Improvements of the Weil bound for Artin-Schreier curves

IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES ANTONIO ROJAS-LEON AND DAQING WAN Abs ac . Fo he A in-Sch eie cu e yq−y= (x) de ined o e a ini e ield Fqo qelemen s, he celeb a ed Weil bound o he numbe o Fq - a ional poin s can be sha p, especially in supe -singula cases and when is di isible. In his pape , we show how he Weil bound can be signi ican ly imp o ed, using ideas om momen L- unc ions and Ka z’s wo k on `-adic monod omy calcu- la ions. Roughly speaking, we show ha in a o able cases (which happens qui e o en), one can emo e an ex a √q ac o in he e o e m. 1. In oduc ion Le k=Fqbe a ini e ield o cha ac e is ic p > 2 wi h qelemen s, and le ∈k[x] be a polynomial o deg ee d > 1. Wi hou loss o gene ali y, we can and will always assume ha dis no di isible by p. Le C be he a ine A in-Sch eie cu e de ined o e kby yq−y= (x). Le be a posi i e in ege , and le N ( ) deno e he numbe o Fq - a ional poin s on C . The genus o he smoo h p ojec i e model o C is gi en by g= (q−1)(d−1)/2. The celeb a ed Weil bound in his case gi es he es ima e |N ( )−q | ≤ (d−1)(q−1)q 2. This bound can be sha p in gene al, o ins ance when C is supe singula and is di isible. I q is no a squa e, Se e’s imp o emen [14] leads o a somewha be e bound: |N ( )−q | ≤ (d−1)(q−1) 2[2q 2], whe e [x] deno es he in ege pa o a eal numbe x. In his pape , we shall show ha i qis la ge compa ed o d(and hus he genus g= (d−1)(q−1)/2 is small compa ed o he ield size q wi h ≥2), hen he abo e Weil bound can be signi ican ly imp o ed in many cases. The ype o heo ems we p o e is o he ollowing na u e. Fo simplici y, we jus s a e one special case. Theo em 1.1. Le ≥1and p > 2. I he de i a i e 0is squa e- ee and ei he is odd o he hype su ace (x1) + ···+ (x )=0in A kis non-singula , hen we ha e he es ima e |N ( )−q | ≤ Cd, q +1 2, The esea ch o An onio Rojas-Leon is pa ially suppo ed by P08-FQM-03894 (Jun a de An- daluc´ıa), MTM2007-66929 and FEDER. The esea ch o Daqing Wan is pa ially suppo ed by NSF. 1 2 ANTONIO ROJAS-LEON AND DAQING WAN whe e Cd, is he cons an Cd, = X a=0 |a−1|d−2 + −a −ad−1 a. No e ha he cons an Cd, is independen o qand i is a polynomial in dwi h deg ee . Thus, o ixed dand , ou esul essen ially emo es an ex a √q ac o om Weil’s bound. The non-singula i y hypo hesis canno be d opped in gene al, as he e a e cases o e en whe e we can ha e |N ( )−(q +q 2+1)| ≤ Cd, q +1 2, see sec ion 4 o mo e de ails. This gi es u he examples ha he q- ac o in he Weil bound canno be eplaced by an O(√q) ac o in gene al. As an ex eme illus a ion, we conside he elemen a y case ha = 1. I is clea ha N1( ) = qn , whe e n is he numbe o dis inc oo s o (x) in Fq which is a mos d. Thus, he bes es ima e in his case should be |N1( )−q| ≤ (d−1)q, which is p ecisely wha ou bound gi es! I is a be e han he Weil bound |N1( )−q| ≤ (d−1)(q−1)√q. Fo = 2, ou bound akes he o m |N2( )−q2| ≤ (d−1)2q3/2, which is be e han he Weil bound |N2( )−q2| ≤ (d−1)(q−1)q as soon as q≥(d−1)2+ 3. Fo = 3, ou bound akes he o m |N3( )−q3| ≤ (d−1)(d2−3d+ 3)q2, which is be e han he Weil bound |N3( )−q3| ≤ (d−1)(q−1)q3/2 as soon as q≥(d2−3d+ 4)2. Ou idea is o ansla e N ( ) o momen exponen ial sums and hen calcula e he associa ed momen L- unc ion as explici ly as possible. Le ψbe a ixed non- i ial addi i e cha ac e o k. Fo ∈k[x], i is clea ha we ha e he o mula N ( ) = X ∈kX x∈k ψ(T ( (x))), whe e k =Fq and T deno es he ace map om k o k. Sepa a ing he e m om = 0, we ob ain (1) N ( )−q =X ∈k?X x∈k ψ(T ( (x))). Now, Weil’s bound o exponen ial sums gi es he es ima e X x∈k ψ(T ( (x)))≤(d−1)q 2 o e e y ∈k?. I ollows ha |N ( )−q | ≤ (q−1)(d−1)q 2. IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 3 In o de o imp o e his bound, we need o unde s and he cancela ion o he ou e sum o (1) o e ∈k?. Heu is ically, one expec s ha he ou e sum con ibu es ano he O(√q) ac o ins ead o he i ial q ac o , i is su icien ly “ andom”. This is in ac wha we shall p o e using he ull s eng h o Deligne’s gene al heo em on Riemann hypo hesis. The double sum in (1) is p ecisely a momen exponen ial sum associa ed o