Drinfeld -- l-adic representations - Harvard Universitypeople.math.harvard.edu/~gaitsgde/Jerusalem_2010/... · tions on Bun2 and elements of V as functions of Flag2. Note that the

TWO-DIMENSIONAL l-ADIC REPRESENTATIONS OF THE FUNDAMENTAL GROUP OF A CURVE OVER A FINITE FIELD

AND AUTOMORPHIC FORMS ON GL(2)

By V. G. DRINFELD

Introduction. Let X be a smooth projective absolutely irreducible curve over Fq, k the field of rational functions on X, W its adele ring. We denote the completion of k at v by kv, the ring of integers of kv by Ov and the order of its residue field by qv. Put 0 = Hv Ov. Fix a prime number 1 which does not divide q.

Definition. An unramified cusp form on GL(2, W) is a function f: GL(2, 2X) -+ Q1 such that

1) f(x-y) = f(x) for all x E GL(2, 2), -y E GL(2, k); 2) f(ux) = f(x) for all x E GL(2, W), u E GL(2, 0);

3) |L/(1(o z))dz = Ofor allx E GL(2, t). For every closed point v E X there are two Hecke operators Tv and Uv

acting on the space of unramified cusp forms on GL(2, 2). Recall that

(Tvf)(x) = s f(g- x)dg gev

(1)

(Uvf Xx) = f( v

r_lx

def Here 7rV denotes a prime element of Ov, Mv {h e Mat(2, Ov)ldet h E 7r 01* }, dg is the Haar measure on GL(2, kv) such that mes GL(2, 0v) = 1. The geometric Frobenius element of xr1(X) corresponding to a closed point v E X is denoted by Frv.

MAIN THEOREM. Let E C Ql be a finite extension of Ql, p an absolutely irreducible two-dimensional representation of ir1(X) overEcontinu- ous in the 1-adic topology. For every closed point v E X put tv = Tr p(Frv), UV = qvj Idet p(Frv). Then there exists a nonzero unramified cuspformf on

85

86 v. G. DRINFELD

GL(2, 2f) such thatfor every closedpoint v eX the equalities Tjf = tjf, Uvf = uvf hold.

Remarks. 1) It follows from Jacquet-Langlands' theory [2] that the formf mentioned in the theorem is unique up to a constant factor.

2) For representations p contained in a compatible system of l-adic representations the theorem was proved by P. Deligne in [3] (not only for representations of irI(X) but also for those of Gal(k /k)). This proof does not work for a single l-adic representation because it has not been proved that the constant appearing in Grothendieck's functional equation for the L-functions of l-adic representations of Gal(k /k) [3] is "correct" (in [3] the "correctness" of the constant was proved only for compatible systems of l-adic representations). P. Deligne managed to prove the "correctness" of the constant in Grothendieck's functional equation for the L-functions of unramified representations of Gal(k /k) (probably the proof has not been published yet). But in order to prove the theorem formulated above it is necessary to show that the constant is "correct" not only in the unramified case but also for representations of Gal(k /k) of the form p 0 X where p is unramified and X is an arbitrary one-dimensional representation.

3) It is shown in [1] that if a nonzero unramified cusp form f on GL(2, W) exists such that Tvf = tvf, Uvf = uvf and uv E Zl*, then there exists a finite extension E of Q, and an absolutely irreducible continuous two-dimensional representation p of irI(X) overE such that Tr p(Frv) = tv, det p(Frv) = qvuv

4) It is easy to deduce from the main theorem that for every irreducible two-dimensional representation p of 7r,(X) over a finite extension of Ql, there is a representation co of dimension 1 such that p 0 X is contained in a compatible system of l-adic representations.

5) The method developed in this paper works also in the case when p is a two-dimensional l-adic representation of Gal(k /k) such that the inertia group of each point of X acts unipotently. Using this fact and the Saito- Shintani-Langlands base change theorem [7] it is possible to prove that every irreducible two-dimensional l-adic representation of Gal(k /k) (rami- fied at finitely many places only) corresponds to a cuspidal automorphic representation of GL(2, W). The complete proof of this theorem will be given elsewhere.

