ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS. Gopal Prasad and Sai-Kee Yeung

Transcription

1 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS Gopa Prasad and Sai-Kee Yeung Dedicated to Robert P. Langands on his 70th birthday 1. Introduction Let n be an integer > 1. A compact Käher manifod of dimension n 1 is caed a fae projective space, or a fae P n 1 C, if it is not isomorphic to Pn 1 C but it has the same Betti numbers as P n 1 C. We wi ca a fae projective space of dimension n 1 an arithmetic fae projective space, or an arithmetic fae P n 1 C, if it is the quotient of the open unit ba B n 1 in C n 1 by a torsion-free cocompact arithmetic subgroup of PU(n 1, 1). Note that B n 1 is the symmetric space of PU(n 1, 1), P n 1 C is its compact dua, and the Euer-Poincaré characteristic of P n 1 C, and so aso of any arithmetic fae P n 1 C, is n. We observe that the Grassmann space Gr m,n of m-dimensiona subspaces in C n is the compact dua of the symmetric space of the group PU(n m, m), and so we wi ca the quotient of the symmetric space of PU(n m, m) by a cocompact torsion-free arithmetic subgroup (of PU(n m, m)), whose Betti numbers are same as that of Gr m,n, an arithmetic fae Gr m.n. It is an immediate consequence of the Hirzebruch proportionaity principe, see [Se1], Proposition 23, that the orbifod Euer-Poincaré characteristic (i.e., the Euer- Poincaré characteristic in the sense of C.T.C. Wa, see [Se1]) of any cocompact discrete subgroup of PU(n 1, 1), for n even, is negative. This impies that if there exists an arithmetic fae P n 1 C, then n is necessariy odd. The purpose of this paper is to determine a irreducibe cocompact torsion-free arithmetic subgroups Γ of a product G of r groups of the form PU(n m, m), m < n, n > 3 odd, whose Euer- Poincaré characteristic χ(γ) is equa to the Euer-Poincaré characteristic χ(x u ) of the compact dua X u of the symmetric space X of G. Let Γ be an irreducibe cocompact torsion-free arithmetic subgroup of G with χ(γ) = χ(x u ). Let G be the connected semi-simpe Lie group obtained by repacing each of the r factors PU(n m, m) of G by SU(n m, m). As the erne of the natura surjective homomorphism G G is a group of order n r, if Γ is the fu inverse image of Γ in G, then Γ is an arithmetic subgroup whose orbifod Euer-Poincaré characteristic is χ(x u )/n r. Therefore, the orbifod Euer-Poincaré characteristic of any arithmetic subgroup of G, which contains Γ, is a submutipe of χ(x u )/n r. Assume, if possibe, that G contains an irreducibe maxima arithmetic subgroup 1

2 2 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS Γ whose orbifod Euer-Poincaré characteristic χ(γ) is a submutipe of χ(x u )/n r. As Γ is an irreducibe maxima arithmetic subgroup of G, there exist a totay rea number fied, an absoutey simpe simpy connected group G defined over, r archimedean paces of, say v j, j = 1,..., r, such that G = r G( v j ), and for a other archimedean paces v of, G( v ) is isomorphic to the compact rea Lie group SU(n), and a principa arithmetic subgroup Λ of G() such that Γ is the normaizer of Λ in G (we identify G with r G( v j ) and use this identification to reaize G() as a subgroup of G), see Proposition 1.4(iv) of [BP]. From the description of absoutey simpe groups of type 2 A n 1 (see, for exampe, [T1]), we now that there exists a quadratic extension of, a division agebra D with center and of degree s = [D : ], s n, D given with an invoution σ of the second ind such that = {x x = σ(x)}, and a nondegenerate hermitian form h on D n/s defined in terms of the invoution σ so that G is the specia unitary group SU(h) of h. It is obvious that is totay compex. In terms of the normaized Haar-measure µ on G = r G( v j ) used in [P] and [BP], and to be used in this paper, χ(γ) = χ(x u )µ(g/γ) (see [BP], 4.2). Thus the condition that χ(γ) is a submutipe of χ(x u )/n r is equivaent to the condition that µ(g/γ) is a submutipe of 1/n r. We sha prove that if n > 7, there does not exist an arithmetic subgroup whose covoume is 1/n r, and if n = 5 or 7, there does not exist an arithmetic subgroup whose covoume is a submutipe of 1/n r. The main resut of this paper impies that arithmetic fae P n 1 C can exist ony if n = 3 or 5, and an arithmetic fae Gr m,n exists, with n > 3 odd, ony if n = 5. The first fae projective pane was constructed by David Mumford in [M] using p-adic uniformization. In [PY] we have constructed seventeen distinct (finite) casses of arithmetic fae projective panes, and have proved that there can exist at most four more. In 5 of this paper we have constructed four distinct 4-dimensiona arithmetic fae projective spaces and four distinct fae Gr 2,5. We wi aso use certain resuts and computations of [PY] to exhibit five smooth compex projective varieties ( fae P 2 C P2 C ) which are not isomorphic to, but have the same Betti numbers as, P 2 C P2 C, and which are not isomorphic to the product of a fae projective pane with either P 2 C or with a fae projective pane. A! these are connected smooth (compex projective) Shimura varieties, and these are the first exampes of fae P 4 C, fae Grassmannians, and irreducibe fae P 2 C P2 C. It was proved by Bruno Kinger and the second author independenty that any fae projective pane is arithmetic. It has just been shown by the second author in [Y] that any fae projective space of dimension 4 is arithmetic. According to Proposition 5 of this paper, the first integra homoogy group of an arithmetic fae P 4 C is aways nonzero. This eads to the foowing very interesting resut: A compact Käher manifod of dimension 4 is isomorphic to P 4 C if it has the same integra homoogy as P 4 C. 2. Preiminaries

3 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS 3 A comprehensive survey of the basic definitions and the main resuts of the Bruhat Tits theory of reductive groups over nonarchimedean oca fieds is given in [T2] Throughout this paper we wi use the notations introduced in the introduction. n wi aways be an odd integer > 3. A unexpained notations are as in [BP] and [P]. Thus for a number fied K, D K denotes the absoute vaue of its discriminant, h K its cass number, i.e., the order of its cass group C(K). We sha denote by h K,n the order of the subgroup (of C(K)) consisting of the eements of order dividing n. Then h K,n h K. We sha denote by U K the mutipicative-group of units of K, and by K n the subgroup of K consisting of the eements x such that for every normaized vauation v of K, v(x) nz. We wi denote the degree [ : Q] of by d, V f (resp. V ) wi denote the set of nonarchimedean (resp. archimedean) paces of. As admits at east r distinct rea paces, see the introduction, d r. For v V f, q v wi denote the cardinaity of the residue fied f v of v. For a parahoric subgroup P v of G( v ), we define e(p v ) by the foowing formua (cf. Theorem 3.7 of [P]): (1) e(p v ) = If v spits in, et and if v does not spit in, then or M v+dim Mv)/2 q(dim v. #M v (f v ) n 1 e (P v ) = e(p v ) (1 1 (n 1)/2 e (P v ) = e(p v ) (1 1 qv 2j qv j+1 ), )(1 + 1 (n 1)/2 e (P v ) = e(p v ) (1 1 qv 2j qv 2j+1 according as v does not or does ramify in. It can be seen that for a v V f, e (P v ) is an integer. It is obvious that e (P v ) < e(p v ) We note that if P v is a hyperspecia parahoric subgroup of G( v ), then the f v -group M v, which in this case is just the reduction mod p of P v, is either SL n or SU n according as v does or does not spit in, and M v = M v. If v ramifies in, then G is quasi-spit over v, and if P v is specia, then M v is isogenous to either SO n or Sp n 1, and so is M v. Now by a direct computation we find that e (P v ) = 1 if either P v is hyperspecia, or v ramifies in and P v is specia Let v be a nonarchimedean pace of which spits in and G spits at v. Then M v is f v -isomorphic to SL n. It can be seen by a direct computation that for any ) ),

