Mathematical Research Letters 6, (1999) SPACE FILLING CURVES OVER FINITE FIELDS. Nicholas M. Katz

Transcription

1 Mathematcal Research Letters 6, (1999) SPACE FILLING CURVES OVER FINITE FIELDS Ncholas M. Katz Introducton In ths note, we construct curves over fnte felds whch have, n a certan sense, a lot of ponts, and gve some applcatons to the zeta functons of curves and abelan varetes over fnte felds. In fact, we found the basc constructon, gven n Lemma 1, of curves n A n whch go through every ratonal pont, as part of an unsuccessful attempt to fnd curves of growng genus over a fxed fnte feld wth lots of ponts n the sense of the Drnfeld-Vladut bound [2]. The dea of applyng that constructon along the lnes of ths note grew out of an August 1996 conversaton wth Ofer Gabber about whether every abelan varety over a fnte feld was a quotent of a Jacoban, durng whch he constructed, on the fly, a proof of that fact. A varant of hs proof appears here n Theorem 11. It s a pleasure to acknowledge my debt to hm. The basc constructons Lemma 1. Let k be a fnte feld, p ts characterstc, k an algebrac closure of k, E/k a fnte extenson nsde k, and n 1 an nteger. There exsts a smooth, geometrcally connected curve C/k and a closed mmerson of k-schemes C A n Z k whch nduces a bjecton of E-valued ponts C(E) =A n (E). Constructon-proof. If n = 1, take C = A n Z k.ifn = r + 1 wth r 1, choose a sequence of r nonzero polynomals n one varable over k, f 1 (X),...,f r (X), wth the followng three propertes: 1) For each, f (x) = 0 for every x E. 2) For each, the degree d of f s prme to p. 3) The degrees are strctly ncreasng: d 1 <d 2 < <d r. [Here s a smple way to make such a choce. Wrte q := #E, and pck a strctly ncreasng sequence of r postve ntegers each of whch s prme to p, say e 1 <e 2 < <e r. Then take each f (X) :=(X q X)X e.] Receved March 4,

2 614 NICHOLAS M. KATZ In A r+1 k wth coordnates X, Y 1,...,Y r, consder the closed subscheme C/k defned by the r equatons (Y ) q Y = f (X), =1,...,r. It s obvous from these equatons that every E-valued pont of A n les n C. We must see that C/k s a smooth curve whch s geometrcally connected. Frst of all, C/k s a smooth curve, for t s the fbre product over A 1 k of r fnte etale galos coverngs E A 1 k, wth E the affne plane curve (Y ) q Y = f (X) na 2 k. It remans to see that C k k s connected. Ths results from Artn-Schreer theory. On A 1 k, or ndeed on any smooth, affne, connected scheme S/ k, the Artn-Schreer sequence relatve to q, f P(f):=f 0 E O q f S O S 0 gves, va the long exact cohomology sequence, an somorphsm of E-vector spaces H 0 (S, O S )/P(H 0 (S, O S )) = Het(S, 1 E) = Hom(π 1 (S),E). Gven f n H 0 (S, O S ), the coverng of S defned by Y q Y = f (n A 1 S) s fnte etale galos wth group E (α n E translates Y ), so s an element Class(f) n Hom(π 1 (S),E). Now return to the case when S s A 1 k and take any nontrval C-valued character ψ of E. If f n k[x] has degree d prme to p, then the composte homomorphsm s known [1, 3.5.4] to have Swan conductor d at. Our C k k s a fnte etale galos coverng of A 1 k wth group E E E = E r, correspondng to the r-tuple (f 1,f 2,...,f r ) va ) r ( k[x]/p( k[x]) = H 1 et (A 1 k, E r ) = Hom(π 1 (S),E r ). The total space C k k of ths coverng s connected f and only f the correspondng homomorphsm Class(f 1,f 2,...,f r ):π 1 (A 1 k) E r s surjectve, or equvalently (Pontrajagn dualty!) f and only f for every nontrval C-valued addtve character (ψ 1,ψ 2,...,ψ r )ofe r, the composte homomorphsm (ψ 1,ψ 2,...,ψ r ) Class(f 1,f 2,...,f r ):π 1 (A 1 k) C, s nontrval. But ths composte s just the product (ψ 1,ψ 2,...,ψ r ) Class(f 1,f 2,...,f r )= L ψ (f ). In ths product, L ψ (f ) s trval f ψ tself s trval, and L ψ (f ) has Swan = d f ψ s nontrval. Because the d are all dstnct, and at least one ψ s nontrval, we have ( ) Swan L ψ (f ) = Sup wth ψ nontrv(d ) > 0.

