Supersingular Abelian Varieties over Finite Fields

Transcription

1 Journal of Number Theory 86, 6177 (2001) do: jnh , avalable onlne a hp: on Supersngular Abelan Varees over Fne Felds Hu June Zhu Deparmen of Mahemacs, Unversy of Calforna, Berkeley, Calforna E-mal: zhualum.calberkely.org Communcaed by K. Rubn Receved Aprl 13, 1999 Le A be a supersngular abelan varey defned over a fne feld k. We gve an approxmae descrpon of he srucure of he group A(k) ofk-raonal pons of A n erms of he characersc polynomal f of he Frobenus endomorphsm of A relave o k. Wre f = >g e for dsnc monc rreducble polynomals g and posve negers e. We show ha here s a group homomorphsm.: A(k) > (Zg (1) Z) e ha s almos'' an somorphsm n he sense ha he szes of he kernel and he cokernel of. are bounded by an explc funcon of dm A Academc Press 1991 Mahemacs Subjec Classfcaons: Prmary 11G15; Secondary 14G10, 11N37. Key Words: supersngular abelan varey; fne feld; Merens heorem. 1. INTRODUCTION Le A be an abelan varey of dmenson d defned over a fne feld k of characersc p wh q elemens. Le f be he characersc polynomal of he Frobenus endomorphsm of A relave o k. An abelan varey A over k s supersngular f each complex roo of f can be wren as - q, he produc of some roo of uny and he posve square roo - q. Ths defnon s equvalen o he sandard ones as n [6] or [5]. The group srucure of raonal pons on an ellpc curve over a fne feld has been well suded (see [9, Chap. V]). We have suded he queson for elemenary supersngular abelan varees n [10]. In hs paper and [10], an elemenary abelan varey means an abelan varey ha s k-sogenous o a power of a smple abelan varey. Here we sudy arbrary supersngular abelan varees. For a fne abelan group G we wre *G for s order. Le log( } ) be he naural logarhm. Wre f => ge for dsnc monc rreducble polynomals g wh neger coeffcens and posve negers e X Copyrgh 2001 by Academc Press All rghs of reproducon n any form reserved.

2 62 HUI JUNE ZHU Theorem 1.1. Le A be a supersngular abelan varey over k of dmenson d2. Wre f => ge as above. Then here exss a group homomorphsm such ha.: A(k) *Ker(.)=*Coker(.) (Zg (1) Z) e log(2d&2))2d f d>4.35_10 7 < {(2 log(100d&100)) 2d f 2d4.35_10 7. If q s a nonsquare, he l-par of *Ker(.) dvdes l 3d&2 f l=2, dvdes l [(2d&2)(l&1)] f l>2, and s rval f l>d orl= p. If q s a square, he l-par of *Ker(.) dvdes l [(2d&2)(l&1)] f l2, and s rval f l>2d orl= p. Le Z[?] be he Z-algebra generaed by he Frobenus? n he endomorphsm rng of A. Le k be an algebrac closure of k. Theorem 1.2. Le A be a supersngular abelan varey over k of dmenson d2. Wre f => ge as above. Le R =Z[?](g (?)) and R be s localzaon a p. There s a surjecve Z[?]-module homomorphsm where.: A(k ) (R R ) e, log(2d&2))2d f d>4.35_10 7 *Ker(.)<{(2 log(100d&100)) 2d f 2d4.35_10 7. Our heorems essenally demonsrae he followng observaon: The group srucure of a supersngular abelan varey over a fne feld s deermned by he characersc polynomal of s Frobenus endomorphsm wh an error erm'' dependng only on dm A, no on he sze of he base feld. The organzaon of hs paper s as follows: Secons 2 and 3 are echncal. Secon 2 conans a lemma (see Lemma 2.1) from analyc number heory whch wll be used for Secon 3. In Secon 3 we wll deermne all possble rreducble facors of he characersc polynomal f and compue her muual resulans so as o gve a useful approxmaon (see Lemma 3.2). In Secon 4, we consder fnely generaed orson-free

