ON THE NUMBER OF RATIONAL POINTS ON CURVES OVER FINITE FIELDS WITH MANY AUTOMORPHISMS

ON THE NUMBER OF RATIONAL POINTS ON CURVES OVER FINITE FIELDS WITH MANY AUTOMORPHISMS ANTONIO ROJAS-LEÓN Abstract. Using Wil scnt, w giv bouns for th numbr of rational points on two familis of curvs ovr finit fils with a larg ablian group of automorphisms: Artin-Schrir curvs of th form y q y = f(x) with f F q r [x], on which th aitiv group F q acts, an Kummr curvs of th form y q 1 = f(x), which hav an action of th multiplicativ group F q. In both cass w can rmov a q factor from th Wil boun whn q is sufficintly larg. 1. Introuction Lt k = F q b a finit fil of charactristic p an C a gomtrically connct smooth curv of gnus g in P 2 k. Th wll known Wil boun givs th following stimat for th numbr of points N r of C rational ovr F q r for vry r 1: N r q r 1 2gq r 2 This boun is sharp in gnral if w fix C an tak variabl F q r, in th sns that lim sup log q N r q r 1 = r r 1 2 an N r q r 1 lim sup = 2g. r 1 q r 2 Howvr, for som curvs it is possibl to improv this boun for larg valus of q if w kp r unr control. In th articl [6] it was provn that this was th cas for th affin Artin-Schrir curv A f fin by y q y = f(x) with f k[x], whos singular mol has gnus ( 1)(q 1)/2 an only on point at infinity. For A f on can gt an stimat of th form N r q r C,r q r+1 2 unr crtain gnric conitions on f (whr N r is now th numbr of points on th affin curv A f ). In this formula C,r is inpnnt of q (mor Partially support by P08-FQM-03894 (Junta Analucía), MTM2007-66929 an FEDER. 1

2 ANTONIO ROJAS-LEÓN prcisly, it is a polynomial in of gr r) so it givs a grat improvmnt of th Wil boun if q is larg. This stimat was obtain by writing N r q r as a sum of aitiv charactr sums N r q r = ψ(tr kr/k(f(x))), t k x k r ach of thm boun by ( 1)q r 2, an thn showing that thr is som cancllation on th outr sum so that th total sum is boun by O q (q r+1 2 ). In this articl w tak a iffrnt approach: sinc N f = q #{x k r Tr kr/k(f(x)) = 0}, using Wil scnt w construct a hyprsurfac in A r k whos numbr of rational points is prcisly th numbr of x k r such that Tr kr/k(f(x)) = 0. Unr crtain conitions th projctiv closur of this hyprsurfac is smooth, so w can us Dlign s boun to stimat its numbr of rational points an uc th boun (1) N r q r ( 1) r q r+1 2. This mtho is crtainly lss powrful than th on us in [6]. In particular, th hypothss w n f to satisfy in orr to gt (1) ar mor rstrictiv than thos in [6, Corollary 3.4, Corollary 4.2], an th constant ( 1) r is also slightly wors (notic that th cofficint of th laing trm of C,r in [6, Corollary 3.4] crass rapily as r grows). On th othr han, this mtho works vn whn f is fin only ovr k r, not just ovr k, thus giving a positiv answr to on of th qustions pos in th introuction of [6]. W also apply th sam procur to stuy th othr xampl propos in th introuction of [6]: Kummr curvs, a particular typ of suprlliptic curvs of th form E f : y q 1 = f(x) whr is( a positiv ) ivisor of q 1. ( Ths ) curvs can hav gnus anywhr btwn q 1 1 ( 2)/2 an q 1 1 ( 1)/2, but in any cas for fix th Wil stimat givs N r q r = O(q r 2 +1 ). Hr w also hav a larg ablian group acting faithfully on E f, namly th multiplicativ group k /µ of non-zro lmnts of k moulo th subgroup of -th roots of unity, so on also xpcts to b abl to rmov a q factor from th boun. W can writ N f = δ + q 1 #{x k r N kr/k(f(x)) = λ} λ =1 whr δ is th numbr of roots of f in k r. Again using Wil scnt w will construct a hyprsurfac (or rathr a on-paramtr family of hyprsurfacs) W λ in A r k such that th numbr of rational points of W λ ovr k is #{x