The works and ideas of Deligne contributed much to this paper. In particular, the idea of using the fact that pn is simply connected, which plays a fundamental role in the proof of the main theorem (see Section 2), was taken from [5], pp. 491-519. The technique for studying symmetric

TWO-DIMENSIONAL 1-ADIC REPRESENTATIONS 87

powers of an l-adic sheaf on a curve used in the present paper is also due to Deligne.

The following feature of this work seems to be essentially new: we have constructed an automorphic form without using L-functions.

Now some remarks as to the words and symbols which will be used in a nonstandard manner. We shall write "scheme" instead of "Noetherian scheme over Fq". Accordingly, if X and Y are schemes then X X Y denotes the product of X and Y over Fq. For every field K the symbol K means the separable closure of K. We denote by Pic X the Picard group of X, while Pic X is used for the Picard scheme of X. The fiber of a coherent sheaf I' on a scheme S at a point x E S is defined as :xl/Mxjix, where 3 x is the stalk of 3: at x and Mx is the maximal ideal of the local ring of x. If I' is an l-adic sheaf on a scheme S, the trace of the geometric Frobenius automorphism on Hi (S 0 Fq, I) is denoted by Tr(Fr, Hi (S (? Fq, 3)). We denote by Tr(Fru, :) the trace of the geometric Frobenius automorphism acting on the stalk of I' at a closed point u E S. The constant etale sheaf on S, whose stalks are equal to A, is denoted by A.

I am greatly indebted to P. Deligne who communicated to me some of his results before their publication. I also wish to thank A. A. Beilinson and Yu. I. Manin for the discussion of the manuscript and helpful remarks.

1. Unramified cusp forms on GL(2, 2f). Let B denote the group of upper triangular matrices over k. Denote by V the space of locally constant functionsf: GL(2, 2f) -- Q, such that

1) f(x-y) = f(x) for x E GL(2, 2f), y E B;

2) V f(x( z

))dz = 0 forx E GL(2, 2f);

3) f(ux) = f(x) for u E GL(2, 0), x E GL(2, 9). Fix a nontrivial locally constant homomorphism T: W/k -- Qj* and denote by W the space of locally constant functions so: GL(2, 2f) -+ Q, such that

1) se(jx(o )) = I'(z)s(x) for x E GL(2, 2f), z E 2f;

2) snjx(a )) = *(x) for a &k*, x E GL(2, 2f);

3) so(ux) = so(x) for u EiGL(2, O), xEiGL(2, 2f). Iff E V the function so: GL(2, 2f) -- Q, defined by

f At T(-z)dz

88 v. G. DRINFELD

belongs to W. It is shown in [2] that this mapping V -+ Wis an isomorphism and the inverse mapping W -+ V is given by

(2) (x k*(a (?1)

The operators in V and W defined by formula (1) are denoted by TV and UV as before. It is clear that the isomorphism W z+ V commutes with Tv and UV_.

Suppose that a homomorphism i: Pic X -+ Qj* and numbers tv , Qt, v eX are given. Put uv = (v) where v- is the divisor class of v. It is proved in [6] that the space

def U = E WlVvTv,, = tvp,, Uv,, uv,,}

has dimension 1. Besides that a nonzero function sp E U is found. To write the formula for this function we shall introduce some notations. For every a E 2{* let us denote the corresponding divisor by div a and the adele norm of a by la . Suppose that the largest 0-submodule of 2f on which T is trivial equals to Ou; then put 6 = - div u. For every closed point v E X we denote the coefficient of zn in the formal series (1 - tz + qvuvzz2)'- by cv(n) (in particular, cv(n) = 0 if n < 0). For every divisor D on X put r(D) = Hv cv(n v), where nv is the multiplicity of D at v. Note that r(D) = 0 if D is not effective. In [6] (Chapter VI, Proposition 6) U is shown to be generated by the function sp whose restriction to the group of upper triangular matrices is given by

(3) (a ob 1oZ)) = h(z) q (div b)-lr(div b + 6 - div a)

Note that so is uniquely defined by formula (3) and the condition sO(ux) = so(x) for u e GL(2, 0).