he wo a iable polynomial (x). Thus, we can use he echniques o momen L- unc ions o ge imp o ed in o ma ion abou he solu ion numbe N ( ). We now b ie ly ou line ou me hod. Le `be a ixed p ime di e en om p. Le G deno e he ela i e `-adic cohomology wi h compac suppo associa ed o he amily o one a iable exponen ial sums a ached o (x), whe e xis he a iable and is he pa ame e on he o us Gm. Applying he `-adic ace o mula ib e by ib e, we ob ain X ∈k?X x∈k ψ(T ( (x))) = −X ∈k? T (F ob q|(G ) ), whe e (G ) is he ib e o G a , and F obqis he geome ic q- h powe F obenius map. Al e na i ely, one can ew i e T (F ob q|(G ) ) = T (F obq|[G ] ), whe e [G ] deno es he - h Adams ope a ion o G . I is a i ual `-adic shea on Gm. Fo example, Ka z [9] used he o mula [G ] = X i=1 (−1)i−1i·Sym −iG ⊗∧iG . We shall use he ollowing op imal o mula om [17] gi en by [G ] = X i=0 (−1)i−1(i−1) ·Sym −iG ⊗∧iG . No e ha he e m i= 0 does no occu in he i s o mula, and he e m i= 1 does no occu in he second o mula as he coe icien becomes ze o o i= 1. The coe icien s o he second o mula a e smalle and hus lead o ewe numbe o ze os and poles o he co esponding L- unc ions. In his way, we ge he smalle cons an Cd, in Theo em 1.1. I ollows ha N ( )−q = X i=0 (−1)i(i−1) ·X ∈k? T (F obq|(Sym −iG ⊗∧iG ) ). This educes ou p oblem o he s udy o he L- unc ion o e Gmo he `-adic shea es Sym −iG ⊗ ∧iG o all 0 ≤i≤ . By gene al esul s o Deligne [3], we deduce ha |N ( )−(q +δ , q 2+1)| ≤ Cd, q +1 2, whe e Cd, comes om he Eule cha ac e is ic o he componen s o he i ual shea [G ] , and δ , = X i=0 (−1)i−1(i−1) ·dimH2 c(Gm,¯ k,Sym −iG ⊗∧iG ). 4 ANTONIO ROJAS-LEON AND DAQING WAN Unde he condi ions o Theo em 1.1, i ollows ha he shea Sym −iG ⊗ ∧iG has no geome ically i ial componen o any 0 ≤i≤ , and hus we deduce ha δ , = 0. Ou main esul is somewha s onge . We de e mine he weigh s, he i ial ac- o s and he deg ees o he L- unc ions o all he shea es Sym −iG ⊗∧iG , hus ob- aining a ai ly comple e in o ma ion abou he associa ed momen L- unc ion, see [5][6] and [13] o he s udy o momen L- unc ions in wo o he examples, namely, he amily o hype -Kloos e man sums and he Dwo k amily o o ic Calabi-Yau hy- pe su aces. Unde sligh ly mo e gene al hypo heses, Ka z’s esul s on monod omy g oup calcula ions [7][8] gi e s onge esul s which lead o u he imp o emen s o Theo em 1.1 (see Co olla ies 4.2 and 4.6). See also [9] o a esul on he a e age numbe o a ional poin s on hype su aces ob ained using a simila app oach. The possibili y o ou imp o emen o he Weil bound in he case o A in- Sch eie cu es is due o he ac ha he cu e has a la ge au omo phism g oup Fq, which is he g oup o Fq- a ional poin s on he g oup scheme A1. We expec ha simila imp o emen s should exis o many o he cu es (o highe dimensional a ie ies) wi h a la ge au omo phism g oup. Fo example, in he las sec ion o his pape we ea he case o A in-Sch eie hype su aces yq−y= (x1, ..., xn). This me hod leads o simila imp o emen s o Deligne’s bound o such hype su - aces in many cases. As an explici new example o y, we would sugges he a ine Kumme cu e o he o m y(q−1) e= (x), whe e eis a ixed posi i e in ege , qis a p ime powe cong uen o 1 modulo e, and (x)∈k[x] is a polynomial o deg ee d. Fo ≥1 and N ( , e) deno ing he numbe o Fq - a ional poin s on he abo e Kumme cu e, we conjec u e ha o ce ain gene ic , he e is he ollowing es ima e |N ( , e)−q | ≤ sd,e, q +1 2, whe e sd,e, is a cons an independen o q. We do no know how o p o e his conjec u e, e en in he case e= 1. To conclude his in oduc ion, we aise ano he open p oblem. In Theo em 1, we assumed ha he cu e C :yq−y= (x) is de ined o e he sub ield Fqo Fq . We belie e ha simila imp o emen is also ue i C is de ined o e he la ge ield Fq . Bu we could no p o e his a p esen . Rema ks. Weil’s es ima e gi es bo h an uppe bound and a lowe bound o he numbe o a ional poin s on a cu e o genus go e he ini e ield Fq. Im- p o emen s o he lowe bound a e in gene al ha de o ge . Imp o emen s o he uppe bound can o en be ob ained by mo e elemen a y means. In ac , he e a e al eady se e al such esul s in he li e a u e o la ge genus cu es. The i s esul along