\changing-invariants" trick. Shimura varieties of this kind are therefore much expected to possess rich structures involving the interplay between p-a

Transcription

1 Arithmetic structure of Mumford's fake projective plane 1 Fumiharu Kato We prove that Mumford's fake projective plane is a connected Shimura variety associated to a certain unitary group. We also describe explicitly the unitary group and the necessary data for dening this Shimura variety, and give a eld of denition. 1. Introduction In this paper we construct explicitly a Shimura surface of PEL-type whose connected components are, as complex surfaces, isomorphic to Mumford's fake projective plane. A fake projective plane is a non-singular projective surface of general type having the same topological Betti numbers as CP. The rst example of fake projective planes was discovered by Mumford in [Mu79], where he constructed it by using the theory of p-adic uniformization. Later in part of [IK98] Masanori Ishida and the author observed other possible fake projective planes according to Mumford's idea, and proved there exist (at least) two more dierent fake projective planes. As far as the author knows, these three are the only known fake projective planes. A fake projective plane has complex unit-ball as the topological universal covering, and in this sense it is among the most interesting classes of algebraic surfaces. However, very little is known about its algebraic or analytic structure (recently Barlow [Ba99] investigated zero-cycles on Mumford's fake projective plane in connection with Bloch conjecture). For instance, topological or even some algebro-geometric properties of them (e.g. fundamental groups, eld of denition, etc.) are quite obscure; even the arithmeticity of the fundamental groups seems unknown. The main reason for this is that the known fake projective planes are, a priori, constructed as surfaces over the eld of -adic numbers Q, not over C, and hence are not quite concrete as complex surfaces. (A fake projective plane over an arbitrary eld may be dened to be a smooth surface enjoying: c 1 = c = 9, p g = q = 0 and the canonical class is ample; if one considers this condition in the complex analytic category, this condition is equivalent to that the surface in question is a fake projective plane in the origial sense.) The theory of p-adic uniformization, on the other hand, involves surprizingly rich algebro-geometric and arithmetic aspects concerning with Shimura varieties. This was rst observed by Cerednik [ Ce76] and Drinfeld [Dr76] for Shimura curves. The generalizations to higher dimensions have been developed over the past two decades; recently, Rapoport-Zink [RZ96], Boutot-Zink [BZ95] and Varshavsky [Va98a][Va98b] gave powerfully extended generalities. These theories give a nice symmetric picture involving p-adic and complex uniformizations of certain Shimura varieties of PEL-type; viz. such a Shimura variety has both p-adic (for very special p) and complex analytic uniformizations, and they can be viewed symmetrically by means of Mathematics Subject Classication. Primary 14G5; Secondary 14J9, 11G15, 11Q5. 1

2 \changing-invariants" trick. Shimura varieties of this kind are therefore much expected to possess rich structures involving the interplay between p-adic and complex analysis. However, as far as the author knows, very few such examples are known in dimension two or high, in contract with numerous explicit examples of Shimura curves of this type (see, e.g. [Ku79][Ku94][An98]). Sketch of Results. In the next section we will construct explicitly a Shimura variety Sh C associated to a certain central division algebra over K = Q( p 07). The Shimura variety Sh C has the canonical model Sh C over K. We will apply the theory by Rapoport-Zink [RZ96], Boutot-Zink [BZ95] and Varshavsky [Va98a][Va98b] to this Shimura variety to show the following results (Theorem 4. and Corollary 4.4): Theorem. The canonical model Sh C of the Shimura variety Sh C has exactly three geometrically connected components dened over Q( 7 ) which are permuted by the action of Gal(Q( 7 )=Q( p 07)). Moreover, for any connected component S of Sh C there exists an isomorphism X Mum Spec Q Spec Q ( 7 ) 0! S Spec Q( 7) Spec Q ( 7 ) over Q ( 7 ), where X Mum is the Mumford's fake projective plane. Corollary. Each connected component of the complex surface Sh C is a fake projective plane, which has a eld of denition Q( 7 ). Moreover, it is an arithmetic quotient of complex unit-ball. The structure of this paper is as follows: In the next section we construct the Shimura variety Sh by presenting the necessary data explicitly. In x we review briey Mumford's construction of the fake projective plane. In the nal section x4 we investigate the relation between the fake projective plane and the Shimura variety Sh, and prove the main theorem. Notation and conventions. The notation A B (absence of the base ring) only occurs when A and B are Q-algebra, and the tensor is taken over Q. For a eld extension F=E, we denote by Res F=E the Weil restriction. For a ring A we denote by A op the opposite ring. By an involution of a ring A we always mean a homomorphism : A! A op such that = id A.. Construction of Shimura variety.1. In this section we construct a Shimura variety which will later be related to Mumford's fake projective plane. Our Shimura variety Sh is of PEL-type related to a certain unitary group. In order to dene it we introduce a central division algebra D over K, a positive involution of D of second kind, and a non-degenerate anti-hermitian form. These data yield the Shimura data (cf. [De71]) of our Shimura variety. Let = 7 be a primitive 7th-root of unity, and set L = Q(); L is a cyclic extension of Q of degree 6 having the intermediate quadratic extension K = Q() ( = Q( p 07)), where = The Galois group of L=K is generated by the

