ON THE NUMBER OF RATIONAL POINTS ON CURVES OVER FINITE FIELDS WITH MANY AUTOMORPHISMS
|
|
- Rafe Mosley
- 6 years ago
- Views:
Transcription
1 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 (Junta Analucía), MTM an FEDER. 1
2 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
3 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 4 ANTONIO ROJAS-LEÓN in which f(x 1, x 2 ) = x 1 x 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 σ j (α r )x r ) = r j=1 σj (f(α 1 x α r x r )) = 0, if an only if Tr(f(α 1 x 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)r ( 1) r 1 ( 1) q r 2
5 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)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 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 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 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 σ j (α r )x r ) = r j=1 σj (f(α 1 x α r x r )) = λ, if an only if N(f(α 1 x 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
7 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 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
9 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 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
11 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 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 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.
13 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 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 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 ),Lctur Nots in Mathmatics 569, Springr-Vrlag. [2] Dlign, P., La conjctur Wil II, Publ. Math. IHES, 52(1980), [3] Katz, N., Estimats for Soto-Anra sums, J. rin angw. Math. 438 (1993), [4] Katz, N., A smicontinuity rsult for monoromy unr gnration, Forum Math. 15 (2003), [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: [math:ag]. Dpartamanto Álgbra, Univrsia Svilla, Apo 1160, Svilla, Spain arojas@us.s
Introduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationThe second condition says that a node α of the tree has exactly n children if the arity of its label is n.
CS 6110 S14 Hanout 2 Proof of Conflunc 27 January 2014 In this supplmntary lctur w prov that th λ-calculus is conflunt. This is rsult is u to lonzo Church (1903 1995) an J. arkly Rossr (1907 1989) an is
More informationFirst order differential equation Linear equation; Method of integrating factors
First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial
More informationY 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall
Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationSection 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.
MAT 444 H Barclo Spring 004 Homwork 6 Solutions Sction 6 Lt H b a subgroup of a group G Thn H oprats on G by lft multiplication Dscrib th orbits for this opration Th orbits of G ar th right costs of H
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit
More informationCombinatorial Networks Week 1, March 11-12
1 Nots on March 11 Combinatorial Ntwors W 1, March 11-1 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α
More informationOn the irreducibility of some polynomials in two variables
ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints
More informationPROBLEM SET Problem 1.
PROLEM SET 1 PROFESSOR PETER JOHNSTONE 1. Problm 1. 1.1. Th catgory Mat L. OK, I m not amiliar with th trminology o partially orr sts, so lt s go ovr that irst. Dinition 1.1. partial orr is a binary rlation
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationLie Groups HW7. Wang Shuai. November 2015
Li roups HW7 Wang Shuai Novmbr 015 1 Lt (π, V b a complx rprsntation of a compact group, show that V has an invariant non-dgnratd Hrmitian form. For any givn Hrmitian form on V, (for xampl (u, v = i u
More informationCOUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM
COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM Jim Brown Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 9634, USA jimlb@g.clmson.du Robrt Cass Dpartmnt of Mathmatics,
More informationThus, because if either [G : H] or [H : K] is infinite, then [G : K] is infinite, then [G : K] = [G : H][H : K] for all infinite cases.
Homwork 5 M 373K Solutions Mark Lindbrg and Travis Schdlr 1. Prov that th ring Z/mZ (for m 0) is a fild if and only if m is prim. ( ) Proof by Contrapositiv: Hr, thr ar thr cass for m not prim. m 0: Whn
More informationEquidistribution and Weyl s criterion
Euidistribution and Wyl s critrion by Brad Hannigan-Daly W introduc th ida of a sunc of numbrs bing uidistributd (mod ), and w stat and prov a thorm of Hrmann Wyl which charactrizs such suncs. W also discuss
More informationFunction Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0
unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More informationHardy-Littlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.