hese lines is due o S a k [15] in he hype ellip ic case, using S epano ’s me hod. Using he explici o mula, D in eld-Vladu and Se e [14] ob ained an uppe bound imp o emen when 2g > q −q /2, which in ou A in-Sch eie se ing becomes (d−1)(q−1) > q −q /2. Fo > 1, his means ha qmus be small compa ed o d. In compa ison, ou imp o emen s apply when qis la ge compa ed o d. Using a geome ic in e sec ion IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 5 a gumen , S ¨ohe -Voloch [16] ob ained ano he uppe bound which in ou case becomes N ( )≤1 2D(D+q −1), whe e D= max(d, q). Acknowledgmen . I is a pleasu e o hank he e e ee o his ca e ul eading o he i s e sion and o his e y help ul commen s. 2. Cohomology o he amily 7→ Pψ(T ( (x))) Le k=Fqbe a ini e ield o cha ac e is ic p, and ∈k[x] a polynomial o deg ee dp ime o p. Le C be he A in-Sch eie cu e de ined on A2 kby he equa ion (2) yq−y= (x) and deno e by N ( ) i s numbe o a ional poin s o e k := Fq . Fix a non- i ial addi i e cha ac e ψ:k→C?. I is clea ha (3) N ( ) = X ∈kX x∈k ψ( ·T ( (x))) = X ∈kX x∈k ψ(T ( (x))) whe e T deno es he ace map k →k. Fix a p ime `6=pand an isomo phism ι:¯ Q`→C. Conside he Galois ´e ale co e o Gm×A1(wi h coo dina es ( , x)) gi en by u−uq= (x), wi h Galois g oup k; and le Lψ( (x)) be he ank 1 smoo h ¯ Q`-shea co esponding o he ep esen a ion o kgi en by ψ−1 ia ι. De ine K = Rπ!Lψ( (x)) ∈ Db c(Gm,k,¯ Q`), whe e π:Gm×A1→Gmis he p ojec ion. The ace o mula implies ha he ace o he ac ion o he - h powe o a local geome ic F obenius elemen a ∈k?on K is gi en by Px∈k ψ(T ( (x))). I is known [3, 3.7] ha K =G [−1] o a smoo h shea G o ank d−1 and punc ually pu e o weigh 1, whose local - h powe F obenius ace a ∈k?is hen gi en by −Px∈k ψ(T ( (x))). The e o e (4) N ( )−q =X ∈k?X x∈k ψ(T ( (x))) =−X ∈k? T (F ob |(G ) ) = −X ∈k? T (F ob |[G ] ) whe e [G ] = X i=0 (−1)i−1(i−1) ·Sym −iG ⊗∧iG is he - h Adams ope a ion on G . The shea G can also be in e p e ed in e ms o he Fou ie ans o m. Conside he shea ?¯ Q`on A1 k. The e is a canonical su jec i e ace map φ: ?¯ Q`= ? ?¯ Q`→¯ Q`, le F be i s ke nel. I is a cons uc ible shea o gene ic ank d−1 on A1 k. Lemma 2.1. I j:Gm,k →A1 kis he inclusion, he shi ed shea j!G [1] is he Fou ie ans o m o F [1] wi h espec o ψ. 6 ANTONIO ROJAS-LEON AND DAQING WAN P oo . Taking Fou ie ans o m in he dis inguished iangle in Db c(A1 k,¯ Q`): F [1] → ?¯ Q`[1] →¯ Q`[1] → we ge a dis inguished iangle: FTψ(F )[1] →FTψ( ?¯ Q`)[1] →(¯ Q`)0(−1)[0] →. whe e (¯ Q`)0is a punc ual shea suppo ed a 0. I µ:A1×A1→A1is he mul ipli- ca ion map, he Fou ie ans o m o ?¯ Q`[1] is gi en by Rπ1!(π? 2 ?¯ Q`⊗µ?Lψ)[2] = Rπ1!(Lψ( (x)))[2], whe e πi:A1×A1→A1a e he p ojec ions. In pa icula , by p ope base change j?FTψ( ?¯ Q`)[1] = j?Rπ1!(Lψ( (x)))[2] = K [2] = G [1]. Applying j? o he iangle abo e we ind quasi-isomo phisms j?FTψ(F )[1] ∼ =G [1] and j!j?FTψ(F )[1] ∼ =j!G [1]. To conclude, i emains o show ha he na u al map j!j?FTψ(F )[1] →FTψ(F )[1] is a quasi-isomo phism. Since i s es ic ion o Gm,k is a quasi-isomo phism, we only need o check ha i induces a quasi-isomo phism on he s alks a (a geome ic poin o e ) 0, ha is, ha FTψ(F )0= 0. By de ini ion o he Fou ie ans o m, FTψ(F )0= RΓc(A1 ¯ k,F ). We conclude by using he long exac sequence o coho- mology wi h compac suppo associa ed o he sequence 0→ F → ?¯ Q`→¯ Q`→0, since Hi c(A1 ¯ k, ?¯ Q`) = Hi c(A1 ¯ k,¯ Q`) = 0 o i6= 2 and H2 c(A1 ¯ k, ?¯ Q`) = H2 c(A1 ¯ k,¯ Q`) = ¯ Q`(−1) is one-dimensional.  We can now use Laumon’s local Fou ie ans o m heo y o de e mine he mon- od omy ac ions a 0 and ∞ o G . Recall ha , o e e y cha ac e χ:k?→¯ Q? `, he e is an associa ed Kumme shea Lχon Gm,k: The (q−1)- h powe map Gm,k →Gm,k is a Galois ´e ale co e wi h Galois g oup canonically isomo phic o k?, and one jus akes he pull-back o he cha ac e ¯χ o π1(Gm,k,¯η)k?. Fo e - e y d|q−1, i [d] deno es he d- h powe map Gm,k →Gm,k we ha e [d]?