4 4 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS nonhyperspecia parahoric subgroup P v of G( v ), e (P v ) is an integer greater than n. Let now v be a nonarchimedean pace of which spits in but G does not spit at v. Then v D = ( v ) D = M n/dv (D v ) σ(m n/dv (D v )), where D v is a division agebra with center v, of degree d v > 1, d v n. Hence, G is v - isomorphic to SL n/dv,d v. Let P v be a maxima parahoric subgroup of G( v ). Then M v is f v -isomorphic to SL n, and M v is isogenous to the product of the norm-1 torus R (1) F v/f v (GL 1 ) and the semi-simpe group R Fv/f v (SL n/dv ), where F v is the fied extension of f v of degree d v. So and hence, /d v n/d v #M v (f v ) = qn2 v q v 1 e (P v ) = q n2 (d v 1)/2d v v n/d v ( 1 1 ( 1 1 q jdv v q jdv v ), ) n 1 ( 1 ) 1 n = (qj v 1) n/dv (qjdv v 1) > v 1)/2d q(n2 2n)(d v v > n. It is obvious that for any parahoric subgroup P v contained in P v, e (P v ) is an integra mutipe of e (P v ) Let Γ be a maxima arithmetic subgroup of G = r G( v j ) such that n r µ(g/γ) 1, see the introduction. Let Λ = Γ G(). Then Γ is the normaizer of Λ in G, and Λ is a principa arithmetic subgroup (see [BP], Proposition 1.4(iv)), i.e., if for a nonarchimedean pace v of, P v is the cosure of Λ in G( v ), then P v is a parahoric subgroup, and Λ = G() v V f P v. Let T be the set of v ( V f ) which spit in and P v is not a hyperspecia parahoric subgroup of G( v ). Let T be the set of v ( V f ) which do not spit in, and either P v is not a hyperspecia parahoric subgroup of G( v ) but a hyperspecia parahoric exists (which is the case if, and ony if, v is unramified over ), or v is ramified in and P v is not a specia parahoric subgroup Let µ n be the erne of the endomorphism x x n of GL 1. Then the center C of G is -isomorphic to the erne of the norm map N / from the agebraic group R / (µ n ), obtained from µ n by Wei s restriction of scaars, to µ n. As n is odd, the norm map N / : µ n () µ n () is onto, µ n ()/N / (µ n ()) is trivia, and hence, the Gaois cohomoogy group H 1 (, C) is isomorphic to the erne of the homomorphism / n / n induced by the norm map. We sha denote this erne by ( / n ) in the seque. By Dirichet s unit theorem, U = {±1} Z d 1, and U = µ() Z d 1, where µ() is the finite cycic group of roots of unity in. Hence, U /U n = (Z/nZ) d 1, and U /U n = µ() n (Z/nZ) d 1, where µ() n is the group of n-th roots of unity in. Now we observe that N / (U ) N / (U ) = U 2, which impies that, as n is odd, q j v

5 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS 5 the homomorphism U /U n U /U n, induced by the norm map, is onto. Therefore, the order of the erne (U /U n) of this homomorphism equas #µ() n. The short exact sequence (4) in the proof of Proposition 0.12 of [BP] gives us the foowing exact sequence: 1 (U /U n ) ( n / n ) (P I n )/P n, where ( n / n ) = ( n / n ) ( / n ), P is the group of a fractiona principa ideas of, and I the group of a fractiona ideas (we use mutipicative notation for the group operation in both I and P). Since the order of the ast group of the above exact sequence is h,n, see (5) in the proof of Proposition 0.12 of [BP], we concude that #( n / n ) #µ() n h,n. Now we note that the order of the first term of the short exact sequence of Proposition 2.9 of [BP], for G = G and S = V, is n r /#µ() n. Using the above observations, together with Proposition 2.9 and Lemma 5.4 of [BP], and a cose oo at the arguments in 5.3 and 5.5 of [BP] for S = V and G of type 2 A n 1, we can derive the foowing upper bound: (2) [Γ : Λ] n r+#t h,n. From this we obtain (3) 1 n r µ(g/γ) µ(g/λ) n #T h,n Now we wi use the voume formua of [P] to write down the precise vaue of µ(g/λ). As the Tamagawa number τ (G) of G equas 1, Theorem 3.7 of [P] (recaed in 3.7 of [BP]), for S = V, gives us for n odd, (4) µ(g/λ) = D (n2 1)/2 where E = v V f e(p v ), with e(p v ) as in 2.1. n 1 (D /D 2 )(n 1)(n+2)/4( ) de, 2.7. Let ζ be the Dedeind zeta-function of, and L be the Hece L-function associated to the quadratic Dirichet character of /. Then ζ (j) = (1 1 v V f qv j ) 1 ; L (j) = 1 (1 ) 1 1 (1 + ) 1, q j v where is the product over those nonarchimedean paces of which spit in, and is the product over a the other nonarchimedean paces v which do not ramify in. Hence the Euer product E appearing in (4) can be rewritten as E = (n 1)/2 ( ζ (2j)L (2j + 1) ) q j v v V f e (P v ).

6 6 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS Since e (P v ) = 1, if v / T T (see 2.2), E = (n 1)/ Using the functiona equations and we find that (5) D (n2 1)/2 ζ (2j) = D 1 2 2j ( ζ (2j)L (2j + 1) ) v T T e (P v ). (( 1) j 2 2j 1 π 2j ) dζ (1 2j), (2j 1)! L (2j + 1) = ( D ) 2j+ 1 ( ( 1) j 2 2j π 2j+1 ) 2 dl ( 2j), D (2j)! n 1 (D /D 2 )(n 1)(n+2)/4( ) d (n 1)/2 ( ζ (2j)L (2j + 1) ) = R := 2 (n 1)d ζ ( 1)L ( 2)ζ ( 3)L ( 4) ζ (2 n)l (1 n) We assume here that the orbifod Euer-Poincaré characteristic χ(γ) of Γ is a submutipe of χ(x u )/n r, see the introduction. Then we have the foowing as χ(λ) = χ(x u )µ(g/λ), χ(γ) = χ(λ) [Γ : Λ] = χ(x u)µ(g/λ). [Γ : Λ] On the other hand, Proposition 2.9 of [BP] appied to G = G and Γ = Γ, impies that any prime divisor of the integer [Γ : Λ] divides n. So we concude that any prime which divides the numerator of the rationa number µ(g/λ) is a divisor of n. It is easy to see, using the voume formua of [P], Theorem 3.7, and (5), that µ(g/λ) is an integra mutipe of R; the atter as in (5). Hence we obtain the foowing proposition. Proposition 1. If the orbifod Euer-Poincaré characteristic of Γ is a submutipe of χ(x u ), then any prime divisor of the numerator of the rationa number R divides n We now (cf. [P], Proposition 2.10(iv), and 2.3 above) that (6) for a v V f, e(p v ) > 1, and for a v T, e(p v ) > n. Now combining (3), (4) and (6), we obtain (7) 1 n r µ(g/γ) > D (n 1)(n+2)/4 D (n 1)/2 h,n ( n 1 ) d. It foows from Brauer-Siege Theorem that for a rea s > 1, (8) h R w s(s 1)Γ(s) d ((2π) 2d D ) s/2 ζ (s),