3 Frobenius map : 7!, and that of K=Q is generated by the complex conjugation z 7! z. Note that the prime decomposes on K such that = gives the prime factorization. The prime 7 ramies on K. We x an innite place (CM-type) ": K,! C by 7! 01+p 07. Note that the class number of K is 1. We set = =: Now the central division algebra D over K (of dimension 9) is dened by D = M i=0 L5 i ; 5 = ; 5z = z 5 (z L):.1.1. Lemma. The L-algebra D K L is isomorphic to the matrix algebra M (L). In particular, we have D " = M (C). This is well-known; for the later purpose we give an isomorphism 8: D K L! M (L) explicitly as follows: Set V = D, which we consider as a left D-module. Then we have V K L 0! M i=0 V L; i L: Looking at the action of D on the factor V L; 0 L subject to the base f1 1; 5 1; 5 1g we obtain a matrix presentation of elements in D determined by (:1:) z 7! 4 z z 5 and 5 7! 4 z which induces the desired isomorphism 8. By taking an embedding : L,! C by 7! exp i 7 we can extend 8 to an isomorphism D "! M (C)..1.. Lemma. We have inv D = 1= and inv D = 01=, while for any nite place ` which does not divide we have inv` D = ; This is clear by the construction. We dene the order O D of D by (:1:4) O D = O L 8 O L 5 8 O L 5 : It is easy to verify that O D = O D OK O K and O D = O D OK O K are maxial orders of D = D K K and D = D K K, respectively... Next we dene an involution (positive of second kind) on D by 5 = 5 and z = z (z L): It is easy to check that the involution is mapped by the isomorphism 8 as above to the standard involution (a ij ) = (a ji ). Set (::1) b = ( 0 ) :

4 The element b is anti-symmetric (i.e. b = 0b); we have (::) 8(b) = : The element b induces an involution, positive of second kind, F: D! D op F = b b 01 and a Q-bilinear form : V V! Q on V by by for x; y V. (x; y) = tr D=Q (ybx ).. Lemma. 1) The bilinear form is non-degenerate and anti-symmetric. Moreover we have (x; y) = (x; y) for D and x; y V. ) There exists P M (C) such that P 8(b)P = diag(0 p 01; 0 p 01; p 01). Proof. 1) Non-degeneracy is clear since b is invertible. The anti-symmetricity follows from the equality b = 0b. The other assertion is clear. The characteristic polynomial of 8(b) is t 0 p 07t 0 15t 0 p 07. By this ) follows immediately..4. Denition. We dene an algebraic group G over Q by G(R) = f (D R) j (x; y) = c() (x; y); c() R ; x; y V Rg = f (D R) j F Rg for any (commutative) Q-algebra R Lemma. The isomorphism 8: D R! M (C) dened by (.1.) induces an isomorphism 8: G R! GU(; 1). This is clear by Consider the homomorphism of algebraic groups over R (:5:1) h: Res C=R G m 0! G R such that 8h( p 01) = diag( p 01; p 01; 0 p 01) and 8h(r) is the scalar matrix r for r R. By [Ko9, 4.1] the pair (G R ; h) satises the conditions (1.5.1), (1.5.) and (1.5.) in [De71], and hence we get the associated Shimura variety; for a suciently small open compact subgroup C of G(A f ) the corresponding Shimura variety Sh C is dened by (:5:) Sh C = G(Q) n X 1 G(A f )=C; where X 1 is the set of all conjugates of h under the action of G(R). Note that X 1 is isomorphic to the open unit-ball in C. The Shimura variety Sh C is a nite disjoint union of quasi-projective manifolds, which are arithmetic quotients of the complex unit-ball. It is easy to see that the Shimura eld E coincides with "(K). 4