3 SUPERSINGULAR ABELIAN VARIETIES 63 modules over a fbre produc of rngs by applyng Goursa's lemma. Fnally, by consderng he l-adc Tae module of A as a orson-free module over Z[?], we apply Secon 4 o our problem and prove he wo heorems. Ths paper s based on a poron of he auhor's Ph.D hess. The auhor hanks Professor Hendrk W. Lensra, Jr., for hs gudance and he Mahemacal Scence Research Insue (Berkeley) for s excellen workng envronmen and suppor whle she was preparng hs paper. The auhor also hanks he referee for many very helpful commens. 2. A VARIATION OF MERTENS'S THEOREM Here we prove a lemma from analyc number heory ha wll be used n Lemma 3.2 n he nex secon. An mmedae consequence s Corollary 2.2 whch was nally conjecured by Lensra (see [7, Sec. 1] for s applcaon). Merens's heorem mples ha when n s large enough we have > ln l 1l < n, where l ranges over all prmes n (see [2, Theorem 425] or [8, (2.5)]). Le,( } ) denoe he Euler ph-funcon. In hs secon we wll prove ha when n s large enough we have > l n l 1(l&1) < log,(n). The subscrp l n denoes ha l ranges over all dsnc prmes dvdng n. Le C be Euler's consan (r0.5772) and p he h prme number. Lemma 2.1. Le n 0 :=2 > 9 p r4.46_10 8. Then l n l 1(l&1) < {log,(n) log(50,(n)) f n>n 0, f 2nn 0. Proof. Gven an neger n2 we fnd he posve neger such ha > p n<> +1 p. Snce n has a mos dsnc prme facors and (log l)(l&1) s a decreasng funcon, : l n log l l&1 : log p p &1. (1) By [8, (2.8) and (3.23)], we have for 12 ha : log p p &1 = : log p + : p m=2 : log p <log p p m + 1 &C. (2) log p Suppose n> 13 p. The wo auxlary funcons F(n):=F 0 (n)+ 1F 0 (n)&c and F 0 (n) :=log log n&log(1&1(log log n&0.7093)) are ncreasng wh respec o n. By Berrand's Posulae (see [2, 22.3]) and

4 64 HUI JUNE ZHU [8, (3.32)], we oban p >p +1 2>(log n)2.0325; hus by [8, (3.16)] we have log p <log log n&log(1&1(log p ))<F 0 (n). Combnng (1) and (2) yelds : l n log l <F(n). (3) l&1 Defne H(n) :=exp(c) log log n+2.5(log log n). Snce nh(n) s an ncreasng funcon for n30, by [8, (3.41)] we ge > p H(> p ) < n <,(n) for all n{ 9 H(n) p. (4) Suppose n> 25 p. I s no hard o show ha n>(h(n)) 21,andso n<,(n) H(n)<,(n)(nH(n)) 120 <,(n) (5) Now F(n)&log log n s decreasng, so F(n)<log log n& Then (3) and (5) yeld l n ((log l)(l&1))<log log n&0.0529<log(1.05 log,(n))& <log log,(n). Suppose > 10 p n<> 25 p ; by explc calculaon for each and by (4) we have l n l 1(l&1) p 1( p &1) <log > p H(> p ) log n <log,(n). H(n) Suppose n<> 10 p. Ths mples ha n has a mos 9 dsnc prme facors. Snce l 1(l&1) >1 and l 1(l&1) s decreasng n l, we have > l n l 1(l&1) > 9 p1( p &1). When 3 > 9 p n<> 10 p, by explc compuaon and (4) we have l n 9 l 1(l&1) p 1( p &1) <log 3 >9 p H(3 > 9 p <log,(n). ) When n 0 <n<3 > 9 p, by smlar compuaon we have l n 8 l 1(l&1) p 1( p 10 &1) 10 Ths proves he frs half of he lemma. p 1( p &1) <log n 0 <log,(n). H(n 0 )

5 SUPERSINGULAR ABELIAN VARIETIES 65 Suppose 30nn 0 and n{> 9 p. By (1), (4), and explc compuaon on each 39, we oban l n l 1(l&1) p 1( p &1) <log 50 > p H(> p ) <log(50,(n)). Ths s easy o verfy for n=> 9 p and 2n<30. By a smlar bu easer calculaon, we can show ha > l n l 1(l&1) <log n for all n so ha p 8 > 6 p <nn 0 and hus for all n>n 0 by he above lemma. Ths gves he followng corollary. Corollary 2.2. <log n. For all n>p 8 > 6 p =570570, we have > l n l 1(l&1) Remark 2.3. The mnmal bounds for n n Lemma 2.1 and Corollary 2.2 are boh sharp. I s no hard o verfy he followng: f n=n 0 =2 > 9 p, hen > l n l 1(l&1) >log,(n); f n= p 8 > 6 p, hen > l n l 1(l&1) >log n. K 3. SUPERSINGULAR POLYNOMIALS In hs secon, we wll quoe algebrac number heory from [1] or [3] whou commen. Recall ha q s a power of he prme p. An algebrac number n C s called a supersngular q-number f s of he form - q, he produc of some roo of uny and he posve square roo of q. Obvously s an algebrac neger. Here we deermne all mnmal polynomals of supersngular q-numbers, calculae her muual resulans, and prove Lemma 3.2. Ths lemma s a core echncal pon for our proof of Theorems 1.1 and 1.2 n Secon 5. Le ( a b ) be he Jacob symbol for an neger a and odd neger b; furher, defne ( a 1)=1 and defne ( a 2)=0 f 2 a and ( a 2)=(&1) (a2&1)8 f 2 % a. Denoe by m he prmve mh roo of uny, exp(2? - &1m). The Galos group Gal(Q(m)Q) consss of he _ defned by _ (m)=m wh coprme o m and 1m. We clam ha - p # Q(m) mples _ (- p)=( p ) - p. Snce hey are boh mulplcave, suffces o show ha _ l (- p)=( p ) - p for each prme l dvdng. Ifl s odd hen _ l l(- p)# (- p) l = p (l&1)2 - p mod l and hus _ l (- p)=( p l ) - p. Suppose l=2. Snce m s odd, our hypohess mples ha Q(- p)q(p)q(m). Denoe by _ 2 he mage of _ 2 n Gal(Q(p)Q), hen _ 2 (- p)=_ 2(- p)=- p or &- p. I equals he former f and only f _ 2 # Gal(Q(p)Q(- p)), ha s, f and only f 2 s a square n (ZpZ)*. Thus _ 2 (- p)=( p ) - p. 2

6 66 HUI JUNE ZHU Le 8 m be he mh cycloomc polynomal. We wre (m 1, m 2 ) for he greaes common dvsor for negers or polynomals m 1 and m 2.LeC(?) be he conjugacy class of? n C. Proposon 3.1. Le g be he mnmal polynomal of a gven supersngular q-number?. I. If q s a squar, hen C(?)=C(m - q) for some m, and g=9 m (X) :=(- q),(m) 8 m\ X - q+. II. If q s a nonsquare, hen C(?)=C(& m - q) for some prmve mh roo of uny & m wh m2 mod 4. Defne G m (X) :=q,(m)(2, m) 8 m(2, m) (X 2 q). (6) () () If Q(?){Q(? 2 ), hen g=g m (X). If Q(?)=Q(? 2 ), hen g=e m,\1 (X) := (, m(2, m))=1\ X \ q + 1m(2, m) m - q +. (7) Proof. Par I s sraghforward. We shall show par II. Wre?=& m - q for some prmve mh roo of uny & m. If 2& m, hen?=&&(m+2)4 m2 - q; bu snce m2 s odd,? s conjugae o + - q for m2 some prmve (m2)h roo of uny + m2. Thus we may assume m2 mod 4 for he res of he proof. Now [Q(?) :Q(? 2 )]=1 or 2. Le 2 denoe he dscrmnan of a number feld exenson. I can be shown ha Q(?)=Q(? 2 ) f and only f 2 Q(- p)q m and 22 Q(- p)q % m (see [10, Lemma 2.6]). Suppose [Q(?):Q(? 2 )]=2. I s no hard o see ha? s a roo of G m and s mnmal polynomal s G m snce G m has degree 2,(m)(2, m) and [Q(?) :Q]= [Q(?) :Q(m(2, m))][q(m) :Q] =2,(m)(2, m). [Q(m) :Q(m(2, m) )] Suppose [Q(?): Q(? 2 )]=1. Then - p # Q(m) and so by he argumen precedng hs proposon we have _ (?)=_ (& - m q)=(q) & m - q for all _ # Gal(Q(m(2, m) )Q). The degree of? s,(m(2, m)), so s mnmal polynomal s E m, ( q )=> (X&(q) & m - q) where he produc ranges over & wh (, m(2, m))=1 and 1m(2, m). K

7 SUPERSINGULAR ABELIAN VARIETIES 67 We nroduce some noaon here. For any prme number l, we wre n l and n (l) for he l-par and he non-l-par of a posve neger n, respecvely. Le E denoe he se of supersngular q-numbers & m- q for some prmve mh roo of uny & m where p % m, p{2, and q s no a square, such ha (I) 4 % m when p#1 mod 4 whle (II) 4 & m when p#3 mod 4. Le Q be he se of supersngular q-numbers & m - q for some prmve mh roo of uny & m such ha eher (I) m=1, 2 or (II) q s a square, (2, p) p % m and ord( p mod m ) s odd. We noe ha? # Q (respecvely, E) f and only f C(?)/Q (respecvely, E). In oher words, hese defnons are ndependen of he choce of? from s conjugacy class. For, 2,...,, le C be conjugacy classes of supersngular q-numbers wh mnmal polynomals g. By Proposon 3.1, C =C(& m - q) where m 2 mod 4 when q s a nonsquare. We order he C 's so ha m 1 }}} m. For, 2,...,, le e be posve negers such ha (I) e e +1 when m =m +1 and (II) e s even when? =& m - q # Q. Under hese condons, he numbers defned by d := 1 e deg(g )2 and d E := 1,? # E e deg(g )2 are posve negers (see [10, Proposon 3.3]). These wo echncally defned numbers wll be used n Secon 5. Le R( }, } ) denoe he resulan of wo polynomals. For any real number r we denoe he larges neger r by [r]. Lemma 3.2. \ 2d E Le he noaon be as above and le d2. Then =2 &1 j=1 R(g, g j ) e + < {(2 log(2d&2))2d (2 log(100d&100)) 2d f d>4.35_10 7 ; f d4.35_10 7. Le l be a prme dfferen from p. If q s a nonsquare, we have \ 2d E =2 &1 j=2 R(g, g j ) e +l 1 f l>d; dvdes{l [(2d&2)(l&1)] f 2<ld; 2 3d&2 f l=2. If q s a square, we have \ 2d E =2 &1 j=1 R(g, g j ) e +l f l>2d; dvdes {1 l [(2d&2)(l&1)] f l2d. Remark 3.3. The sraegy of our proof s frs o compue he resulans of cycloomc polynomals (n Lemma 3.4) and hen o reduce our problem o he cycloomc case (see Lemma 3.5). Fnally, we apply Lemma 2.1 o approxmae our desred bounds.

8 68 HUI JUNE ZHU Lemma 3.4. For any posve negers m>n, we have R(8 m, 8 n )= {(&1),(n),(m) l,(n) 1 f mn s a power of a prme l, oherwse. Proof. Le l be a prme number. Wre m=m (l) l : and n=n (l) l ;, hen 8 m (X)= 8 m (l) (X l : ) 8 m(l) (X l :&1 ) # 8 (X) l : m(l) 8 m(l) (X) :&1=8 l m (l) (X),(m),(m (l)) mod l. Hence, l R(8 m, 8 n ) f and only f m (l) =n (l), ha s, mn # l Z. Thus we have R(8 m, 8 n ) =1 f mn s no a prme power. Now assume m (l) =n (l). Then R \8 n(x), X m &1 X ml &1+ =R(8 n(x), 8 l (X ml ))= (, n)=1 8 l (ml n )=l,(n). Accordng o he facorzaon (X m &1)(X ml &1)=8 m (X) > s 8 ms (X) where s ranges over dvsors of m (l) ha are no equal o 1, we ge l,(n) = } R \ X m &1 X ml &1, 8 n(x) +}= R(8 m, 8 n ) R(8 ms, 8 n ). s The las produc s rval snce n(ms) s no a prme power. Ths proves our asseron up o a sgn. I remans o show ha R(8 m, 8 n ) s posve f and only f m{2. Indeed, f m3 hen complex conjugaon s conaned n Gal(Q(m)Q) and so R(8 m, 8 n )=N Q(m)Q(8 n (m)) s posve. If m=2 hen s rval o see ha R(8 2, 8 1 )=&2. Ths fnshes our proof. K Le E be he complemen of E n he se of all supersngular q-numbers wh nonsquare q. Lemma 3.5. Le he noaon be as n Lemma 3.2, \ R(g, g j ) + e > j dvdes 2 [ : e deg( g )2]? # E l [e deg(g)(l&1)]. 2, l 2m Proof. (a) Denoe by F he se of? 's wh m =m +1.Lel be a prme dfferen from p. For any fxed?, by Lemma 3.5, he produc > R(8 m, 8 mj ) l over all j wh 1 j&1,? j F aans s maxmum

9 SUPERSINGULAR ABELIAN VARIETIES 69 when each m j =m l & j. In hs case we have &1 j=1,(m j),(m l)+,(m l 2 )+ }}} +,(m l &1 ),(m )(l&1). Hence 1 j&1,? j F R(8 m, 8 mj ) l dvdes l,(m )(l&1) for prme l m. (8) (b) Assume q s a square. Snce F s empy, by Proposon 3.1(I), > j R(g, g j ) e = > j R(9 m, 9 mj ) e = > j R(8 m, 8 mj ) e, whch dvdes > 2, l m l [e deg(g )(l&1)] by (8). Ths proves he lemma n he case. (c) Assume q s a nonsquare. Then? # F f and only f he pars?,? +1 have mnmal polynomals g =E m,\1 and g +1 =E m, 1. Snce he produc > > j R(g, g j ) e dvdes? &1 # F R(E m,1, E m,&1) e R(g, G mj ) e > j,? j F suffces o show ha he wo dvsbles? &1 # F R(E m,1, E m,&1) e dvdes 2 [ : e deg( g )2]? # E (9) > j,? j F R(g, G mj ) e dvdes l[e deg(g )(l&1)] (10) hold. We frs prove (9): Le _ and $ range over he embeddngs of Q(? ) n C. By (7) we have? &1 # F R(E m,1, E m,&1) e = ( p ) _,? _ &(&? ) $ e.? &1 # F $ Splng he produc no wo pars accordng o _=$ and _{$, hey are =? &1 # F, _ 2? _ e }? &1 # F, _{$ } [ : e deg( g )]? =2 &1 # F }? &1 # F} 2 2 Z[? ]Z? 2_ &? 2$? _ &? $ } 2 Z[? ]Z } e. e

10 70 HUI JUNE ZHU The las produc s rval snce he ncluson chan Z[? 2 ]= Z[qm (2, m )]Z[? ]Z[m (2, m )] has p-power ndex. Noe ha? &1 # F mples? &1,? # E ; bu e &1 e by our hypohess, so we have : e deg(g ) :? &1 # F? # E e deg(g )2. Then (9) follows. Second, we prove (10): Le n (2, p) denoe he non-2 and non-p par of he neger n. Now we clam ha for any > j,? j F R(g, G mj ) dvdes 2 deg(g )? j F R(8 m, 8 mj ) deg(g ),(m ) (2, p). By Proposon 3.1(II), suffces o consder he followng wo cases: Case 1. Suppose g =G m. By he defnon n (6), we have R(G m, G mj ) = R(8 m (X 2(2, m ) ), 8 mj (X 2(2, m j ) )). Noe ha 8 m (X 2 )=8 m (X) 8 2m (X) when 2 % m. Furher calculaons va Lemma 3.4 and (8) yeld ha? j F R(G m, G mj ) dvdes 2 deg(g m )? j F R(8 m, 8 mj ) 2(2, m ) (2, p). Noe ha deg(g m ),(m )=2(2, m ). Thus (11) holds. Case 2. Suppose g =E m,\1. From (6) and (7), G m =E m,1e m,&1 and R(E m,1, G mj ) = R(E m,&1, G mj ), so we have R(E m,\1, G mj ) = R(G m, G mj ) 12. Bu deg(e m,\1)=deg(g m )2, hus (11) follows from Case 1. By (11) and (8), he dvsbly n (10) follows. Ths fnshes our proof. K Proof of Lemma 3.2. If =1, hen 2 d E >> j R(g, g j ) e =2 d E dvdes 2 d snce d E d. In hs case s sraghforward o verfy our asseron. For he res of he proof we assume ha 2. We shall prove he local bound frs. Below le l{ p. (I) Le q be a nonsquare. Le l>d2. We clam ha > > j R(g, g j ) e l =1. Suppose he conrary. By Lemma 3.5, we have ha l m for some. Suppose m =l or 2l, hen Q(? ){Q(? 2 ) and so g =G m by Proposon 3.1(III). Thence ddeg(g )2+1=,(m )+1=l, whch conradcs our assumpon ha l>d. Suppose m 3l, hen l,(m )2+1 deg(g )2+1d whch s absurd.

11 SUPERSINGULAR ABELIAN VARIETIES 71 Le q be a square. Le l>2d. We clam ha > > j R(g, g j ) e l =1. Suppose he conrary, ha here are and j such ha m m j =l s for some neger s>0. Then 2d,(m )+,(m j )=,(l s m j )+,(m j )l, whch leads o a conradcon. Second, f l>2 or q s a square, hen by Lemma 3.5 he l-exponen of 2 d E >> j R(g, g j ) e [(2 e deg(g ))(l&1)][(2d&2)(l&1)] snce e 1 deg(g 1 )2. Smlarly, by Lemma 3.5, he 2-exponen of 2 d E >> j R(g, g j ) e 2 s less han or equal o? # E e deg(g )2+? # E e deg(g )2+ e deg(g ) 3d&2. (II) Now we prove he global bound. Le m$ be he non-2-par of m. Le he noaon be as n Lemma 3.5; (2 d E >> j R(g, g j ) e ) dvdes 2 [( e deg(g ))2] l [(e deg(g ))(l&1)] l 2m$ <2 d \ 1(l&1)+ e deg(g ) l. l 2m$ Noe ha,(2m$ )=,(m$ )2d&2, so by Lemma 2.1 we have Thus l 1(l&1) <log(50,(2m$ ))<log(100d&100). l 2m$ \ 2d E R(g, g j ) e <2 > j +( d (log(100d&100)) e deg(g ) p) (2 log(100d&100)) 2d. Now assume ha d>4.35_10 7.If2m$ >n 0 hen Lemma 2.1 mples ha > l 2m$ l 1(l&1) <log(2d&2). Oherwse, by nequaly (1) n he proof of he same lemma and explc compuaon, > l 2m$ l 1(l&1) > 9 p1( p &1) < log(2d&2). Therefore, 2 d \ 1(l&1)+ e deg(g ) l <2 d (log(2d&2)) 2d <(2 log(2d&2)) 2d. l 2m$ Ths fnshes our proof. K Example 3.6. Those local upper bounds n Lemma 3.2 are sharp. The second bound s acheved n he followng example: Le q be a square. Le l be an odd prme dfferen from p. Le? =l &1 - q and e be even posve negers for,...,. Then we have 2 d E >> j R(g, g j ) e l =l(2d&2)(l&1).

12 72 HUI JUNE ZHU Here s a nonrval example n whch he hrd bound s approached very closely: Consder? 1 =3-3,? 2 =12-3,? 3 = I can be checked ha? # E, so 2 d E >> j R(g, g j ) e j =2 2e 2 +4e 3, whle 2 3d&2 = 2 6e 1 +3e 2 +3e 3 &2. 4. TORSION-FREE MODULES AND FIBRE PRODUCTS All rngs are commuave wh 1. Le 2. Le a be an deal of a rng R for,...,. Inducvely he fbre produc R 1 _ R2 a 2 R 2 _}}}_ R a R s R$ &1 _ R a R :=[(r$ &1, r )#R$ &1 _R # (r$ &1 +a$ &1 )=r +a ], where R$ &1 =R 1 _ R2 a 2 R 2 _}}}_ R&1 a &1 R &1 has an deal a$ &1 such ha here s an somorphsm R$ &1 a$ &1 w # R a. Gven an R -module M wh a submodule N $a M for,...,, defne he fbre produc of modules M 1 _ M2 N 2 M 2 _}}}_ M N M analogously as M$ &1 _ M N M :=[(x$ &1, x )#M$ &1 _M % (x$ &1 +N$ &1 )=(x +N )], where M$ &1 =M 1 _ M2 N 2 M 2 _}}}_ M&1 N &1 M &1 has a submodule N$ &1 $a &1 M &1 such ha here s a # -lnear somorphsm % : M$ &1 N$ &1 M N. (Noe ha # -lnear means ha % r$ &1 =(# r$ &1 ) % for every r$ &1 # R$ &1.) Then we see ha M 1 _ M2 N 2 M 2 _}}}_ M N M s a module over R 1 _ R2 a 2 R 2 _}}}_ R a R. We have he followng Goursa's Lemma for rngs (also see [4, Exercse 5, p. 75] for Goursa's Lemma for groups). Lemma 4.1. Le R 1,..., R be rngs. Suppose R s a subrng of > R such ha he projecons R w \ R are surjecve. Le R$ be he mage of he projecon R > R j=1 j. Denoe he projecon maps from R$ o R$ &1 and R by \$ &1 and \ ", respecvely. We may denfy a =Ker(\$ &1 ) and a$ &1 =Ker(\ ") wh deals n R and R$ &1, respecvely. We oban somorphsms R$ &1 a$ &1 w # R a for =2,..., whch defne an somorphsm R$R 1 _ R2 a 2 R 2 _}}}_ R a R. As abelan groups, (R$ &1 _R )R$ $R a for =2,...,. Proof. From he nducve defnon of he fbre produc, suffces o prove he lemma for =2. I s clear ha a$ 1 =R & (R 1 _[0]), and by

13 SUPERSINGULAR ABELIAN VARIETIES 73 assumpon can be denfed wh an deal n R 1. Smlarly, we denfy a 2 wh an deal n R 2. Thus a$ 1 _a 2 s he larges deal of R 1 _R 2 ha s also an deal n R. The naural map %: R R 1 a$ 1 _R 2 a 2 defnes an somorphsm #: R 1 a$ 1 R 2 a 2 whose graph s he mage of R. In fac, f wo elemens (r 1, r 2 ), (r 1, r 3 )#R 1 _R 2 le n R, hen (0, r 2 &r 3 )#R. Hence r 2 &r 3 # a 2. Ths shows ha # s well-defned. Usng he same argumen, we see ha # s njecve and surjecve. From our consrucon R 1 _ R2 a 2 R 2 s exacly he pullback of he map % and hence s dencal o R. K We have an analogous Goursa's Lemma for modules. Lemma 4.2. Le R be as n Lemma 4.1. Le M be an R -module and M be an R-submodule of > M such ha he projecons M w * M are surjecve. Le M$ denoe he mage of he projecon M > M j=1 j. Denoe he projecon maps from M$ o M$ &1 and M by *$ &1 and *", respecvely. We may denfy N =Ker(*$ &1 ) and N$ &1 =Ker(* ") wh submodules of M and M$ &1, respecvely. We oban # -lnear somorphsms M$ &1 N$ &1 w % M N whch defne an R-module somorphsm M$M 1 _ M2 N 2 M 2 _}}}_ M N M. Remark 4.3. Any subrng R of > R wh surjecve projecons R R s somorphc o a fbre produc of R 1,..., R as defned n Lemma 4.1. For he res of he paper we defne he fbre produc R=R 1 _ R2 a 2 R 2 _}}}_ R a R by he projecons R w \ R. Smlarly, we defne a fbre produc of R -modules M by he projecons M w \ M. Assume ha all modules are fnely generaed. Le l be a prme. Suppose K s a fne-dmensonal separable Q l -algebra. Le R be an Z l -order n K, ha s, a Z l -algebra ha spans K over Q l.anr-module M s orson-free f :m{0 for all non-zerodvsor : # R&[0] and m # M&[0]. (IfR s a doman hen hs s equvalen o he sandard noaon.) If M s a orsonfree R-module, hen here s a naural njecve map M M R K; f moreover M R K$K e for some neger e hen we say ha M s of rank e. See [10, Lemma 3.6] for he proof of he followng auxlary lemma. Lemma 4.4. Le R, K be as above. Le r # R&[0] be a nonzero dvsor. Le MM$ be orson-free R-modules of rank e, hen *MrM= (*(RrR)) e. There exs homomorphsms \: MrM M$rM$ and \$: M$rM$ MrM wh *Ker(\)=*Coker(\) and *Ker(\$)=*Coker(\$) dvdng *(M$M). Proposon 4.5. For,...,, le R be a Z l -order n a separable Q l -algebra K. Le a be an deal n R such ha R=R 1 _ R2 a 2

14 74 HUI JUNE ZHU R 2 _}}}_ R a R. Le M be a orson-free R-module, and denoe by M he mage of he njecon M (M Zp Q p ) K K. The projecons M M defne an R-module somorphsm M $ M 1 _ M2 N 2 M 2 _}}}_ M N M for some R -submodules N n M. Furher, f M s of rank e hen *((> M )M) dvdes > =2 *(R a ) e. Proof. By hypohess, M> M. We use nducon on o show ha M s he desred fbre produc and *((> M )M)= > *(M =2 N ). Suppose =2. By Lemma 4.2, M$M 1 _ M2 N 2 M 2 for some submodule N 2. Wre a=a 1 _a 2. Snce M 1 _ M2 N 2 M 2 =M$aM= a(r 1 _R 2 ) M=a(M 1 _M 2 )=a 1 M 1 _a 2 M 2, we ge a 1 M 1 N 1, a 2 M 2 N 2 and *((M 1 _M 2 )M)=*(M 2 N 2 ). Denoe by M$ he mage of he projecon M > j=1 M j. Suppose here are R -submodules N n M such ha, for =2,..., &1, we have M$ =M 1 _ M2 N 2 M 2 _}}}_ M N M, and *(> &1 j=1 M j )M$ &1 => &1 j=2 (*M j N j ). Then M$M$ &1 _ M N M and * \\ &1 M +< M + =* \\ M +< &1+ M$ } * \\ M$ &1_M +< M + = \ &1 = =2 =2 *(M N ) + } *(M N ) *(M N ). Ths fnshes our nducon. Bu we have *(M N ) *(M a M )= *(R a ) e by Lemma 4.4, so our asseron follows. K Below s an explc example of a fbre produc of rngs. Proposon 4.6. Le g 1,..., g # Z[X] be arbrary monc polynomals n one varable such ha (g, g j )=1 n Q[X] for { j. Denoe by? and? he mages of X n he Z-algebras Z[X](> g ) and Z[X](g ), respecvely. Le R=Z[?] l, R =Z[? ] l, and a =(> &1 g j=1 j(? )) R. The naural projecons R w \ R defne an somorphsm R$R 1 _ R2 a 2 R 2 _}}}_ R a R such ha *R a => &1 R(g j=1, g j ) l for all 2. Proof. Sendng? o (? 1,...,? ) defnes a rng homomorphsm R > R. I s njecve snce (g, g j )=1 for all { j. For each, hs map nduces surjecve projecons R R. The assered somorphsm follows from nducon on by nvokng Lemma 4.1. Thus *R a = *R > &1 g j=1 j(? ))=> &1 N j=1 Q(? )Q(g j (? )) l => &1 R(g j=1 j, g ) l. K