As V - W, the space {f E VIVvTvf = tvf, Uvf = u vf} has dimension 1 and is generated by the functionf whose restriction to the set of upper triangular matrices is given by (2) and (3). We have

(4) f((a b) 1 = h

f7(divb)F

* r(div h + div b + 6- div a)'I(hz) hek*


Note that T may be represented as T(z) = TI0 ()wwhere 0o:Fq Qj* is a non-trivial character, wo is a rational differential form on X (the symbol < , > denotes the canonical pairing between W/k and the space of rational differentials Qk). 6 is nothing else but the divisor of wo. Therefore

f((a 0)@ Z))

= |-| aj(div bh)' E r(div X + div b - div a)YTo() a COd2k*

where Qk = Qk\{?}. The number r(div X + div b - div a) does not change if X is multiplied by an element of Fq*. Hence

(5) f( 0) (I z)

- q(div b)-'(q r(div X + divb - div a) a 'WEok*lFk*, =0

- S r(div X + divb-diva)) WEok*/Fq*

Now we are going to give a geometric interpretation of formula (5).

Definition. Let ?C be a sheaf of Ox-modules. An invertible subsheaf (i C ? is said to be maximal if it is not contained in any larger invertible subsheaf.

Denote by Bun2 the set of isomorphism classes of two-dimensional locally free sheaves of Ox-modules. We shall denote by Flag2 the set of isomorphism classes of pairs (2, (i) consisting of a two-dimensional locally free sheaf ?C and a maximal invertible subsheaf (i C 2. There are canonical bijections

GL(2, O)\GL(2, 2)/GL(2, k) z+ Bun2, GL(2, O)\GL(2, 2f)/B z+ Flag2.

Therefore unramified cusp forms on GL(2, 2f) may be considered as functions on Bun2 and elements of V as functions of Flag2. Note that the bijection GL(2, O)\GL(2, 2f)/B z+ Flag2 associates to

C 1 z)

\O b/\O 1

90 V. G. DRINFELD

the pair iC, (a where (i is the invertible sheaf corresponding to div a, ?C is the two-dimensional locally free sheaf of


PROPOSITION 1.1. Letf: Flag2 -_+Qj be an eigenfunction of Hecke operators. Suppose thatfor some N E Z thefollowing condition is satisfied: f(43, (a) = f(4, C, ')for all 43, (a, a' such that the degrees of (a and (' are less than h(2) - N. Then there exists a function g: Bun2 -_+Q0 such that f(4, (t) = g(43).

Proof. Putg(2) =f(4, () where (t is a maximal invertible subsheaf of 43 such that deg (a < h(3) - N. In order to show thatg is well defined we have to prove the following lemma.

LEMMA 1. Let 4 be a two-dimensional locally free sheaf of Ox-modules. Then there exist maximal invertible subsheaves of S whose degree is arbitrarily small.

Proof. The set of degrees of invertible subsheaves of 4 is bounded from above because dim H 0 (X, 43) < oo. The set of invertible subsheaves of 43 having a fixed degree is finite since Pic?X is finite and for every (a E PicX the set Hom((a, 43) is finite. On the other hand the set of all maximal invertible subsheaves of 43 is infinite because such subsheaves correspond bijec- tively to one-dimensional vector subspaces of the generic fiber of 4. C

Let v be a closed point of X, t,, uV E QI, Tvf = tvf, Uvf = uvf. Then TVg = tvg, Uvg u Vg.

LEMMA 2. Let (4, (a) E Flag2 and let 43' be an upper modification of 4 at v such that (a is maximal as a subsheaf of 43'. Suppose thatfor every two-dimensional locallyfree Ox-module sheaf 4 D 43' such that a is maxi- malas a subsheaf of 43, the equalityf(4, (a) = g(S) holds. Thenf(S3, () = g(S3).

