusp forms of weight of defined with respect to the space , ([4] No. 36, \S 5-8). Writing $H_{w,p}(u)=\prod_{j}(1-\beta_{j}u)$ we put $S_{w}(\Gamma(n))$

$n$

$w\geqq 2$

.

$(f\geqq 1)$

in characNow we let $X=E_{p}^{*}$ , the elliptic modular surface for level teristic , defined in A vii); put $q_{0}=p^{f}$ . It follows from the results of Eichler-Shimura-Igusa ([23], [6]) that the zeta function of the base curve (considered over ) is given by $n$

$p$

$\Delta=M_{n}^{*}\otimes F_{p}$

$F_{q_{0}}$

$\zeta(\Delta, T)=H_{2,q_{0}}(T)/(1-T)(1-q_{0}T)$

.

Hence we have $P_{1}(X, T)=H_{2,q_{0}}(T)$

,

is isomorphic to the Jacobian variety of . To consider $P_{2}(X, T)$ , we denote by $NS^{0}(X)$ the subgroup of the N\’erongenerated by the curves of Severi group

since the Picard variety of

$X$

$\Delta$

$\overline{X}$

$NS(\overline{X})$

(0),

$C_{u_{0}},$

$\Theta_{v,t}$

$(v\in\Sigma_{p}, 1\leqq i\leqq m_{v}-1)$

,

of $NS^{0}(X)$ is equal in the notation of Theorem 1.1. Note that the rank of $B(n)$ in characteristic zero, cf. (5.4): to the Picard number $\rho_{0}$

$\rho(n)$

$\rho_{0}=2+\sum_{v}(m_{v}-1)=2+(n-1)\mu(n)/n$

.

by A viii). We also note that all the elements of $NS^{0}(X)$ are rational over so that all the elements in $NS^{0}(X)$ are defined over THEOREM. Take . Then $F_{q_{0}^{2}}$

$q=q_{0^{d}}$

$F_{q}$

$P_{2}(X\otimes F_{q}, T)=(1-qT)^{\rho 0}H_{3,q}(T)$

.

We shall indicate two proofs for this theorem. The first proof is based on the results of Deligne [2]. As a special case $(w=3)$ of the construction of l-adic representations in [2], there exists a : Galois submodule $W$ of $H^{2}(\overline{X}, Q_{\iota})$

$W=_{n}^{1}W_{\iota}\subset H^{2}(\overline{X}, Q_{t})$

,

T. SIIIODA

54

such that i) ii)

$\dim_{Q_{l}}(W)=2\dim_{C}S_{3}(\Gamma(n))$

;

$\det(1-\varphi_{2,l}T)|_{W}=H_{3,q}(T)$ ,

induced by the Frobenius endois the endomorphism of ; morphism of iii) denotes the endoacts on $W$ by multiplication by $m$ , where $X$ $m$ on the generic fibre, $m$ morphism of over inducing multiplication by is the endomorbeing an integer with $m\equiv 1mod n$ and $m\neq- 0mod p$ , and (cf. the proof of Theorem 3.1 and [2] induced by phism of Lemme 5.3). On the other hand, we can easily prove:

where

$H^{2}(\overline{X}, Q_{\iota})$

$\varphi_{2,l}$

$\varphi$

$X\otimes F_{q}$

$\psi_{m}$

$\psi_{m}^{*}$

$\Delta$

$\psi_{m}^{*}$

$H^{2}(\overline{X}, Q_{\iota})$

iv)

$\psi_{m}^{*}(C_{u})=C_{u},$

$\psi_{m}$

$\psi_{m}^{*}(\Theta_{v,i})=\Theta_{v,i}(v\in\Sigma_{p} ; i\geqq 0)$

$\psi_{m}^{*}(D_{0})\approx m^{2}D_{0}$

If we consider

, where

$D_{0}=2(0)+(p_{a}+1)C_{u}$

as a

$NS(X)\otimes Q_{\iota}$

subspace of

;

.