3 SPACE FILLING CURVES OVER FINITE FIELDS 615 Hence L ψ (f ) must be nontrval. Lemma 2. Let k be a fnte feld, X/k projectve (resp. quas-projectve), smooth, and geometrcally connected of dmenson n 1. LetE/k be a fnte extenson. There exsts an affne (resp. quas-affne) open set U X whch contans all the E-valued ponts of X,.e., U(E) =X(E). Proof. To fxdeas, say X P N k. We need only construct an affne open set U n P N k whch contans all the E-valued ponts of P N k, for then X U s the desred affne (resp. quas-affne) open set of X. To do ths, denote by K/E the feld extenson of degree N + 1, and pck a bass α 0,α 1,...,α N of K/E. Denote by H the form of degree N +1 n X 0,...,X N wth coeffcents n E defned by H(X s) := Norm K/E (α 0 X α N X N ). Then H s nonzero at every E-valued pont of P N. For each σ n Gal(E/k), the form H σ has the same property (ndeed, f we extend σ to an element σ n Gal(K/k) whch nduces σ, then σ(α 0,α 1,...,α N ) s another bass of K/E, and H σ s ts norm form to E). So Norm E/k (H) s a form wth coeffcents n k whch s nonzero at every E-valued pont of P N. We may take for U the affne open set (P N k) [ 1/Norm E/k (H) ]. Lemma 3. Let k be a fnte feld, U/k a quas-affne, smooth, and geometrcally connected of dmenston n 1. Let E/k be a fnte extenson. There exsts an open set V U whch contans all the E-valued ponts of U and whch admts an etale map to A n k. Proof. Say U s open n the affne scheme Ū. Frst vew U(E) as a fnte closed subscheme Z of U, by groupng ts ponts nto orbts under Gal(E/k). More precsely, Z s the dsjont unon of the fntely many closed ponts of U the degree over k of whose resdue felds dvdes deg(e/k), wth ts reduced structure. Thus, Z s a closed subscheme of U whch s fnte etale over k. Ths same Z s closed n Ū, snce we may descrbe t as the dsjont unon of the fntely many closed ponts of Ū whose resdue feld degrees over k dvde deg(u/k) and whch le n U. Denote by A the coordnate rng of Ū, I A the deal defnng Z. At each pont P n Z, pass to the local rng O Ū,P of P n Ū, and pck n elements f 1,P,f 2,P,...,f n,p whch form a k(p )-bass of m/m 2, m the maxmal deal. The rng A/I 2 s just the product rng P Z OŪ,P /m 2. So, we can fnd functons f 1,...,f n n A such that, for each and each P, f nduces f,p n O Ū,P /m 2. Restrct each functon f to U, and vew (f 1,...,f n ) as a map π of U to A n. Ths map π s etale at each pont P n Z by constructon. Thus, the set V of ponts of U at whch π s etale s open, and contans Z. Lemma 4. Let k be a fnte feld, V/k smooth and geometrcally connected of dmenson n 1, and π : V A n k

4 616 NICHOLAS M. KATZ an etale map of k-schemes. For each nteger r 1, denote by k r the extenson feld of k nsde k of degree r over k. For each r 1, apply Lemma 1 wth E := k r to produce a closed mmerson r : C r /k A n k, wth C r /k a smooth, geometrcally connected curve such that C r (k r )=A n (k r ). Form the fbre product D r := C r A n k V V π r A n k. C r 1) For every r, D r /k s a smooth curve, space-fllng n V for k r,.e., va the closed mmerson : D r := C r An k V V, we have D r (k r )=V(k r ). 2) For all suffcently large r, D r /k s geometrcally connected. Proof. 1) s obvous from the cartesan dagram defnng D r, n whch π s etale, C r /k s a smooth curve, and r s surjectve on k r -valued ponts. To prove 2), we argue as follows. The etale map π need not be fnte etale, but there s a dense open set j : W A n k over whch π s fnte etale (just because π s fnte etale over the generc pont of A n k). Take the entre dagram D r := C r A n k V C r V π r A n k. n the category of A n k-schemes, and pull t back to the open set W,.e., base change t by j : W A n k. We get a dagram D r,w C r,w j r C r W r,w VW π W j r A n k

