Weil Cojecture I Yichao Tia Morigside Ceter of Mathematics, AMSS, CAS [This is the sketch of otes of the lecture Weil Cojecture I give by Yichao Tia at MSC, Tsighua Uiversity, o August 4th, 20. Yuaqig Cai(ycai7@jhu.edu) is resosible for takig ad tyig these otes.] Solvig Diohatie equatio is oe of the mai roblem i umber theory for a log time. It is very difficult but woderful. For examle, it took over 300 years to see that X + Y = Z has o otrivial itegers solutio whe 3. We would like to cosider a easier roblem: solvig the Diohatie equatio modulo, where is a rime umber. We exect that this roblem is easy eough to hadle, but still is ot too trivial. Two examles Let C be the affie curve C : x 2 + y 2 =, ad cosider oits o F 2. We deote the set of solutios o F 2 by C(F ). First ote that we have a trivial solutio (, 0). Draw a lie through (, 0), with sloe t F. This lie will itersect with C(F ) with aother oit ( t2 2t, ) (if it +t 2 +t 2 exists). So we ca defie a ma t : F {t F t 2 + 0 mod } C(F ) {(, 0)} This ma is early a bijectio. Note that + t 2 0(mod ) has two solutios if (mod 4); ad it has o solutios if 3(mod 4). Hece, we have ( ) {, whe (mod 4) (.) C(F ) = = +, whe 3(mod 4) The result is ot as beautiful as we exect. Now we cosider the rojective curve C : X 2 + Y 2 = Z 2, X, Y, Z is the homogeeous coordiates. Whe Z 0, we ca get the origial affie curve ( ) X 2 ( ) Y 2 + =, Z Z
which we call this the affie art of C. Whe Z = 0, we get X 2 + Y 2 = 0. Because X, Y, Z ca t be all zero, we get ( ) X 2 Y = i F. Whe (mod 4), this equatio has two solutios. Whe 3(mod 4), this equatio has o solutios. Hece we have C(F ) = + regardless of. We get a more beautiful result. This may exlai why we cosider rojective curves istead of affie curves. We have aother way to get the umber of solutios to x 2 + y 2 = i F. We have C(F ) = x F (( x 2 = + (( )) x 2 x F Combiig with (.), we get a iterestig equality ( ) ( ) x 2 =, x F ) ) + which is ot so trivial to be roved directly. We cosider aother examle y 2 + x 3 =, ad its corresodig rojective curve Y 2 Z + X 3 = Z 3. Whe Z = 0, we have X = 0, hece there is oly oe solutio at ifiity. Let s we deote by N(y 2 = a) the umber of solutios of y 2 = a i F for a F, ad similarly for N(x 3 = b) for b F. We have C(F ) = + C(F ) = + N(y a+b= a,b F 2 = a)n(x 3 = b) We have N(y 2 = a) = + ( a ) as i the revious examle. Note that F is a cyclic grou of order. If 3 ( ), we have N(x 3 = b) = for ay b F. If 3 ( ), the we have two cases: if b is a cubic of some elemets i F, we have N(X 3 = b) = 3; otherwise, N(X 3 = b) = 0. I summary, oe obtais {, whe 2(mod 3) N(X 3 = b) = + χ(b) + χ 2 (b), whe (mod 3) 2
where χ is a multilicative character of F of order 3. Therefore, C(F ) = + ( ( )) a + ( + χ(b) + χ 2 (b)) where J(λ, χ) = a+b= a,b F = + a+b= a,b F = + + ( + a+b= a,b F + χ(b) + χ 2 (b) + χ(b) + = + + J(λ, χ) + J(λ, χ 2 ) = + + J(λ, χ) + J(λ, χ) a+b= a,b F a+b= a,b F χ 2 (b) χ(b) + ( ) ) a χ 2 (b) ( a ) χ(b) is a Jacobi sum. From the theory of Jacobi sum ad Gauss sums, we have J(λ, χ) =. Therefore, we obtai which is called the Hasse-Weil boud. C(F ) ( + ) = J + J = 2 Re(J) 2, 2 Zeta fuctio of a rojective variety over a fiite field Recall that a fiite field is a field with fiite elemets. The field F = Z/Z is a examle of such fields. Theorem 2. (Galois). For every iteger N, there exists a uique field extesio over F of degree. We deote this field by F. Moreover, every fiite field is of this form, ad F /F is a Galois extesio with Galois grou Gal(F /F ) Z/Z. Let q = f, the Galois grou Gal( F q /F q ) Ẑ is commutative. It has a caoical toological geerator σ q defied by σ q (x) = x q. We call σ q the arithmetic Frobeius of F q. I geometric alicatios, it s more coveiet to use the geometric Frobeius F r q = σq. We cosider a rojective variety X 0 over F q, which ca be roughly viewed as the commo zeros of a set of homogeeous olyomials with coefficiets i F q : f (T 0,, T N ) = 0, f 2 (T 0,, T N ) = 0, (2.) f r (T 0,, T N ) = 0. 3