5 We therefore obtain the canonical model Sh C, which is a quasi-projective variety over E = Q( p 07).. Mumford's fake projective plane.1. In this section we briey review the Mumford's construction of a fake projective plane. We mainly focus on the construction of the discrete group 0 Mum which gives the uniformization of the fake projective plane. For more details the reader should consult Mumford's original paper [Mu79]. We use the notation as in.1. Regarding L as a -dimensional K-vector space we dene a hermitian form h: L L! Q by h(x; y) = tr L=K (xy) for x; y L. Subject to the basis 1; ; of L the form h is represented by the matrix (:1:1) H = 4 5 ; which is a positive denite hermitian matrix of determinant 7. exists a decomposition over Q (:1:) H = W W with W = : Note that there Let M (K) be the set of all matrices with entries in K, regarded as an algebra over Q. We denote by : M (K)! M (K) op the standard involution (a ij ) = (a ji ). We dene another involution (positive of second kind) y: M (K)! M (K) op by A y = HA H Denition. We dene an algebraic group I over Q by for any (commutative) Q-algebra R. I(R) = f (M (K) R) j y Rg.. Lemma. We have the following isomorphisms: 1) I(R) = GU(). In particular, I ad (R) is a compact Lie group. ) I(Q ) = f(x; y) GL (Q ) GL (Q ) op j xy Q g. Hence in particular we have I ad (Q ) = PGL (Q ). Proof. 1) is clear by denition. To prove ) we rst note the isomorphism M (K) Q = M (Q ) M (Q ) op such that the involution acts on the right-hand side by interchanging the factors, i.e. (x; y) = (y; x). We have (x; y) y = (HyH 01 ; H 01 xh). Then ) follows easily from the existence of the decomposition (.1.)..4. Let O K (resp. O L ) be the integer ring of K (resp. L). We consider O L as a free O K -module of rank. For a nite set S of prime numbers we denote by A S f 5

6 the prime-to-s part of the nite adele ring of Q, and set bz S = Ỳ 6S Z`: For S as above and a prime p we dene the maximal open compact subgroups C S max = f I(A S f ) j (O L Z b ZS ) O L Z b ZS g, C max p = f I(Q p ) j (O L Z Z p ) O L Z Z p g of I(A S f ) and I(Q p), respectively. We are going to dene an open compact subgroup C 7 of C7 max. Since the prime 7 ramies in K we have (:4:1) O L Z Z 7 = e Z e Z7 1 8 e Z7 1 ; where Z7 e is the ramied quadratic extension of Z 7. We can therefore consider the modulo p 07 reduction of C7 max, which takes values in GL (F 7 ). Now the matrix (H mod p 07) is of rank 1 and has dimensional null space, to which the action of C7 max can be restricted. We therefore obtain a homomorphism (:4:) $: C max 7 0! GL (F 7 ): We take a -Sylow subgroup P of the subgroup GL (F 7 ) T f j det = 61g and put C 7 = $ 01 (P )..5. Denition. Set C Mum = C ;7 max C 7, which is an open compact subgroup of I(A f ). We dene the subgroup 0 Mum of I ad (Q ) = PGL (Q ) to be the image of I(Q) T (I(Q ) C Mum) 0! I ad (Q ) induced by the rst projection, where the intersection on the left-hand side is taken in I(A f )..6. Theorem (Mumford [Mu79]). The subgroup 0 Mum is a uniform lattice in PGL (Q ) which acts transitively on the vertices of the Bruhat-Tits building attached to PGL (Q ). Moreover, the quotient (:6:1) X Mum = 0 Mum n Q of the Drinfeld symmetric space Q is algebraized to a fake projective plane X Mum. Proof. We need to show that the subgroup 0 Mum coincides with the one constructed by Mumford in [Mu79]. The Mumford's group is the image of I 1 (Q) T (I(Q ) C Mum) 0! I ad (Q ), where I 1 (Q) = f I(Q) j y = 1g. Hence it suce to see that every element I(Q) T (I(Q ) C Mum) can be written as = k with k Q and I 1 (Q) T (I(Q ) C Mum). Set k = ( y ) 01 det 6