ON THE NUMBER OF RATIONAL POINTS ON CURVES OVER FINITE FIELDS 3 k r N kr/k(f(x)) = λ}. Th hyprsurfacs W λ ar highly singular at infinity, so this cas rquirs a tail stuy of th cohomology of this family, which taks most of th lngth of this articl. Th scnt mtho works surprisingly wll in this cas, an w gt th stimat N f q r δ + 1 r( 1) r (q 1)q s 1 2 unr th only hypothsis that f is squar-fr of gr prim to p. Th fact that th scnt mtho works wll in th Kummr cas an not so wll in th Artin-Schrir cas has an xplanation: for Artin-Schrir curvs, w can writ N r q r as a sum of aitiv charactr sums, paramtriz by th st of non-trivial aitiv charactrs of k. Upon choosing a non-trivial charactr ψ, this st can b intifi with th st of k-points of th schm G m = A 1 {0}, an th corrsponing xponntial sums ar th local Frobnius tracs of th r-th Aams powr of som gomtrically smisimpl l-aic shaf on G m. In orr to gt a goo stimat (i.., of th form O(q r+1 2 )), w n (all componnts of) this Aams powr to not hav any invariants whn rgar as rprsntations of π 1 (G m ). Whn oing Wil scnt, what w ar rally looking at is th invariant spac of th (Frobnius twist) r-th tnsor powr of this shaf, which is a much largr objct. In particular, w may gt som unsir aitional invariants. In this cas th monoromy group is smisimpl, an thrfor its trminant has som finit orr N. Thn its N-th tnsor powr is finitly going to hav non-zro invariant spac, which (in gnral) woul not b prsnt if w just consir th Aams powr. On th othr han, for th Kummr cas w can writ N r q r as a sum of multiplicativ charactr sums, paramtriz by th st of all non-trivial multiplicativ charactrs χ of k of orr ivisibl by q 1. Evn though it is not possibl to raliz ths sums as th Frobnius tracs of an l-aic shaf on a schm, rcnt work of Katz ([5], spcially rmark 17.7) shows that ths sums ar approximatly istribut lik tracs of ranom lmnts on a compact Li group. For gnric f, this group is th unitary group U 1. In particular, all tnsor powrs of this rprsntation (th stanar ( 1)-imnsional rprsntation of U 1 ) hav zro invariant spac, an this maks our mtho work wll. W conjctur that on shoul gt a similar stimat for Kummr hyprsurfacs of th form y q 1 = f(x 1,..., x n ) whr f k r [x 1,..., x n ] is in som Zariski opn st, namly on shoul hav N r q nr C n,,,r q nr+1 2 for som C n,,,r inpnnt of q. Howvr, th conitions in this cas shoul ncssarily b mor rstrictiv, as shown by th xampl y q 1 = x 1 x 2 + 1

4 ANTONIO ROJAS-LEÓN in which f(x 1, x 2 ) = x 1 x 2 + 1 is as smooth as it can b but it is asy to chck that N r = q 2s + (q 2)q r for vry o q an vry r. Th author woul lik to thank Daqing Wan for pointing out som mistaks in an arlir vrsion of this articl. 2. Th Artin-Schrir cas Lt k = F q b a finit fil of charactristic p, k r = F q r th xtnsion of k gr r insi a fix algbraic closur k, an f k r [x] a polynomial of gr prim to p. Lt A f b th Artin-Schrir curv fin in A 2 k r by th quation (2) y q y = f(x) an not by N f its numbr of k r -rational points. Th group of k-rational points of th affin lin A 1 acts on A f (k r ) by λ (x, y) = (x, y + λ). By th gnral Artin-Schrir thory, an lmnt z k r can b writtn as y q y for som y k r if an only if Tr kr/k(z) = 0, an in that cas thr ar xactly q such y s. Thrfor N f = q #{x k r Tr(f(x)) = 0} whr Tr = Tr kr/k is th trac map k r k. Lt us rcall th Wil scnt stup (cf. for instanc [3]). Fix