I this talk, all the varieties are uderstood to be reduced ad coected. We defie the dimesio of X 0 as dim(x 0 ) = N rk(jac), where Jac = ( f i T j ) i r,0 j N is the Jacobi matrix ad the T 0,, T N are cosidered as ideedet variables. We say the variety X 0 is smooth, if for every geometric oit x 0 X 0 ( F q ), we have rk(jac(x 0 )) = rk(jac). Note that Gal( F q /F q ) has a atural actio o X 0 ( F q ). A orbit of X 0 ( F q ) uder this actio is called a closed oit of X 0. We deote by X the set of closed oits of X 0. For all x 0 X, we defie deg(x 0 ) to be the cardiality of the orbit x 0. For ay iteger, we defie N to be the umber of solutios i F q of the defiig solutios (2.) of X 0. Defiitio 2.2. Let X 0 be a rojective variety over F q. The zeta fuctio of X 0 is defied as the formal ower series ( N t ) Z(X 0, t) = ex. N= Lemma 2.3. The Zeta fuctio ca be writte as a Euler roduct Proof. Note first that By defiitio, we have log Z(X 0, t) = = Z(X 0, t) = N (X 0 ) = = N t = m= = m= deg(x) = t deg(x). deg(x) tm m deg(x). deg(x) t m deg(x) m = Here, i the last ste we used the well kow equality log x = = x. deg(x)t log t deg(x). 4
Theorem 2.4 (Weil cojecture). Let X 0 be a rojective smooth variety of dimesio d. The () the zeta fuctio Z(X 0, t) is ratioal, i.e., we have Z(X 0, t) Q(t); (2) the zeta fuctio satisfies the fuctioal equatio Z(X 0, t) = ±q dχ(x 0 ) 2 t χ(x0) Z(X 0, q d t ), where χ(x 0 ) is the Euler-Poicaré characristic of X 0 ; (3) we ca write more recisely Z(X 0, t) = P P 3 P 2d P 0 P 2 P 2d, where P i = + a t + + a bi t b i = b i ( α j t) is a olyomial with coefficiets i Z, ad α j C = q i 2. j= The last art is called the Riema Hyothesis for fiite fields, ad the umber b i = deg(p i ) is called the i-th Betti umber of X 0. Usig these umbers, the Euler-Poicaré characteristic i (2) is exressed as χ(x 0 ) = 2d ( )i b i. Here are some history of the Weil cojecture. Weil first calculated the zeta fuctios of some curves usig Gauss sums ad Jacobi sums, the raised the cojecture i 949. Later o, he roved this cojecture i the case of curves usig Jacobias. Dwork roved statemet () i 960 usig -adic cohomology. I 965, Grothedieck roved (), (2) ad the art of (3) before where by usig the owerful machie of l-adic cohomology that he develoed. I 974, Delige roved the remaiig art of statemet (3). I 987, Laumo gave a differet roof by usig l-adic Fourier trasform. Fially, Kedlaya develoed i 2006 the theory of -adic Fourier trasform, ad gave a urely -adic roof of Weil cojecture. Examle 2.5. () Let X 0 be the curve over F q defied by the homogeeous equatio X 2 + Y 2 = Z 2. By a similar argumet as i sectio, we see easily that X 0 (F q ) = + q. Therefore, oe obtais Z(X 0, t) = ex( = q + t ) = ( t)( qt). (2) Let X 0 be the ellitic curve X 3 + Y 2 Z = Z 3 over F. I geeral, the zeta fuctio of a ellitic curve over F has the form Z(X 0, t) = a t + t 2 ( t)( t), 5