Proof. Denote by S the set of lower modifications of 43' at v different from 4. We have

f(S, (a) + E f(S3", a n 43) = tvf(3', a() = tvg(4')

g(4) + , g(4") = tvg(4')

On the other hand, if 4 " E S then fl n43" = a(-v). Hence we may put S = 4 "(v). Thereforef(S ", a n s ") = uvf(43 "(v), t) = uvg(43 "(v)) = g(4 ") for every 4 " E S, whence it follows thatf(43, () = g(S). OI

92 v. G. DRINFELD

LEMMA 3. Let (2, i) E Flag2, deg ?C > 2h(4C). Then there exists an upper modification ?V' of ?C at v such that h(V ') > h(G$) and (i is maximal as a subsheaf of i'.

Proof. Let YZ C ?C 0 Fq be an invertible subsheaf of the highest degree. Then deg YZ = deg ?C - h(G$) > h(G$). There exists an upper modification ? ' D ? such that (i and YZ are maximal as subsheaves of ? ' and ? ' 0 Fq, respectively. We are going to show that h(G$') > h(G$). As- sume that there is an invertible quotient sheaf l of ? '0 Fq such that deg l c h(4C). The superposition YZ ? ' 0& Fq -- l must be equal to zero.

Hence l = ( '& Fq)/MZ and deg 4' = deg YZ + deg ? c deg i, which is impossible. C

Fix (i E Pic X and denote by P(m, n) the following statement: "For every two-dimensional locally free sheaf ?C on X containing (i as a maximal invertible subsheaf and such that h(4C) 2 m, deg ?C 2 n the equality f(G$, (i) = g(G$) holds." By the condition of the proposition, P(m, n) is true for m > deg (i + N. It follows from Lemma 2 that Vm, n(P(m, n + 1) X P(m, n)). Lemmas 2 and 3 show that vmvn > 2m(P(m + 1, n) = P(m, n)). Thus P(m, n) is true for all m, n E Z. C

2. Derivation of the main theorem from vanishing cycle theorems. Let E, p, t V, uV denote the same objects as in the main theorem. If the restriction of p to r1 (X 0 Fq) is not absolutely irreducible then after replacingE by its finite extension, p becomes induced by a one-dimensional representation of ir I(X 0 Fq2). The existence of a cusp form corresponding to such a representation p is well-known ([2], Theorem 12.2). So we shall suppose that the restriction of p to r1 (X 0 Fq) is absolutely irreducible.

According to class field theory the homomorphism det p: r1 (X 0 Fq) Q1* corresponds to a homomorphism A: Pic X -- Ql*. Define : Pic X

Q1* by (63) = A(,)q-deg '. Then uv = q(v). Therefore the results of Sec- tion 1 are applicable. Thus, in order to prove the main theorem we have to show that the function f: Flag2 -_ Q, defined by (6) is the pullback of a function Bun2 -_ Q,.

Let us give a cohomological interpretation of formula (6). Denote by ? the locally constant sheaf onX corresponding to p. If l'i is an l-adic sheaf on a scheme Yi then I 1 1X 2 will denote the following sheaf on Y1 X Y2: 3 1 X def 32= prl 3*1 0 pr2* 32 where pri is the projection Yi X Y2 -- Yi. Put

n times

TWO-DIMENSIONAL l-ADIC REPRESENTATIONS 93

Put Sym' X = XI/Sn, FJ() =(p* Fn g)S1 where so:X' -- Sym' X is the natural mapping. Let (2, (i) E Flag2. In formula (6) H(G$, (i) denotes a projective space over Fq considered as a set. Denote by 3C(C, (i) "the same" space considered as a scheme over Fq. We have 3C(4, (i) C Symm X, where m = deg ?C - 2 deg (i + 2g - 2. The restriction of ?(m) to 3C(G$, a) will be denoted by (i).