15 SUPERSINGULAR ABELIAN VARIETIES ARBITRARY SUPERSINGULAR ABELIAN VARIETIES In hs secon, we shall prove Theorems 1.1 and 1.2. We denoe by A[n] he subgroup of A(k ) conssng of all pons of order dvdng n. Lel be a prme { p. Le T l :=T l A be he l-adc Tae module of A and V l := T l Zl Q l. There s a k-sogeny A w # > A, where A s an elemenary abelan varey wh characersc polynomal g e as n Secon 1. Le Q[?] be he Q-subalgebra generaed by? n he endomorphsm algebra of A. Wre R :=Z[?] andr :=Z[?]g (?) Z[?]. Le R l and R l, be he l-adc compleons of R and R, respecvely. The sogeny # gves an somorphsm of Q[?]-modules, V l [ > V l(a ), and an njecve map of R-modules, T l w # > T l(a ). The mage of # n T l (A ), denoed by T l,, s an R l, -submodule of fne ndex. We assume ha A has been chosen n such a way ha # maps surjecvely ono T l (A ), ha s R l, =T l (A ). Ths can be seen from an elemenary lemma below. Lemma 5.1. For every Z[?] l -submodule M of fne ndex n T l A, here s an abelan varey A$ over k and a k-sogeny A$ w : A such ha :T l A$=M. Proof. Choose n so large ha l n T l AM. Le G be he mage of Ml n T l A n he somorphsm T l Al n T l A w \ A[l n ]. Snce G has a Gal(k k)- module srucure and has order dvdng l n (coprme o p), deermnes a fne e ale subgroup scheme G of A over k wh G(k )=G. LeA$ :=AG. So he sogeny A w l n A facors hrough A w ; A$ and we have A w ; A$ w : A wh :;=l n. Noe ha :T l A$$l n T l A. I s clear ha : maps T l A$;T l A ono :T l A$l n T l A, whose mage n \ s exacly G=Ker(;)(k ). Therefore, we have :T l A$=M. K Clearly, T l s a orson-free R l -module. Le? be he mage of? n Q[?](g (?)). Then Q[?](g (?))=Q(? ) s acually a feld, and we fx her embeddng n C. Snce V l (A )$ Q(? ) l Q(? ) e, we noe ha T l l, s a orson-free R l, -module of rank e for each. Lemma 5.2. Le he noaon be as above. Le r # R be a non-zerodvsor. There s an R l -module homomorphsm. l : T l rt l w : l (T l, rt l, ) w ; l (R l, rr l, ) e wh *Ker(. l )=*Coker(. l ) dvdng (2 d E > =2 >&1 j=1 R(g, g j ) e )l.

16 76 HUI JUNE ZHU Proof. By Proposons 4.5 and 4.6, R l =R l,1 _ Rl, 2 a 2 R l,2 _}}}_ Rl, a R l, and T l $T l,1 _ Tl, 2 N l,2 T l,2 _}}}_ Tl, N l, T l, for some R l, -submodules N l, n T l, such ha * \\ T l, +< l+ T dvdes =2 &1 j=1 R(g, g j ) e l. (12) Applyng Lemma 4.4., here s a map : l wh *Ker(: l )=*Coker(: l ) dvdng > =2 >&1 R(g j=1, g j ) e l. On he oher hand, by [10, Proposon 3.11], we have *(T l, R e l, ) 2e deg(g )2 f (l,? )#[2]_E: equals 1 oherwse. Applyng Lemma 4.4 agan, we ge a map ; l wh *Ker(; l )= *Coker(; l ) dvdng *(T l, R e l, )=? # E 2 e deg(g )2 l =2 d E l. (13) Hence he composon map. l =; l } : l has *Ker(. l )=*Coker(. l ) dvdng he produc of he las numbers of (12) and (13). K Remark 5.3. If we order he e such ha e 1 e 2 }}}e and denoe by R$ l, he mage of he projecon R l R l,1 _}}}_R l,, hen he dvsbly n (12) s acually equaly f T l, $ Rl, R e and l, T l $ Rl R e 1 &e 2 l, 1 _(R$ l,2 ) e 2 &e 3 _(R$l, 3 ) e 3 &e 4 _}}}_(R$l, &1 ) e &1 &e _R e l. Proof of Theorem 1.1. Suppose d=dm A2. Nong ha he l-sylow subgroup of A(k) s somorphc o T l (?&1) T l and ha he p-sylow subgroup s rval, we defne. :=> l{ p. l, wh he. l as n Lemma 5.2. Our asseron follows from Lemmas 5.2 and 3.2. K The proof of Theorem 1.2 s almos dencal o ha of [10, Theorem 1.2]. We provde a skech of s proof. For any neger n coprme o p we fnd an R-module homomorphsm A[n] > (R nr ) e wh kernel and cokernel bounded as n he asseron. These bounds do no depend on n. Afer akng he suable njecve lm on boh sdes over n we ge he desred homomorphsm. wh he same bounds. REFERENCES 1. H. Cohen, A Course n Compuaonal Algebrac Number Theory,'' Graduae Texs n Mahemacs, Vol. 138, Sprnger-Verlag, New YorkBerln, G. H. Hardy and E. M. Wrgh, An Inroducon o he Theory of Numbers,'' 5h ed., Oxford Scence Publ., Oxford, UK, 1979.

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