$H^{2}(\overline{X}, Q_{t})$

$W\cap NS^{0}(X)\otimes Q_{\iota}=\{0\}$

Hence, by comparing dimensions (cf. A ix)),

we

, iii) and iv) imply that

. get

$H^{2}(\overline{X}, Q_{\iota})=(NS^{0}(X)\otimes Q_{\iota})\oplus W$

;

this is analogous to the Hodge decomposition of Proposition 6.4. the assumption on , it is immediate that

By ii) and

$q$

$P_{2}(X\otimes F_{q}, T)=(1-qT)^{\rho 0}H_{3,q}(T)$

.

This completes the first proof. The second proof is based on Ihara’s theory of congruence monodromy problems [9]. This method has been used by Morita [16] following a suggestion of Ihara (cf. [8] Introduction) for establishing the relation between Hecke polynomials for even weights and the zeta functions of (incomplete) fibre varieties whose fibres are self-product of even number of elliptic curves. To apply the same method to our case, we note: 1) The trace formula of Hecke operators is also available for odd weight, cf. Shimizu [21]. , the fibre 2) For any closed point is an elliptic of of $X$ over is rational over . Therecurve such that all points of order on satisfy congruence: the its zeta function of reciprocal roots fore the $u$

$\Delta^{\prime}$

$C_{u}$

$n$

$u$

$C_{u}$

$F_{q_{0}}(u)$

$\pi^{\prime}$

$\pi,$

$\pi\equiv\pi^{\prime}\equiv 1$

$mod n$

.

, which was Since $n\geqq 3$ , this eliminates any ambiguity of sign of main reason why the odd weight case (or fibre varieties whose fibres self-products of an odd number of elliptic curves) had to be excluded in and [16]. The rest of the proof is similar to that of [16], at least for $\pi^{\prime}$

$\pi,$

the

are [8]

the

Elliptic modular

case

$q=p^{2}$

surfaces

55

, in which Ihara’s theory [9] is directly applicable.

for our surface $X=E_{p}^{*}$ . We shall consider the conjectures stated in COROLLARY 1. Conjecture 1 (Weil) for the surface $X=E_{p}^{*}$ is equivalent to the Petersson conjecture for the eigenvalues of Hecke opera $torT_{p}$ acting on $B$

$S_{3}(\Gamma(n))$

.

COROLLARY 2. If $H_{3,q}(q^{-1})\neq 0$ , then the rank of $NS(X)$ is equal to , $X=E_{p}^{*}$ (Tate) true Conjecture the is over . 2 and for surface be the multiplicity of as the reciprocal roots in $P_{2}(X, T)$ . PROOF. Let . On the other hand, the above theorem implies Then $\rho$

$\rho_{0}$

$F_{q}$

$q$

$\rho^{\prime}$

$\rho^{\prime}\geqq\rho$

$\rho^{\prime}=\rho_{0}\leqq\rho$

.

. Hence COROLLARY 3. If $H_{3,q}(q^{-1})\neq 0$ , then the group of sections of over sections of order . consists of This follows from the above Corollary 2 and Corollary 1.5 of \S 1 (cf. Theorem 5.5). Moreover we can give explicit values for quantities in Conjecture 3 (Artin-Tate) under the assumption that $H_{3,q}(q^{-1})\neq 0$ . By Corollary 1.7 of \S 1, we have $\rho=\rho_{0}=\rho^{\prime}$

$E_{p}^{*}$

$n^{2}$

$M_{p}^{*}\otimes F_{p}$

$n$

$\frac{|\det((D_{i}D_{j}}{|NS(X)_{tor}}))\underline{|}|^{2}=\frac{n^{t(n)}}{(n^{2})^{2}}=n^{t(n)- 4}$

,

since $E(K)\cong(Z/n)^{2}$

,

$m_{v}^{(1)}=n$

,

$|\Sigma|=t(n)=\mu(n)/n$

,

in the notations used there. Hence COROLLARY 4. If Conjecture 3 (Artin-Tate) is true, then the order is given by the formula: Brauer group $Br(X\otimes F_{q})$ of

of the

$X\otimes F_{q}$

$|Br(X\otimes F_{q})|=q^{\alpha(X)}H_{3,q}(q^{-1})/n^{t(n)- 4}$

provided that $H_{3,q}(q^{-1})\neq 0$ . By [27] Theorem 5.2 and Corollary 2 above, this formula is true up to a factor of a -power. Therefore, using [27] Theorem 5.1, we can restate Corollary 4 as follows: COROLLARY 5. If $H_{3,q}(q^{-1})\neq 0$ , then the integer $q^{p}eH_{3,q}(q^{-1})$ is of the form: $p$

$q^{p_{g}}H_{3,q}(q^{-1})=\pm a\cdot b^{2}\cdot n^{t(n)- 4}$

where are integers and is a p-power or twice a p-power. REMARK 6. The values of $t(n)=\mu(n)/n$ for small $n\geqq 3$ are given as $a,$

$b$

$a$

follows: $\frac{n}{t(n)}|\frac{34567891011}{4612122424363660}$

56

T. SHIODA

Thus Corollary 5 implies rather remarkable divisibility properties of the integer associated with the Hecke polynomials for weight 3. In the above discussions, we assumed that . In the general case, we put $q^{p_{g}}H_{s,q}(q^{-1})$

$H_{\epsilon,q}(q^{-1})\neq 0$

$H_{\epsilon,q}(u)=(1-qu)^{r_{P}}R(u)$

,

$R(q^{-1})\neq 0$

,

with a non-negative integer . Then we have COROLLARY 7. The following statements are equivalent: 1) Conjecture 2 (Tate) is true for . 2) The Picard number is equal to of $r_{p}$

$X\otimes F_{q}$

$X\otimes F_{q}$

$\rho$

$\rho=\rho_{0}+r_{p}$

.

(or equithe group of sections of over group valently, the rank of the of rational points of the generic fibre ) is equal to . over the function field of The case $r_{p}>0$ actually occurs, as is shown by the example below. , which is the REMARK 8. Recall that our $X$ is a compactification of structure in characteristic universal family of elliptic curves with level $Cor$ ollary 7 Thus, ). if the statements (over the moduli scheme will admit sections of infinite order are true in the case $r_{p}>0$ , then

3)

The rank

of

$X\otimes F_{q}$

$M_{n}^{*}\otimes F_{q}$

$M_{n}^{*}\otimes F_{q}$

$r_{p}$

$E\otimes F_{p}$

$n$

$p$

in.

$M_{n}\otimes F_{p}$

$E\otimes F_{p}$

over

$M_{n}\otimes F_{p}$

.

EXAMPLE 9 (level 3

case).

$g=p_{g}=0$ ,

Let $mod

Let

$n=3$

.

Then we have (cf. Example 5.4)

$\rho_{0}=b_{2}=10$

,

and put $q=p$ or be a prime number 3$ . Then the zeta function of $X=E_{p}^{*}$ (over $\neq 3$

$p$

$t(3)=4$ $p^{2}$

$F_{q}$

.

according as ) is given by

$\zeta(X, T)=1/(1-T)(1-qT)^{10}(1-q^{2}T)$

$P\equiv 1$

or

$-1$

.

Hence Conjectures 1 and 2 are trivial and Conjecture 3 implies $|Br(X)|=1$ , which is compatible with the fact that $X$ is a rational surface (cf. Remark

5.6

(ii)).

EXAMPLE 10 (level 4 case). Let $n=4$ . In this case, $p_{g}=\dim S_{3}(\Gamma(4))=1$ is given by being the cusp and a non-trivial element of (cf. $SL(2, Z)$ 7.7). Example By weight Schoeneberg [25] . 12 for form of 181, we have $S_{3}(\Gamma(4))$

$\Delta(z)^{1,/4},$

$\Delta(z)$

$p$

$H_{s,p}(u)=$

are integers in $Z[\sqrt{}^{-}-1]$ such that where . according as or $-1mod 4$ ; we can take $q=q_{0}$ in the Let $q_{0}=p$ or theorem, since the elliptic modular surface of level 4 has a model, classically known as Jacobi quartic: $\pi^{\prime}$

$p=\pi\pi^{\prime},$

$\pi,$

$p^{2}$

$p\equiv 1$

$\pi\equiv 1mod 2\sqrt{}\overline{-1}$

57

surfaces

Elliptic modular

$A:y^{2}=(1-\sigma^{2}x^{2})(1-\sigma^{-2}x^{2})$

over

$K=F_{q}(\sigma),$

$\sigma$

a variable over

being

$F_{q}$

.

,

The zeta function of

$X=E_{p}^{*}$

(for

is given by

$n=4)$

$\zeta(X, T)=1/(1-T)(1-qT)^{20}H_{s,q}(T)(1-q^{2}T)$

.

Therefore we get the following (cf. [24] I. Introduction): i) Conjecture 1 (Weil) is true, since . ii) If $p\equiv 1mod 4$ , then Conjecture 2 (Tate) is true. iii) If $p\equiv-1$ mod4, then Conjecture 2 is true if and only if the rank of group of K-rational points of the elliptic curve $A$ is equal to $r_{p}=2$ . (We do not know whether or not $A$ has a K-rational point of infinite order in the $|\pi^{2}|=|\pi^{\prime 2}|=p$

case

$p\equiv-1mod 4.$ )

If

iv)

$p\equiv 1mod 4$ ,

we

put

$\pi=a+2b\sqrt{-1},$

$a,$

.

$b\in Z$

$pH_{3,p}(p^{-1})=-(\pi-\pi^{\prime})^{2}=(4b)^{2}$

By Corollary 4, the conjectured value of

$|Br(X)|$

.

is equal to ;

$(t(4)=6)$

$pH_{3,p}(p^{-1})/4^{\epsilon- 4}=b^{2}$

Then

a square integer ! Let

v)

of

$E_{0}^{*}$

over

$E_{0}^{*}(n=4)$

be

$k=Q(\sqrt{-1})$

as in the part A vi). The Hasse-Weil zeta function is given as follows:

$\prod_{\mathfrak{p}+2}\zeta(E_{p}^{*}, N\mathfrak{p}^{-s})\sim\zeta_{k}(s)\zeta_{k}(s-1)^{20}\zeta_{k}(s-2)D_{4}(s)^{2}$

Here

$\zeta_{k}(s)$

is the Dedekind zeta function of

$Q(\sqrt{-1})$

.

, and

$D_{4}(s)=\prod_{p\neq 2}H_{s,p}(p^{-s})^{-1}$

is a zeta function of $Q(\sqrt{-1})$ with a Gr\"ossencharacter, cf. [25] Formula (10); . indicates equality up to a factor of a rational function of $K3$ vi) The Picard number equal is to 20. On the of the surface other hand, we have $2^{-s}$

$\sim$

$E_{0}^{*}$

$\rho_{0}$

$D_{4}(2)\neq 0$

e. g. by

,

[28] p. 288, Theorem

11. Hence the Picard number of is equal to the order of the pole at $s=2$ of its Hasse-Weil zeta function (cf. Tate [26] $E_{0}^{*}$

\S 4, Conjecture 2). REMARK 11. The Hasse-Weil zeta function of for an arbitrary level can be obtained in the same way. REMARK 12. There is no doubt that the arithmetic theory of elliptic modular is meaningful also for certain groups other than surfaces . For example, let be the group considered in Example 5.8. For $q=7,11$ consisting or 19, Hecke ([4] No. 41, pp. 906-910) constructs a basis of $E_{0}^{*}$

$n$

$\Gamma$

$B_{\Gamma}$

$\Gamma(n)$

$\Gamma=\Gamma_{0}^{\prime}(q)$

$S_{3}(\Gamma)$

58

T. SHIODA

of eigenfunctions of the Hecke operators. For $q=7$ or 11, we have $\dim S_{3}(\Gamma)$ $=1$ and the Hecke polynomial $H_{3,p}(u)$ is similar to that of . For $q=19$, $p_{g}=\dim S_{3}(\Gamma)=3$ . Using the result of Hecke, we see that $\Gamma(4)$

$p^{3}H_{s,p}(p^{-1})=19$

. (square)

if

$p^{3}[H_{3,p}(u)/(1-pu)]_{u\Rightarrow p^{-1}}=2p\cdot 13$

$(-19^{-)}p=1$

. (square)

,

if

$(-19^{-)}p=-1$

.

In view of Conjecture 3 (Artin-Tate), this seems to suggest that there might be some connection between the field of eigenvalues of Hecke operator for weight 3 (e.g. $Q(\sqrt{}=13)$ ) and the discriminant of the intersection matrix of being assumed to be the N\’eron-Severi group $NS(X)$ of $X=B_{\Gamma}mod p,$ ( defined over ) or the order of the Brauer group $Br(X)$ of $X$. $B_{\Gamma}$

$Q,$

University of Tokyo

References [1]

J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc., 41 (1966), 193-291.

[2]

P. Deligne, Formes modulaires et repr\’esentations 1-adiques, S\’em. Bourbaki, F\’ev. 1969, exp. 355,1-33. M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Z., 67 (1957),

[3]

267-298. [4]

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[12]

E. Hecke, Mathematische Werke, G\"ottingen, 1959 (esp. No. 36 (1937), 672-707; No. 41 (1940), 789-918). J. Igusa, Fibre systems of Jacobian varieties (III. Fibre systems of elliptic curves), Amer. J. Math., 81 (1959), 453-476. J. Igusa, Kroneckerian model of fields of elliptic modular functions, Amer. J. Math., 81 (1959), 561-577. J. Igusa, Modular forms and projective invariants, Amer. J. Math., 89 (1967), 817-855. Y. Ihara, Hecke polynomials as congruence functions in elliptic modular case, Ann. of Math., 85 (1967), 267-295. $\zeta$

Y. Ihara, On congruence monodromy problems I, II, Lecture notes, Univ. of Tokyo, 1968-69. Y. Kawada, Theory of automorphic functions of one variable I, II, Lecture notes, Univ. of Tokyo, 1963-64, (in Japanese). K. Kodaira, On compact analytic surfaces II-III, Ann. of Math., 77 (1963), 563626; 78 (1963), 1-40, (cited as [K]). K. Kodaira, On the structure of compact complex analytic surfaces, I, Amer.

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Math.,

86

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59

surfaces

S. Lang, Diophantine geometry, Intersc. Tracts, No. 11, 1962. Y. Morita, Hecke polynomials of modular groups and congruence zeta functions of fibre varieties, J. Math. Soc. Japan, 21 (1969), 607-637. A. N\’eron, Mod\‘eles minimaux des vari\’et\’es ab\’eliennes sur les corps locaux et globaux, Publ. I. H. E. S., No. 21, 1964. A. P. Ogg, Cohomology of abelian varieties over function fields, Ann. of Math., 76 (1962), 185-212. A. P. Ogg, Elliptic curves and wild ramification, Amer. J. Math., 89 (1967), 1-21. I. R. afarevi , Principal homogeneous spaces defined over a function field, Amer. Math. Soc. Trans., (2), 37 (1964), 85-114. H. Shimizu, On traces of Hecke operators, J. Fac. Sci. Univ. of Tokyo, Sec. I, 10 (1963), 1-19. G. Shimura, Sur les int\’egrales attach\’ees aux formes automorphes, J. Math. Soc. Japan, 11 (1959), 291-311, (cited as [S]). de courbes alg\’ebriG. Shimura, Correspondances modulaires et les fonctions ques, J. Math. Soc. Japan, 10 (1958), 1-28. T. Shioda, Elliptic modular surfaces, I, II, Proc. Japan Acad., 45 (1969), 786790, 833-837. B. Schoeneberg, \"Uber den Zusammenhang der Eisensteinschen Reihen und Thetareihen mit der Diskriminante der elliptischen Funktionen, Math. Ann., 126 (1953), 177-184. J. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry, Harpar and Row, 1965, p. 93-110. J. Tate, On the conjecture of Birch and Swinnerton-Dyer and a geometric analog, S\’em. Bourbaki, F\’ev. 1966, exp. 306, 1-26. A. Weil, Basic number theory, Springer, New York, 1967. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japan 11, 1971. $\zeta$