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Modular forms of weight 3m and elliptic modular surfaces.

Let [GAMMA] be a subgroup of [SL.sub.2](Z) of finite index which does not contain--1. In [3], Shioda introduced the elliptic modular surface [pi] : [S.sub.[GAMMA]] [right arrow] [X.sub.[GAMMA]] over the (compactified) modular curve [X.sub.[GAMMA]] associated to [GAMMA], and proved, among other things, that the space of cusp forms of weight 3 with respect to [GAMMA] is canonically isomorphic to that of holomorphic 2-forms on the surface [S.sub.[GAMMA]] ([3] Theorem 6.1). Our purpose is to extend this correspondence to that between modular forms of weight 3m and certain rational m-pluricanonical forms on [S.sub.[GAMMA]]. This is parallel to the situation for weight 2m where modular forms of weight 2m correspond to m-pluricanonical forms on the curve [X.sub.[GAMMA]].

To state the result, let [DELTA][subset] [X.sub.[GAMMA]] be the (reduced) cusp divisor. A cusp is called irregular if its stabilizer in [GAMMA] contains an element conjugate to [mathematical expression not reproducible] in [SL.sub.2](Z), and regular otherwise. We write [DELTA] = [[DELTA].sub.reg] + [[DELTA].sub.irr] for the corresponding decomposition. The singular fibres over [DELTA] are divided accordingly, which we write [pi]* [DELTA] = [D.sib.reg] + [D.sub.irr]. By [3], [D.sub.reg] consists of type [I.sub.n] fibres, and [D.sub.irr] consists of type [I*.sub.n] fibres (n depends on the cusps). We consider the Q-divisor

D = [D.sub.reg] + 1/2 [D.sub.irr].

We write [M.sub.k]([GAMMA]) for the space of [GAMMA]-modular forms of weight k (cf. [1]).

Our main result is the following

Theorem 0.1. We have a natural isomorphism of graded rings

[mathematical expression not reproducible]

Here [mathematical expression not reproducible] (mD) should be understood as [mathematical expression not reproducible]. Explicitly, the isomorphism (0.1) is given by associating to f ([tau]) [member of] [M.sub.3m]([GAMMA]) the m-canonical form [f ([tau])(d[tau] [LAMBDA] dz).sup.[cross product]m] where z is the uniformizing coordinate on the smooth fibres C/(Z T Z[tau]) of [pi].

Independently of our proof of Theorem 0.1, we can check that the m-th components of both sides of (0.1) indeed have the same dimension, which is given by

(0-2) (3m - 1)(g - 1)T [me.sub.3] + [[epsilon].sub.reg] + [3n/2] [[member of].sub.irr].

Here g is the genus of [X.sub.[GAMMA]] and [[epsilon].sub.3] [[epsilon].sub.reg], [[epsilon].sub.iri], are the number of elliptic points, regular cusps and irregular cusps respectively. All elliptic points are of order 3 because -1 [??]2 [GAMMA]. It is classical that dim [M.sub.3m]([GAMMA]) is given by (0.2) (see [1] [section]3.5 and [section]3.6). On the other hand, [mathematical expression not reproducible]) can be calculated by expressing [mathematical expression not reproducible] as the pullback of a line bundle [L.sub.m] on [X.sub.[GAMMA]] by the canonical bundle formula, and then computing [h.sup.0]([L.sub.m]) by the Riemann-Roch formula on [X.sub.[GAMMA]]. This also gives (0.2). Anyway, our proof of Theorem 0.1 is conceptual and does not use this equality.

By imposing the vanishing condition mD on both sides of (0.1), we also obtain an isomorphism between the canonical ring [mathematical expression not reproducible] of [S.sub.[GAMMA]] and the following subring of [[direct sum].sub.m][M.sub.3m]([GAMMA]):

{[mathematical expression not reproducible] [greater than or equal to] m at every cusp s}

Here we measure the vanishing order of / at a cusp s = [gamma](i[infinity]), [gamma] [member of] [SL.sub.2](Z), by the parameter [e.sup.2[pi]i[tau]/N]] where N is the smallest positive integer such that [mathematical expression not reproducible] is contained in [[gamma].sup.-1] [GAMMA][gamma]. (For irregular cusps, this is square root of local coordinate.) When m = 1, this is the Shioda isomorphism [mathematical expression not reproducible].