¯ Q`=LLχ, whe e he sum is aken o e all cha ac e s o k?such ha χdis i ial. Assume ha kcon ains all d- h oo s o uni y. The shea F is smoo h on he complemen Uo he se o he c i ical alues o in A1. Since dis p ime o p, in a neighbo hood o in ini y he map x7→ (x) = adxd(1 + ad−1 adx+··· +a0 adxd) is equi alen ( o he ´e ale opology) o he map x7→ adxd(jus by making he change o a iable x7→ αx, whe e αd= 1 + ad−1 adx+···+a0 adxd). In pa icula , he decomposi ion g oup D∞a in ini y ac s on he gene ic s alk o ?¯ Q` h ough he di ec sum o he ame cha ac e s (ad)?Lχ o all non- i ial cha ac e s χo k?such ha χd=1, whe e (ad) : Gm,k →Gm,k is he mul iplica ion by admap. Since (ad)?Lχ= (a−1 d)?Lχ= ¯χ(ad)deg ⊗ Lχ, we conclude ha D∞ac s on he gene ic s alk o F ia he di ec sum L¯χ(ad)deg ⊗Lχ aken o e all non- i ial cha ac e s χo k?such ha χdis i ial. P oposi ion 2.2. Suppose ha kcon ains all d- h oo s o uni y. The ac ion o he decomposi ion g oup D0a 0on G is ame and semisimple, and i spli s as a di ec sum L(χ(ad)g(¯χ, ψ))deg ⊗ Lχo e all non- i ial cha ac e s χo k?such ha χd=1, whe e g(¯χ, ψ) := −P ¯χ( )ψ( )is he Gauss sum. IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 7 P oo . By ([12, P oposi ion 2.5.3.1],[8, Theo em 7.5.4]), he local monod omy a 0 o G can be ead om he local monod omy a in ini y o F . Mo e p ecisely, we ha e LFT(∞,0)(L¯χ(ad)deg ⊗Lχ) = LLFT(∞,0)(¯χ(ad)deg ⊗Lχ). Now, o e e y χ, since he Fou ie ans o m commu es wi h enso ing by an un ami ied shea (by he p ojec ion o mula, since π? 1(αdeg) = αdeg and µ?(αdeg) = αdeg o π1and µ:A1 k×A1 k→A1 k he p ojec ion and mul iplica ion) we ha e LFT(∞,0)(¯χ(ad)deg ⊗ Lχ) = ¯χ(ad)deg ⊗LF T (∞,0)Lχ= ¯χ(ad)deg ⊗g(χ, ψ)deg ⊗ L¯χby [12, P oposi ion 2.5.3.1] (no e ha Lχco esponds o V¯χas a ep esen a ion o D∞and o V0 χas a ep esen a ion o D0in he no a ion o [12] due o he choice o uni o mize s).  Fo simplici y, we will assume om now on ha 0is squa e- ee and p > 2. Suppose ha kcon ains all oo s o 0(and he e o e all c i ical alues o ). Le s∈kbe a c i ical alue o . The polynomial s:= −shas a wo s double oo s and kcon ains all i s double oo s. Le gsbe he squa e- ee pa o s(i.e. sdi ided by he p oduc o all i s monic double linea ac o s), which lies in k[x]. Le S0be he henseliza ion o A1 ka s,z1, . . . , ze∈k he double oo s o s,Sj he henseliza ion o A1 ka zj o j= 1, . . . , e and T he union o he henseliza ions o A1 ka he closed poin s o he subscheme de ined by gs= 0. We ha e a ca esian diag am (`jSj)`T−−−−→ A1 k   y(`jhj)`h  y S0−−−−→ A1 k whe e he map hj:Sj→S0is isomo phic ( o he ´e ale opology) o he map x7→ bj(x−zj)2(whe e bjis s(x)/(x−zj)2e alua ed a zj, ha is, 00(zj)/2) ia he change o a iable mapping he local coo dina e x−zj o α(x−zj), whe e α∈Sjis a squa e oo o s(x)/bj(x−zj)2(which exis s by Hensel’s lemma, since i s image in he esidue ield kis 1), and h:T→S0is ini e ´e ale. In pa icula , he decomposi ion g oup Dsa sac s on he gene ic s alk o ?¯ Q` h ough he di ec sum Lj(1⊕(bj)?Lρ)LL=Lj(ρ(bj)deg ⊗Lρ)L(e·1⊕L) whe e Lis un ami ied and ρ= ¯ρ:k?→¯ Q? `is he quad a ic cha ac e . P oposi ion 2.3. Suppose ha p > 2, 0is squa e- ee and all i s oo s a e in k. The ac ion o he decomposi ion g oup D∞a in ini y on G spli s as a di ec sum Lz(ρ(bz)g(ρ, ψ))deg ⊗Lρ⊗Lψ (z)whe e he sum is aken o e he oo s o 0, bz= 00(z)/2,ρ:k?→¯ Q? `is he quad a ic cha ac e and g(ρ, ψ) = −P ρ( )ψ( ) he co esponding Gauss sum. P oo . By ([12],[8, Theo em 7.5.4]), he local monod omy a in ini y o G can be ead om he local monod omies o F . Mo e p ecisely, he pa o slope >1 co esponds o he slope >1 pa o he local monod omy a in ini y o F , so i anishes. The pa o slope ≤1 is a di ec sum, o e all c i ical alues so , o Lψs enso ed wi h he local Fou ie ans o m LFT(0,∞)applied o he ac ion o Is on he gene ic s alk o F modulo i s Is-in a ian space. Using [12, 2.5.3.1] and he ac ha Fou ie ans o m commu es wi h enso ing by un ami ied shea es, o e e y oo zo 0LF T (0,∞)(ρ(bz)deg ⊗Lρ) = ρ(bz)deg ⊗ g(ρ, ψ)deg⊗Lρ. So each c i ical alue scon ibu es a ac o L (z)=s(ρ(bz)g(ρ, ψ))deg⊗ Lρ⊗Lψs o he monod omy o G a in ini y.  