5 SPACE FILLING CURVES OVER FINITE FIELDS 617 In ths dagram, both W and V W are smooth over k and geometrcally connected, π s fnte etale, and r,w : C r,w W s spacefllng for k r.nowc r,w s open n C r, so t s ether dense and open n C r and tself geometrcally connected, or t s empty. For large r, C r,w s not empty, because W (k r ) s nonempty for large r (by Lang-Wel, because W/k s geometrcally rreducble), and r,w : C r,w W s spacefllng for k r. Let us temporarly admt the truth of Lemma 5. Let k be a fnte feld, E/k and W/k two smooth, geometrcally connected k-schemes of the same dmenson n 1, and E π W a fnte etale k-morphsm. Suppose gven an nteger r 0 1, and for all ntegers r r 0, a smooth, geometrcally connected curve C r /k and a closed k-mmerson r : C r W whch s spacefllng for k r,.e., C r (k r )=W(k r ). Form the fbre product D r C r r,e E π r,w W Then for r suffcently large, the curve D r /k s geometrcally connected. Applyng ths lemma to our stuaton (E s V W, C r s C r,w ), we fnd that for large r, D r,w s geometrcally connected. We wsh to nfer that D r /k tself s geometrcally connected. If t s not, then D r k k s a unon of two or more connected components, each of whch s etale over C r k k. But as etale maps are open, the mage of each connected component meets the dense open set C r,w k k, and hence Dr,W k k s not connected, contradcton. QED for Lemma 4 modulo Lemma 5. Proof of Lemma 5. Fxa geometrc pont ω n W k k, and vew the fnte etale coverng π : E W as an acton of the group π 1 (W, ω) on the fnte set S := π 1 (ω),.e., a homomorphsm ρ : π 1 (W, ω) Aut(S). The geometrc connectedness of E means precsely that va ths acton, the subgroup π geom 1 (W, ω) :=π 1 (W k k, ω) π1 (W, ω) acts transtvely on S. Recall the short exact sequence

6 618 NICHOLAS M. KATZ 1 π geom 1 (W, ω) π 1 (W, ω) degree Gal( k/k) 1 Denote by Γ geom Γ Aut(S) the mages n Aut(S) ofπ geom 1 (W, ω) and of π 1 (W, ω) respectvely under ρ. The quotent Γ/Γ geom s cyclc, say of order N, generated by ρ(f ) for any fxed element F n π 1 (W, ω) of degree 1. For each n Z/N Z, denote by Γ() Γ the set of elements whose degree mod N s,.e., Γ() s the coset ρ(f )Γ geom. By Chebotarev (cf., [5, ]) for every r 0, we have: ( r, E/W ) The mages under ρ of all degree r Frobenus elements n π 1 (W, ω),.e., all elements n all Frobenus conjugacy classes Frob kr,w n π 1 (W, ω) attached to k r -valued ponts w of W, fll the coset Γ(r). We wll show that for any r r 0 large enough that ( r, E/W ) holds, D r s geometrcally connected. To see ths, pck a geometrc pont c r n C r, take for ω ts mage n W, and consder the composte homomorphsm π 1 ( r,w ) ρ π 1 (C r,c r ) π 1 (W, ω) Γ Aut(S), whch we label ρ r : π 1 (C r,c r ) Γ Aut(S). Now D r /k s geometrcally connected f and only f the subgroup ρ r (π geom 1 (C r,c r )) Aut(S) acts transtvely on S. A suffcent condton for ths transtvty s that ( r) ρ r (π geom 1 (C r,c r )) = Γ geom, (because the geometrc connectedness of E means that Γ geom acts transtvely). A suffcent condton for ρ r (π geom 1 (C r,c r )) = Γ geom, s that the condton ( r, D r, C r ) hold: ( r, D R, C r ) The mages under ρ r of all the Frobenus elements of degree r n π 1 (C r,c r ) fll Γ(r). Indeed, every element n Γ geom := Γ(0) s of the form A 1 B wth A and B n Γ(r) =ρ(f r )Γ geom, and hence every element of Γ geom wll be the mage under ρ r of a rato (Frob kr,x) 1 (Frob kr,y) for two ponts x and y n C r (k r ). Such a rato les n π geom 1 (C r,c r ). But C r (k r )=W(k r ) by assumpton, so every degree r Frobenus element n π 1 (W, ω) s the mage under π 1 ( r,w ) of a degree r Frobenus element n Ẑ