Hardy-Littlwood Conjctur and Excptional ral Zro JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that Hardy-Littlwood
More informationDeift/Zhou Steepest descent, Part I
Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n,
More informationMathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination
Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Tim-spac: 9:-: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts
More informationMultiple Short Term Infusion Homework # 5 PHA 5127
Multipl Short rm Infusion Homwork # 5 PHA 527 A rug is aministr as a short trm infusion. h avrag pharmacokintic paramtrs for this rug ar: k 0.40 hr - V 28 L his rug follows a on-compartmnt boy mol. A 300
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More informationKernels. ffl A kernel K is a function of two objects, for example, two sentence/tree pairs (x1; y1) and (x2; y2)
Krnls krnl K is a function of two ojcts, for xampl, two sntnc/tr pairs (x1; y1) an (x2; y2) K((x1; y1); (x2; y2)) Intuition: K((x1; y1); (x2; y2)) is a masur of th similarity (x1; y1) twn (x2; y2) an ormally:
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationLINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM
Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 1072-6691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL
More informationThe Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function
A gnraliation of th frquncy rsons function Th convolution sum scrition of an LTI iscrt-tim systm with an imuls rsons h[n] is givn by h y [ n] [ ] x[ n ] Taing th -transforms of both sis w gt n n h n n
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )
More informationWeek 3: Connected Subgraphs
Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationOn spanning trees and cycles of multicolored point sets with few intersections
On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More informationBSc Engineering Sciences A. Y. 2017/18 Written exam of the course Mathematical Analysis 2 August 30, x n, ) n 2
BSc Enginring Scincs A. Y. 27/8 Writtn xam of th cours Mathmatical Analysis 2 August, 28. Givn th powr sris + n + n 2 x n, n n dtrmin its radius of convrgnc r, and study th convrgnc for x ±r. By th root
More informationSearching Linked Lists. Perfect Skip List. Building a Skip List. Skip List Analysis (1) Assume the list is sorted, but is stored in a linked list.
3 3 4 8 6 3 3 4 8 6 3 3 4 8 6 () (d) 3 Sarching Linkd Lists Sarching Linkd Lists Sarching Linkd Lists ssum th list is sortd, but is stord in a linkd list. an w us binary sarch? omparisons? Work? What if
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability
More informationMinimum Spanning Trees
Yufi Tao ITEE Univrsity of Qunslan In tis lctur, w will stuy anotr classic prolm: finin a minimum spannin tr of an unirct wit rap. Intrstinly, vn tou t prolm appars ratr iffrnt from SSSP (sinl sourc sortst
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationConstruction of asymmetric orthogonal arrays of strength three via a replacement method
isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy
More informationν a (p e ) p e fpt(a) = lim
THE F -PURE THRESHOLD OF AN ELLIPTIC CURVE BHARGAV BHATT ABSTRACT. W calculat th F -pur thrshold of th affin con on an lliptic curv in a fixd positiv charactristic p. Th mthod mployd is dformation-thortic,
More informationa 1and x is any real number.
Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars
More informationAdditional Math (4047) Paper 2 (100 marks) y x. 2 d. d d
Aitional Math (07) Prpar b Mr Ang, Nov 07 Fin th valu of th constant k for which is a solution of th quation k. [7] Givn that, Givn that k, Thrfor, k Topic : Papr (00 marks) Tim : hours 0 mins Nam : Aitional
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More information1 N N(θ;d 1...d l ;N) 1 q l = o(1)
NORMALITY OF NUMBERS GENERATED BY THE VALUES OF ENTIRE FUNCTIONS MANFRED G. MADRITSCH, JÖRG M. THUSWALDNER, AND ROBERT F. TICHY Abstract. W show that th numbr gnratd by th q-ary intgr part of an ntir function
More informationIndeterminate Forms and L Hôpital s Rule. Indeterminate Forms
SECTION 87 Intrminat Forms an L Hôpital s Rul 567 Sction 87 Intrminat Forms an L Hôpital s Rul Rcogniz its that prouc intrminat forms Apply L Hôpital s Rul to valuat a it Intrminat Forms Rcall from Chaptrs
More informationExamples and applications on SSSP and MST
Exampls an applications on SSSP an MST Dan (Doris) H & Junhao Gan ITEE Univrsity of Qunslan COMP3506/7505, Uni of Qunslan Exampls an applications on SSSP an MST Dijkstra s Algorithm Th algorithm solvs
More informationUnit 6: Solving Exponential Equations and More
Habrman MTH 111 Sction II: Eonntial and Logarithmic Functions Unit 6: Solving Eonntial Equations and Mor EXAMPLE: Solv th quation 10 100 for. Obtain an act solution. This quation is so asy to solv that
More informationA. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).