7 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS 7 where h is the cass number and R is the reguator of, and w is the order of the finite group of roots of unity contained in. Using the ower bound R 0.02w e 0.1d due to R. Zimmert [Z], we get (9) h,n h s(s 1) Now from bound (7) we obtain (10) 1 > D (n 1)(n+2)/4 D (n 1)/2 D s/2 ζ (s) 0.02 s(s 1) ((2π) s e 0.1 Γ(s) ((2π) s e 0.1 Γ(s) ) d 1 D s/2 ζ (s). ) ( n 1 d ) d. Letting s = 1 + δ, with δ in the interva [1, 10], and using D D 2, and the obvious bound ζ (1 + δ) ζ(1 + δ) 2d, we get (11) D 1/d D 1/2d < [ n 1 Γ(1 + δ)ζ(1 + δ)2 { (2π) 1+δ e 0.1 We wi now prove the foowing simpe emma. Lemma 1. For every integer j 2, ζ (j) 1/2 L (j + 1) > 1. } {50δ(1+δ)} 1/d] 2/(n 2 2δ 3). Proof. The emma foows from the product formua for ζ (j) and L (j + 1) and the foowing observation. For any positive integer q 2, (1 1 q j )(1 + 1 q j+1 )2 = 1 q 2 2q 1 qj+1 q 2j+2 1 < 1. q3j+2 The above emma impies that for every integer j 2, ζ (j)l (j+1) > ζ (j) 1/2 > 1. Aso we have the foowing obvious bounds for any number fied of degree d over Q, where, as usua, ζ(j) denotes ζ Q (j). For every positive integer j, From this we obtain the foowing: Lemma 2. Let E 0 = (n 1)/2 1 < ζ(dj) ζ (j) ζ(j) d. ( ζ (2j)L (2j+1) ). Then E 0 > E 0 := (n 1)/2 ζ(2dj) 1/ To find restrictions on n and d, we wi use the foowing three bounds for the reative discriminant D /D 2 obtained from bounds (3), (4), (8) and (9), and Lemma 2. (12) D /D 2 < p 1(n, d, D, δ) := [ 50δ(1 + δ) Γ(1 + δ)ζ(1 + δ)2 { E 0 D (n2 2δ 3)/2 (2π) 1+δ e 0.1 n 1 (13) D /D 2 < p 2(n, d, D, R /w, δ) } d] 4/(n 2 +n 2δ 4).

8 8 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS := [ δ(1 + δ) Γ(1 + δ)ζ(1 + δ)2 { (R /w )E 0 D (n2 2δ 3)/2 (2π) 1+δ (14) D /D 2 < p 3(n, d, D, h,n ) := [ n 1 h,n { E 0 n 1 } d D (n2 1)/2] 4/(n 1)(n+2). } d] 4/(n 2 +n 2δ 4). Simiary, from bounds (3), (4), (8), and Lemma 2 we obtain the foowing: (15) D 1/d D 1/2d < ϕ(n, d, R /w, δ) := [ Γ(1 + δ)ζ(1 + δ)2 { (2π) 1+δ n 1 δ(1 + δ) } { } 1/d] 2/(n2 2δ 3). (R /w )E 0 3. Determining 3.1. We define M r (d) = min K D 1/d K, where the minimum is taen over a totay rea number fieds K of degree d. Simiary, we define M c (d) = min K D 1/d K, by taing the minimum over a totay compex number fieds K of degree d. The precise vaues of M r (d), M c (d) for ow vaues of d are given in the foowing tabe (cf. [N]). d M r (d) d M c (d) d We aso need the foowing proposition which provides ower bounds for the discriminant of a totay rea number fied in terms its degree. Proposition 2. (a) Let be a totay rea number fied of degree d, Q. Then D 1/d 5 > (b) If d 3, then D 1/d 49 1/3 > (c) If d 4, then D 1/d 725 1/4 > (d) If d 5, then D 1/d /5 > 6.8. Proof. Let g(x, d) and x 0 be as in 6.2 of [PY]. Let N(d) = im sup x x0 g(x, d). It has been observed in [PY], Lemma 6.3, that N(d) is an increasing function of d, and it foows from the estimates of Odyzo [O] that M r (d) N(d). We see by a direct computation that g(2, 9) > 9.1. Hence, M r (d) N(d) N(9) g(2, 9) > 9.1, for a d 9. For 1 d 7, the expicit vaues of M r (d) isted above satisfies M r (d) M r (d + 1). (a) (d) now foow from the expicit vaues of M r (d), for d 8, and the above bound for M r (d) for d 9.

9 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS We note here for atter use that except for the totay compex sextic fieds with discriminants 9747, 10051, 10571, 10816, 11691, 12167, and the totay compex quartic fieds with discriminants 117, 125, 144, R /w > 1/8 for every number fied, see [F], Theorem B. For r 2 = d = 2, we have the unconditiona bound R /w , see Theorem B and Tabe 3 in [F] For d a positive integer, n an odd positive integer, and δ 0.02, denote by f(n, d, δ) the expression on the extreme right of bounds (11) i.e., f(n, d, δ) = [ n 1 Γ(1 + δ)ζ(1 + δ)2 { (2π) 1+δ e 0.1 } {50δ(1 + δ)} 1/d] 2/(n 2 2δ 3). For fixed n and δ, f(n, d, δ) ceary decreases as d increases. Furthermore, for a given d and δ, f(n, d, δ) decreases as n increases provided n a, where a is any positive integer such that (a 1)! > (2π) a. It is easy to see that (a 1)! > (2π) a for a a 19. So we concude that f(n, d, δ) is a decreasing function of d for fixed n and δ, and a decreasing function of n for n 19 if d and δ are fixed. We obtain by a direct computation the foowing upper bound for the vaue of f(n, 2, 3) for sma n. n f(n, 2, 3) < From the above tabe, Proposition 2, and the fact that f(n, 2, 3) is decreasing in n for n 19, we concude that if n 13, then d = 1, i.e., = Q Now we wi investigate the restriction on the degree d of for n 11 imposed by bound (11). We get the foowing tabe by evauating f(n, d, δ), with n given in the first coumn, d given in the second coumn, and δ given in the third coumn n d δ f(n, d, δ) < Taing into account the upper bound in the ast coumn of the above tabe, Proposition 2 impies the foowing: If n = 11, d 2.