PROPOSITION2.1. Let (2, t) E Flag2, deg 4 -2 deg (a > 2g -2. Put JC = 3C(4C, (), 3: = 3k$, (). Thenf (42, (t) = q l+dega (det i,) - 1 t((i) X Ej (- 1)i Tr(Fr, Hi(3C (8) Vq , fl)

Proof. According to the Lefschetz-Grothendieck formula

1 (-1)' Tr(Fr, Hi(3C 0 Fq' i))

= S Tr(FrD, i) = 3 Tr(FrD, 6(m)) DeH(JC,a) DEH(G ,a)

It is easy to show that Tr(FrD, ?(m)) = r(D). It remains to prove the equality

(7) , r(D) = 0 DeP(4C,a)

Let X be a homomorphism PicX -- Q,*. We shall also denote by co the corresponding homomorphism ir1 (X) --+Q,1. Denote by deg v the degree over Fq of the residue field of a closed point v E X.

According to Grothendieck's theorem [3] the formal series

t)def -te ( L(p 0(A), t)-IH det(t - vtdcgv(v)p(Fr )) 1 v

is a polynomial in t of degree 4g - 4. Since m = deg 42 - 2 deg (i + 2g - 2 > 4g - 4 the coefficient of tm in L(p (8 c, t) is equal to zero. On the other hand, it is equal to EDes r(D)o,(D), where S is the set of effective divisors of degree m. Thus EDc r(D)X(D) = 0 for all w. Therefore EDeT r(D) = 0 for every complete linear system T of degree m. In particular, the equality (7) holds. C

Fix a locally free (9x-module sheaf 42 of dimension 2. Let us try to understand how Hi (3C(2, (i) (0 Fq, 3(4, (i)) depends on (i. To do this we need some notations. Denote by Vn the moduli scheme of invertible subsheaves of 42 having degree n. The scheme Vn is projective. There are canonical morphisms SPrn: Vr+n X SymrX -_ Vn (they act on Fq -points as

94 V. G. DRINFELD

follows: if (i is an invertible subsheaf of ?C, D is an effective divisor onX, deg = r + n, degD = r, then (Prn(a, D) = (i(-D)). Put Vn = Vn\s01n(Vn+

X X), Vn = Vn\s02n(Vn+2 X Sym2X). Vn and Vn are open subsets of V,t. Note that Vn C Vn. Vn is the moduli scheme of maximal invertible subsheaves of ? having degree n. The morphism (Pln maps Vn+j X X isomorphically onto a closed subscheme An C Vn such that Supp An = Vn\Vn. There is a canonical morphism Vn -_ Picn X (associating to an invertible subsheaf (i C ? its isomorphism class). Denote by g the genus of X.

PROPOSITION 2.2. If n < h(4C) - 2g + 2the morphism Vn -_ PiCnX is a locally trivial bundle whose fibers are isomorphic to pN, N = deg ?- 2n + 1 - 2g.

Proof. From now on we denote by ?J and Q the inverse images of ?C and Q on X (D Fq. It is enough to prove that for every (i E Pic(X (D Fq) the equalities dimH0(X(?Fq, (i 82) = N + 1,H1(X(?Fq, G7-1?4) = 0 hold. Indeed, dimH1(X?(Fq, (- 1(D2)=dimHO(X?(Fq,2*?(?(D Q) = dim Hom(42, (a (?Q) = O because deg (a (?Q = n + 2g -2 < h(42). It remains to use the Riemann-Roch theorem. C

COROLLARY. If n < h(G) - 2g, then dim(Vn\Vn) = dim Vn - 2, dim An = dim Vn - 1, An is an irreducible smooth variety.

From now on n will denote an integer less than h(42) - 2g.

PROPOSITION 2.3. deg 42 - 2h(42) 2 -2g.