7 SPACE FILLING CURVES OVER FINITE FIELDS 619 π 1 (C r,c r ). Therefore ( r, D r /C r ) s equvalent to ( r, E/W ). In partcular, for large r, ( r, D r /C r ) and hence ( r) hold. Wth an eye to later applcatons, we extract from the proof of Lemma 5 the followng varant. Lemma 6. Let k be a fnte feld, W/k a smooth, geometrcally connected k- scheme, and w a geometrc pont of W. Suppose gven an nteger r 0 1, and, for each nteger r r 0, a smooth geometrcally connected k-scheme C r /k and a k-morphsm f r : C r W whch s surjectve on k r -valued ponts. For each r r 0, pck a geometrc pont c r n C r, and a chemn from f r (c r ) to w. Suppose that G s ether 1) a fnte group, or, 2) GL(n, O λ ) for some postve nteger n and for O λ the rng of ntegers n a fnte extenson of Q l, for some prme number l. 3) GL(n, Q l ) for some n and some prme l. Suppose gven a contnuous group homomorphsm ρ : π 1 (W, w) G. We denote ρ r : π 1 (C r,c r ) G the composte homomorphsm π 1 (C r,c r ) f π1 (W, f(c r )) chemn π 1 (W, w) ρ G. Then we have: a) For r suffcently large, we have an equalty of mages of geometrc fundamental groups ρ r (π geom 1 (C r,c r )) = ρ(π geom 1 (W, w)) (equalty nsde G). b) Suppose n addton that, for each r r 0, f r s also surjectve on k s -valued ponts for all dvsors s of r. Then for r suffcently large and suffcently dvsble, we have an equalty of mages of fundamental groups ρ r (π 1 (C r,c r )) = ρ(π 1 (W, w)) (equalty nsde G). Proof. In case 1), G fnte, we put Γ := ρ(π 1 (W, w)), Γ geom := ρ(π geom 1 (W, w)), denote by N the order of the cyclc group Γ/Γ geom, and denote by Γ() the set of elements n Γ of degree mod N. By Chebotarev, for r 0, the Froben of k r -valued ponts of W fll the coset Γ(r), hence by the surjectvty of the map f r on k r -valued ponts, so do the Froben of k r -valued ponts of C r for r 0. For these r, the A 1 B argument shows that ratos A 1 B of such Froben fll Γ geom, whence a).