The isomorphism (0.1) looks analogous to the classical correspondence between modular forms of even weight and pluricanonical forms on [X.sub.[GAMMA]] (see [1] [section]3.5), but some coefficients are different. Specifically, there is no contribution from the elliptic points to the pole condition, and the contributions from the regular and irregular cusps have different weights. The main reason is that around the singular fibres over the elliptic points and irregular cusps, after contracting some (-2)-curves, the projection from the universal family is unramified in codimension 1.

The proof of Theorem 0.1 is given in [section]1. In [section]2 we explain the interpretation of the Hecke operators on [M.sub.3m]([GAMMA]) and the Petersson inner product on [S.sub.3]([GAMMA]) in terms of the pluricanonical forms.

1. Proof. Before proceeding to the proof, let us first explain why modular forms and pluricanonical forms on elliptic modular surfaces correspond, at the level of period domain. We prefer to view the upper half plane as the open set of [P.sup.1]

D = {[w] [member of] [P.sup.1](C) | [square root of (-1([omega], [bar.[omega]) > 0},

where (,) is the symplectic form (([x.sub.1], [y.sub.1]), ([x.sub.2], [y.sub.2])) = -[x.sub.1][y.sub.2] + [y.sub.1][x.sub.2] on [C.sup.2]. Let [Laplace] = [Op.sup.1] [(-1)|.sub.D] be the tautological bundle over D, endowed with the natural action of the group GL^ (R) of 2 x 2 matrices of determinant > 0. Modular forms of weight k for [GAMMA] < [SL.sub.2] (Z) are T-invariant sections of [[Laplace].sup.[cross product]k] with cusp condition.

Remark 1.1. The relation with the more traditional definition ([1]) is as follows. Pick a primitive vector l [member of] [Z.sup.2], which corresponds to the rational boundary point [l] [member of] [P.sup.1](Q) of D. Let [s.sub.l] be the rational section of [Op.sub.1] (-1) defined by the equation ([s.sub.l]([[omega]]), l) = 1, where sl([[omega]]) [member of] C[omega] [subset] [C.sup.2]. Then [s.sub.l] has a pole of order 1 at [l]. A choice of a vector m [member of] [Z.sup.2] with (m, l) = 1 induces an isomorphism [L.sub.l] : H [right arrow] [s.sub.l](D) [equivalent] D defined by [tau] [??] [tau]l + m.

By the frame [s.sup.[cross product]k.sub.l] of [[Laplace].sup.[cross product]k] and this coordinate [tau], we can identify sections of [[Laplace].sup.[cross product]k] with functions on H. This gives the Fourier expansion of modular forms of weight k at the cusp [l], and defines the condition of holomorphicity there.

For example, when l = (1,0) and m = (0,1), we have [L.sub.l]([tau]) = ([tau], 1), so the action of [gamma] = (a b D coincides with [gamma]([tau]) = (a[tau] + b)/(cT + d) on H via [L.sub.l], and the factor of automorphy of [s.sub.l] by [gamma] is c[tau] + d.

Let [C.sup.2] = [C.sup.2] x D and [Z.sup.2] = [Z.sup.2] x D be the local systems over D. Then [Laplace] is naturally a sub line bundle of the vector bundle [member of] = [O.sup.[cross product]2.sub.D] corresponding to [C.sup.2]. The universal marked elliptic curve [??]: S [right arrow] D over D is the quotient

S = E/([Laplace] + [Z.sup.2]) [equivalent] [[Laplace].sup.v]/[Z.sup.2v].