We can now compu e he de e minan o G : 8 ANTONIO ROJAS-LEON AND DAQING WAN Co olla y 2.4. Suppose ha kcon ains all d- h oo s o uni y. I dis odd, he de e minan o G is he Ta e- wis ed A in-Sch eie shea Lψs((1 −d)/2), whe e s=s1+··· +sd−1is he sum o he c i ical alues o and ψs( ) = ψ(s ). I dis e en, he de e minan o G is Lρ⊗ Lψs⊗(ρ(ad)g(ρ, ψ))deg((2 −d)/2), whe e ρis he mul iplica i e cha ac e o o de 2,g(ρ, ψ) = −P ρ( )ψ( )is he co esponding Gauss sum, adis he leading coe icien o ,= 1 i d≡0o 2 mod 8 and = (−1)(q−1)/d i d≡4o 6 mod 8. P oo . The de e minan o G is a smoo h shea o ank one on Gm,k. A 0, i is isomo phic by p oposi ion 2.2 o he p oduc Nχd=1,χ6=1(χ(ad)g(¯χ, ψ))deg ⊗Lχ. Fo any χwe ha e (χ(ad)g(¯χ, ψ))deg ⊗Lχ⊗(¯χ(ad)g(χ, ψ))deg ⊗L¯χ= (g(¯χ, ψ)g(χ, ψ))deg = (χ(−1)q)deg. I dis odd, he non- i ial cha ac e s wi h χd=1can be g ouped in conjuga e pai s. Mo eo e , χ(−1) = χ((−1)d) = χd(−1) = 1. We conclude ha he de e minan a 0 is he un ami ied cha ac e (qd−1 2)deg =¯ Q`(1−d 2). A in ini y, i is geome ically isomo phic by p oposi ion 2.3 o he p oduc Nz(Lρ⊗Lψ (z)) = Lψs ( he hypo hesis ha kcon ains all oo s o 0is no needed o he geome ic iso- mo phism, since i is always sa is ied in a su icien ly la ge ini e ex ension o k). So de (G )⊗Lψ−sis e e ywhe e un ami ied and he e o e geome ically cons an . Looking a he F obenius ac ion a 0, i mus be ¯ Q`(1−d 2), so de (G ) = Lψs(1−d 2). I dis e en, he ac o a 0 co esponding o he quad a ic cha ac e ρs ays un- ma ched, so as a ep esen a ion o D0 he de e minan is (q d−2 2)deg⊗(ρ(ad)g(ρ, ψ))deg⊗ Lρ, whe e =Q(d−2)/2 i=1 χi(−1) o a ixed cha ac e χo exac o de d. A ∞i is geome ically isomo phic o Lρ⊗Lψs, so de (G )⊗Lρ⊗Lψ−sis e e ywhe e un am- i ied and he e o e geome ically cons an . Looking a he F obenius ac ion a 0, i mus be (q d−2 2ρ(ad)g(ρ, ψ))deg, so de (G ) = Lρ⊗Lψs⊗(ρ(ad)g(ρ, ψ))deg(2−d 2). I emains o compu e he alue o . We ha e = (d−2)/2 Y i=1 χi(−1) = χd(d−2)/8(−1) = χ((−1)d(d−2)/8). I d≡0 o 2 mod 8, d(d−2)/8 is e en and he e o e = 1. I d≡4 o 6 mod 8, d(d−2)/8 is odd so =χ(−1) = (−1)(q−1)/d. 3. The momen L- unc ion o G . Recall he de ini ion [4] o he momen L- unc ion o he shea G . Fo a ixed ≥1, le L ( , ψ, T ) := Y ∈|Gm,k| 1 de (1 −F ob Tdeg( )|(G ) ), whe e |Gm,k|deno es he se o closed poin s o Gm,k. I is known ([4, Theo em 1.1]) ha L ( , ψ, T ) is a a ional unc ion, and we ha e he o mula (5) L ( , ψ, T ) = de (1 −F obkT|H1 c(Gm,¯ k,[G ] )) de (1 −F obkT|H2 c(Gm,¯ k,[G ] )) = =Q i=0 de (1 −F obkT|H1 c(Gm,¯ k,Sym −iG ⊗∧iG ))(−1)i−1(i−1) Q i=0 de (1 −F obkT|H2 c(Gm,¯ k,Sym −iG ⊗∧iG ))(−1)i−1(i−1) . Thus, we ge a decomposi ion IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 9 (6) L ( , ψ, T ) = Q(T)P0(T)P∞(T) P(T)P0(T). We now desc ibe each o he ac o s in his decomposi ion. Fi s , Q(T) = Y i=0 de (1 −F obkT|H1(P1 ¯ k, j?(Sym −iG ⊗∧iG )))(−1)i−1(i−1) is he non- i ial ac o . No ice ha he dual o G is G− (1), since F is sel -dual and D◦FTψ=FT ¯ ψ◦D(1) [11, Co ollai e 2.1.5] and FT ¯ ψF [1] = [ 7→ − ]?G [1] = G− [1]. The e o e he dual o Sym −iG ⊗∧iG is Sym −iG− ⊗∧iG− ( ), so he dual (in he de i ed ca ego y) o j?Sym −iG ⊗∧iG [1] is j?Sym −iG− ⊗∧iG− [1]( +1), c . [2, 2.1]. Since P1is p ope , by [2, Th´eo `eme 2.2] we ge a pe ec pai ing H1(P1 ¯ k, j?(Sym −iG ⊗∧iG )) ×H1(P1 ¯ k, j?(Sym −iG− ⊗∧iG− )) 7→ ¯ Q`(− −1) o e e y i= 0, . . . , . In pa icula , we ge a unc ional equa ion ela ing he polynomial Qi(T) := de (1 −F obkT|H1(P1 ¯ k, j?(Sym −iG ⊗∧iG ))) = si Y j=1 (1 −γijT). and he co esponding polynomial Q? i(T) o − . The unc ional equa ion is gi en by Q? i(T) = si Y j=1 (1 −q +1γ−1 ij T) = =Tsiq( +1)si (−1)siγi1···γisi si Y j=1 (1 −γijq−( +1)T−1) = Tsiq( +1)si csi Qi(q−( +1)T−1) whe e csiis he leading coe icien o Qi(T). The e o e, Q?(T) := Y i=0 Q? i(T)(−1)i−1(i−1) =Tsq( +1)s cs Q(q−( +1)T−1) whe e sis he deg ee o he a ional unc ion Q(T) and csi s leading coe icien (i.e. he a io o he leading coe icien s o he nume a o and denomina o ). By [3, Th´eo `eme 3.2.3], all ecip ocal oo s and poles o Q(T) a e pu e Weil in ege s o weigh + 1. The o he ac o s o L ( , ψ, T ) a e he “ i ial ac o s”: P(T) = Y i=0 de (1 −F obkT|H0(P1 ¯ k, j?(Sym −iG ⊗∧iG )))(−1)i−1(i−1) and P0(T) = Y i=0 de (1 −F obkT|H2(P1 ¯ k, j?