10 10 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS If n = 9, d 2. If n = 7, d 3. If n = 5, d 4. We wi now prove the foowing theorem by a case-by-case anaysis. Theorem 1. If n > 7 and the orbifod Euer-Poincaré characteristic of Γ is χ(x u )/n r, then d = 1, i.e., = Q. If n = 7 or 5, and the orbifod Euer-Poincaré characteristic of Γ is a submutipe of χ(x u )/n r, then again = Q. Proof. (i) First of a, we wi show that if n = 11, then d cannot be 2. A direct computation shows that f(11, 2, 1.8) < 2.6. Hence, if n = 11 and d = 2, then D < < 46. However, from the tabe in 3.1, we see that the smaest discriminant of a totay compex quartic is 117. So we concude that if n = 11, then d = 1. (ii) Let us now consider the case n = 9. We wi rue out the possibiity that d = 2 using bound (15). Note that we can use the ower bound R /w , see 3.2. We see by a direct computation that ϕ(9, 2, , 1.5) 4 < 97. Hence, D < 97 from bound (15). As M c (4) 4 = 117, d = 2 cannot occur. Hence, if n = 9, then d = 1. (iii) We now consider the case n = 7. We need to rue out the possibiities that d is either 3 or 2. We see from a direct computation that f(7, 2, 1.2) < 4.3 and f(7, 3, 1.4) < 4.14, where f(n, d, δ) is as in 3.3. Consider first the case where d = 3 (and n = 7). As D 1/6 < f(7, 3, 1.4) < 4.14, D < < This eads to a contradiction since according to the tabe in 3.1, a ower bound for the absoute vaue of the discriminant of a totay compex sextic fieds is Hence, it is impossibe to have d = 3 if n = 7. Consider now the case where n = 7 and d = 2. As mentioned above, f(7, 2, 1.2) < 4.3, and hence, D 1/2 D 1/4 < f(7, 2, 1.2) < 4.3. It foows that D < < There are then the foowing five cases to discuss. (a) D = 5, = Q( 5) (b) D = 8, = Q( 2) (c) D = 12, = Q( 3) (d) D = 13, = Q( 13) (e) D = 17, = Q( 17). Case (e): We wi use bound (13). As R /w (see 3.2), D /D 2 < p 2(7, 2, 17, , 1.26) < 1.1, which impies that D = D 2 = 172. From the tabe of totay compex quartics in [1], we find that there does not exist a totay compex quartic with discriminant Case (d): D /D 2 < 4.34 /13 2 < 2.1. Hence, D /D 2 = 1 or 2. So D = 169 or 338. From the tabe of totay compex quartics in [1], we see that neither of these two numbers occurs as the discriminant of such a fied. Therefore we concude that case (d) does not occur.

11 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS 11 Case (c): D /D 2 < 4.34 /12 2 < 2.4. Hence D /D 2 = 1 or 2 and D = 144 or 288. Again, from the tabe of totay compex quartics in [1], we now that there is no compex quartic with discriminant 288. Moreover, there is a unique totay compex quartic, namey = Q[x]/(x 4 x 2 +1) = Q( 1, 3), whose discriminant equas 144. It ceary contains = Q( 3). We wi now eiminate this case using Proposition 1 (whenever we use Proposition 1 in the seque, we wi assume that the orbifod Euer-Poincaré characteristic of Γ is a submutipe of χ(x u )/n r ). In this case, we have the foowing data. ζ ( 1) = 1/6, ζ ( 3) = 23/60, ζ ( 5) = 1681/126, L ( 2) = 1/9, L ( 4) = 5/3, L ( 6) = 427/3. (Observe that for a positive integer j, ζ ( (2j 1)) and L ( 2j) are rationa numbers according to we-nown resuts of Siege and Kingen. The denominators of these rationa numbers can be estimated. In this paper, we have used the software PARI together with their functiona equations to obtain the actua vaues of the Dedeind zeta and Hece L-functions. These vaues have been recheced using MAGMA. This software provides precision up to more than 40 decima paces!) Therefore, µ(g( vo )/Λ) is an integra mutipe of 2 12 ζ ( 1)L ( 2)ζ ( 3)L ( 4)ζ ( 5)L ( 6) = / As the numerator of this number is not a power of 7, according to Proposition 1 this case cannot occur. Case (b): D /D 2 < 4.34 /8 2 < 5.4. Hence, D /D 2 = c and D = 64c, where c is a positive integer 5. As D M c (4) 4 117, the possibe vaues of D are 128, 192, 256, 320. According to the tabes in [1], the ony possibiities are: D = 256: is obtained by adjoining a primitive 8-th root of unity to Q; the cass number of this fied is 1. D = 320: is obtained by adjoining a root of the poynomia x 4 2x to Q, the cass number of this fied is aso 1. Now, as p 3 (7, 2, 8, 1) < 3.1, from bound (14) we find that D = 192. So neither of the above two cases can occur. Case (a): As D = 5, D is an integra mutipe of 25. We wi now use bound (13) to find an upper bound for D /D 2, maing use of the estimate of Friedman [F] mentioned in 3.2 that R /w > 1/8 if D 125. We find that D /D 2 < p 2(7, 2, 5, 1/8, 1.3) < 8.7. So D = 25c, where c is a positive integer 8. Since the smaest discriminant of a totay compex quartic is 117, c 5. Hence, 5 c 8. The possibe vaues of D are therefore 125, 150, 175, 200. From the tabes in [1] we see that there is no totay compex quartic fied with discriminant 150, 175 or 200, whereas the fied obtained by adjoining a primitive 5th root of unity to Q is the unique totay compex quartic fied with D = 125. It is a cycic extension of Q, and it contains = Q( 5). We wi use Proposition 1 to eiminate this case. In this case, we have the foowing data. ζ ( 1) = 1/30, ζ ( 3) = 1/60, ζ ( 5) = 67/630,