The second isomorphism is induced by the symplectic form (,). If [omega]] = ([tau], 1), the fibre [(C[omega]).sup.v]/[([Z.sup.2]).sup.v] over [[omega]] is isomorphic to C/(Z + Z[tau]) by the pairing with [omega]. The local systems [R.sup.1][??] Z, [R.sup.1] [??] C are identified with [Z.sup.2], [C.sup.2] respectively, and the Hodge bundle of S [right arrow] D is identified with [Laplace]. Let [K.sub.[??] be the relative canonical bundle of S/D. Since [K.sub.[??]|F] [equivalent] [K.sub.F] is trivial at each fibre F, [??]* [K.sub.[??]] is an invertible sheaf on D, and the natural homomorphism ([??] [K.sub.[??]]).sup.[cross product]] [right arrow] [??] ([K.sup.[cross product].sub.[??]]) is an isomorphism for any m.

We have two fundamental [SL.sub.2](Z)-equivariant isomorphisms:

(1.1) [mathematical expression not reproducible]

The first isomorphism is just [K.sub.D] = [Kp.sub.1|D] [equivalent] [[Op.sup.1] (-2)|.sub.d]. The second isomorphism is given by the period integral: integration of the 1-forms in [H.sup.0]([K.sub.[??]|F)] = [H.sup.0] ([K.sub.F]) along the 1-cycles in each fibre F gives [H.sup.0] ([K.sub.F]) [equivalent] [H.sup.1,0](F). Combining these isomorphisms, we obtain

(1.2) [mathematical expression not reproducible]

This is the source of (0.1). To summarize, modular forms correspond to two types of differential forms: (local) weight 2 forms to (local) base differentials, and (local) weight 1 forms to fibre differentials. Then (0.1) comes from the combination of these two correspondences.

Now let T < [SL.sub.2](Z) be a finite-index subgroup not containing -1. Taking the quotient of S [right arrow] D by [GAMMA] and resolving the [A.sub.2]-singularities arising from the fixed points, we obtain an elliptic fibration over [Y.sub.[GAMMA]] = [X.sub.[GAMMA]] - [DELTA]. The elliptic modular surface [pi]: [S.sub.[GAMMA]] [right arrow] [X.sub.[GAMMA]] is the nonsingular, relatively minimal extension of this fibration over [X.sub.[GAMMA]]. (See [3] [section]4 for the construction of [S.sub.[GAMMA]].) Let us abbreviate [Y.sub.[GAMMA]], [X.sub.[GAMMA]], [S.sub.[GAMMA]] as Y, X, S respectively. We shall prove Theorem 0.1 in two steps.

1.1. Semi-stable case. We first consider the case where [GAMMA] has no elliptic point nor irregular cusp. In this case [GAMMA] acts on D freely and [pi] has only singular fibres of type [I.sub.n] over the regular cusps ([3] p. 35). For instance, P(N) with N > 2 and more generally neat subgroups of [SL.sub.2](Z) satisfy this condition.

The [GAMMA]-equivariant bundles [K.sub.D], [??] [K.sub.[??]] and [Laplace descend to line bundles on Y. We extend them over X as follows. The first one, [K.sub.Y], is extended to [K.sub.X]. The second one is extended to [pi]* [K.sub.[pi]] where [K.sub.[pi]] is the relative canonical bundle of [pi]. Recall that local sections of [K.sub.[pi]\F] at a singular fibre F are identified with local 1-forms on F\Sing(F) that have a pole of order [less than or equal to] 1 at each node and for which the sum of its residues is zero. Hence [K.sub.[??]|F] [equivalent] [O.sub.F], so that [pi]* [K.sub.[pi]] is still invertible and we have ([pi]* [K.sub.[pi]]).sup.[cross product]m] [equivalent] n* ([K.sup.[cross product]m.sub.pi]]).

The third one, the descent of [Laplace], is extended as follows. Let l be a primitive vector of [Z.sup.2] and [s.sub.l] be the frame of [Laplace] as in Remark 1.1. Since [s.sub.l] is invariant under the stabilizer of l in P, it descends to a local frame of the descent of [Laplace] near the cusp [l]. The extension over the cusp is defined so that this punctured local frame extends as a local frame. We write L for the extended line bundle on X. By construction, a local P-invariant section s of [Laplace] extends holomorphically over the cusps as a local section of L if and only if (s([[omega]]), l) does not diverge as [[omega]] [right arrow] [l]. By Remark 1.1, this coincides with the usual cusp condition for modular forms. Thus we have [M.sub.k]([GAMMA]) = [H.sup.0]([L.sup.[cross product]k]).