an basis B = {α 1,..., α r } k r of k r ovr k, an consir th polynomial S(x 1,..., x r ) = r j=1 f σj (σ j (α 1 )x 1 + +σ j (α r )x r ) k r [x 1,..., x r ], whr σ Gal(k r /k) is th Frobnius automorphism an f σj mans applying σ j to th cofficints of f. Sinc th cofficints of S ar invariant unr th action of Gal(k r /k), S k[x 1,..., x r ]. Lt V b th subschm of A r k fin by th polynomial S. Notic that a point (x 1,..., x r ) k r is in V (k) if an only if r j=1 f σj (σ j (α 1 )x 1 + + σ j (α r )x r ) = r j=1 σj (f(α 1 x 1 + + α r x r )) = 0, if an only if Tr(f(α 1 x 1 + + α r x r )) = 0. Sinc {α 1,..., α r } is a basis of k r ovr k, w conclu that (3) N f = q #{x k r Tr(f(x)) = 0} = q #V (k). On th othr han, V k r is isomorphic, unr a linar chang of variabl, to th hyprsurfac fin by f σ (x 1 ) + f σ2 (x 2 ) + + f σr (x r ) = 0 in A r k r. Sinc is prim to p, V has at worst isolat singularitis, an its projctiv closur has no singularitis at infinity. In particular, w gt: Thorm 2.1. Lt f k r [x] b a polynomial of gr prim to p. If th hyprsurfac fin in A r k by f σ (x 1 ) + f σ2 (x 2 ) + + f σr (x r ) = 0 is nonsingular, th numbr N f of k r -rational points on C f satisfis th stimat N f q r ( 1)r+1 ( 1) r ( 1) q r+1 2 + ( 1)r ( 1) r 1 ( 1) q r 2

ON THE NUMBER OF RATIONAL POINTS ON CURVES OVER FINITE FIELDS 5 ( 1) r q r+1 2. Proof. If V is th projctiv closur of V in P r k an V 0 = V V, w hav #V (k) q r 1 = # V (k) #V 0 (k) (#P r 1 (k) #P r 2 (k)) = so = (# V (k) #P r 1 (k)) (#V 0 (k) #P r 2 (k)) N r q r = q #V (k) q r 1 q ( # V (k) #P r 1 (k) + #V 0 (k) #P r 2 (k) ) ( 1)r+1 ( 1) r ( 1) q r+1 2 + ( 1)r ( 1) r 1 ( 1) q r 2 sinc V an V 0 ar non-singular of gr an imnsion r 1 an r 2 rspctivly. As not in [6, n of sction 3], th non-singularity conition is gnric in vry linar spac of polynomials of gr that contains th constants: if λ k r is such that Tr kr/k(λ) is not a critical point of f σ (x 1 ) + + f σr (x r ), thn f λ satisfis th conition. Th orr of magnitu of th constant is polynomial in of gr r, ssntially th sam as in [6]. Howvr, th laing cofficint thr crass rapily with r, whras hr it is always 1. Th sam procur can b appli to Artin-Schrir hyprsurfacs. Lt f k r [x 1,..., x n ] b a Dlign polynomial, that is, its gr is prim to p an its highst homognous form fins a non-singular projctiv hyprsurfac. Lt B f b th Artin-Schrir hyprsurfac fin in A n+1 k r by th quation (4) y q y = f(x 1,..., x n ) an not by N f cas, w hav its numbr of k r -rational points. Lik in th prvious N f = q #{(x 1,..., x n ) k n r Tr(f(x 1,..., x n )) = 0} whr Tr is th trac map k r k. Lt S k r [{x i,j 1 i n, 1 j r}] b th polynomial ) r r σ j (α i )x 1,i,..., σ j (α i )x n,i j=1 f σj ( r i=1 which has cofficints in k, an V th subschm of A nr k fin by S. Again N f = q #V (k), an V k r is isomorphic to th hyprsurfac fin by f σ (x 1,1,..., x n,1 ) + + f σr (x 1,r,..., x n,r ) = 0. Sinc this hyprsurfac is non-singular at infinity, w gt i=1

6 ANTONIO ROJAS-LEÓN Thorm 2.2. Lt f k r [x 1,..., x n ] b a Dlign polynomial of gr prim to p. If th hyprsurfac fin in A nr k by f σ (x 1,1,..., x n,1 ) + + f σr (x 1,r,..., x n,r ) = 0 is non-singular, th numbr N f of k r -rational points on C f satisfis th stimat N r q nr ( 1)nr+1 ( 1) nr ( 1) q nr+1 ( 1) nr q nr+1 2. 