(Sym −iG ⊗∧iG )))(−1)i−1(i−1) a e a ional unc ions o he same deg ee and pu e o weigh and +2 espec i ely, and anish i Sym −iG ⊗∧iG has no in a ian s o he ac ion o π1(Gm,¯ k) o any i. The o he wo a e he local ac o s a 0: P0(T) := de (1 −F ob0T|([G ] )I0) = 16 ANTONIO ROJAS-LEON AND DAQING WAN Sp(V) = G. In pa icula , all F obenii ac i ially on Wi( /2), and he e o e hey ac by mul iplica ion by q 2on Wi⊆N V. The e o e Y i=0 de (1 −F obkT|H2 c(Gm,¯ k,Sym −iG ⊗∧iG ))(−1)i−1(i−1) = de (1 −F obkT|W −1(−1))(−1) −2( −2) de (1 −F obkT|W (−1))(−1) −1( −1) = (1 −q 2+1T)(−1) −2( −2)+(−1) −1( −1) = (1 −q 2+1T)(−1) −1= (1 −q 2+1T)−1 since −1 = d−2 is odd. In he case whe e b6= 0, G/Sp(V)∼ =µpac s on Wi. Le A= diag(ζp, . . . , ζp)∈G be a scala ma ix, whe e ζp∈¯ Q`is a p- h oo o uni y. Then he class o Agene a es G/Sp(V), so G ixes Wii and only i Adoes. Bu Aac s on Wiby mul iplica ion by ζ p, so his ac ion is i ial i and only i ζ p= 1, ha is, i and only i pdi ides . In ha case, Sym −iG ⊗∧iG = (Sym −iG −b/2⊗∧iG −b/2)⊗L⊗ ψb/2= Sym −iG −b/2⊗∧iG −b/2, so we can apply he b= 0 case and we ge again Y i=0 de (1 −F obkT|H2 c(Gm,¯ k,Sym −iG ⊗∧iG ))(−1)i−1(i−1) = = (1 −q 2+1T)(−1) −1= (1 −q 2+1T)−1. We conclude as in co olla y 4.2.  Again, he hypo hesis o p oposi ion 4.5 can be checked om he coe icien s o : A e adding a cons an , we may assume ha b= 0. Le A 0be he companion ma ix o 0, and B= (A 0). The eigen alues o he (d−1) ×(d−1) ma ix Ba e s1, . . . , sd−1, and i s ace is s=b(d−1) 2. Cons uc he (d−1)2×(d−1)2 ma ix B⊗Id−1−Id−1⊗B, whose eigen alues a e all di e ences si−sj. I s cha ac e is ic polynomial is hen o he o m Td−1h(T/2)g(T)2, whe e h(T) is he cha ac e is ic polynomial o B, since all non-ze o oo s di e en om si−sd−i= 2si o i= 1, . . . , d −1 appea in pai s. The hypo hesis o p oposi ion 4.5 is equi alen o he disc iminan o h(T/2)g(T) being non-ze o. 5. Gene aliza ion o A in-Sch eie hype su aces In his sec ion we will ex end co olla y 3.4 o highe dimensional hype su aces. Since he p oo s a e e y simila , we will only ske ch hem, indica ing he di e ences whe e necessa y. Le ∈k[x1, . . . , xn] be a polynomial o deg ee dp ime o p,C he A in- Sch eie hype su ace de ined on An+1 kby he equa ion (7) yq−y= (x1, . . . , xn). Deno e by N ( ) i s numbe o a ional poin s o e k . We ha e again a o mula (8) N ( )−qn =X ∈k?X x∈kn ψ( ·T ( (x))) = X ∈k?X x∈kn ψ(T ( (x))) whe e T deno es he ace map k →k. Assume ha is a Deligne polynomial, ha is, he leading o m o de ines a smoo h p ojec i e hype su ace o deg ee IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 17 dno di isible by p. Applying Deligne’s bound [3] o he abo e inne sum, one deduces ha |N ( )−qn | ≤ (q−1)(d−1)nqn 2. This is p ecisely Weil’s bound in he case n= 1. Ou pu pose o his sec ion is o imp o e he abo e bound and ob ain he es ima e o he ollowing o m |N ( )−qn | ≤ Cd, qn +1 2, o some cons an Cd, depending only on d, and n. De ine K = Rπ!Lψ( (x)) ∈ Db c(Gm,k,¯ Q`), whe e π:Gm×An→Gmis he p ojec ion. The ace o mula implies ha he ace o he ac ion o he - h powe o a local F obenius elemen a ∈k?on K is gi en by Px∈kn ψ(T ( (x))). Suppose om now on ha he homogeneous pa do highes deg ee o de ines a non-singula hype su ace. Then by [3, 3.7], K is a single smoo h shea G placed in deg ee n, o ank (d−1)nand pu e o weigh n. The e o e N ( )−qn = (−1)nX ∈k? T (F ob |(G ) )=(−1)nX ∈k? T (F ob |[G ] ) whe e [G ] = X i=0 (−1)i−1(i−1) ·Sym −iG ⊗∧iG is he - h Adams ope a ion on G . We can gi e an in e p e a ion o G in e ms o he Fou ie ans o m like we did in he one-dimensional case. Exac ly as in lemma 2.1, we can show Lemma 5.1. The objec G [1] ∈ Db c(Gm,¯ Q`)is he es ic ion o Gmo he Fou ie ans o m o R !¯ Q`[n]wi h espec o ψ. We compac i y ia he map ˜ :X→A1 k, whe e X⊆Pn×A1is de ined by he equa ion F(x0, x1, . . . , xn) = xd 0,Fbeing he homogeniza ion o wi h espec o he a iable x0, and ˜ he es ic ion o he second p ojec ion o X. Suppose ha he subscheme o An kde ined by he ideal h∂ /∂x1, . . . , ∂ /∂xniis ini e ´e ale o e k, and he images o i s ¯ k-poin s unde a e dis inc . Then o e e y s∈¯ k, he ib e Xshas a wo s one isola ed non-degene a e quad a ic singula i y, which is loca ed on he a ine pa (since he pa a in ini y is de ined o e e y ib e by d(x) = 0 and is he e o e non-singula ). We ha e a dis inguished iangle R !¯ Q`→R˜ ?¯ Q`→R( ˜ |X0)?