12 12 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS L ( 2) = 4/5, L ( 4) = 1172/25, L ( 6) = 84676/5. Hence µ(g( vo )/Λ) is an integra mutipe of 2 12 ζ ( 1)L ( 2)ζ ( 3)L ( 4)ζ ( 5)L ( 6) = / Again, as the numerator of this number is not a power of 7, according to Proposition 1 this case cannot occur. (iv) Finay we tae-up the case n = 5. We wi rue out the possibiities that d = 4, 3 or 2. (1) Consider first the case where n = 5 and d = 4. Bound (11) with δ = 1 eads to D 1/8 < f(5, 4, 1) < 6.4. Now from Tabe 2 of [F] we find that R /w Next we use bound (15) to concude that D 1/4 D 1/8 < ϕ(5, 4, , 1.2) < As < 1340, D < From the ist of quartics with sma discriminants given in [1], we see that the ony integers smaer than 1340 which are the discriminant of a totay rea quartic are 725 and Moreover, for any of these two integers, there is a unique totay rea quartic fied whose discriminant is that integer. Each of these fieds has cass number 1. If D = 1125, D /D 2 < / < 2. So D /D 2 = 1. This impies that D = = If D = 725, D /D 2 < /725 2 < 4. Hence D = c725 2 with c 3. In particuar, D At our request, Gunter Mae has shown by expicit computation 1 that there is exacty one pair of number fieds (, ) with (D, D ) among the four possibities above. (resp., ) is obtained by adjoining a root of x 4 x 3 4x 2 + 4x + 1 (resp., a primitive 15th root of unity which is a root of x 8 x 7 + x 5 x 4 + x 3 x + 1) to Q. For this pair D = 1125, D = = , and the cass number of is 1. We wi now empoy Proposition 1 to eiminate this case. We have the foowing vaues of the Dedeind zeta and Dirichet L-functions for this pair (, ). ζ ( 1) = 4/15, ζ ( 3) = 2522/15, L ( 2) = 128/45, L ( 4) = /75. From which we concude that µ(g( vo )/Λ) is an integra mutipe of 2 16 ζ ( 1)L ( 2)ζ ( 3)L ( 4) = / As the numerator of this number is not a power of 5, Proposition 1 rues out this case. (2) We wi consider now the case where n = 5 and d = 3. As is a totay compex sextic fied, from 3.2 we now that R /w > 1/8 uness is a totay 1 Mae used the foowing procedure in his computation. Any quadratic extension of is of the form ( α), with α in the ring of integers o of. As the cass number of any totay rea quartic presenty under consideration is 1, o is a unique factorization domain. Now using factorization of sma primes and expicit generators of the group of units of, he isted a possibe α moduo squares, and then for each of the α, the discriminant of ( α) coud be computed.

13 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS 13 compex sextic fied whose discriminant equas one of the six negative integers isted in 3.2. Now using this ower bound for R /w, we deduce from (15) that D D 1/2 < ϕ(5, 3, 1/8, 1) 3 < < 243. On the other hand, if is a totay compex sextic fied whose discriminant equas one of the six negative integers isted in 3.2, then D /2 < 111. Hence, if n = 5, d = 3, then D < 243. From Tabe B.4 in [Co] of discriminants of totay rea cubic number fieds we infer that D must equa one of the foowing five integers: 49, 81, 148, 169, and 229. If D = 229, D /D 2 < /229 2 < 1.2. Hence, D = = There are however no such totay compex sextic fieds according to [1]. If D = 169 or 148, D D > 12167, and hence R /w > 1/8, see 3.2. We wi now use bound (13). As p 2 (5, 3, 169, 1/8, 1.1) < 1.9, and p 2 (5, 3, 148, 1/8, 1.1) < 2.3, D must equa cd 2 for some c 2. None of these appear in the tabe t of totay compex sextics in [1]. If D = 81, then 81 2 D, but none of the six negative integers isted in 3.2 are divisibe by Hence, R /w > 1/8. We wi again use bound (13). We see by a direct computation that p 2 (5, 3, 81, 1/8, 1.1) < 6.2. Therefore, if D = 81, then D = cd 2, with 1 c 6. But from the tabe t in [1] we see that there is no totay compex sextic fied the absoute vaue of whose discriminant equas c81 2, with 1 c 6, except for c = 3. Thus D can ony be = Let be the fied obtained by adjoining a root of x 3 3x 1 to Q, and its totay compex quadratic extension obtained by adjoining a primitive 9th root of unity to Q. Then (resp., ) is the unique totay rea cubic (resp., totay compex sextic) fied with D = 81 (resp., D = 19683). In this case, we have the foowing data on the vaues of the zeta and L-functions. ζ ( 1) = 1/9, ζ( 3) = 199/90, L ( 2) = 104/27, L ( 4) = 57608/9. From which we concude that µ(g( vo )/Λ) is an integra mutipe of 2 12 ζ ( 1)L ( 2)ζ ( 3)L ( 4) = / As the numerator of this rationa number is not a power of 5, according to Proposition 1 this case cannot occur. If D = 49, then D is divisibe by 49 2, but none of the six negative integers isted in 3.2 are divisibe by So R /w > 1/8. We appy bound (13) to obtain D /D 2 < p 2(5, 3, 49, 1/8, 1.2) < Hence D = c49 2, with 1 c 14. On the other hand, the tabe in 3.1 impies that c > 9747/49 2 > 4. Therefore, we need ony consider 5 c 14. From the tabe t in [1] we see that among these ten integers, = is the ony one which equas D of a totay compex sextic. This is obtained by adjoining a primitive 7th root of unity to Q and it contains the totay rea cubic fied obtained by adjoining a root of x 3 x 2 2x + 1 to Q. It is easy to see that D = 49 in this case. We have the foowing data on the vaues of the zeta and L-functions. ζ ( 1) = 1/21, ζ( 3) = 79/210, L ( 2) = 64/7, L ( 4) = /7.

14 14 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS From which we concude that µ(g( vo )/Λ) is an integra mutipe of 2 12 ζ ( 1)L ( 2)ζ ( 3)L ( 4) = / Again, as the numerator of this rationa number is not a power of 5, according to Proposition 1 this case cannot occur. (3) We wi consider now the case n = 5, d = 2. We reca the ower bound R /w from 3.2. From bound (15) we obtain that D 1/2 D 1/4 < ϕ(5, 2, , 1) < 6.7. Since < 45, D 44. It foows that the discriminant D of the rea quadratic fied can ony be one of the foowing integers, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44. If D 37, D /D 2 < 6.74 /37 2 < 2. In these cases, D = D 2 is one of the foowing integers 1369, 1600, 1681, Of these ony 1600 and 1936 appear as the discriminant of a totay compex quartic, chec [1]. Moreover, there is a unique totay compex quartic with D = 1600 (resp., D = 1936). The cass number of both of these quartics is 1. Now we wi use bound (14). Since p 3 (5, 2, 40, 1) < 0.6 < 1 and p 3 (5, 2, 44, 1) < 0.5 < 1, if either D = 40 or 44, D /D 2 < 1, which is impossibe. If D = 33, then D 33 2 = 1089, and hence R /w > 1/8, see 3.2. Now from bound (13) we obtain that D /D 2 < p 2(5, 2, 33, 1/8, 1) < 2. Hence, D = D 2 = There is a unique totay compex quartic whose discriminant is Its cass number is 1. Now we appy bound (14), 1 D /D 2 < p 3(5, 2, 33, 1) < 0.77, to reach a contradiction. If D = 29, D /D 2 < 6.74 /29 2 < 3. Hence, D /D 2 = 1 or 2. Therefore, D = 29 2 = 841 or None of these integers is the discriminant of a totay compex quartic ([1]). If D = 17 or 13, then D 169, and hence R /w > 1/8 from 3.2. Now we wi use bound (13). As p 2 (5, 2, 17, 1/8, 1) < 4.7, and p 2 (5, 2, 13, 1/8, 1) < 7.2, D = c17 2, with 1 c 4, or D = c13 2, with 1 c 7. But of these eeven integers none appears as the discriminant of a totay compex quartic fied. To eiminate the remaining cases (namey, where D = 5, 8, 12, 21, 24 or 28), we wi use Proposition 1. Let us assume in the rest of this section that D is one of the foowing six integers: 5, 8, 12, 21, 24, 28. As D is an integra mutipe of D 2, we concude from 3.2 that uness D = 125 or 144, R /w > 1/8. We wi now use upper bounds (13) and (14) for D /D 2 to mae a ist of the pairs (, ) which can occur. (i) As p 2 (5, 2, 28, 1/8, 1) < 2.1, if D = 28, D = c28 2, with c = 1 or 2. We see from [1] that the cass number of any totay compex quartic with D = 28 2 or is 1. Now we note that p 3 (5, 2, 28, 1) < 1.1. Hence D can ony be 28 2 = 784. The corresponding quartic fied is = Q[x]/(x 4 3x 2 +4) = Q( 1, 7), which contains = Q( 7). We sha denote this pair (, ) by C 1. (ii) As p 2 (5, 2, 24, 1/8, 1) < 2.6, if D = 24, D = c24 2, with 1 c 2. Of these integers, ony 24 2 = 576 is the discriminant of a totay compex quartic. There are