A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts
More informationSome remarks on Kurepa s left factorial
Som rmarks on Kurpa s lft factorial arxiv:math/0410477v1 [math.nt] 21 Oct 2004 Brnd C. Kllnr Abstract W stablish a connction btwn th subfactorial function S(n) and th lft factorial function of Kurpa K(n).
More informationINCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS. xy 1 (mod p), (x, y) I (j)
INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS T D BROWNING AND A HAYNES Abstract W invstigat th solubility of th congrunc xy (mod ), whr is a rim and x, y ar rstrictd to li
More informationy cos x = cos xdx = sin x + c y = tan x + c sec x But, y = 1 when x = 0 giving c = 1. y = tan x + sec x (A1) (C4) OR y cos x = sin x + 1 [8]
DIFF EQ - OPTION. Sol th iffrntial quation tan +, 0
More informationAlpha and beta decay equation practice
Alpha and bta dcay quation practic Introduction Alpha and bta particls may b rprsntd in quations in svral diffrnt ways. Diffrnt xam boards hav thir own prfrnc. For xampl: Alpha Bta α β alpha bta Dspit
More information4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.
. 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More information2. Finite Impulse Response Filters (FIR)
.. Mthos for FIR filtrs implmntation. Finit Impuls Rspons Filtrs (FIR. Th winow mtho.. Frquncy charactristic uniform sampling. 3. Maximum rror minimizing. 4. Last-squars rror minimizing.. Mthos for FIR
More informationPHYS ,Fall 05, Term Exam #1, Oct., 12, 2005
PHYS1444-,Fall 5, Trm Exam #1, Oct., 1, 5 Nam: Kys 1. circular ring of charg of raius an a total charg Q lis in th x-y plan with its cntr at th origin. small positiv tst charg q is plac at th origin. What
More informationSPH4U Electric Charges and Electric Fields Mr. LoRusso
SPH4U lctric Chargs an lctric Fils Mr. LoRusso lctricity is th flow of lctric charg. Th Grks first obsrv lctrical forcs whn arly scintists rubb ambr with fur. Th notic thy coul attract small bits of straw
More information10. EXTENDING TRACTABILITY
Coping with NP-compltnss 0. EXTENDING TRACTABILITY ining small vrtx covrs solving NP-har problms on trs circular arc covrings vrtx covr in bipartit graphs Q. Suppos I n to solv an NP-complt problm. What
More informationCase Study 4 PHA 5127 Aminoglycosides Answers provided by Jeffrey Stark Graduate Student
Cas Stuy 4 PHA 527 Aminoglycosis Answrs provi by Jffry Stark Grauat Stunt Backgroun Gntamicin is us to trat a wi varity of infctions. Howvr, u to its toxicity, its us must b rstrict to th thrapy of lif-thratning
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationON RIGHT(LEFT) DUO PO-SEMIGROUPS. S. K. Lee and K. Y. Park
Kangwon-Kyungki Math. Jour. 11 (2003), No. 2, pp. 147 153 ON RIGHT(LEFT) DUO PO-SEMIGROUPS S. K. L and K. Y. Park Abstract. W invstigat som proprtis on right(rsp. lft) duo po-smigroups. 1. Introduction
More informationUNTYPED LAMBDA CALCULUS (II)
1 UNTYPED LAMBDA CALCULUS (II) RECALL: CALL-BY-VALUE O.S. Basic rul Sarch ruls: (\x.) v [v/x] 1 1 1 1 v v CALL-BY-VALUE EVALUATION EXAMPLE (\x. x x) (\y. y) x x [\y. y / x] = (\y. y) (\y. y) y [\y. y /
More informationMSLC Math 151 WI09 Exam 2 Review Solutions
Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationSlide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS
Slid DIGITAL SIGAL PROCESSIG UIT I DISCRETE TIME SIGALS AD SYSTEM Slid Rviw of discrt-tim signals & systms Signal:- A signal is dfind as any physical quantity that varis with tim, spac or any othr indpndnt
More informationThe Equitable Dominating Graph
Intrnational Journal of Enginring Rsarch and Tchnology. ISSN 0974-3154 Volum 8, Numbr 1 (015), pp. 35-4 Intrnational Rsarch Publication Hous http://www.irphous.com Th Equitabl Dominating Graph P.N. Vinay