3. Th Kummr cas 2 + ( 1)nr ( 1) nr 1 ( 1) Fix a positiv intgr which ivis q 1. Lt E f b th Kummr curv fin in A 2 k r by th quation (5) y q 1 = f(x) an not by N f its numbr of k r -rational points. Th group k of k- rational points of th torus G m acts on E f (k r ) by λ (x, y) = (x, λ y). A non-zro lmnt z k r can b writtn as y q 1 for som y k r if an only if N kr/k(z) = 1, an in that cas thr ar xactly q 1 such y s. Thrfor N f = #Z(k r ) + q 1 #{x k r N(f(x)) = 1} whr N is th norm map k r k an Z is th subschm of A 1 k r fin by f = 0. If w apply th Wil scnt mtho to intify th st {x k r N(f(x)) = 1} with th st of k-rational points on a schm ovr k lik w i in th Artin-Schrir cas w gt a schm gomtrically isomorphic to th on fin by (f σ (x 1 ) f σr (x r )) = 1, which is highly singular at infinity. In particular, its highr cohomology groups o not vanish. Howvr, ths cohomology groups ar rlativly asy to control as w will s. Fix an basis B = {α 1,..., α r } k r of k r ovr k, an consir th polynomial T (x 1,..., x r ) = r j=1 f σj (σ j (α 1 )x 1 + + σ j (α r )x r ) k r [x 1,..., x r ], whr σ Gal(k r /k) is th Frobnius automorphism an f σj mans applying σ j to th cofficints of f. Th cofficints of T ar invariant unr th action of Gal(k r /k), so T k[x 1,..., x r ]. For any λ k, lt W λ b th subschm of A r k fin by T = λ. A point (x 1,..., x r ) k r is in W λ (k) if an only if r j=1 f σj (σ j (α 1 )x 1 + + σ j (α r )x r ) = r j=1 σj (f(α 1 x 1 + + α r x r )) = λ, if an only if N(f(α 1 x 1 + + α r x r )) = λ. Sinc {α 1,..., α r } is a basis of k r ovr k, w conclu that (6) N f = #Z(k r ) + q 1 #{x k r N(f(x)) = 1} = = #Z(k r )+ q 1 #{x k r N(f(x)) = λ} = #Z(k r )+ q 1 λ =1 #W λ (k). λ =1 q nr 2

ON THE NUMBER OF RATIONAL POINTS ON CURVES OVER FINITE FIELDS 7 Now W λ k r is isomorphic, unr a linar chang of variabls, to th hyprsurfac fin by f σ (x 1 )f σ2 (x 2 ) f σr (x r ) = λ. This hyprsurfac is highly singular at infinity, so in gnral w ar not going to obtain goo bouns for its numbr of rational points. For instanc, in th simplst cas f(x) = x, th hyprsurfac is a prouct of r 1 tori. In particular, it has non-zro cohomology with compact support in all grs btwn r 1 an 2r 2. In orr to unrstan th cohomology of ths hyprsurfacs, it will b convnint to consir th ntir family f σ (x 1 ) f σr (x r ) = λ paramtriz by λ an stuy th rlativ cohomology shavs. W will o this in a mor gnral stting. Lt f 1,..., f s k r [x] b polynomials of gr, an lt F s : A s k r A 1 k r b th map fin by F s (x 1,..., x s ) = f 1 (x 1 ) f s (x s ). Fix a prim l p an an isomorphism ι : Q l C, an lt K s := RF s! Ql D b c(a 1 k r, Q l ) b th rlativ l-aic cohomology complx with compact support of F s. For imnsion rasons, H j (K s ) = 0 for j < 0 an j > 2s 2. Lmma 3.1. Suppos that f i is squar-fr for vry i = 1,..., s. Thn (1) H j (K s ) Gm = 0 for j < s 1. (2) If s 2, H 2s 2 (K s ) Gm is th Tat-twist constant shaf Q l (1 s). (3) H j (K s ) Gm is gomtrically constant of wight 2(j s + 1) for s j 2s 3. (4) H s 1 (K s ) Gm contains a subshaf F s which is th xtnsion by irct imag of a smooth shaf on an opn subst V G m of rank s( 1) s, pur of wight s 1, unipotnt at 0 an totally ramifi at infinity, such that th quotint H s 1 (K s ) Gm /F s is gomtrically constant of rank s ( 1) s an wight 0. (5) H 1 c(g m, F s ) is pur of wight 0 an imnsion ( 1) s. If all f i split compltly in k r on can rplac gomtrically constant by Tat-twist constant vrywhr an Gal( k r /k r ) acts trivially on H 1 c(g m, F s ). Proof. W will proc by inuction on s, as in [1, Théorèm 7.8]. For s = 1, (1), (2) an (3) ar mpty, so w only n to prov (4) an (5). In this cas, K 1 = f 1! Ql [0]. Thr is a natural trac map f 1! Ql Q l, lt F 1 b its krnl. Sinc is