8 620 NICHOLAS M. KATZ For b), we argue as follows. For each nteger n [0,N 1] pck an nteger d mod N and suffcently large that the Froben of k d -valued ponts of W fll the coset Γ(). Then for any r r 0 whch s dvsble by d, the Froben of the ponts on C r wth values n k d for =0, 1,...,N 1 fll Γ. For case 2), put K := the mage ρ(π geom 1 (W, w)) n GL(n, O λ ). By Pnk s Lemma [4, ], there exsts an nteger d 1 such that a closed subgroup H of K s equal to K f and only f H and K have the same mage n GL(n, O λ /l d O λ ). For each nteger r r 0, put H r := the mage ρ r (π geom 1 (C r,c r )) n GL(n, O λ ). Thus H r s a closed subgroup of K. By case 1), appled to the reducton mod l d of ρ, for r 0, H r and K have the same mage n GL(n, O λ /l d O λ ). So by Pnk s Lemma H r = K for all such r. For b), apply Pnk s Lemma to L := the mage ρ(π 1 (W, w)) n GL(n, O λ ) and the subgroups J r := the mage ρ r (π q (C r,c r )) n GL(n, O λ ) to reduce b) to case 1). For case 3), use the fact [5, 9.0.7] that any compact subgroup of GL(n, Q l ), n partcular the mage ρ(π 1 (W, w)), s conjugate to a closed subgroup of GL(n, O λ ) for O λ the rng of ntegers n some fnte extenson E λ of Q l to reduce to case 2). As an mmedate consequence of case 3) of Lemma 6, we get the followng result of Bertn type. Corollary 7. Let k be a fnte feld, W/k a smooth, geometrcally connected k- scheme, and w a geometrc pont of W. Suppose gven an nteger r 0 1, and, for each nteger r r 0, a smooth, geometrcally connected k-scheme C r /k and a k-morphsm f r : C r W, whch s surjectve on k r -valued ponts. For each r r 0, pck a geometrc pont c r n C r, and a chemn from f r (c r ) to w. Letl be a prme number, and F a lsse Q l -sheaf on W of rank denoted n, correspondng to a contnuous homomorphsm ρ : π 1 (W, w) GL(n, Q l ). Denote by G geom,f on W the Zarsk closure of ρ(π geom 1 (W, w)) n GL(n) Q l. Then for r suffcently large, the pullback sheaf (f r ) (F) on C r has the same G geom : G geom, (fr ) F on C r = G geom, F on W. Moreover, f F on W has the property that ρ(π 1 (W, w)) les n G geom, F on W ( Q l ), then for r suffcently large the pullback sheaf (f r ) (F) on C r has the same property, that ρ(π 1 (C r,c r )) les n G geom, (fr ) F on C r ( Q l ). Theorem 8. Let k be a fnte feld, X/k smooth and quas-projectve and geometrcally connected, of dmenson n 1. Let E/k be a fnte extenson. There exsts a smooth, geometrcally connected curve C 0 /k, and an mmerson π : C 0 X whch s bjectve on E-valued ponts.

9 SPACE FILLING CURVES OVER FINITE FIELDS 621 Proof. Frst apply Lemmas 2 and 3 to fnd an open set V n X whch contans all the E-valued ponts and whch admts an etale map π to A n k. Let d := degree(e/k), so E s k d. For each r 1, use Lemma 1 to fnd a smooth, geometrcally connected curve C rd /k n A n k whch s spacefllng for k rd. Take D rd /k n V to be the fbre product D rd := C rd A n k V. By Lemma 4, for large r ths closed subscheme D rd of V s a smooth, geometrcally connected curve over k whch s spacefllng for k rd. Takng the Gal(k rd /k d )-nvarants on both sdes of the equalty D rd (k rd )=V(k rd ), we get D rd (k d )=V(k d ), or n other words D rd s spacefllng n V for E. The composte ncluson D rd V X s the desred mmerson. Corollary 9. Let k be a fnte feld, X/k projectve, smooth, and geometrcally connected, of dmenson n 1. LetE/k be a fnte extenson. There exsts a proper, smooth, geometrcally connected curve C/k, and a k-morphsm π : C X whch s surjectve on E-valued ponts. Moreover, 1) there s an open dense set U n C such that π U : U X s bjectve on E-valued ponts, 2) π s bratonally an somorphsm of C wth ts mage π(c) taken wth the nduced reduced structure. Proof. Apply Theorem 8 to get π : C 0 X, and then take C/k to be the complete nonsngular model of C 0 /k. Take U to be C 0. Because X/k s proper, the map π extends to a k-morphsm π : C X wth all the asserted propertes. Queston 10. Gven X/k projectve, smooth, and geometrcally connected of dmenson n 2, and E/k a fnte extenson, s there always a closed subscheme Y n X, Y X, such that Y (E) =X(E) and such that Y/k s smooth and geometrcally connected? What, f any, s the obstructon to the exstence of such Y? For example, take for X an odd dmensonal projectve space P 2n+1, n 1 wth homogeneous coordnates X and Y for =1,...,n + 1. Wrte q := Card(E) and take for Y the smooth hypersurface Hyp(2n +1,q) of degree q +1: Hyp(2n +1,q): (X (Y ) q (X ) q Y )=0. But what to do for P 2n? Take the easy case k = E (= F q ). One dea s to vew P 2n as an F q -ratonal hyperplane secton L =0ofP 2n+1, and then take ts Y to be L Hyp(2n + 1,q). Ths dea does not work, because the Gauss map for Hyp(2n +1,q)s (X,Y ) s ((Y ) q, (X ) q ) s = Frob q ((Y, X ) s). The map (X,Y ) s (Y, X ) s

10 622 NICHOLAS M. KATZ s an nvoluton of Hyp(2n+1,q). Thus Hyp(2n+1,q) s ts own dual varety, cf., [8, XVII, 3.4]. Exactly because Hyp(2n +1,q) contans all the F q -valued ponts n P 2n+1, there are no F q -ratonal hyperplanes L n P 2n+1 whch are transverse to Hyp(2n +1,q)! The smplest form of the queston s ths: n P 2 /F q, s there a smooth plane curve C/F q whch goes through all the F q -ponts of P 2? Applcatons to abelan varetes and to zeta functons of curves Theorem 11. Let k be a feld, A/k an abelan varety of dmenson g 1. There exsts a proper, smooth, geometrcally connected curve C/k, ak-valued pont O C n C(k), and a k-morphsm π : C A, whch maps the pont O C on C to the orgn O A on A, and whose Albanese map Alb(π) : Alb(C/k,0 C ) A Jac(C/k) s surjectve. Moreover, f the feld k s nfnte, there exsts such data wth π a closed mmerson. Proof. We frst treat the well known case when the feld k s nfnte. The proof we gve n ths case (cf., [6, 10.1] for a varant) s qute smple. We gve t both for the reader s convenence and because t concevably could be made to work over a fnte feld as well, see Queston 13 below. It depends on the followng geometrc fact: Lemma 12. In P N over an nfnte feld k, let X/k be a closed subscheme whch s smooth and geometrcally connected, of dmenson n 1. Gven an pont P n X(k) and an nteger d 2, there exsts a hypersurface H/k of degree d n P N such that P les on H and such that X H s smooth of dmenson n 1. Proof. Denote by H the projectve space of all degree d hypersurfaces n P N. Insde H, we have two subvaretes of partcular nterest: 1) the dual varety ˇX (of X for the d-fold Segre embeddng, cf., [8, XVII, 2.4]), consstng of those degree d hypersurfaces H such that X H fals to be smooth of dmenson n 1. 2) the hyperplane ˇP consstng of those degree d hypersurfaces whch contan P. We clam that ˇP ˇP ˇX has a k-pont. Snce ˇP ˇP ˇX s open n the projectve space ˇP and the feld k s nfnte, ˇP ˇP ˇX s ether empty or t has a k-pont. [Ths comes down to the fact that f a k-polynomal n some number m of varables vanshes on k m then t s the zero polynomal, provded k s nfnte.] If ˇP ˇP ˇX s empty, then ˇP ˇX. But the dual varety s rreducble of codmenson at least one, cf., [8, XVII, 3.1.4], so ˇP = ˇX. Take homogeneous