Now the isomorphisms (1.1) descend to Y as

[K.sub.Y] [equivalent] [L.sup.[cross product]2]|.sub.Y], [pi]*[K.sub.[pi]]Y [equivalent] [L|.sub.Y].

These isomorphisms extend over X to

[K.sub.X] [equivalent] [L.sup.[cross product]2]|.sub.Y], (-[DELTA]), [pi]*[K.sub.[pi]]Y [equivalent] L.

Indeed, let l [member of] [Z.sup.2] be a primitive vector and F be the singular fibre over the cusp [l]. The first isomorphism holds because the frame [s.sup.[cross product]2.sub.l] of [[Laplace].sup.[cross product]2] corresponds to the frame d[tau] of [K.sub.H] up to constant via [L.sub.l] (both have a pole of order 2 at [l]), and we have D[tau] = [q.sup.-1] dq up to constant for the local parameter q = exp(2[pi]i[tau]/N) around the cusp [l]. Next, for the second isomorphism, note that in the period map around [l], the vanishing cycle near a node of F corresponds to the vector l [member of] [Z.sup.2] (up to [+ or -] 1) because both are invariant under the stabilizer of l in [GAMMA] (= local monodromy around the cusp). The integral of a generator of [H.sup.0]([K.sub.[pi]|F]) along the vanishing cycle is equal to its residue at the node, whence nonzero. Therefore a local frame of [pi]*[K.sub.[pi]] corresponds to a local frame of L under the isomorphism [pi]* [K.sub.[pi]|Y] [equivalent] L|Y.

To sum up, we obtain

[mathematical expression not reproducible]

Here recall that D = [D.sub.reg] = [pi]*[DELTA]. Taking global sections give (0.1). Compatibility of the multiplications is obvious.

1.2. General case. We next study the general case. Let P < [SL.sub.2] (Z) be a finite-index subgroup not containing -1. We choose a normal subgroup [GAMMA]' [??] [GAMMA] of finite index that has no elliptic point nor irregular cusp. We will abbreviate the elliptic modular surfaces [S.sub.[GAMMA]] [right arrow] [X.sub.[GAMMA]] and [S.sub.[GAMMA]'] [right arrow] [X.sub.[GAMMA]] as [pi]: S [right arrow] X and [pi]': S' [right arrow] X' respectively. The quotient group [GAMMA] = [GAMMA]/[GAMMA]' acts on S' biregularly. S'/X' is the nonsingular, relatively minimal elliptic surface birational to the base change X' [x.sub.X] S of S/X by the projection f: X' [right arrow] X. We observe this process of birational transformation around each singular fibre of [pi], with emphasis on the relation between the canonical divisors.

Let p [member of] X be either an elliptic point or a cusp, and F = [pi]* p be the singular fibre over p. Choose an arbitrary point p' [member of] [f.sup.-1] (p) and let F' = ([pi]')* p' be the fibre over p'. Take a small neighborhood V [subset] X of p and let V' C X' be the connected component of [f.sup.-1](V) that contains p'. We write U = [[pi].sup.-1](V) and U' = [([pi]').sup.-1] (V'). The stabilizer G [subset] [GAMMA] of p' is cyclic since it is embedded in GL([T.aub.p'] X') [equivalent] [C.sup.x]. We have V [equivalent] V'/G. Write d = [absolute value of G..

(1) When p is a regular cusp, F is a type [I.sub.n] fibre ([3] p. 35). The fibre product U" = V' [X.sub.V] U is normal and has [A.sub.d-1]-singularities ([2] [section]7.5) at the n nodes of the central fibre F" of U" [right arrow] V' (which is isomorphic to F). Then U' is the minimal resolution of those [A.sub.d-1]-points, and so F' is of type [I.sub.dn]. Let [phi]: U' [right arrow] U" be the resolution map and f: U" [right arrow] U be the base change map. We have [K.sub.U'](F') [equivalent] [phi]* ([K.sub.U"] (F")) because F' = [phi]* F" and [K.sub.U]' [equivalent] [phi]* [K.sub.U]", where the second holds since [A.sub.d-1] points are canonical singularities ([2] [section]7.5). On the other hand, we have [K.sub.U]" (F") [equivalent] f* ([K.sub.u] (F)) by the ramification formula for f (see [2] [section]6.1). It follows that

(1.3) [mathematical expression not reproducible]

The last isomorphism holds because f is the quotient map by G.

(2) When p is an irregular cusp, F is a type [I*.sub.n] fibre ([3] p. 35). Then d is an even number, say d = 2d', because the generator [gamma] ~ [mathematical expression not reproducible] the stabilizer of p in T must satisfy [[gamma].sup.d]

(Here ~ means conjugation in [SL.sub.2](Z).) Let G' be the subgroup of G of order d' and let G' = G/G" [equivalent] Z/2. We set V" = V'/G'. Below we will consider the following factorization:

Let [phi]: U [right arrow] [bar.U] be the contraction of the four components of F of multiplicity 1. The central fibre of [bar.U] [right arrow] V is a multiple fibre of multiplicity 2, say 2[bar.F]. Here F is a Weil divisor with 2[bar.F] Cartier. We take the fibre product [bar.U]" = V" [X.sub.V] and write g: [bar.U]" [right arrow] [bar.U] for the natural map. Let U" be the normalization of U".

Claim 1.2. U" is nonsingular, the composition f" : U" [right arrow] U" is a double covering unramified outside the four Appoints, and the central fibre F" of j" [right arrow] V" is of type [I.sub.2n]. In particular, U"/V" is the nonsingular relatively minimal model of U"/V".

Proof. Let p be a point in the central fibre of [bar.U]". When g(p) is a nonsingular point of [bar.F], the local equation of U" around p is given by [t,spu.2] = [x.sup.2] in [C.sup.3.sub.(x,y,t)]. So [bar.u]" consists of two smooth branches around p, which are separated by the normalization. The same picture holds in case g(p) is a node of F but is nonsingular at [bar.U]", where the local equation of [bar.U]" is given by [t.sup.2] = [x.sup.2][y.sup.2] in [C.sup.3.sub.(x,y,t)]. Finally, when g(p) is an [A.sub.1]-point of j, the germ of ([bar.U], g(p)) is defined by [x.sup.2] = ys in [C.sup.3.sub.(x,y,s) with [bar.U]" [right arrow] V given by (x,y,s) [right arrow] s. So the germ of ([bar.U]", p) is defined by [x.sup.2] = [yt.sup.2] in [C.sup.3.sub.(x,y,t)]. Its normalization is given by [C.sip.2] [right arrow] [C.sup.3], (u, v) [??] (uv, [u.sup.2], v). The claim follows from these calculation of normalization.

Now by the same argument as in case (1), we obtain

[K.sub.U"] + F" = (f") * ([K.sub.u] + [bar.F]), [phi]* (K.sub.U] + [bar.F] = [K.sub.U] + 1/2 F. '*(Kj + F) = KV + - F.

Here (f") * [bar.F] and [phi]* [bar.F] mean pullback of Q-Cartier divisor. Since f" is the quotient map by G", this gives

[mathematical expression not reproducible]

On the other hand, the relation of U'/V' and U"/V" is the same as described in case (1). Hence we have a natural isomorphism

(1.4) [H.sup.0](U', [K.sup.[cross product]m.sub.U]' [(mF')).sup.G] = [H.sup.0] (U, [K.sup.[cross product]m.sub.u] ((m/2)F)).