where a = + N (X 0 ). Recall that i sectio, oe saw that N (X 0 ) = X 0 (F ) = + + J + J with J = ( a a F ) χ( a) is a Jacobi sum. Hece, we get a = 2Re(J), ad ( + Jt)( + Jt) Z(X 0, t) = ( t)( t). 3 Sketch of Proof of Theorem 2.4 For a algebraic variety X over a algebraically closed field k, Grothedieck defied two cohomology theory with coefficiets i Q l : H (X et, Q l ) ad Hc (X et, Q l ). Here, l is a rime umber differet from the characteristic of k; H (X et, k) is a aalogue of classical sigular cohomology, ad Hc (X et, Q l ) is aalogous to the comactly suorted cohomology grous. More geerally, we ca cosider a costructible Q l -sheaf E ad defie similarly the cohomology grous H (X et, E) ad H (X et, E). These grous are fiite dimesioal Q l -vector saces, ad H, Hc = 0 whe > 2 dim(x) or < 0. Let X 0 be a variety over F q, E be a costructible sheaf o X 0, ad ut X = X 0 Fq F q. Alyig Grothedieck s costructio to X, we get H (X et, E) ad Hc (X et, E). These grous are moreover equied with atural actios of Gal( F q /F q ). A imortat igrediet i the roof of Weil cojecture is the followig Theorem 3. (Grothedieck-Lefschetz Trace Formula). Uder the assumtios above, we have 2d Tr(F r q, E x ) = ( ) i Tr(F r q, Hc(X i et, E)). x X 0 (F q ) Now we idicate how Theorem 3. imlies 2.4() ad first art of 2.4(3). If E = Q l, the the left had side of the Grothedieck-Lefschetz trace formula is N (X 0 ), ad the right had side is 2d ( ) i Tr(F r q, Hc(X i et, Q l )). Therefore, oe gets log Z(X 0, t) = = = 2d 2d ( ) i ( ) i Tr(F r q, H i c(x et, Q l ))t = Tr(F r q, H i c(x et, Q l ))t Now we have a elemetary lemma i liear algebra, whose roof is left to the reader. Lemma 3.2. Let V be a fiite dimesioal vector sace over a field, ad A : V V be a edomorhism of V. The we have a idetity of formal ower series i t: = Tr(A ) t = log det( At). 6
Alyig this lemma, oe obtais Hece, log Z(X 0, t) = 2 dim(x 0 ) ( ) i log det( F r q t, H i c(x et, Q l ). (3.) Z(X 0, t) = det( F r q t, Hc(X i et, Q l )) ( )i+ i det( F r q t, Hc(X i et, Q l )) i odd = det( F r q t, Hc(X i et, Q l )) Q l(t) i eve We ut P i = det( F r q t, H i c(x et, Q l )). Sice the zeta fuctio is ideedet from the choice of l, we actually have Z(X 0, t) Q(t). This roves () ad the first art of (3) excet for P i Z[t]. We ote that this is true for all varieties over F q without assumig that X 0 is smooth ad rojective. To rove statemet 2.4(2), we have to use Poicaré duality. Assume X 0 is smooth. Grothedieck showed that the atural cu roduct H i (X et, Q l ) H 2d i c (X et, Q l ) H 2d (X et, Q l ) Q l ( d). is o-degeerate, where Q l ( d), called the d-th Tate twist of Q l, is a oe-dimesioal Q l -vector sace where F r q acts as multilicatio by q d. This is called the Poicaré duality for X. Moreover, if X 0 is rojective, we have H i = H i c. Therefore, if X 0 is both smooth ad rojective, we have a erfect airig H i (X et, Q l ) H 2d i (X et, Q l ) Q l ( d). It follows that H 2d i (X et, Q l ) H i (X et, Q l ) ( d), where H i (X et, Q l ) meas the dual of H i (X et, Q l ) (equied with the iduced actio of Gal( F q /F q ).) So if α is a eigevalue of F r q o H i, the q d /α is a eigevalue of F r q o H 2d i. The fuctioal equatio i 2.4(2) follows immediately from this fact ad (3.). To ed this take, I d like to say a few words o Delige s roof of the Riema Hyothesis art of 2.4. There are several key ideas i the roof. The first oe is the geeralizatio, due to Delige ad Katz, to l-adic cohomology of classical Lefschetz ecils. The secod idea is the estimatio of eigevalues usig a method i aalytic umber theory iveted by Raki. Fially, a exlicit comutatio of the moodromy grou of certai reresetatios of the algebraic fudametal grou of curves layed also a very crucial role. c 7