11 SPACE FILLING CURVES OVER FINITE FIELDS 623 coordnates X 0,...,X N n whch the pont P s (1, 0, 0,...,0). The hypersurface (X 0 ) d = 0 les n ˇX but not n ˇP, contradcton. To exhbt a g-dmensonal abelan varety A over an nfnte feld k as the quotent of a Jacoban, embed A n projectve space, pck g 1 ntegers d 2, and successvely ntersect A wth general hypersurfaces of degrees d defned over k whch each contan the orgn 0 A, to obtan a smooth curve C/k n A, defned over k, whch contans 0 A. The weak Lefschetz theorem [7, VII, 7.1] on hypersurface sectons tells us that for any prme l nvertble n k, the restrcton map H (A k k, Ql ) H (C k k, Ql ), s bjectve for = 0, soc/k s geometrcally connected, and njectve for = 1. Ths njectvty for = 1 mples that the Albanese map Alb(C, 0 A ) A s surjectve. The proof we gve below, over a fnte feld, s due to Ofer Gabber. We do not know f the proof gven above n the nfnte feld case can be made to work over a gven fnte feld, say by takng the degrees d qute large, cf., Queston 13 below. Pck a prme number l p, and a fnte extenson E/k such that each of the l 2g ponts n A( k) of order dvdng l les n A(E). Apply the prevous corollary to produce a proper smooth geometrcally connected curve C/k, anopenset U C, and a k-morphsm π : C A such that π U s bjectve on E-valued ponts: U(E) = A(E) byπ. Takng Gal(E/k)-nvarants, we see that U(k) = A(k) byπ. Take 0 C n U(k) tobe (π U) 1 (0 A ). The mage of Alb(C/k,0 C )na s an abelan subvarety B A. SoB( k) sa subgroup of A( k). Hence B( k) A( k)[l] =B( k)[l]. But by constructon we have A( k)[l] A(E) = π(u(e)) π(c( k)) B( k). Therefore B( k)[l] =A( k)[l], hence #(B( k)[l]) = l 2g. Therefore B has dmenson g, so t must be all of A. Queston 13. Suppose we are n the settng of Lemma 12, but over a fnte feld k. Thus n P N over k, we are gven a closed subscheme X/k whch s smooth and geometrcally connected, of dmenson n 1. Gven a pont P n X(k), does there exst an nteger d 2 and a hypersurface H/k of degree d n P N such that P les on H and such that X H s smooth of dmenson n 1? Does ths hold for all d 0? Corollary 14. Gven a fnte feld k, and an abelan varety A/k, there exsts a proper, smooth, geometrcally connected curve C/k such that the characterstc polynomal of Frobenus on (H 1 of) A/k dvdes the characterstc polynomal of Frobenus on (H 1 of) C/k.

12 624 NICHOLAS M. KATZ Proof. Once the Albanese map s surjectve, for l p we have a Gal( k/k)- equvarant ncluson H 1 (A k k, Ql ) H 1 (Alb(C/k,0 C ) k k, Ql )=H 1 (C k k, Ql ), whence a dvsblty of characterstc polynomals det(1 TF k H 1 (A k k, Ql ) det(1 TF k H 1 (C k k, Ql )). Corollary 15. Suppose we are gven an nteger r 1, a lst of r Wel numbers α for q := #k (each α s an algebrac nteger whch has all ts archmedean absolute values equal to Sqrt(q)), and a lst r postve ntegers n. There exsts a proper, smooth, geometrcally connected curve C/k whose zeta functon has a zero of multplcty at least n at the pont T =1/α for each =1,...,r. Proof. By Honda-Tate ([3, 9]), there exsts an abelan varety A /k on whch α s an egenvalue of Frobenus. Apply the prevous corollary to the product abelan varety (A ) n. References [1] P. Delgne, Applcaton de la formule des traces aux sommes trgonométrques, n Cohomologe Étale (SGA 4 1 ), Sprnger Lecture Notes n Mathematcs, 569, pp , 2 Sprnger-Verlag, Berln-New York, [2] V. Drnfeld, and S.G. Vladut, The number of ponts of an algebrac curve, Functonal Anal. App. 17 (1983), [3] T. Honda, Isogeny classes of abelan varetes over fnte felds, J. Math. Soc. Japan 20 (1968), [4] N. Katz, Exponental sums and dfferental equatons, Annals of Mathematcs Studes, 124, Prnceton Unversty Press, Prnceton, NJ, [5] N. Katz and P. Sarnak, Random matrces, Frobenus egenvalues, and monodromy, Amercan Mathematcal Socety Colloquum Publcatons, 45, Amercan Mathematcal Socety, Provdence, RI, [6] J.S. Mlne, Jacoban varetes, narthmetc Geometry, (G. Cornell and J.H. Slverman, eds.), pp , Sprnger, New York, [7] Sémnare de Géométre Algébrque du Bos-Mare , Cohomologe l-adque et Fonctons L, Sprnger Lecture Notes n Mathematcs, 589, Sprnger, New York, [8] Sémnare de Géométre Algébrque du Bos-Mare , Groupes de Monodrome en Géométre Algébrque, Sprnger Lecture Notes n Mathematcs, 340, Sprnger, New York, [9] J. Tate, Classes d Isogéne des varétés abélennes sur un corps fn (d après T. Honda), Exposé 352, Sémnare Bourbak 1968/69. Fne Hall, Department of Mathematcs, Prnceton Unversty, Prnceton NJ E-mal address: nmk@math.prnceton.edu