7 and = k 01. Since ( mod p 07) P and (y mod p 07) is trivial we have (k mod p 07) P. Hence we have ( mod p 07) P. The property y = 1 follows from the fact that for any I(Q) we have det det = ( y ). 4. Relation between X Mum and Sh 4.1. Proposition. Let I and G be the Q-algebraic groups dened in. and.4, respectively. Then we have the following: (a) I is an inner form of G. (b) I(A f ) = G(A f ). Moreover, the algebraic group I is uniquely determined up to Q-isomorphisms by (a), (b) and the following condition: (c) I(Q ) = f(x; y) GL (Q ) GL (Q ) op j xy Q g. Proof. First we note that, since G(Q ) 6 = I(Q ), the condition (a), (b) and (c) imply that I ad (R) is a compact Lie group. By this the uniqueness of I will follow if we prove that the group G satises the Hasse principle; for this we refer the argument by Kottwitz [Ko9, x7]. Our situation belongs to the case (A) in the notation as in [Ko9, x5]. Kottwitz showed that validity of the Hasse principle for G depends only on that for the center Z(G) of G. In our case we have Z(G) = Res K=Q G m, which clearly satises the Hasse principle, since K is a quadratic extension of Q. Next we prove that I is an inner form of G. It suces to show that the Q-algebra with involution (M (K); y) is an inner form of (D;F). Set R = M (K) L; and consider the action of the Galois group Gal(L=Q) on R via the second factor. The invariant part R is isomorphic to M (K). Note that the Galois group Gal(L=Q) is generated by and complex conjugation (cf..1). Now we consider the matrix Q = 8(5) = M (K); and dene another Galois action 0 by 0 (complex conj.) = (complex conj.) and 0 ()(A z) = (Q 01 AQ) z for A M (K) and z L Claim. The invariant part R 0 is isomorphic to D. Looking at the isomorphism R = M (L) M (L) op such that (resp. complex conjugation) acts on M (L)M (L) op diagonally (resp. by interchanging the factors), we deduce that R is isomorphic to 0 D 0 = fa M (L) j Q 01 A Q = Ag: Obviously D 0 contains 8(D). Comparing Q-ranks we have D 0 = 8(D), thereby the claim Lemma. Let A be a ring. For u A with u u 01 Z(A) we dene the involution i u on the ring A A op by (x; y) iu = (uyu 01 ; u 01 xu) 7

8 for x; y A. Then the ring with involution (AA op ; i u ) is isomorphic to (AA op ; i 1 ). This is well-known. In fact, the conjugation by (1; u) gives the desired isomorphism. By the lemma we deduce in particular that the L-algebras with involution (R; i H ) and (R; i 8(b) ) are isomorphic to each other. Since H M (K), the involution i H commutes with the Galois action, and hence it induces an involution on R = M (K), which coincides with y. Also, since 8(b) M (K) and 8(b)Q = Q8(b), the involution i 8(b) commutes with the Galois action 0 and can be restricted to R 0 = D, which is nothing but F. We therefore deduce that (M (K); y) is an inner form of (D;F), thereby (a). Finally we prove (b). Let ` 6= be a rational prime. By.1. we know that D` = M (K) Q`. Hence it suces to show that the involutions y and F are Q`isomorphic to each other. If the prime ` splits in K, the assertion follows from Suppose that ` does not split in K. In order to show that y and F are isomorphic it suces to show that the hermitian forms H and H 0 = ( 0 )8(b) are isomorphic. We can do this by the well-known fact (cf. [MH7, App. ]): An isomorphism classe of hermitian forms on a xed vector space over a local eld F (with involution ) is determined by the residue class of determinant modulo N F=F 0(F ), where F 0 is the xed eld of. We know that det H = 7 and det H 0 = 7. If ` 6= 7 then K`=Q` is an unramied quadratic extension. Since both det H and det H 0 are unit integer and since every unit integer of Q` is a norm of an element in the unramied quadratic extension, we have the desired result in this case. If ` = 7 then K 7 = Q 7 ( p 07), and hence both det H and det H 0 are obviously norms, thereby the result. Now we have proved (b), and hence the proof of the proposition is nished. 4.. We x an isomorphism ': I(A f ) = G(A f ) as in 4.1 (b). Let Cmax G(Q ) be a maximal open compact subgroup dened by C max = f G(Q ) j (O D Z Z ) O D Z Z g: We dene an open compact subgroup C G(A f ) by (4::1) C = C max '(C Mum) (cf..5). Let us x an embedding : Q,! Q. Let E be the -adic completion of the Shimura eld E (= "(K)), and E the maximal unramied extension of E. By [RZ96, 6.51] (see also [BZ95, 1.1][Va98a][Va98b]) and the uniqueness of I as in 4.1 we have an isomorphism of formal schemes (4::) I(Q)n(b Q Spf Z Spf O E ) G(A f )=C 0! Sh^C Spf O E Spf O E ; where Sh^C is the formal completion of a model of Sh C over Spec O E along the closed ber. 4.. Theorem. The canonical model Sh C of the Shimura variety Sh C has exactly three geometrically connected components dened over Q( 7 ) which are permuted 8

9 by the action of Gal(Q( 7 )=Q( p 07)). Moreover, for any connected component S of Sh C there exists an isomorphism of schemes (4::1) X Mum Spec Q Spec Q ( 7 ) 0! S Spec Q( 7) Spec Q ( 7 ) over Q ( 7 ), where X Mum is the Mumford's fake projective plane (cf..6). Proof. First we note that by [RZ96, 6.51] we can take the isomorphism as in (4..) such that the natural descent datum on the right-hand side induces on the left-hand side the natural descent datum on the rst factor multiplied with the action of (5; 5 01 ) G(Q ) D D op on G(A f )=C. Since 5 O D and 5 01 = 5 O D we have (5; 5 01 ) C max. Therefore the action of (5; 5 01 ) is trivial on G(A f )=C. Hence the isomorphism (4..) descends to an isomorphism (4::) I(Q)nb Q G(A f)=c 0! Sh^C over Z = OE. Next we note that the left-hand side of (4..) is the disjoint union of formal schemes of form 0 n(b Q Spf Z Spf O E ); T where 0 = the image of I(Q) (I(Q ) (CMum 01 )) in I ad (Q) for G(A f ). Note also that 0 1 = 0 Mum (cf..5). By these observations it suces to prove that the canonical model Sh C of the Shimura variety has three connected components as in the theorem. To do this we consider the torus T over Q dened by T = f(k; f) Res K=Q G m;k G m;q j kk = f g = Res K=Q G m;k and dene an epimorphism #: G! T by 7! (N D op ; c()), where =K N D op is the =K reduced norm (the last = is given by (k; f) 7! kf 01 ). The kernel of # is the derived group of G. By [De71, x] we know that the morphism # induces the canonical morphism of Shimura varieties Sh C = G(Q) n X 1 G(A f )=C 0! T (Q) + n T (A f )=#(C) which induces the Galois-equivariant bijection between the sets of all connected components of canonical models of each side. To calculate #(C) we have only to look at the component '(C 7 ) at Claim. Under the canonical identication T (Q 7 ) = e Q 7, where e Q 7 is the ramied quadratic extension of Q 7, we have where e Z 7 is the integer ring of e Q 7. #('(C 7 )) = 61 + p 07 e Z 7 ; 9

10 First we note that the map # ' is described by # ': C 7 7! ( y ) 01 det e Z 7 : Since Z 7 C 7 we immediately deduce that #('(C 7 )) contains Z 7. Moreover, since 61 + p 07 Z e 7 C 7, #('(C 7 )) contains the element k k 01 for any k 61 + p 07 Z e 7 C 7 ; but, since kk #('(C 7 )), we deduce that (61 + p 07 e Z 7 ) = 61 + p 07 e Z 7 #('(C 7)): Hence the claim follows if we check that (#('(C 7 )) mod p 07) = f61g. By an argument similar to that in the proof of.6 any C 7 has a decomposition = k in C 7 with y = 1. Since det det y (det ) 1 mod p 07 we have #('()) 6k. But, since k = 01 mod p 07 belongs to P, the order of k is either one or two; hence k 61 mod p 07. We therefore get #('()) 61 mod p 07 for any C 7, which proves the claim. Now in view of the theory of complex multiplication we nd that the Shimura variety T (Q) + nt (A f )=#(C) has three connected components having eld of denition L = Q() which are permuted by Gal(L=K), since the ray class eld corresponding to C is easily checked to be L. We therefore proved the all statements in the theorem Corollary. Each connected component of the complex surface Sh C is a fake projective plane, which has a eld of denition Q( 7 ). Moreover, it is an arithmetic quotient of complex unit-ball. Acknowledgments. The author's viewpoint of this work owes much to suggestions by Professor Yves Andre. The author expresses gratitude to him for valuable discussions. The author thanks Professor Yoichi Miyaoka for his interest in this work and several valuable suggestions on the manuscript. The author thanks Max- Planck-Institut fur Mathematik Bonn for the nice hospitality. References [An98] Andre, Y.: p-adic orbifolds and p-adic triangle groups, RIMS Kyoto proceedings (Surikaisekikenkyusho Kokyuroku) No. 107, proceedings of the conference \rigid geometry and group action" Kyoto, December 1998, preprint, Institut de Math. Jussieu (1998). [Ba99] Barlow, R.N.: Zero-cycles on Mumford's surface, Math. Proc. Cambridge Philos. Soc. 16 (1999), no., 505{510. [BZ95] Boutot, J-F., Zink, Th.: The p-adic Uniformization of Shimura Curves, preprint 95{107, Universitat Bielefeld (1995). [ Ce76] Cerednik, I.V.: Uniformization of algebraic curves by discrete arithmetic subgroups of PGL (k w ) with compact quotient spaces, Mat. Sd., 100 (1976), 59{88. 10

11 [De71] Deligne, P.: Travaux de Shimura, Sem. Bourbaki Expose 89, Lect. Notes in Math. 44 (1971), Springer, Heidelberg. [Dr76] Drinfeld, V.G.: Coverings of p-adic symmetric regions, Funct. Anal. and Appl. 10 (1976), 9{40. [IK98] Ishida, M., Kato, F.: The strong rigidity theorem for non-archimedean uniformization, T^ohoku Math. J., 50 (1998), 57{555. [Ko9] Kottwitz, R.E.: Points on some Shimura varieties over nite elds, Journal Amer. Math. Soc. 5 (199), 7{444. [Ku79] Kurihara, A.: On some examples of equations dening Shimura curves and the Mumford uniformization, J. Fac. Sci. Uni. Tokyo, Sec. IA 5 (1979), 77{00. [Ku94] Kurihara, A.: On p-adic Poincare series and Shimura curves, Internat. J. Math. 5 (1994), 747{76. [MH7] Milnor, J., Husemoller, D.: Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 7, Springer-Verlag, Berlin, Heidelberg, New York (197). [Mu79] Mumford, D.: An algebraic surface with K ample, (K ) = 9; pg = q = 0, in Contribution to Algebraic Geometry, Johns Hopkins Univ. Press (1979), {44. [RZ96] Rapoport, M., Zink, Th.: Period Spaces for p-divisible Groups, Annals of Math. Studies 141 Princeton University Press (1997). [Va98a] Varshavsky, Y.: p-adic uniformation of unitary Shimura varieties, Inst. Hautes Etudes Sci. Publ. Math. 87 (1998), 57{119. [Va98b] Varshavsky, Y.: p-adic uniformization of unitary Shimura varieties. II, J. Dierential Geom. 49 (1998), no. 1, 75{11. Graduate School of Mathematics, Kyushu University, Hakozaki Higashi-ku, Fukuoka , Japan. fkato@math.kyushu-u.ac.jp 11