More information2.3 Matrix Formulation
23 Matrix Formulation 43 A mor complicatd xampl ariss for a nonlinar systm of diffrntial quations Considr th following xampl Exampl 23 x y + x( x 2 y 2 y x + y( x 2 y 2 (233 Transforming to polar coordinats,
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationAnalysis of Algorithms - Elementary graphs algorithms -
Analysis of Algorithms - Elmntary graphs algorithms - Anras Ermahl MRTC (Mälaralns Ral-Tim Rsach Cntr) anras.rmahl@mh.s Autumn 00 Graphs Graphs ar important mathmatical ntitis in computr scinc an nginring
More informationA RELATIVISTIC LAGRANGIAN FOR MULTIPLE CHARGED POINT-MASSES
A RELATIVISTIC LAGRANGIAN FOR MULTIPLE CHARGED POINT-MASSES ADRIAAN DANIËL FOKKER (1887-197) A translation of: Ein invariantr Variationssatz für i Bwgung mhrrr lctrischr Massntilshn Z. Phys. 58, 386-393
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationCramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter
WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)
More information(Upside-Down o Direct Rotation) β - Numbers
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg
More informationCS 491 G Combinatorial Optimization
CS 49 G Cobinatorial Optiization Lctur Nots Junhui Jia. Maiu Flow Probls Now lt us iscuss or tails on aiu low probls. Thor. A asibl low is aiu i an only i thr is no -augnting path. Proo: Lt P = A asibl
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationMA1506 Tutorial 2 Solutions. Question 1. (1a) 1 ) y x. e x. 1 exp (in general, Integrating factor is. ye dx. So ) (1b) e e. e c.
MA56 utorial Solutions Qustion a Intgrating fator is ln p p in gnral, multipl b p So b ln p p sin his kin is all a Brnoulli quation -- st Sin w fin Y, Y Y, Y Y p Qustion Dfin v / hn our quation is v μ
More informationComplex Powers and Logs (5A) Young Won Lim 10/17/13
Complx Powrs and Logs (5A) Copyright (c) 202, 203 Young W. Lim. Prmission is grantd to copy, distribut and/or modify this documnt undr th trms of th GNU Fr Documntation Licns, Vrsion.2 or any latr vrsion
More informationWhere k is either given or determined from the data and c is an arbitrary constant.
Exponntial growth and dcay applications W wish to solv an quation that has a drivativ. dy ky k > dx This quation says that th rat of chang of th function is proportional to th function. Th solution is
More information[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then
SYSTEM PERFORMANCE Lctur 0: Stady-tat Error Stady-tat Error Lctur 0: Stady-tat Error Dr.alyana Vluvolu Stady-tat rror can b found by applying th final valu thorm and i givn by lim ( t) lim E ( ) t 0 providd
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
More informationAnalysis of Algorithms - Elementary graphs algorithms -
Analysis of Algorithms - Elmntary graphs algorithms - Anras Ermahl MRTC (Mälaralns Ral-Tim Rsarch Cntr) anras.rmahl@mh.s Autumn 004 Graphs Graphs ar important mathmatical ntitis in computr scinc an nginring
More informationMATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES JESSE BURKE AND MARK E. WALKER Abstract. W study matrix factorizations of locally fr cohrnt shavs on a schm. For a schm that is projctiv ovr an affin schm,
More informationDISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P
DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P Tsz Ho Chan Dartmnt of Mathmatics, Cas Wstrn Rsrv Univrsity, Clvland, OH 4406, USA txc50@cwru.du Rcivd: /9/03, Rvisd: /9/04,
More informationRevisiting Wiener s Attack New Weak Keys in RSA
Rvisiting Winr s Attack w Wak Kys in RSA Subhamoy Maitra an Santanu Sarkar Inian Statistical Institut, 0 B T Roa, Kolkata 700 08, Inia {subho, santanu r}@isicalacin Abstract In this papr w rvisit Winr
More information