prim to p, th inrtia group I at infinity acts on F 1 via th irct sum of all its non-trivial charactrs of orr ivisibl by. In particular, F 1 is totally ramifi at infinity, an is clarly pur of wight 0. Now from th xact squnc 0 F 1 f 1! Ql Q l 0 w gt H 1 c(g m, F 1 ) H 1 c(g m, f! Ql ) = H 1 c(u 1, Q l ) = H 0 c(z 1, Q l ) which is pur of wight 0, whr Z 1 A 1 is th subschm fin by f 1 = 0 an U 1 = A 1 Z 1. Morovr, im H 1 c(g m, F 1 ) = im H 1 c(g m, f 1! Ql ) im H 1 c(g m, Q l ) = im H 1 c(u 1, Q l ) im H 1 c(g m, Q l ) = 1 sinc f 1 has istinct roots in k. If f 1 splits compltly in k r, thn U 1 (k r ) = U 1 ( k r ) an thrfor Gal( k r /k r ) acts trivially on H 1 c(u 1, Q l ) an a fortiori on H 1 c(g m, F 1 ). From now on lt us not K(f 1 ) = K 1 an F(f 1 ) = F 1 in orr to kp track of th polynomial from which thy aris. W mov now to th

8 ANTONIO ROJAS-LEÓN inuction stp, so suppos th lmma has bn prov for s 1. Sinc F s is th composition of F s 1 f s an th multiplication map µ : A 2 k r A 1 k r, w gt K s = Rµ! (A 1 A 1, K s 1 K(f s )). In particular, K s Gm = Rµ! (G m G m, K s 1 K(f s )). From th istinguish triangls an F(f s )[0] K(f s ) Q l [0] F s 1 [2 s] K s 1 L s 1 whr L s 1 is th constant part of K s 1, w gt th istinguish triangls (7) Rµ! (K s 1 F(f s )[0]) K s Gm Rµ! (π 1K s 1 ), (8) Rµ! (π 1F s 1 )[2 s] Rµ! (π 1K s 1 ) Rµ! (π 1L s 1 ) an (9) Rµ! (F s 1 F(f s ))[2 s] Rµ! (K s 1 F(f s )[0]) Rµ! (L s 1 F(f s )[0]) whr π 1, π 2 : G m G m G m ar th projctions. Lt σ : G m G m G m G m b th automorphism givn by (u, v) (u, uv). Thn µ = π 2 σ an π 1 = π 1 σ, so Rµ! (π 1F s 1 ) = Rπ 2! (π 1F s 1 ) = RΓ c (G m, F s 1 ) whr th last objct is sn as a gomtrically constant objct (in fact constant if f 1,..., f s 1 split in k r ) in D b c(g m, Q l ). By part (4) of th inuction hypothsis, w hav H 1 c(g m, F s 1 ) = 0 for i = 0, 2, so RΓ c (G m, F s 1 )[2 s] = H 1 c(g m, F s 1 )[1 s]. Similarly, using th automorphism (u, v) (uv, v) w gt Rµ! (L s 1 F(f s )) = RΓ c (G m, L s 1 F(f s )) an Rµ! (π 1L s 1 ) = RΓ c (G m, L s 1 ) which ar both gomtrically constant (an constant if f 1,..., f s split in k r ). With ths ingrints w can now start proving th lmma. W hav alray sn that RΓ c (G m, F s 1 )[2 s] only has non-zro cohomology in gr s 1. By inuction, L s 1 only has non-zro cohomology in grs s 2. Sinc a constant objct has obviously no punctual sctions in G m, w uc that RΓ c (G m, L s 1 F(f s )) an RΓ c (G m, L s 1 ) only hav cohomology in grs s 1. For th first trm in th triangl (9) w hav Rµ! (F s 1 F(f s )) = Rπ 2! ((π 1 σ 1 ) F s 1 (π 2 σ 1 ) F(f s )) = = Rπ 2! (π 1F s 1 (π 2 σ 1 ) F(f s )) Its fibr ovr a gomtric point t G m is RΓ c (G m, F s 1 σt F(f s )), whr σ t (u) = t/u is an automorphism of G m. Sinc F s 1 σt F(f s ) has no punctual sctions, it os not hav cohomology in gr 0, an thrfor

ON THE NUMBER OF RATIONAL POINTS ON CURVES OVER FINITE FIELDS 9 Rµ! (F s 1 F(f s ))[2 s] only has cohomology in grs s 1. Using th istinguish triangls 7, 8 an 9 this provs (1). Sinc F s 1 is totally ramifi at infinity, H 2 c(g m, F s 1 ) = 0, so Rµ! (πi F s 1)[2 s] = RΓ c (G m, F s 1 )[2 s] has no cohomology in gr s (an in particular in gr 2s 2). On th othr han, sinc F(f s ) is totally ramifi at infinity, so ar all cohomology shavs of L s 1 F(f s ). Sinc L s 1 only has cohomology in grs 2s 4, th spctral squnc H i c(g m, H j (L s 1 ) F(f s )) H i+j c (G m, L s 1 F(f s )) implis that L s 1 F(f s ) only has nonzro cohomology in grs 2s 3. Finally, sinc F(f s ) is smooth at 0 (bcaus f s is squar-fr an thrfor étal ovr 0), σt F(f s ) is unramifi at infinity an thrfor F s 1 σt F(f s ) is totally ramifi at infinity. In particular, H 2 c(g m, F s 1 σt F(f s )) = 0 an Rµ! (F s 1 F(f s ))[2 s] has no cohomology in gr s (in particular in gr 2s 2). From th triangls 7 an 8 w gt thn isomorphisms H 2s 2 (K 2 Gm ) = R 2s 2 µ! (π 1K s 1 ) = R 2s 2 µ! (π 1L s 1 ) = = H 2 c(g m, H 2s 4 (L s 1 )) = H 2 c(g m, Q l (2 s)) = Q l (1 s) by th inuction hypothsis an th spctral squnc H i c(g m, H j (L s 1 )) H i+j c (G m, L s 1 ), whr th last two objcts ar rgar as constant shavs on G m. This provs (2). For (3), w hav alray sn that th lft han si of triangl 9 only has cohomology in gr s 1. Similarly, th lft han si of triangl 8 RΓ c (G m, F s 1 )[2 s] = H 1 c(g m, F s 1 )[1 s] only has cohomology in gr s 1. Sinc th othr two ns of 8 an 9 ar gomtrically constant, w conclu that H j (K) Gm is gomtrically constant for j s using triangl 7. Lt s j 2s 3. For any gomtrically constant objct L, w hav RΓ c (G m, L) = L RΓ c (G m, Q l ) = L[ 1] L( 1)[ 2]. In particular H j (RΓ c (G m, L s 1 )) = H j 1 (L s 1 ) H j 2 (L s 1 )( 1) is pur of wight 2(j s + 1) by inuction. Similarly H j (RΓ c (G m, L s 1 F(f s ))) = H j (L s 1 RΓ c (G m, F(f s ))) = H j 1 (L s 1 ) H 1 c(g m, F(f s )) is pur of wight 2(j s + 1) sinc H 1 c(g m, F(f s )) is pur of wight 0. Using triangl 7 this provs that H j (K) Gm is pur of wight 2(j s + 1). From triangls 7 an 9 w gt xact squncs (10) 0 R s 1 µ! (K s 1 F(f s )[0]) H s 1 (K s Gm ) an R s 1 µ! (π 1K s 1 ) R s µ! (K s 1 F(f s )[0]) 0 R 1 µ! (F s 1 F(f s )) R s 1 µ! (K s 1 F(f s )[0]) H s 1 c (G m, L s 1 F(f s )) 0. W hav alray shown that R s µ! (K s 1 F(f s )[0]) is pur of wight 2(s s + 1) = 2. On th othr han, from triangl 8 w gt an xact

10 ANTONIO ROJAS-LEÓN squnc H 1 c(g m, F s 1 ) R s 1 µ! (π 1K s 1 ) H s 1 (RΓ c (G m, L s 1 )) whr th lft han si has wight 0 by part (5) of th inuction hypothsis an th right han si H s 1 (RΓ c (G m, L s 1 )) = H s 1 (L s 1 [ 1] L s 1 ( 1)[ 2]) = H s 2 (L s 1 ) H s 3 (L s 1 )( 1) = H s 2 (L s 1 ) also has wight 0 by part (4) of th inuction hypothsis. Thrfor R s 1 µ! (π 1 K s 1) is pur of wight 0, an th last arrow in squnc (10) is trivial: 0 R s 1 µ! (K s 1 F(f s )[0]) H s 1 (K s Gm ) R s 1 µ! (π 1K s 1 ) 0 Lt F s := R 1 µ! (F s 1 F(f s )) (th multiplicativ convolution of F(f 1 ),..., F(f s )). Thn F s H s 1 (K s Gm ), an th quotint sits insi an xact squnc 0 H s 1 c (G m, L s 1 F(f s )) H s 1 (K s Gm )/F s R s 1 µ! (π 1K s 1 ) 0 whos xtrms ar alray known to b gomtrically constant by triangl 8. Th rank of this quotint is im H s 1 c (G m, L s 1 F(f s )) + im R s 1 µ! (π 1K s 1 ) = (im H s 2 (L s 1 ))(im H 1 c(g m, F(f s )))+im H 1 c(g m, F s 1 )+im H s 1 c (G m, L s 1 ) = ( s 1 ( 1) s 1 )( 1)+( 1) s 1 +im H s 2 (L s 1 )+im H s 3 (L s 1 ) = s s 1 ( 1) s + ( 1) s 1 + ( s 1 ( 1) s 1 ) = s ( 1) s by parts (4) an (5) of th inuction hypothsis. By [4, Corollary 6 an Proposition 9], H s 1 (K s ) (an in particular its subshaf F s ) os not hav punctual sctions in A 1. Lt j 0 : G m A 1 b th inclusion. W claim that H 1 c(a 1, j 0 F s ) = 0. This will prov both that F s is th xtnsion by irct imag of its rstriction to any opn st j V : V G m on which it is smooth an that it is totally ramifi at infinity, sinc from th xact squncs 0 j 0 F s j 0 j V j V F s Q := j V j V F s /F s (punctual) 0 an 0 j! j 0 F s j j 0 F s Fs I 0 whr j : A 1 P 1 is th inclusion, w gt injctions Q H 1 c(a 1, j 0 F s ) an Fs I H 1 c(a 1, j 0 F s ). Lt i 0 : {0} A 1 b th inclusion. From th xact squnc 0 j 0! F s j 0 F s i 0 i 0j 0 F s 0 an th fact that F s has no punctual sctions w gt 0 F I 0 s H 1 c(g m, F s ) H 1 c(a 1, j 0 F s ) 0 whr F I 0 s is th invariant spac of F s as a rprsntation of th inrtia group I 0. So it suffics to show that im F I 0 s im H 1 c(g m, F s ) (an thn w will automatically hav quality). By finition of F s, H 1 c(g m, F s ) = H 2 c(g m G m, F s 1 F(f s )) = H 1 c(g m, F s 1 ) H 1 c(g m, F(f s )). Thrfor

ON THE NUMBER OF RATIONAL POINTS ON CURVES OVER FINITE FIELDS 11 H 1 c(g m, F s ) is pur of wight 0 an imnsion ( 1) s 1 ( 1) = ( 1) s by inuction, thus proving (5). If f 1,..., f s split in k r thn H 1 c(g m, F s ) is a trivial Gal( k r /k r )-moul, also by inuction. On th othr han, H s 1 (K s ) Gm contains F s plus a gomtrically constant part of imnsion s ( 1) s. So im H s 1 (K s ) I 0 = im F I 0 s + ( s ( 1) s ). Sinc H s 1 (K s ) has no punctual sctions, thr is an injction H s 1 (K s ) 0 H s 1 (K s ) I 0, so im H s 1 (K s ) I 0 im H s 1 (K s ) 0. By bas chang, H s 1 (K s ) 0 = H s 1 c ({f 1 (x 1 ) f s (x s ) = 0}, Q l ) = H s c({f 1 (x 1 ) f s (x s ) 0}, Q l ) = H s c(u 1 U s, Q l ) = H 1 c(u 1, Q l ) H 1 c(u s, Q l ), whr U i A 1 is th opn st fin by f i (x) 0 (sinc th U i only hav nonzro cohomology in grs 1 an 2), so im H s 1 (K s ) 0 = s. W conclu that im F I 0 s = im H s 1 (K s ) I 0 ( s ( 1) s ) im H s 1 (K s ) 0 ( s ( 1) s ) = ( 1) s = im H 1 c(g m, F s ). To prov (4) it only rmains to show that F s V is pur of wight s 1 an rank s( 1) s an has unipotnt monoromy action at 0. Lt t G m b a gomtric point which is not th prouct of a non-smoothnss point of F s 1 an a non-smoothnss point of F(f s ). Th fibr of F s ovr t is H 1 c(g m, F s 1 σt F(f s )). By th choic of t, at vry point of G m at last on of F s 1, σt F(f s ) is smooth. Thrfor if F s 1 σt F(f s ) is smooth in th opn st j W : W G m, j W jw (F s 1 σt F(f s )) = (j W jw F s 1) (j W jw σ t F(f s )) = F s 1 σt F(f s ). Givn that F s 1 (rspctivly σt F(f s )) is pur of wight s 2, unipotnt at 0 an totally ramifi at (rsp. pur of wight 0, unramifi at an totally ramifi at 0), F s 1 σt F(f s ) is pur of wight s 2 an totally ramifi at both 0 an, so H 1 c(g m, F s 1 σt F(f s )) = H 1 (P 1, j jw (F s 1 σt F(f s ))) is pur of wight s 1, whr j : W P 1 is th inclusion. As for th rank, sinc F s 1 σt F(f s ) has no punctual sctions an is totally ramifi at 0 an, im H 1 c(g m, F s 1 σt F(f s )) = χ(g m, F s 1 σt F(f s )). By th Ogg-Shafarvic formula, for ach of F s 1, σt F(f s ) its Eulr charactristic is ( 1 tims) a sum of local trms for th points of P 1 whr thy ar ramifi. Th local trms at 0, ar th Swan conuctors, which gt multipli by D upon tnsoring with a unipotnt shaf of rank D. Th local trms corrsponing to ramifi points in G m (Swan conuctor plus rop of th rank) ar multipli by D upon tnsoring with an unramifi shaf of rank D. Sinc at vry point of G m at last on of F s 1, σt F(f s ) is unramifi, w conclu that χ(g m, F s 1 σt F(f s )) = ( 1)χ(G m, F s 1 ) (s 1)( 1) s 1 χ(g m, F(f s )) = ( 1)( 1) s 1 + (s 1)( 1) s 1 ( 1) = s( 1) s. Finally, sinc F I 0 s = H 1 c(g m, F s ) has wight 0 an F s is pur of wight s 1, for vry Frobnius ignvalu of F I 0 s thr is a unipotnt Joran block of siz s for th monoromy of F s at 0 by [2, Sction 1.8]. Sinc its rank is s( 1) s, ths Joran blocks fill up th ntir spac, an thrfor th I 0 action is unipotnt. This finishs th proof of (4) an of th lmma.

12 ANTONIO ROJAS-LEÓN Now lt T : A r k r A 1 k r b th map fin by th polynomial T, an K r := RT! Ql Dc(A b 1 k, Q l ). Aftr xtning scalars to k r, K r bcoms isomorphic to K r for f j = f σj, j = 1,..., r. Sinc th rsults of th lmma ar invariant unr finit xtnsion of scalars, thy also hol for K r. In particular, for vry r 1 j 2r 2 thr xist β j,1,..., β j,j C of absolut valu 1, whr j = rank H j (K r ) (or th rank of th constant part if j = r 1) such that for vry finit xtnsion k m of k of gr m an vry λ km (11) #{(x 1,..., x r ) k r m T (x 1,..., x r ) = λ} = = 2r 2 j=r 1 j ( 1) j q m(j r+1) βj,l m + ( 1)r 1 Tr(Frob km,λ F r ). Taking th sum ovr all λ k m an using th trac formula: = = 2r 2 j=r 1 2r 2 j=r 1 j #{(x 1,..., x r ) k r m T (x 1,..., x r ) 0} = ( 1) j (q m 1)q m(j r+1) βj,l m + ( 1)r 1 j λ k m Tr(Frob km,λ F r ) = ( 1) j (q m 1)q m(j r+1) βj,l m + ( 1)r Tr(Frob km H 1 c(g m, F r )). Lt b b th gr of a splitting fil of f ovr k r. Thn th (gomtrically constant) cohomology shavs of K r bcom constant aftr xtning scalars to k br. In particular all β j,l ar br-th roots of unity. If m is any positiv intgr congrunt to 1 moulo br w hav thn = 2r 2 j=r 1 = j #{(x 1,..., x r ) k r m T (x 1,..., x r ) 0} = ( 1) j (q m 1)q m(j r+1) β j,l + ( 1) r Tr(Frob km H 1 c(g m, F r )) = 2r 2 β 2r 2,l q mr + 2r 2 j=r 1 j 1 j ( 1) j 1 β j 1,l + +( 1) r Tr(Frob km H 1 c(g m, F r )). β j,l q m(j r+1) + Sinc m is prim to r, B is a basis of k mr ovr k m, an thus T (x 1,..., x r ) = N kmr/k m (f(α 1 x 1 + + α r x r )). Thrfor an in particular #{(x 1,..., x r ) k r m T (x 1,..., x r ) 0} = = #{x k mr N kmr/k m (f(x)) 0} = #{x k mr f(x) 0} #{(x 1,..., x r ) k r m T (x 1,..., x r ) 0} q mr.

ON THE NUMBER OF RATIONAL POINTS ON CURVES OVER FINITE FIELDS 13 Substituting in th formula abov, w gt 2r 2 2r 2 j 1 j β 2r 2,l 1 q mr + ( 1) j 1 β j 1,l + j=r 1 +( 1) r Tr(Frob km H 1 c(g m, F r )) β j,l q m(j r+1) + Ltting m an using that Tr(Frob km H 1 c(g m, F r )) ( 1) r is boun by a constant, w conclu that 2r 2 β 2r 2,l = 1 an j 1 j β j 1,l + β j,l = 0 for vry r j 2r 2, so j β j,l = ( 1) j for vry r 1 j 2r 2. Thorm 3.2. Lt f k r [x] b a squar-fr polynomial of gr prim to p an q 1. Thn th numbr N f of k r -rational points on th curv satisfis th stimat y q 1 = f(x) N f q r δ + 1 r( 1) r (q 1)q r 1 2 whr 0 δ is th numbr of roots of f in k r. Proof. Substituting th comput valus for j β j,l in quation 11 for m = 1 w gt = so 2r 2 j=r 1 #W λ (k) = #{(x 1,..., x r ) k r T (x 1,..., x r ) = λ} = r 1 q j r+1 + ( 1) r 1 Tr(Frob k,λ F r ) = q j + ( 1) r 1 Tr(Frob k,λ F r ). So, by quation 6, w hav N f = #Z(k r ) + q 1 #W λ (k) = λ =1 = δ + q 1 r 1 q j + ( 1) r 1 Tr(Frob k,λ F r ) = λ =1 j=0 = δ + (q r 1) + ( 1) r 1 q 1 N f q r δ + 1 q 1 λ =1 j=0 Tr(Frob k,λ F r ), λ =1 r( 1) r q r 1 2 = r( 1) r (q 1)q r 1 2 sinc F r is pur of wight r 1 an gnric rank r( 1) r by th lmma, an its rank can only rop at ramifi points.

14 ANTONIO ROJAS-LEÓN Rmark 3.3. Th conition that f is squar-fr is ncssary, as shown by th xampl y q 1 = x in which N r = 1 + #{x k r N kr/k(x ) = 1} = 1 + #{x k r N kr/k(x) = t} = t k,t =1 = 1 + µ (q s 1 + q r 2 + + q + 1) whr µ 1 is th numbr of -th roots of unity in k. Rfrncs [1] Dlign, P., Applications la formul s tracs aux somms trigonométriqus, in Cohomologi Étal, Séminair Géométri Algébriqu u Bois-Mari (SGA 4 1 2 ),Lctur Nots in Mathmatics 569, Springr-Vrlag. [2] Dlign, P., La conjctur Wil II, Publ. Math. IHES, 52(1980), 137-252. [3] Katz, N., Estimats for Soto-Anra sums, J. rin angw. Math. 438 (1993), 143 161. [4] Katz, N., A smicontinuity rsult for monoromy unr gnration, Forum Math. 15 (2003), 191 200. [5] Katz, N., Sato-Tat Thorms for Finit-Fil Mllin Transforms, prprint (2010) [6] Rojas-Lón, A. an Wan, D., Big improvmnts of th Wil boun for Artin-Schrir curvs, prprint (2010), arxiv:1004.2224 [math:ag]. Dpartamanto Álgbra, Univrsia Svilla, Apo 1160, 41080 Svilla, Spain E-mail: arojas@us.s