Proof. Let YZ be an invertible sheaf on X (0 Fq of degree h(42) - 1. Then Hom(42, MZ) = Ho(X (O Fq, 4? (0 MZ) = 0. Hence x(4*0 (M tZ) < 0, i.e.2(h(42)- 1)-deg42+2-2g 0. D

Put m = deg JC - 2n + 2g - 2. By Proposition 2.3, m 2 2(h(42) - n - 1) 2 4g (recall that n c h(42) - 2g - 1). Denote by Pn the morphism Vn- Picm X mapping x E Vn (Fq) to a7-2 OQO det L E Picm (X & Fq), where (i is the isomorphism class of the invertible subsheaf of 42 corresponding to x. Denote by jacm the canonical morphism Symm X -- Picm X. Consider the Cartesian square

7rn

(Sn Vn

jacm Sym'm X -~ Pic'm X


Since m 2 4g > 2g - 2, jacm and, therefore, Wr, are locally trivial bundles whose fibers are isomorphic to pm-g. Put 5,n = 0,F, *(m) (n = rn-'(Vn)V )n = n-1(V)Cn), = irn'( n). Therestrictionsof gn to(P n Sn and WnA will be denoted by gn, 5n and ?nA- 1rn induces morphisms 7rn: (Sn _+Vn 9 f*n (S)n __ Vn 9 7rAn (Sn A -+

Let u E Vn (Fq) and (i be the corresponding invertible subsheaf of ? . Put 63 = J/(t. As (a is maximal we have 63 = (i- det C. So 71n- (u) is the variety of one-dimensional vector subspaces of Hom((a, (3 (0 Q). Note that Hom($, (i) = 0 because deg (i = n < h(G$). Therefore the exact sequence 0 (- JC -( '3 -O 0 does not split. Thus we obtain a nonzero element X E Ext((3, (i) = Hom((a, (3 (0 Q)*. We shall denote by 3Cu the hyperplane in the projective space irjn'(u) defined by the functional X: Hom((a, (3 &0 Q) - Fq. It is clear that there exists a subscheme JCn C (S)n such that the morphism JCn -_ Vn is a locally trivial bundle and its fiber over every point u E Vn(Fq) is equal to 3Cu. Put iJn = t*t*?Rn where't is the embedding 3Cn 9 n

PROPOSITION 2.4. Let (i C ?J be a maximal invertible subsheaf of degree n, denote by u the corresponding Fq -point of Vn. Then

f(4C t) = q 1+deg ai,(det ?C)-' (a) (-1)iTr(Fru, Ri(i7rn)*In)

Proof. Using Proposition 2.3 and the inequality n < h(L) - 2g we get deg ?J - 2 deg (i > 2g. Therefore Proposition 2.1 is applicable. It is clear that 3C($, (i) is the fiber of 3Cn over u and the restriction of in to 7r1 (u) is equal to the direct image of '(W$, (i). It remains to use the base change theorem. C

Let us explain the general scheme of the proof of the main theorem. We are going to formulate two "vanishing cycle theorems" (their proof will be given in Sections 3, 4). It follows from these theorems that the sheaves Ri(1rn )*in are locally constant. Moreover, Ri(1rn )*Oin is the restriction of a locally constant sheaf on Vn . Since the fibers of the morphism Vn -+ Picn X are connected and simply connected (they are projective spaces), RJ(1rn )*in is the inverse image of a locally constant sheaf YmnJ on Pica X. Taking into account Proposition 2.4 we see thatf(C, (i) =f(G, (i') if (i - (' and deg (i < h(4) - 2g. In order to show thatf(C, (i) does not depend on the isomorphism class of (i we shall prove that the sheaves SMnj are "almost constant" (the difference from the constant sheaf compensates the factor qdeg aA((i) in the formula forf(G, (i)).

96 V. G. DRINFELD

VANISHING CYCLE THEOREM 1. The sheaves R'(r,)e*ln are locally constant.

Let 0n denote the henselization of the local ring of the generic point of An considered as a point of Vn. The strict henselization of the same ring is denoted by 0n,. Denote by Kn the field of fractions of 0n and by Kn the field of rational functions on An . The residue fields of 0n and On are equal to Kn and kn. Denote by Wn the fiber product of (6n and Spec 0n over Vn. Let ('IK and (S,I" denote the fibers of (S'0 over SpecK, and Spec K,n. Denote by i3,K the inverse image of in on (6n K. