Fermat Varieties of Hodge Witt Type

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1 JOURNAL OF ALGEBRA 180, ARTICLE NO Ferat Varetes of HodgeWtt Type Kesuke Tok Koyaa 3-5-5, Nera-ku, Tokyo 176, Japan Councated by Walter Fet Receved January 31, INTRODUCTION Let X be a sooth proectve varety over a perfect feld k of characterstc p 0, and W be the rng of Wtt vectors on k. Then there are two spectral sequences. The one s the Hodge to de Rha spectral sequence E, H X, H Ž Xk., 1 X DR and the other s the slope spectral sequence E, H X, W H Ž XW.. 1 X crs By eans of the unversal coeffcent theore Žcf., II4, Ž , we can descrbe the process of connectng the crystallne cohoology group H Ž XW. wth the de Rha cohoology group H Ž Xk. crs DR. But, the Ž relatonshp between Hodge cohoology groups H X,. X and Ž HodgeWtt cohoology groups H X, W. X s ntrcate. However, f X s ordnary n the sense of Ž 4.1. n Chaptre IV of 4 then Hodge and HodgeWtt cohoology groups are concsely connected. Ž Let X be ordnary. Then, f H XW s torsonless, H X,. crs X are Ž soorphc to the reducton of H X, W. od p for all Ž,. Ž X cf. 4, Chap. IV, Sect. 4.. Moreover, f X s ordnary then t s known that Ž H X,W. are of fnte type over W for all Ž,. X and the HodgeWtt decoposton Ž X. H n XW H X, W crs n holds n any n Žcf. 3, Now we consder a noton of HodgeWtt type soewhat wder than the noton ordnary. If X s of HodgeWtt type n all degrees n the sense of Ž 4.6. n Chaptre IV of 4, that s, $18.00 Copyrght 1996 by Acadec Press, Inc. All rghts of reproducton n any for reserved. 136

2 FERMAT VARIETIES OF HODGEWITT TYPE 137 Ž H X,W. are of fnte type over W for all Ž,. X, then we say that X s Ž of HodgeWtt type. We note that the cohoology groups H X, W. X are odules over the Deudonne rng W F, V Žcf.. 3. These two notons are closely connected as follows: Ž. a If X s ordnary then X s of HodgeWtt type Žcf. 3, Ž b. For two sooth proectve varetes X, Y over k, f X and Y are ordnary then X Y s ordnary Žcf. 4, Chap. IV, Ž , and f X s of HodgeWtt type and Y s ordnary then X Y s of HodgeWtt type Žcf. 4, Chap. IV, Ž or 3, Corollary Ž c. If X X s of HodgeWtt type then X s ordnary Žcf. 4, Chap. IV, Ž or 3, Proposton Consequently, two notons, ordnary and of HodgeWtt type, for the self-product X X are equvalent. For soe specal cases of X, we shall revew known condtons for the ordnary-ness or the HodgeWtt type-ness of X. Frst, let X be a curve. Then X s of HodgeWtt type Žcf. 7;, Chap. II.. Second, let X be an abelan varety. Then X s ordnary f and only f the p-rank of X s equal to d X Žcf., II-7, 7.1; 4, Chap. IV, Ž , and X s of HodgeWtt type f and only f the p-rank of X s equal to d X or d X 1 Žcf. 3, Corollary Thrd, let X be a K 3 surface. Then X s ordnary f and only f the heght of BrX k s equal to one, and X s of HodgeWtt type f and only f the heght of BrX k s of fnte type Žcf., II-7, 7... Fourth, let X be a sooth coplete ntersecton n a proectve space. Then, f the nveau de Hodge of X n the sense of 1 or 6 s at ost one, we know, through 9, that X s of HodgeWtt type. Accordng to 6, when we consder the sooth coplete ntersecton V Ž,,...,. n 1 d of d hypersurfaces of degrees 1,,..., n the general poston n an Ž n d. d -densonal proectve nd space Ž n 0, d 0., the coplete ntersecton V Ž,,...,. n 1 d wth the nveau de Hodge 1 s as follows: V Ž,,. Ž n odd 3,. n VŽ,.Ž n,v. Ž, 3., V Ž.Ž n 0,V. 3,VŽ. 3,VŽ. 4,or VŽ.Ž n 3 n cf. 6, Sect., Table 1.. In partcular, we are concerned wth the sooth hypersurface S of degree 0 defned by the equaton a x ax a x n1 n1 Ž the a are n k, and not 0. over a fnte feld k of characterstc p 0 n1 p n the hoogeneous coordnates of whch are x 0, x 1,..., x n1. Then, over an algebrac closure of k, the hypersurface S s soorphc to the Ferat varety Fn,, p of denson n 0, degree 0 over the pre feld of characterstc p 0 Ž p. defned by x x x n1

3 138 KEISUKE TOKI n n1. As for the HodgeWtt type-ness of S, t s suffcent to consder the HodgeWtt type-ness of F n,, p. And we ay alternate ts HodgeWtt type-ness wth ts beng of HodgeWtt type n degree n Žcf.. 9. Fro now on, we shall be concerned anly wth the Ferat varety F wth n,, p4 Ž n0, 0, p. n,, p. Then, consderng the above stateents we know the followng: Case Ž. 1. n,, p4 n, 1, p4 or n,, p 4; Fn,1, p and Fn,, p are ordnary and hence of HodgeWtt type. Case Ž.. n,, p4 1,, p 4; F1,, p s of HodgeWtt type. Case Ž. 3. n,,p4 wth p 1 od ; Fn,, p s ordnary and hence of HodgeWtt type. Case Ž. 4.,3, p4 wth p od 3, 3,3, p4 wth p od 3, n,, p4 3,4, p4 wth p 3 od 4, 5,3, p4 wth p od 3; F, 3, p, F 3, 3, p, F 3, 4, p, and F5, 3, p are of HodgeWtt type. Addtonally, through Suwa s crteron Ž see Secton 1., the followng result has been obtaned. THEOREM. Let the trplet n,, p4 of ntegers n 1,, and a pre nuber p wth p and p 1 Ž od. be gen. Then we hae the asserton F s of HodgeWtt type n,, p f and only f n,, p4 s n Case Ž 4., aboe, or n Case Ž. 5. n, 7, and p, 4 Ž od 7.. A proof of the theore wll be gven n Secton. Here, we gve the strategy of ts proof: Lea 1 n Secton 1 becoes a bass. As for the f part, snce t s known that F n Case Ž. n,, p 4 are of HodgeWtt type, we shall show t for n,, p4 n Case Ž. 5 only. For that reason, t wll be noted that n Lea 1 holds. As for the only f part, t wll be shown that n Lea 1 does not hold n all n,, p4 except for those n Cases Ž. 4 and Ž. 5. For splcty s sake we transfer all n,, p4 excluded fro Cases

4 FERMAT VARIETIES OF HODGEWITT TYPE 139 Ž. 4 and Ž. 5 to Cases 1 10 through Lea Ž see Secton.. Moreover we dvde these 10 cases nto the case n, 3 and 9 and the other case. In the forer case, by usng Leas 3, 4, 5 Ž see Secton., t wll be shown that n Lea 1 does not hold. And, n the latter case, t wll be shown by ndvdual verfcaton that n Lea 1 does not hold. Accordng to the theore, for n 1 and, t s seen that the case of non-ordnary and HodgeWtt type for F Žor S n n,, p4. n,, p s able to appear only n Case Ž. 4 or Case Ž. 5 of n,, p 4. N. Suwa assued that the asserton for n, 4 n the theore s probably vald, and he proposed deternng Ferat varetes F n,,p, asn the ttle, for any denson n. Moreover, he showed the author such known facts as those stated above concernng the HodgeWtt type-ness for sooth proectve varetes over a perfect feld of postve characterstc. The presentaton of the frst proof of the theore gven by the author was proved by N. Suwa, and the author apprecates hs perttng h to wrte the presentaton of the proof here Ž cf. Secton.. The author expresses hs sncere grattude to Professor N. Suwa.. PRELIMINARIES Throughout the present paper, n,, p4 denotes the trplet of ntegers such that n 0, 0, wth p a pre nuber, p. The notaton eans the potency of a set. n For n and w w, w,...,w, let the nteger w 0 1 n1 be defned by n1 w w, where denotes the set of all ntegers. Moreover, n n and, we set 0 4 n W w ;0w Ž 0,1,,..., n 1., w0 od, 4 W w W ; w Ž 1. Ž 0,1,,..... Then we have n W W. 0 Let be an nteger relatvely pre to. We consder the well known acton on W,.e., for w W, w w 4, w 4,..., w 4, 0 1 n1

5 140 KEISUKE TOKI where each w 4 denotes the reander of dvdng w by. Then, for two ntegers, relatvely pre to, we have Ž f od., Ž. on W. Let the nteger w be defned by Then we obtan that w w 1 for w W n n and. w W f and only f w. The followng lea s ndspensable for our proof of the theore. LEMMA 1 Ž Suwa s crteron.. Let n,, p4 wth n 1,, p, and p 1 Ž od. be gen. Let W be the set n n and as aboe. Then the followng condtons are equalent: Ž. The Ferat arety Fn,, p s of HodgeWtt type. The trplet n,, p4 satsfes the asserton At each eleent w n W, we hae or or p w w p ww 0, 14 for all non-negate ntegers for all non-negate ntegers and p w w 1 for soe poste nteger p ww 0,14 for all non-negate ntegers and p w w 1 for soe poste nteger. Proof. We shall present a proof due to N. Suwa. Put X F. n,, p Moreover, accordng to 5, we put h,n 0 d H n Ž X, X., n, where denotes the Kronecker delta. For 0, let the ntegers, be defned by rank H n X, W Ž tor s V., W X rank H n X, W Ž tor s F.. W X

6 FERMAT VARIETIES OF HODGEWITT TYPE 141 Then the HodgeWtt type-ness of X,.e., X beng of HodgeWtt type n degree n, s characterzed by the condton h 0,n 0 0 and h,n 0 1 for 1 n, Ž C. Ž. through the theory of Hodge and Newton polygons cf. 3, Sect. 6. On the other hand, we know that h,n W 0 for 0 n Žcf. 5, Introducton; 8, Sect. 3.. Therefore the condton Ž. n Lea 1 s characterzed by the condton W0 0 and W 1 for 1 n. Ž C. Let f be the order of p4 n the ultplcatve group. For w W, we put f1 AŽ w. AŽ w. p w and Ž w.. f 0 Then we know, through the theory of Hodge and Newton polygons, that 1 Ž w. and Ž w. for 0, w where the suaton s taken over all w W wth Ž w. 1. Frst, we assue that the condton Ž. s satsfed. Then, through the nducton on, we shall show that the condton holds n W for 0. In the followng, f Ž w. denotes the nuber of tes appears n the set p w ;0f4 for w W. Then f Ž w. 0 for 0 and f f Ž w., AŽ w. f Ž w. for any w W. 0 0 Let n; W 4 0. Then W 0 for 0 0. For w W, we have the followng asserton: Ž w. 1 f and only f p w for any and p w 0 for soe. w

7 14 KEISUKE TOKI Then, by usng C, we have the relaton W w:0ž w. 01 Ž 1. faž w. f Ž 1. faž u. Ž 1. fa 0 0 Ž Ž 1. f AŽ w.. 0 f u f f w Ž 1. f Ž u. 0 0 Ž 1. f 0 0 Ž 1. f Ž w. 0 f u f f w W, where u,,...,w are eleents of W taken n the suaton satsfyng the followng condton: the sets p u;0f 4, p ;0f 4,..., p w;0f4 are dsont. Ž.1. Therefore, n ths relaton, f Ž u. f f Ž w. 0 for 0 and W f Ž u. f f Ž w Then we get the dentty 4 W w W ; w, p w, 14 for any, whch shows that the condton holds n W 0. Let. Suppose that s satsfed n W Ž the nducton hypothess.. Then, wth w W, the assertons Ž.., Ž.3. hold: 1 Ž w. f and only f p w 1, 4 for any and p w 1 for soe. Ž.. f Ž w. 1 then p w for any. Ž.3.

8 FERMAT VARIETIES OF HODGEWITT TYPE 143 Now we consder the subsets, N of W defned by 1 4 w W ;w satsfes the second asserton of Ž.., 1 Then N 4 N w W ;w satsfes the second asserton of Ž and, under the nducton hypothess, we have 4 w W ; w, p w1 for soe. 1 And oreover, through., we have the relaton AŽ w. Ž 1. f f 1 w:1ž w. Ž AŽ u. Ž 1. f. Ž A Ž 1. f. Ž AŽ w. Ž 1. f. fž u. f fž w., 1 Ž.4. where u,,...,w are eleents of W taken n the suaton satsfyng Ž.1., and through Ž.3., we have the relaton Ž 1. f AŽ w. f w:ž w. 1 Ž Ž 1. faž u.. Ž Ž 1. fa. Ž Ž 1. f AŽ w.. fž u. f fž w. Ž 1. f Ž u. Ž 1. f Ž 1. f Ž w. fž u. f fž w. N, Ž.5. where u,,...,w are eleents of W taken n the suaton satsfyng Ž.1.. And then, by Ž.4. and Ž.5., we obtan N W

9 144 KEISUKE TOKI Moreover, by usng C, we obtan N, and hence 1 1 WN 1. Ž.7. Consderng the relaton Ž.5., under N, we get f Ž u. f f Ž w. 0 for. Therefore we have and hence N fž u. f fž w. w W ; w, p w,14 for any 4 N w W ; w, p w,14 for any 4. Therefore Ž.7. shows that the condton holds n W. Second, we assue that the condton s satsfed. Then f we take w W, the asserton Ž.. holds n 1 n, at w. Thereby we have and and hence, by.6, Then, snce we know that 1 Ž w. W h 0,n, w:0ž w. 1, N for 1 n. 1 1 W h,n for 1 n. 1 0 n Ž. h,n n through 3, Proposton 6..7, we obtan h 0,n, h,n for 1 n, whch shows that C holds,.e., that the condton holds. In next secton, a proof of the theore wll be perfored by usng Lea 1 as a bass.

10 FERMAT VARIETIES OF HODGEWITT TYPE PROOF OF THE THEOREM Throughout ths secton, for n,, p 4, we assue that n 1,, p, and p 1 Ž od. and let W be the set n n, 4 as n Secton Proof of the f Part n the Theore Let n, 7, and p, 4 od 7. Then we have W0 Ž 1,1,1,4., Ž 1,1,,3., Ž 1,,,., perutatons of the respectvely 4, W1 Ž 1,1,6,6., Ž 1,,5,6., Ž 1,3,4,6., Ž 1,3,5,5., Ž 1,4,4,5., Ž,,4,6., Ž,,5,5., Ž,3,3,6., Ž,3,4,5., Ž,4,4,4., Ž 3,3,3,5., Ž 3,3,4,4., perutatons of the respectvely 4, W Ž 3,6,6,6., Ž 4,5,6,6., Ž 5,5,5,6., perutatons of the respectvely 4, and the order of Ž or 4. n the group Ž 7. becoes 3. It s then easly verfed that p w w Ž 0, 1,,.... satsfy the condton n Lea 1 for each w W W0 W1 W. Slarly, t s also verfed that the condton n Lea 1 s satsfed for each n,,p4 n Case Ž. 4. Therefore we obtan the f part n the theore fro Lea Proof of the Only f Part n the Theore We shall prove that condton Ž. or condton n Lea 1 s not satsfed for all trplets n,, p4 except for those of Cases Ž.Ž. 4, 5 n the theore. We set the stateent Nn, Ž, p:. The Ferat varety F n,, p s not of HodgeWtt type. Then we obtan the followng lea. LEMMA. We hae the asserton: If NŽ n,, p. holds then NŽ n,, p. holds.

11 146 KEISUKE TOKI Proof. Let W, W be sets n n, 4, n, 4, respectvely, as n Secton 1. Assue that Nn, Ž, p. holds. Then, by Lea 1, there exsts an eleent w n W such that Ž. a p ww for soe postve nteger, or Ž b. p ww for soe postve nteger. For such an eleent w Ž w, w,...,w. n W, we put 0 1 n1 w Ž w 0,w 1,...,w n1,1,1.. Then we have w W and w w 1. In the case Ž. a, snce we obtan n1 4 0 p w p w p w, p w Ž p w1. Ž w3. Ž w.. Therefore, by p w 0 Ž od., we have and hence p w w. In the case Ž b., snce p w Ž w3. p w p w Ž 1. Ž p w1. Ž 1., we obtan p w Ž w1. Ž 1. Ž w1. w. Therefore, by p w 0 Ž od., we have p w Ž w1. and hence p w w. Thus the condton n Lea 1 s not satsfed at w, and hence Nn, Ž, p. holds. It s easly seen, by eans of Lea, that Nn,,p holds n all trplets n,, p4 except those of cases Ž. 4, Ž. 5 n the theore, f t s

12 FERMAT VARIETIES OF HODGEWITT TYPE proved that Nn,,p holds n all trplets n,, p n Cases 1 10 : 1 n, 4, p 3 Ž od 4. n, 5, p, 3, 4 Ž od 5. 3 n, 6, p 5 Ž od 6. 4 n, 7, p 3, 5, 6 Ž od 7. 5 n, 8, p 1 Ž od. 6 n3, 5, p 1 Ž od. 7 n4, 3, p Ž od 3. 8 n4, 7, p, 4 Ž od 7. 9 n5, 4, p 3 Ž od n 7, 3, p Ž od 3.. Fro now on, for each n,, p4 n these 10 cases, we shall show that Nn,,p holds, that s, that the condton n Lea 1 does not hold n n,, p 4. Case 1. Take the eleent w Ž 1, 1, 1, 1. n W. Then w 0 and 3w. Case. Take the eleent w Ž 1, 1, 1,. n W. Then w 0 and w3 w4w. Case 3. Take the eleent w Ž 1, 1, 1, 3. n W. Then w 0 and 5w. Case 4. Take the eleents Ž 1,,,., w Ž 1, 1, 1, 4. n W. Then w0 and 3 5 w 6 w. Case Ž Take the eleents u Ž,,,., Ž 1, 1, 3, 3., wž 1, 1, 1, 5. n W. Then u w 0 and 3 u 5 7w. Case Ž Take the eleent w Ž 1, 1, 1, 1, 1. n W. Then w0 and w 3 w 4 w 3. Case Ž Take the eleent w Ž 1, 1, 1, 1,. n W. Then w0 and 5 w 3. Case Ž Take the eleent w Ž 1, 1, 1, 1, 3. n W. Then w0 and w 3 4 w 4 w 5 w, 6 w 3. Case Ž Take the eleents u Ž 1, 1,,,., Ž1, 1, 1,, 3., w Ž 1, 1, 1, 1, 4. n W. Then u w 0 and 3 u 5, 7 w 3.

13 148 KEISUKE TOKI Case 7. Take the eleent w Ž 1, 1, 1, 1, 1, 1. n W. Then w 1 and w 3. Case 8. Take the eleent w Ž 1, 1, 1, 1, 1,. n W. Then w 0 and w 4 w. Case 9. Take the eleent w Ž 1, 1, 1, 1, 1, 1,. n W. Then w 1 and 3 w 4. Case 10. Take the eleent w Ž 1, 1, 1, 1, 1, 1, 1, 1, 1. n W. Then w and w 5. As for Cases 5, 6 wth 9, our proof wll be carred out as follows. In the followng,,. denotes the set of all real nubers x satsfyng x. Moreover denotes the greatest nteger not exceedng the real nuber. And, for the gven n,, p 4, let be an nteger relatvely pre to satsfyng 1, p Ž od.. Then cases of, 3, 3 do not occur Case of n, 9 Case a. 3. In ths case, the followng lea s useful. LEMMA 3. For 9, we hae / / 3,,, 3k k kž 1. 3 where k. Proof. Let x 3,. be gven. If 3 x 3 then there exsts an nteger k satsfyng 3 x k x, because x 3 x 1. Then, for these x, k, we have k 3 Žhence k Ž and x 3k, k..if 3x then x 3k, k. for k. Let 9 and 3,. be gven. By Lea 3, there exsts an nteger c wth c Ž 1. 3 Ž hence 0 c 3. such that 3c c. For such an arbtrary nteger c, the followng asserton holds: Take the eleent w Ž c, c, c, 3c. n W. Then we have w0 and w.