(3) When p is an elliptic point, F is of type IV*, F' is smooth of j-invariant 0, and d = 3 (see [3] p. 35). In this case it is more convenient to look from F'. The group G [equivalent] Z/3 acts on U' with three isolated fixed points in F'. The action around the fixed points are locally given by ([tau], z) [right arrow] ([[zeta].sup.2.sub.3][tau], [[zeta].sub.3]z) where [[zeta].sub.3] = exp(2[pi]i/3). So the quotient U' = U'/G has three A2-singularities ([2] [section]7.5), and the central fibre of U" [right arrow] V has multiplicity 3. Resolving these [A.sub.2]-points, we obtain j. Let f: U' [right arrow] U" be the quotient map by G and [phi]: U [right arrow] U" be the resolution map. Since f is unramified in codimension 1, we have [K.sub.U]' [equivalent] f* [K.sub.U]". We also have [phi]* [K.sub.U]" [equivalent] [K.sub.U] because [A.sub.2]-points are canonical singularities ([2] [section]7.5). As in case (1), these isomorphisms give (1.5) [mathematical expression not reproducible]

Now we can complete the proof of Theorem 0.1. Let D' be the sum of the singular fibres of [pi]'. By [section]1.1 we have an isomorphism

[mathematical expression not reproducible]

which is [bar.[GAMMA]-equivariant by construction. We take the [bar.[GAMMA]-invariant part. On the one hand, we have

[M.sub.3m]([[GAMMA]).sup.[??]] = [M.sub.3m]([GAMMA]).

On the other hand, by the local analysis (1.3), (1.4), (1.5), we see that

[H.sup.0]([K.sup.[cross product]m.sub.S] [(mD')).sup.[bar.[GAMMA]] [equivalent] [h.sup.0]([K.sup.[cross product]m.sub.S] (mD)).

This gives (0.1).

2. Remarks. Let [GAMMA] be a finite-index subgroup of [SL.sub.2](Z) not containing -1. The Hecke operators on [M.sub.3m](T) (for T congruence subgroup) and the Petersson scalar product on [S.sub.3](T) have natural interpretations in terms of the pluricanonical forms. The weight 3 case must be well-known, but we have included it here since we could not find a suitable reference.

2.1. Hecke operators. Assume that T is a congruence subgroup, and let [alpha] be a 2 x 2 matrix with integral coefficients and det(a) > 0. Putting [[GAMMA].sub.[alpha]] = [GAMMA] [intersection] [alpha] [GAMMA][[alpha].sup.-1]] and [[GAMMA].sup.[alpha]] = [GAMMA] [intersection] [[alpha].sup.-1] [GAMMA][alpha], we have the self correspondence

(2.1) [mathematical expression not reproducible]

of [X.sub.[GAMMA]], where [[pi].sub.i] are the projections and [alpha] is the isomorphism induced from the [alpha]-action on D. This induces the Hecke operator (see [1] [section]5.1)

[mathematical expression not reproducible]

on [M.sub.k]([GAMMA]). Here [alpha]* is the pullback by the [alpha]-action on [[Laplace].sup.[cross product]k], namely the k-th power of the [alpha]-action on [Laplace] induced from the natural action of [alpha] on [C.sup.2].

On the other hand, the curve correspondence (2.1) lifts to the rational correspondence of the elliptic modular surfaces

(2.2) [mathematical expression not reproducible]

Here [[??].sub.i] are the base change maps; [??] is induced from the natural [alpha]-action on the bundle [C.sup.2] and gives an isogeny of degree det([alpha]) at each smooth fibre. The indeterminacy points and ramification divisors of [??], [[??].sub.1], [[??].sub.2] are contained in the singular fibres and their preimage. The correspondence (2.2) induces the endomorphism [[??].sub.2*] [??]* o [[??].sub.1*] on rational pluricanonical forms on [S.sub.[GAMMA]]. Here [[??].sub.2*] takes the fibre sum of rational pluricanonical forms on [S.sub.[GAMMA][alpha]] by the finite dominant rational map [[??].sub.2].

