Moduli of abelian surfaces, and regularpolyhedral groups
Shigeru MUKAI

3May, 2003

Abstract : Let Ad be the moduli space of polarized abelian surfaces oftype (1; d). For d = 2; 3 and 4, the Satake compacti cation of Ad is isomorphic to the quotient of P3 by an action of PSL(2; Z=d) 2 P SL(2; Z=d). Let
PSL(2; Z=5) =: G5  P GL(2) be the icosahedral group and P GL(2)  P3 thenatural embedding into the projective space of 2 2 2 matrices. A small resolution of the Satake compacti cation of A5 (at the poit cusps) is isomorphic to~ 3 at the 60 points (' G5 ) by an action of G5 2 G5 .
the quotient of the blow-up PLet X (d) be the moduli space of elliptic curves E with a full level d structure,i.e., a symplectic isomorphism between the standard symplectic module

2[Z=d)] := Z=d 8 Z=d;

0 101 0

and the group Ed of d-torsion points with the Weil pairing. The modular curve
X (d) is rational if and only if d  5. In particular, the nite group P SL(2; Z=d)
is a regular polyhedral group Gd  P GL(2) for d = 2; 3; 4 and 5. The compacti ed modular curve X (d) is identi ed with the circumscribing Riemann sphereof the regular polyhedron Pd with t vertices, where t is the number of cusps.
The order of Gd is equal to dt.

d2
345Pd triangle tetrahedron octahedron icosahedront
34612GdS3A4S4A5
Regular polyhedral groups are also closely related with Hilbert modular surfaces of small discriminant.
3 Supported in part by the JSPS Grant-in-Aid for Exploratory Research ? and ? (Germannside) ?.

1pExample Let Op5 be the ring of integers of the quadratic eld Q( 5) andput
p0 = SL(2; Op5 : 5) =a bc dpThen 0 acts on the product H 2 H of two copies of the upper half planes.
Let 0~ be the group generated by 0 and the switch involution of H 2 H . Then theHilbert modular surface Y0 := 0~ nH 2 H added with 6 point cusps is isomorphic
to the projective plane P2. This Y0 has an action of the icosdahedral group G5 .
Moreover, X0 := 0nH 2 H is the double cover of Y0 with branch a G5-invariant
plane curve of degree 10.
In the 3-dimensional case the wreath product 2kGd plays the role of Gd . Let
P1 2P1  P3 be the Segre embedding. The ambient space is the projectivizationof the space of 2 by 2 matrices and the quadric P1 2 P1 parametrizes therank one marices (modulo constant multiplication). Hence the complement isnaturally identi ed with P GL(2). This P3 is an equivariant compacti cationof the algebraic group P GL(2) and the polyhedral group Gd acts
 on itfromboth sides. Let  be the involution of this P3 interchanging ac db andd 0b . The xed locus is the union of the constant0 c a

a0matrices 0 a and the traceless ones ac 0ba .  interchanges the twofactors of P1 2 P1. So the bipolyhedral group 2kGd acts on P3 .
For a polarized abelian surface (X; L) of type (1; d), a (symplectic) isomorphism between the standard symplectic module 2[Z=d] and the groupits cofactor matrix

 12 mod 5; a; b; c; d; 2 Op5; ad 0 bc = 1 :K (L) := fx 2 X j Tx3 L ' Lg
with the Weil pairing is called a canonical level structure. By a canonical colevelstructure, we mean a (symplectic) isomorphism between 2[Z=d] and the quotientgroup Xd =K (L), which has a symplectic structure as the complement of K (L)in Xd . We denote the moduli space of polarized abelian surfaces with canonicallevel and colevel structure by A(1; d) and A(d; 1), respectively. The forgetful
morphisms
A(1; d) 0! Ad and A(d; 1) 0! Ad
are both Galois coverings with Galois group P SL(2; Z=d). The bre productAwbl
d := A(1; d) 2A A(d; 1)
is called the moduli space of abelian surfaces with a weak bilevel structure.
For a polarized abelian surface (X; L) of type (1; d), its dual X^ has a natural polarization L^ of the same type such that L^ L = dX . The colevel2^ L^ ) andstructure of (X; L) is equivalent to the level structure of its dual (X;wblvice versa. Therefore, the moduli space Ad has an action of wreath product2kPSL(2; Z=d).

Remark 1 The moduli space Ad is the quotient of the Sigel upper half spaceof degree 2 by the full paramodular group

10Z Z Z dZ
B dZ Z dZ dZ C
14 + BA \ Sp4(Q):@ Z Z Z dZ C
1ZZ
 K (L) and : 2[Z=d] !
 K (L^ ) isA pair ( ; ) of isomorphisms : 2[Z=d] !
called a canonical bilevel structure of (X; L). The moduli space Abld of polarizedabelian surfaces of type (1; d) with bilevel structure (X; L; ; ) is the quotientby the subgroup
ZdZ01dZ
dZ dZ
B dZ dZ dZ d2Z CC14 + B@ dZ
dZ dZ A \ Sp4 (Z):dZblThe moduli space Awbl
d is the quotient of Ad by the involution(X; L; ; ) 7! (X; L; ; 0 );which corresponds to the element

01
B01
B@101

1CC
A 2 Sp4 (Q):Theorem (1) For d = 2; 3 and 4, the Satake compacti cation of Awbl
d is(2kGd )-equivariantly isomorphic to the projective 3-space P(M222 C).
(2) There exists a (2kG5 )-equivariant morphismwbl~ 3 0! Ad:P

onto the Satake compacti caiton and contracts the strict transforms of the 72~ 3 is the blow-up of P3
special lines (see below) to the 72 point cusps, where P
with center G5 . (The normal bundles of the strict transforms are isomorphic
to O (04) 8 O(04).) is an isomorphism elsewhere. Moreover, the exceptional
divisors over the 60 points G5 are the Hilbert modular surface Y0 in Example andparametrize the Comesatti surfaces, i.e., abelian surfaces with real multiplication
by Op5 .

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(3) In both cases (1) and (2), P1 2 P1  P(M222 C) parametrizes theproducts ow two elliptic curves (of degree 1 and d).
Let p1 ; : : : ; pt be the cusps of the elliptic modular curve X (d), d = 2; 3; 4; 5.
Then by the theorem, the 2t lines pi 2 P1 and P1 2 pi , 1  i  t, on P1 2 P1 arewbl
wbl
1-dimensional (Satake) boundaries of Awbl
d . A2 and A3 are the complement
wbl , we need the following:of these 2t lines in P3. In order to describe Awbl
andA45De nition A line in P3 joining two points [g1 ] and [g2] of Gd  P GL(2) isspecial if g1 g201 2 Gd is of order d.
The number of special lines is iqual to 9, 16, 18 and 72 for d = 2; 3; 4 and 5.
In the case d = 2; 3, the special lines parametrize the polarized abeian surfaces(X; L) which have symplectic automorphism of order d.
Proposition (1) The moduli space Awbl
4 is the complement of 12 lines pi 2
P1 ; P1 2 pi and the 18 special lines in P3 .
(2) The moduli space Awbl
5 is the complement of the strict trransform of 12~ 3.
lines pi 2 P1 ; P1 2 pi and the 72 special lines in the blow-up P
Remark 2 Let K4 be the klein's subgroup of the octahedral group G4. The
action of K4 2 K4 on P3 is the projectivization of the Schrodinger representationof the Heisenberg group. Each of the 15 involutions in K4 2 K4 has the union oftwo skew lines as xed point locus. The 18 and 12 lines in (1) of the proposition
coincide with these 30 xed lines.
Acknowledgement This note is essentially [?]. But a confusion among thebilevel structure and the weak one is corrected. This was found in the discussionof the author with Prof. Klaus Hulek in his stay at Universitat Hannover in2003. He is very grateful for its hospitality.

References[1] Alu, P. and Faber, C.:Linear orbits of d-tuples of points in P1 , J. f. Reine
Angew. Math., 445 (1993), 205{220.
[2] Freitag, E.: Zur theorie der Modulformen zweiten Grades, Gottingen
Nachrichten, 11(1965).
[3] | : Modulformen zweiten Grades zum rarionalen und Gaussen Zhalkorper,Sitz. der Heidelberger Akad. Wiss. (1967), 3-49.
[4] Geer, G. van der : Hilbert modular surface, Springer Verlag.
4[5] Gritsenko, V.A. and Nikulin, V.V. : Automorphic forms and LorentzianKac-Moody Algebras, Part II, Internat. J. Math. 9(1998), 201-275.

p[6] Gundlach, K.B.: Funktionen zur Modulgruppe von Q( 5), Math. Ann.
152(1963), 226-256.
[7] Horrocks, G. and Mumford, D.: A rank 2 vector bundles on P4 with 15; 000
symmetries, Topology, 12 (1973), 63-81.
[8] Hulek, K., Kahn, C. and Weitraub, S.H. : Moduli Spaces of Abelian Surfaces:
Compacti cation, degenerations, and Theta Functions, de Gruyter, Berlin,
1993.
[9] Ibukiyama, T. and Onodera, F.: On the graded ring of modular forms ofthe Siegel paramodular group of level 2, Abh. Math. Sem. Univ. Hamburg
67(1977), 297-305.
[10] Igusa, J.: On Siegel modular forms of genus two, Amer. J. Math. 84(1962),
175-199.
[11] Mukai, S.: New developments in the theory of Fano 3-folds: Vector bundle
method and moduli problem, Sugaku, 47(1995), 125{144.: English translation, Sugaku Expositions, Amer. Math. Soc., ?(2003), ?{?.
[12] |: Moduli of abelian surfaces, and regular polyhedral groups, in `Moduli ofAlgebraic Varieties and The Monster', Proceedings, Sapporo, January 1999,
ed. I. Nakamura, Hokkaido Univ., 1999, pp. 68-74.
[13] | and Umemura, H.: Minimal rational threefolds, in Algebraic Geometry,
Proceedings, Tokyo/Kyoto 1982, Lecture Notes in Math. 1016, SpringerVerlag, 1983, pp. 490{518.
[14] Sarti, A.: Pencils of symmetric surfaces in P3 , J. Algebra, 246(2001), 429{452.
Research Institute for Mathematical Sciences
Kyoto University
Kitashirakawa Oiwake-cho, Sakyo-ku
Kyoto 606-8502, Japane-mail address : mukai@kurims.kyoto-u.ac.jp

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