14 FERMAT VARIETIES OF HODGEWITT TYPE 149 In fact, the forer s trval. And the latter s due to 0 c, 3c w 3 1 4, w 0 od. Case b. 1. In ths case, the followng lea s useful. LEMMA 4. For 9, we hae Ž 3k 1. Ž k1. 1,,, 6k k / / 1kŽ 1. 3 where k. Proof. Let x 1,. be gven. If 3 x then we have x Ž 3k 1. 6k, Ž k 1. k. for k 1. Assue that 1 x 3. Snce we have the equvalence Ž 3k 1. Ž k1. x 6k k k 6x3 x Ž for an nteger k 0 and a real nuber x, we consder the set 5 1 KŽ x. ½k; k, k. 6x3 x 3 We shall show that K Ž x. for 9 and x 1, 3.. If 9 and x 1 then Ž 1. 3 K Ž x. by 1 6x3 6 3 x for such, x. If 9 and 1 x Ž 3. then 6 1 K Ž x. by 1 1 6x x for such, x. If 9 and Ž 3. x 3 then K Ž x. by x3, Ž x 6x3 for such, x.

15 150 KEISUKE TOKI If 9 then we have Ž Thus Lea 4 has been shown. Reark. Let 9, 1 x 3 be gven. Consder the K Ž x. as above for such, x. Then we reark the followng: Ž. nk; k K Ž x.4, because Ž 6 x 3. 1 for x 3. K Ž 1. for 9, because both Ž , Ž 1. 3K Ž 1. by 6ŽŽ and Ž 1. 3 for 9. K Ž x. for 9 and Ž 3. x 3, by Ž Ž v. x; Ž. x Ž Let 9an 1,. be gven. If 9, 10 then 3. Case b We ay assue 11, here. By Lea 4 and Reark, Ž v., we take an odd nteger k satsfyng Ž 3k 1. Ž k1. Ž k k wth 1 k Ž Then 3 k 3 by Reark, Ž.. For such an arbtrary odd nteger k, the followng asserton holds: Take the eleent w Ž k, k, k, 3k. n W. Then we have w 0 and w. Obvously, we have w W and w 0. We shall show w. Frst, we ay wrte Then we obtan the nequaltes k 3 t 0 t. Ž by the equvalence for x, k k t Ž

16 FERMAT VARIETIES OF HODGEWITT TYPE 151 Let c be the postve nteger defned by c. Ž Then we have k Ž 1 t. ck Ž od., where s defned by. Ž In fact, by Ž , Ž , Ž , we have k k ck k Ž 1t. Ž 1t.. Moreover t s seen that k4 Ž 1t. c k. Ž In fact, by Ž , Ž , Ž , we have c Ž 1t. c k Ž 3c. Ž c. t 3 3c3. c Therefore we obtan 0 Ž 1 t. ck Ž 3c. Ž c. t 3 Ž 3c. Ž c. ž / Ž 3c. 3c3

17 15 KEISUKE TOKI fro Ž , Ž , and the second nequalty of Ž Hence we have Ž On the other hand, by Ž , Ž , we have 59 59c c 3 Therefore, by usng , we obtan k 1t ck Ž 3c. Ž c. t 4 59c9 Ž 3c. Ž c. 6c 3 ž / ž Ž. ž // ž / 1 Ž 3c. 59c c 3 59c fro , , and the frst nequalty of Hence we have 4 3 k Therefore, by w Ž k 4, k 4, k 4, Ž 3k.4. and Ž , we have and hence w3 Ž 1. 4 w3 by w 0 Ž od.. Consequently, w has been shown.

18 FERMAT VARIETIES OF HODGEWITT TYPE 153 Case b-. 3. In ths case, the followng asserton holds: Take the eleent w Ž 1,1,1, 3. n W. Then we have w 0 and w. In fact, the forer s trval. And the latter s due to 3 w and w 0 od. Case c.. Snce s also relatvely pre to and satsfes 3 by 9, we apply Case a to. Case d. 1 Ž od. and Ž 1.. Let 4. Then holds and s relatvely pre to. Moreover t s seen that 1 for 5. In fact, when s odd, we 1 1 have Ž 1. Ž1 Ž 1.. Ž Ž 1. Ž od. and 0 Ž 1.. Hence 1 Ž 1. and 1 for 5. And, when s even, we 3 3 have Ž 1. Ž Ž od. and Hence 11. Therefore Ž,. 1, 1,. And then belongs to Case a or Case b or Case c Case of n 3, 9 Case e. 3. In ths case, the followng lea s useful. LEMMA 5. For 9, we hae 3,,, k k. / 1kŽ 1. 4 where k. Proof. Let x 3,. be gven. If x then we have x k, k. for k 1. Assue that 3 x. Snce we have the equvalence x k k k x x

19 154 KEISUKE TOKI for an nteger k 0 and a real nuber x 0, we consder the set 1 LŽ x. ½k; k, k 5. x x 4 We shall show that L Ž x. for 9 and 3 x. If 4 x then there exsts an nteger k wth x k x, because x x 1. Then k x 4 Žhence k Ž for such k, and hence k L Ž x.. If 3 x 4 then, snce x 6 4 x, we have k x x for k Ž 1. 4 n the case of 9 1. for k 6 1 Ž 1. 4 n the case of 13. Therefore k L Ž x. for these k. If x 3 then we have 3 k, k. for k Ž 1. 4, because t s easly seen that f 9 then 3k Ž and 6k for kž Let 9 and 3,. be gven. By Lea 5, there exsts an nteger c wth 1 c Ž 1. 4 Ž hence 0 c 4. such that c c. For such an arbtrary nteger c, the followng asserton holds: Take the eleent w Ž c, c, c, c, 4c. n W. Then we have w0 and w. In fact, the forer s trval, and the latter s due to 0 c, 4cw4Ž 1. 5 and w 0 Ž od.. Case f.. Snce s also relatvely pre to and satsfes 3 by 9, we apply Case e to. Thus t has been shown that the condton n Lea 1 s not satsfed n each n,, p4 except for those of cases Ž.Ž. 4, 5 n the theore. Therefore the only f part n the theore has been proved. REFERENCES 1. P. Delgne, Cohoologe des ntersectons copletes, ` n SGA 7 II, Expose XI, pp. 3961, Lecture Notes n Matheatcs, Vol. 340, Sprnger-Verlag, BerlnNew York, L. Illuse, Coplexe de de RhaWtt et cohoologe crstallne, Ann. Sc. Ecole Nor. Sup. 1 Ž 1979.,

20 FERMAT VARIETIES OF HODGEWITT TYPE L. Illuse, Fnteness, dualty and Kunneth theore n the cohoology of the de RhaWtt coplex, n Proceedngs of JapanFrance Conference n Algebrac Geoetry, Tokyo and Kyoto, 198, pp. 07, Lecture Notes n Matheatcs, Vol. 1016, Sprnger-Verlag, BerlnNew York, L. Illuse et M. Raynaud, Les sutes spectrales assocees au coplexe de de RhaWtt, Publ. Math. IHES 57 Ž 1983., N. Kobltz, P-adc varaton of the zeta-functon over fales of varetes defned over fnte felds, Coposto Math. 31 Ž 1975., M. Rapoport, Copleent `a l artcle de P. Delgne La conecture de Wel pour les surfaces K3, Inent. Math. 15 Ž 197., J.-P. Serre, Sur la topologe des varetes algebrques en caracterstque p, n Proceedngs, Syposu Internaconal de Topologa Algebraca, 1958, pp T. Shoda and T. Katsura, On Ferat varetes, Tohoku ˆ Math. J. 31 Ž 1979., N. Suwa, HodgeWtt cohoology of coplete ntersectons, J. Math. Soc. Japan 45 Ž 1993.,