15 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS 15 two totay compex quartics with discriminant 576, but ony one of them contains = Q( 6). This quartic is = Q[x]/(x 4 2x 2 +4) = Q( 3, 6). We sha denote this pair (, ) by C 2. (iii) As p 2 (5, 2, 21, 1/8, 1) < 3.3, if D = 21, D = c21 2, with 1 c 3. Of these three integers, ony 21 2 = 441 is the discriminant of a totay compex quartic. This quartic is = Q[x]/(x 4 x 3 x 2 2x + 4) = Q( 3, 7), and it ceary contains = Q( 21). We sha denote this pair (, ) by C 3. (iv) As p 2 (5, 2, 12, 1/8, 1) < 8.3, if D = 12, D = c12 2, with 1 c 8. Among these, ony for c = 1, 3, 4, and 7, there exists a totay compex quartic with D = c12 2, and a these quartics have the cass number 1. Now we note that p 3 (5, 2, 12, 1) < 4.4, which impies that c 4; i.e., c = 1, 3, or 4. The quartics corresponding to c = 3 and 4 do not contain Q( 3). As we observed whie deaing with Case (c) in (iii) above, there is a unique totay compex quartic, namey = Q[x]/(x 4 x 2 + 1) = Q( 1, 3), whose discriminant equas 12 2 = 144. It contains = Q( 3). The pair ( Q( 3), Q( 1, 3) ) wi be denoted by C 4. (v) As p 2 (5, 2, 8, 1/8, 1) < 16.2, if D = 8, D = c8 2, with 1 c 16. Among these, ony for c = 4, 5, 8, 9, and 13, there exists a totay compex quartic fied, and a these quartics have the cass number 1. Now we observe that p 3 (5, 2, 8, 1) < 8.7, which impies that c = 4, 5 or 8. There is ony one totay compex quartic fied containing = Q( 2), with discriminant as above. This is = Q[x]/(x 4 + 1) = Q( 1, 2) (with D = 256). The corresponding pair (, ) wi be denoted by C 5. (vi) As p 2 (5, 2, 5, 1/8, 1) < 35.5, and D 117, see 3.1, if D = 5, D = c25, with 5 c 35. Among these, ony for c = 5, 9 and 16, there exists a totay compex quartic fied. Thus the possibe vaues of D are 125, 225 and 400. There are precisey three totay compex quartic fieds containing = Q( 5) and with discriminant in {125, 225, 400}. These are = Q[x]/(x 4 x 3 + x 2 x + 1) (= the fied obtained by adjoining a primitive 5th root of unity to Q, its discriminant is 125), = Q[x]/(x 4 x 3 + 2x 2 + x + 1) = Q( 3, 5) (with discriminant 225), and = Q[x]/(x 4 + 3x 2 + 1) = Q( 1, 5) (with discriminant 400). The corresponding pairs (, ) wi be denoted by C 6, C 7 and C 8 respectivey. We observe that in a the above cases, the concusion of Proposition 1 is vioated, see the ast coumn of the tabe beow, where R = 2 8 ζ ( 1)L ( 2)ζ ( 3)L ( 4) is as in (5) for n = 5 and d = 2. Hence none of these cases can occur. We have thus competey proved Theorem 1.

16 16 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS (, ) ζ ( 1) ζ ( 3) L ( 2) L ( 4) R C 1 2/3 113/15 8/ /3 2 7 C 2 1/2 87/20 2/ /2 9 5 C 3 1/3 77/30 32/63 64/ /3 5 5 C 4 1/6 23/60 1/9 5/3 23/ C 5 1/12 11/120 3/2 285/ /2 15 C 6 1/30 1/60 4/5 1172/25 293/ C 7 1/30 1/60 32/9 1984/3 31/ C 8 1/30 1/ / Restrictions on and the main resut 4.1. We sha assume in the seque that = Q. (We have proved in the preceding section that this is the case if n > 7, or if n = 7 or 5 and the orbifod Euer-Poincaré characteristic of Γ is a submutipe of χ(x u )/n r.) Hence, = Q( a) for some square-free positive integer a. By setting d = 1 and D = 1 in bound (10) we obtain 1 > D(n 1)(n+2)/4 ζ (s) D s/ s(s 1) (2π)s e 0.1 n 1 Γ(s). Using the obvious bound ζ (s) ζ(s) 2, and by setting s = 1 + δ, we derive from the above that n 1 (16) D < {50δ(1 + δ)e 0.1 Γ(1 + δ)(2π) 1 δ ζ(1 + δ) 2 } 4/(n2 +n 2δ 4) Denote by κ(n, δ) the right hand side of the above bound. We see, as in 3.3, that for a fixed vaue of δ, κ(n, δ) decreases as n increases provided n 19. We obtain the foowing upper bound for κ(n, δ) for n isted in the first coumn and δ isted in the second coumn of the foowing tabe: n δ κ(n, δ) < The bound for D given by the bound for κ(n, δ) in the above tabe restricts the possibiities for n and. In particuar, since an imaginary quadratic fied has discriminant at east 3, we deduce from the above tabe and the monotonicity of κ(n, δ) for a fixed δ that it is impossibe for n to be arger than 15.

17 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS 17 We reca that for = Q( a), where a is a square-free positive integer, D = a if a 3 (mod 4), and D = 4a otherwise. Now we see that the foowing proposition enumerates a the possibe n and. Proposition 3. (a) n 15. (b) The ony possibiities for the number fieds and are = Q and = Q( a), where, for a given n 15, the possibe a are isted in the foowing tabe: n a , , 3 9 1, 2, 3, 7 7 1, 2, 3, 7, 11, , 2, 3, 5, 6, 7, 11, 15, 19, 23, 31, It is nown that the cass number of the fieds appearing in the above tabe is 1, except when a = 5, 6, 15, or 35, in which cases has the cass number 2, or a = 23, or 31, in which cases has the cass number 3. Hence from (7) we get the foowing bound: n 1 (17) D < [h,n ] 4/(n 1)(n+2), where h,n can ony be 1 or 3 since n is odd. Let λ(n, h,n ) be the function on the right hand side of the above bound. Direct computation yieds the foowing tabe. n λ(n, 3) < λ(n, 1) < Using the above tabe, and upper bound (17) for D, we concude the foowing. Proposition 4. The ony possibiities for the number fied = Q( a) are those isted in the foowing tabe. n a , , 3 9 1, 3, 7 7 1, 2, 3, 7, , 2, 3, 7, 11, In the considerations so far we did not have to use that Γ is cocompact. Now we wi assume that Γ is cocompact and use the description of G given in the introduction. Let, the division agebra D, and the hermitian form h be as in there.

18 18 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS If D =, then h is an hermitian form on n and its signature over R is (n 1, 1). The hermitian form h gives us a quadratic form q on the 2n-dimensiona Q-vector space V = n defined as foows: q(v) = h(v, v) for v V. The quadratic form q is isotropic over R, and hence by Meyer s theorem it is isotropic over Q (cf. [Se2]). This impies that h is isotropic, and hence so is G/Q, and then by Godement s compactness criterion, Γ is noncocompact, which is contrary to our hypothesis. We concude therefore that D, and so it is a nontrivia centra simpe division agebra over. From the cassification of centra simpe division agebras over, which admit an invoution of the second ind, we now that the set T 0 of rationa primes p which spit in, but the group G does not spit over Q p, is nonempty. Note that T 0 T, where T is as in 2.4, and p T 0 if, and ony if, Q p Q D = (Q p Q ) D is isomorphic to M r (D p ) M r (D o p ), where D p is a nontrivia centra division agebra over Q p, D o p is its opposite, and r is a positive integer. We sha denote the degree of D p by d p in the seque Now we wi use the Euer product E appearing in the voume formua (4) to eiminate a but the pair (n, a) = (5, 7) appearing in Proposition 4. Reca from 2.7 that E = (n 1)/2 = E 1 E 2 E 3, ( ζ(2j)l Q (2j + 1) ) p T T e (P p ) where E 1 = E 2 = E 3 = (n 1)/2 (n 1)/2 ζ(2j) ζ(2j + 1), ζ (2j + 1), p T T e (P p ). In the above we have used the simpe fact that L Q (j) = ζ (j)/ζ(j).

19 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS Ceary, E 2 > 1 since each factor in the product formua for ζ (2j + 1), for j > 0, is greater than 1. From (3), (4) and (6) we obtain D ( h,n n #T E n 1 < ( n #T n 1 h,n E 1 E 3 (h,n p T 0 ) 4/(n 1)(n+2) ) 4/(n 1)(n+2) (n 1)/2 n e (P p ) n 1 ζ(2j + 1) ζ(2j) ) 4/(n 1)(n+2). We now reca from 2.3 that for p T 0, e (P p ) is an integra mutipe of n (pj 1) n/dp (pjdp 1), where d p > 1 and d p n. Let q be the argest prime beonging to T 0. Then e (P p ) 1 n n n (qj 1) n/dq p T 0 (qjdq 1), which impies that where for any divisor d of n, L(n, d, q, h,n ) = ( nh,n D < L(n, d q, q, h,n ), n/d (qjd (n 1)/2 1) n (qj 1) ζ(2j + 1) ζ(2j) n 1 ) 4/(n 1)(n+2). Note that L(n, d, q, h,n ) is decreasing in q if the other three arguments are fixed. Aso note that L(n, d q, q, h,n ) L(n, d, q, h,n ), where d is any divisor of d q. Let a be a square-free positive integer. We reca now the foowing we-nown fact (cf. [BS]). Lemma 3. (a) An odd prime p spits in = Q( a) if, and ony if, p does not divide a, and a is a square moduo p. (b) 2 spits in if, and ony if, a 1 (mod 8). (c) A prime p ramifies in if, and ony if, p D. As q T 0, q spits in. Thus if p = p a is the smaest prime spitting in = Q( a), then q p. Hence, D < L(n, d q, q, h,n ) L(n, d q, p, h,n ). We easiy see using Lemma 3 that the smaest prime spitting in = Q( a) for a = 1, 2, 3, 7, 11 and 15 are respectivey 5, 3, 7, 2, 3 and 2. The cass number h of = Q( a), for a = 1, 2, 3, 7, 11 and 15 are 1, 1, 1, 1, 1 and 2 respectivey. Now we see by a simpe computation that for the pairs (n, a) appearing in Proposition 4, L(n, d, p, h,n ) < D, for any prime divisor d of n, except for (n, a) = (5, 7).

20 20 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS Moreover, L(5, 5, 2, 1) > D Q( 7) = 7, but for any q > 2, L(5, 5, q, 1) < 7. concude therefore the foowing. Theorem 2. The ony possibiities for, n and T 0 are = Q( 7), n = 5 and T 0 = {2}. In particuar, PU(n m, m) can contain a cocompact arithmetic subgroup whose orbifod Euer-Poincaré characteristic is χ(x u )/n, where X u is the compact dua of the symmetric space of PU(n m, m), ony if n = 3 or 5. We 5. Four arithmetic fae P 4 C and four arithmetic fae Gr 2, Let now = Q( 7) and D be a division agebra with center and of degree 5 such that for every pace v of not ying over 2, v D is the matrix agebra M 5 ( v ), and the invariant of D at v is a/5 and at v it is a/5, where v and v are the paces of ying over 2, and a is a positive integer ess than 5. Let m = 1 or 2. Then D admits an invoution σ of the second ind such that if G is the simpy connected simpe agebraic Q-group with G(Q) = {x D xσ(x) = 1 and Nrd x = 1}, then G(R) is isomorphic to SU(5 m, m). We note that by varying D, and for a given m, varying σ, we get exacty two distinct groups G up to Q-isomorphism. Now in the group G(A f ) of finite adèes of G, we fix a maxima compact-open subgroup P = P q, where for a q 2, 7, P q is a hyperspecia parahoric subgroup of G(Q q ), P 2 = G(Q 2 ), and P 7 is a specia maxima parahoric subgroup of G(Q 7 ) (we note that there are exacty two such parahoric subgroups containing a given Iwahori subgroup of G(Q 7 ) and they are nonisomorphic as topoogica groups, cf. [T2]). Let Λ = G(Q) P. Then Λ, considered as a subgroup of G(R), is a principa arithmetic subgroup. The foowing emma impies that Λ is torsion-free. Lemma 4. Let D be a division agebra of degree 5 with center = Q( a), where a is a square-free positive integer different from 11. Let τ be an invoution of D of the second ind. Then the subgroup H of D consisting of the eements x such that xτ(x) = 1, and Nrd (x) = 1, is torsion-free. Proof. Let x H be a nontrivia eement of finite order. Since the reduced norm of 1 in D is 1, x 1, and therefore the Q-subagebra K := Q[x] of D generated by x is a nontrivia fied extension of Q. If K =, then x ies in the center of D, and hence it is of order 5. However, a nontrivia fifth-root of unity cannot be contained in a quadratic extension of Q and so we concude that K. Then K is an extension of Q of degree 5 or 10. As no extension of Q of degree 5 contains a root of unity other than 1, K must be of degree 10, and hence, in particuar, it contains = Q( a). Now we note that the ony roots of unity which can be contained in an extension of Q of degree 10 are the 11th and 22nd roots of unity. But the ony quadratic extension contained in the fied extension generated by either of these roots of unity is Q( 11). Since K Q( a), and, by hypothesis, a 11, we have arrived at a contradiction.

21 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS We sha now compute the covoume and the Euer-Poincaré characteristic of the principa arithmetic subgroup Λ. Let X be the symmetric space of G(R), X u be the compact dua of X, and F = X/Λ. We note that if m = 1, G(R) = SU(4, 1) and X u = P 4 C ; if m = 2, G(R) = SU(3, 2) and X u = Gr 2,5 ; F is a connected smooth compex projective variety. As = Q, = Q( 7), the voume formua of [P], Theorem 3.7, taing into account the vaue of the Euer-product E determined in 2.7, gives us 4 µ(g(r)/λ) = D 7 E = D 7 4 ζ(2)l Q(3)ζ(4)L Q (5) From the functiona equation for the L-function we obtain L Q (3) = 2π 3 D 5/2 L Q ( 2), L Q (5) = 2π5 3 D 9/2 L Q ( 4). v T e (P v ). The foowing vaues of zeta and L-functions have been obtained using the software PARI. ζ(2) = π2 π4, ζ(4) = 6 90, L Q( 2) = 16 7, L Q( 4) = 32. Note aso that for the subgroup Λ under consideration, T = {2}, and d 2 = 5, so that (2.3) 5 e (P v ) = (2j 1) 4 (2 5 = (2 j 1). 1) v T Substituting a this in the above, we obtain 4 µ(g(r)/λ) = (( 4π 14 ) L Q( 2)L Q ( 4) ) 4 (2 j 1) = 9 512π 14 ( 4π14 ) ( 16 ) = 1. 7 Therefore, χ(λ) = χ(x u ). Theorem 3.2 of [C] impies that H j (Λ, R) vanishes for a odd j. Aso, there is a natura embedding of H (X u, R) in H (Λ, R), see, for exampe, [B], 3.1 and Now since χ(f) = χ(λ) = χ(x u ), and for a odd j, H j (F, R) (= H j (Λ, R)) vanishes, we concude that F is an arithmetic fae P 4 C if m = 1, and is an arithmetic fae Gr 2,5 if m = 2. Thus we have proved the foowing. Theorem 3. There are at east four arithmetic fae P 4 C, and at east four arithmetic fae Gr 2,5. There does not exist any arithmetic fae projective space of dimension > 4, or an arithmetic fae Gr m,n, with n > 5 odd.

22 22 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS We next prove the foowing interesting proposition. Proposition 5. The first integra homoogy group of any arithmetic fae P 4 C, or an arithmetic fae Gr 2,5, is nonzero. Proof. Let F be either an arithmetic fae P 4 C, or an arithmetic fae Gr 2,5. Let Π be its fundamenta group. Then H 1 (F, Z) = Π/[Π, Π]. It foows from Theorem 2 that Π is a cocompact torsion-free arithmetic subgroup of G(R), where G is as in 5.1, and G is its adjoint group. Proposition 1.2 of [BP] impies that Π is actuay contained in G(Q). We wi view it as a subgroup of G(Q 2 ). Let D and σ be as in 5.1. Since Q 2 Q D = (Q 2 Q ) D = D D o, where D is a division agebra with center Q 2, of degree 5, D o is its opposite, and σ(d) = D o, G(Q 2 ) equas the group SL 1 (D) of eements of reduced norm 1 in D, and G(Q 2 ) equas D /Q 2. We now observe that G(Q 2) = D /Q 2 is a pro-sovabe group, i.e., if we define the decreasing sequence {G i } of subgroups of G := G(Q 2 ) inductivey as foows: G 0 = G, and G i = [G i 1, G i 1 ], then G i is trivia; to see this use Theorem 7(i) of [Ri]. This impies that for any subgroup H of G, [H, H] is a proper subgroup of H. In particuar [Π, Π] is a proper subgroup of Π. This proves the proposition.

23 ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS Five arithmetic fae P 2 C P2 C We wi now use certain resuts and computations of [PY] to construct five irreducibe fae P 2 C P2 C. Let ζ 3 be a primitive cube-root of unity, and et the pair (, ) of number fieds be one of the foowing three: C 2 = (Q( 5), Q( 5, ζ 3 )), C 10 = (Q( 2), Q( )), C 18 = (Q( 6), Q( 6, ζ 3 )). Let v be the unique pace of ying over 2 if the pair is C 2 or C 10, and the unique pace of ying over 3 if the pair is C 18. For a given pair (, ), et q v be the cardinaity of the residue fied of the competion v of at v. Let D be a cubic division agebra with center whose oca invariants at the paces of ying over v are nonzero and negative of each other, and whose oca invariants at a the other paces of is zero. Then v D = ( v ) D = D D o, where D is a cubic division agebra with center v, and D o is its opposite. D admits an invoution of the second ind with being the fixed fied in. We fix such an invoution σ so that if G is the simpe simpy connected -group with G() = {z D zσ(z) = 1 and Nrd(z) = 1}, then G( v ) = SU(2, 1) for any rea pace v of. As σ(d) = D o, G( v ) is the compact group SL 1 (D) of eements of reduced norm 1 in D. Let (P v ) v Vf, be a coherent coection of maxima parahoric subgroups P v of G( v ), v V f, such that P v is hyperspecia whenever G( v ) contains such a subgroup. Let Λ = G() v V f P v. Let v and v be the two rea paces of and et G = G( v ) G( v ). Then G = SU(2, 1) SU(2, 1). Let G be the adjoint group of G. Let X be the symmetric space of G and X u its compact dua. Then X u = P 2 C P2 C, and hence, χ(x u) = 9. We wi view Λ as a diagonay embedded arithmetic subgroup of G. Then, in terms of the normaized Haar measure µ on G used in [P], we see using the voume formua given in that paper (cf. aso [PY]) that µ(g/λ) = µe (P v ) = µ(q v 1) 2 (q v + 1), where the vaues of µ and q v are as given in the tabe in section 9.1 of [PY]. Moreover, according to the resut in section 4.2 of [BP], the orbifod Euer-Poncaré characteristic χ(λ) of Λ equas χ(x u )µ(g/λ) = 9µ(q v 1) 2 (q v + 1). Now using the vaues of µ and q v given in the tabe in section 9.1 of [PY] we find that χ(λ) = 3 if (, ) is either C 2 or C 18, and χ(λ) = 9 if (, ) = C 10. We now observe that Lemma 9.2 of [PY] hods for the group G() described above (the proof of this emma given in [PY] remains vaid), i.e., G() is torsion-free if (, ) = C 10, and in case (, ) is either C 2 or C 18, the ony nontrivia eements of finite order of G() are centra, and hence are of order 3. Let Λ be the image of Λ in G. Then Λ is a torsion-free cocomact irreducibe arithmetic subgroup of G. Moreover, the natura homomorphism Λ Λ is an isomorphism if (, ) = C 10, and its erne is of order 3 if (, ) is either C 2 or C 18. Hence, for each of the three pairs