¯ Q`→ whe e X0=X An∼ =Y×A1,Ybeing he smoo h hype su ace de ined in Pn−1 by d= 0. Since R( ˜ |X0)?¯ Q`is jus he cons an objec RΓ(Y, ¯ Q`), i s Fou ie ans o m is suppo ed a 0. So G [1] ∼ =(FTψR !¯ Q`[n])|Gm,k ∼ =(FTψR˜ ?¯ Q`[n])|Gm,k . P oposi ion 5.2. Suppose p > 2. Unde he p e ious hypo heses, le z1, . . . , z(d−1)n∈ An ¯ kbe he dis inc poin s such ha ∂ ∂xi(zj) = 0 o all i= 1, . . . , n, and le si= (zi). The ac ion o he ine ia g oup I∞a in ini y on G decomposes as a di ec sum LLψsii nis e en, and L(Lρ⊗ Lψsi)i nis odd, whe e ρis he unique cha ac e o I∞o o de 2. 18 ANTONIO ROJAS-LEON AND DAQING WAN P oo . We will ob ain, o e e y i, a ac o Lψsi( esp. Lρ⊗ Lψsi) in he local monod omy o G a in ini y. Since he ank is (d−1)nand hese cha ac e s a e pai wise non-isomo phic, his will de e mine he ac ion o I∞comple ely. Le S={si|i= 1,...,(d−1)n}, and U=A1 S. Since ˜ is p ope and smoo h o e U, Ri˜ ?¯ Q`is smoo h on U o e e y i. Since Xscon ains one isola ed non- degene a e quad a ic singula i y o each s∈S, by [1, 4.4] he shea es Ri˜ ?¯ Q`a e smoo h on A1 o i6=n−1, n. In pa icula , hei Fou ie ans o ms a e suppo ed a 0. We conclude ha he e is a dis inguished iangle (FTψRn−1˜ ?¯ Q`[1])|Gm,k → G [1] →(FTψRn˜ ?¯ Q`[0])|Gm,k → and he e o e an exac sequence o shea es (9) 0 → H−1(FTψRn−1˜ ?¯ Q`[1])|Gm,k → G → H−1(FTψRn˜ ?¯ Q`[0])|Gm,k → → H0(FTψRn−1˜ ?¯ Q`[1])|Gm,k →0 since FTψRn˜ ?¯ Q`[0] can only ha e non-ze o cohomology shea es in deg ees 1, 0 and −1. Fu he mo e H0(FTψRn−1˜ ?¯ Q`[1]) is punc ual, so his induces an exac sequence o I∞- ep esen a ions (10) 0 → H−1(FTψRn−1˜ ?¯ Q`[1]) → G → H−1(FTψRn˜ ?¯ Q`[0]) →0. Le Vbe he gene ic s alk o Rn−1˜ ?¯ Q`. Suppose ha nis odd, and le s∈S. Then by [1, 4.3 and 4.4], he ine ia g oup Isac s on Vwi h in a ian space VIso codimension 1 ( he o hogonal complemen o he ’ anishing cycle’ δ) and on he quo ien V/VIs ia i s quad a ic cha ac e ρ. Mo eo e , Rn−1˜ ?¯ Q`is isomo phic a s o he ex ension by di ec image o i s es ic ion o he gene ic poin . By Laumon’s local Fou ie ans o m [8, Sec ion 7.4], he ac ion o he ine ia g oup I∞on H−1(FTψRn−1˜ ?¯ Q`[1]) (and hus on G by (10)) con ains a subcha ac e isomo phic o Lρ⊗Lψs. Suppose now ha nis e en, and le s∈S. By [1, 4.3 and 4.4], he e a e wo pos- sibili ies: i he ’ anishing cycle’ δis non-ze o, he ine ia g oup Isac s on Vwi h in a ian space VIso codimension 1 ( he o hogonal complemen o δ) and i ially on he quo ien V/VIs. Mo eo e , Rn−1˜ ?¯ Q`is isomo phic a s o he ex ension by di ec image o i s es ic ion o he gene ic poin . By Laumon’s local Fou ie ans- o m [8, Sec ion 7.4], he ac ion o he ine ia g oup I∞on H−1(FTψRn−1˜ ?¯ Q`[1]) (and hus on G by (10)) con ains a subcha ac e isomo phic o Lψs. I δ= 0, hen Isac s i ially on V, and he e is an exac sequence o shea es: 0→(¯ Q`)s→Rn˜ ?¯ Q`→js?j? sRn˜ ?¯ Q`→0 whe e (¯ Q`)sis he punc ual objec ¯ Q`suppo ed on sand js:A1− {s},→A1is he inclusion. Taking Fou ie ans o m, we deduce a dis inguished iangle Lψs[1] →FTψRn˜ ?¯ Q`[0] →FTψjs?j? sRn˜ ?¯ Q`[0] → and in pa icula an injec ion 0→ Lψs→ H−1(FTψRn˜ ?¯ Q`[0]). By (10), his gi es a subcha ac e isomo phic o Lψsin he monod omy o G a in ini y.  Fo comple eness, we de e mine also he monod omy o G a 0. IMPROVEMENTS OF THE WEIL BOUND FOR ARTIN-SCHREIER CURVES 19 P oposi ion 5.3. The ine ia g oup I0a 0ac s on G as a di ec sum LnχLχ whe e he sum is aken o e all cha ac e s χo I0such ha χdis i ial, nχ= 1 d((d−1)n−(−1)n)i χis non- i ial and nχ= (−1)n+1 d((d−1)n−(−1)n)i χ is i ial. P oo . We will show ha , o e e y χ, he ac ion o I0on G con ains nχJo dan blocks o he cha ac e χ. Since hese numbe s add up o (d−1)n, which is he dimension o he ep esen a ion G , his will p o e ha he ac ion is semisimple and de e mine i comple ely. Le χbe non- i ial such ha χd=1. Since adding a cons an a o co esponds o enso ing G wi h he A in-Sch eie shea Lψaand his does no change he monod omy a 0, we can assume ha G is o ally wild a ∞(o equi alen ly, ha he hype su ace (x) = 0 is non-singula ). Then so is G ⊗ L¯χ. The numbe o Jo dan blocks associa ed o Lχin he ep esen a ion o I0gi en by G is he dimension o he I0-in a ian subspace o G ⊗L¯χ. I j:Gm,¯ k→A1 ¯ kand i:{0} → A1 ¯ ka e he inclusions, we ha e an exac sequence 0→j!(G ⊗L¯χ)→j?(G ⊗L¯χ)→i?i?j?(G ⊗L¯χ)→0 and he e o e 0→(G ⊗L¯χ)I0→H1 c(Gm,¯ k,G ⊗L¯χ)→H1 c(A1 ¯ k, j?(G ⊗L¯χ)) →0. Since G ⊗L¯χis o ally wild a ∞, he la e cohomology g oup is pu e o weigh n+ 1. So he dimension o (G ⊗L¯χ)I0is he dimension o he weigh ≤npa o H1 c(Gm,¯ k,G ⊗L¯χ). By he p ojec ion o mula, G ⊗L¯χ= (Rnπ!Lψ( (x)))⊗L¯χ∼ =Rnπ!(Lψ( (x)) ⊗L¯χ( )), so H1 c(Gm,¯ k,G ⊗L¯χ)=Hn+1 c(Gm,¯ k×An ¯ k,Lψ( (x)) ⊗L¯χ( )) since Riπ!Lψ( (x)) = 0 o i6=n. Le Z⊂An kbe he closed subse de ined by (x) = 0 and Ui s open complemen . The shea Lψ( (x)) is i ial on Gm×Z, so H? c(Gm,¯ k×Z, Lψ( (x)) ⊗L¯χ( )) = H? c(Gm,¯ k×Z, L¯χ( )) = H? c(Gm,¯ k,L¯χ)⊗H? c(Z⊗ ¯ k, ¯ Q`) = 0 since χis non- i ial. By excision we ge an isomo phism Hn+1 c(Gm,¯ k× An ¯ k,Lψ( (x)) ⊗L¯χ( ))∼ =Hn+1 c(Gm,¯ k×U, Lψ( (x)) ⊗L¯χ( )). Conside he au omo phism φ:Gm×U→Gm×Ugi en by φ( , x)=( (x), x). Then φ?(Lψ( (x)) ⊗L¯χ( )) = Lψ( )⊗L¯χ( / (x)) =Lψ( )⊗L¯χ( )⊗Lχ( (x)). So Hn+1 c(Gm,¯ k×U, Lψ( (x)) ⊗L¯χ( ))∼ =Hn+1 c(Gm,¯ k×U, Lψ( )⊗L¯χ( )⊗Lχ( (x))) which, by K¨unne h, is isomo phic o H1 c(Gm,¯ k,Lψ⊗L¯χ)⊗Hn c(U⊗¯ k, Lχ( )) (since Hi c(Gm,¯ k,Lψ⊗L¯χ) = 0 o i6= 1). The i s ac o is one-dimensional and pu e o weigh 1, so we wan he dimension o he weigh ≤n−1 pa o Hn c(U⊗¯ k, Lχ( )). By [10, Theo em 2.2], his dimension is nχ=1 d((d−1)n−(−1)n). Simila ly, i χ=1is he i ial cha ac e , he sea ched dimension is he di- mension o he weigh ≤npa o Hn+1 c(Gm,¯ k×An ¯ k,Lψ( (x))). F om he exac sequence . . . →Hn c({0}×An ¯ k,¯ Q`)→Hn+1 c(Gm,¯ k×An ¯ k,Lψ( (x)))→ →Hn+1 c(A1 ¯ k×An ¯ k,Lψ( (x)))→Hn+1 c({0}×An ¯ k,¯ Q`)→. . . 20 ANTONIO ROJAS-LEON AND DAQING WAN we ge an isomo phism Hn+1 c(Gm,¯ k×An ¯ k,Lψ( (x)))∼ =Hn+1 c(A1 ¯ k×An ¯ k,Lψ( (x))). Now le π:A1×An→Anbe he p ojec ion, by he base change heo em we ha e R2π!Lψ( (x)) =i?¯ Q`(−1), whe e i:Z→Anis he inclusion o he closed se whe e (x) = 0, and Riπ!Lψ( (x)) = 0 o i6= 2. So we need he dimension o he weigh ≤n−2 pa o Hn−1 c(Z, ¯ Q`). Le Zbe he p ojec i e closu e o Zand Z0=Z Z, we ha e an exac sequence . . . →Hn−2(Z, ¯ Q`)→Hn−2(Z0,¯ Q`)→Hn−1 c(Z, ¯ Q`)→Hn−1(Z, ¯ Q`)→. . . Since Zis smoo h, Hn−1(Z, ¯ Q`) is pu e o weigh n−1, and he e o e he weigh ≤ n−2 pa o Hn−1 c(Z, ¯ Q`) is he coke nel o he map Hn−2(Z, ¯ Q`)→Hn−2(Z0,¯ Q`), ha is, he p imi i e pa P imn−2(Z0,¯ Q`) o he middle cohomology g oup o Z0, which has dimension n1= (−1)n+1 d((d−1)n−(−1)n).  Co olla y 5.4. Le s=P(d−1)n i=1 si. O e ¯ k, he de e minan o G is he A in- Sch eie shea Lψsi n(d−1) is e en, and he p oduc Lρ⊗ Lψsi n(d−1) is odd. P oo . The de e minan is a smoo h shea on Gmo ank 1. A 0, i s monod omy is he p oduc o χnχ o all cha ac e s χo I0such ha χdis i ial. Since he non- i ial cha ac e s (excep o he quad a ic one) appea in conjuga e pai s, he p oduc is i ial i dis odd, and comes down o ρnρ, which is ρo 1depending on he pa i y o nρ=1 d((d−1)n−(−1)n), which is cong uen o nmod 2, i dis e en. A in ini y, i s monod omy is he p oduc o he Lψsi( esp. o he Lρ⊗Lψsi) i nis e en ( esp. i nis odd), which is Lψs( esp. Lρ⊗Lψs) i n(d−1) is e en ( esp. i n(d−1) is odd). We conclude as in Co olla y 2.4.  We now gi e he highe dimensional analogue o Co olla y 3.4: Co olla y 5.5. Le ∈k[x1, . . . , xn]be a polynomial o deg ee dp ime o pand a posi i e in ege . Suppose ha p > 2, he highes deg ee homogeneous pa o de ines a non-singula hype su ace, he subscheme o An kde ined by he ideal h∂ /∂x1, . . . , ∂ /∂xniis ini e ´e ale o e kand he images o i s ¯ k-poin s unde a e dis inc . I n is e en, suppose addi ionally ha he hype su ace de ined by (x1,1, . . . , x1,n)+···+ (x ,1, . . . ,x ,n) = 0 in An k= Spec k[xi,j|1≤i≤ , 1≤j≤ n]is non-singula . Then he numbe N ( )o k - a ional poin s on he hype su ace yq−y= (x1, . . . , xn) sa is ies he es ima e |N ( )−qn | ≤ Cd, qn +1 2 whe e Cd, = X i=0 |i−1|(d−1)n+ −i−1 −i(d−1)n i is independen o q. The p oo is iden ical o he one o Co olla y 3.4, using P oposi ion 5.2. In he ne en case we need he non-singula i y hypo hesis o any , since he Kumme ac o does no appea in he monod omy a in ini y. 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