Proposition 2.1. Via the isomorphism (0.1), the weight 3m Hecke operator [[[GAMMA] [alpha] [GAMMA]].sub.3m] agrees with the endomorphism det[([alpha]).sup.2m-1] [[??].sub.2*] o [??]* o [[??].sub.1] * of [mathematical expression not reproducible].

Proof. It is clear that the pullbacks [[pi]*.sub.1], [[??].sub.1]* agree. Also [[pi].sub.2] and [[pi].sub.2*] agree because the push-forward from elliptic modular surfaces to modular curves (cf. (1.2)) translates the trace map [[??].sub.2]* for surfaces to the trace map [[pi].sub.2]* for curves. We verify that the pullback [??]* of rational m-canonical forms by a corresponds to the det ([alpha]).sup.m] multiple of the pullback [alpha]* of weight 3m modular forms by a. Since both operators are locally defined and compatible with multiplication, we only have to check this for m = 1 and at the level of period domain D. Under the isomorphism [[Laplace].sup.[cross product]2 [equivalent] [K.sub.D], the pullback of weight 2 modular forms agrees with the pullback of 1-forms on D. On the other hand, under [Laplace] [equivalent] [??]]K [K.sub.[??]], the pullback of weight 1 modular forms agrees with the det [([alpha]).sup.-1] multiple of the pullback of 1-forms on the fibres by the isogeny [alpha] : [S.sub.[[omega]]] [right arrow] [S.sub.[[omega][alpha]]], because [alpha] multiplies the symplectic form by det([alpha]).

2.2. Petersson inner product in weight 3.

Let [(,).sub.[Laplace]] be the [SL.sub.2](R) -invariant Hermitian metric on [Laplace] that corresponds to (half) the Hodge metric

[mathematical expression not reproducible]

in each fibre [[Laplace].sub.[[omega]] = [H.sup.0]([K.sub.F]) where F = [S.sub.[[omega]]. Let 2[OMEGA] be the Kahler form of the metric induced on [[Laplace].sup.[cross product]-2]] [equivalent] [T.sub.D]. In the upper half plane model, [OMEGA] is expressed as [y.sup.-2] dx [LAMBDA] dy. The Petersson inner product on [S.sub.3] ([GAMMA]) is defined by

[mathematical expression not reproducible]

Via the trivialization of [Laplace] given by the frame [l.sub.(1,0)]: H [right arrow] L (see Remark 1.1), this agrees with the classical definition ([1] [section]5.4)

[mathematical expression not reproducible]

Proposition 2.2. Via the Shioda isomorphism [S.sub.3]([GAMMA]) [equivalent] [H.spu.0] ([K.sub.S[GAMMA]]), the Petersson inner product agrees with the (1/4-scaled) Hodge metric on [H.sup.0]([K.sub.S[GAMMA]])

(2.3) [mathematical expression not reproducible]

Proof. We have [mathematical expression not reproducible] for 1-forms [[eta].sub.i] on D. If f, g [member of] [S.sub.3]([GAMMA]) correspond to 2-forms which locally can be written in the form [??]* [[eta].sub.1] [cross product] [[omega].sub.1] [??]* [eta].sub.2] [cross product] [[omega],sub.2], we locally have the equality of (1, 1)-forms

[mathematical expression not reproducible]

Integration of the right-hand side over [Y.sub.[GAMMA]] gives (2.3). (Here the minus sign cancels when exchanging [bar.[omega].sub.2] and [[eta].sub.1].)

DOI: 10.3792/pjaa.95.31

Acknowledgement. The author is partially supported by JSPS KAKENHI 15H02051 and 17K14158.

References

[1] F. Diamond and J. Shurman, A first course in modular forms, Graduate Texts in Mathematics, 228, Springer-Verlag, New York, 2005.

[2] S. Ishii, Introduction to singularities, Springer, Tokyo, 2014.

[3] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20 59.

By Shouhei MA Department of Mathematics, Tokyo Institute of Technology, 2-12-1, Ookayama, Meguro-ku, Tokyo 152-8551, Japan (Communicated by Masaki KASHIWARA, M.J.A., March 12, 2019)
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Date:Apr 1, 2019
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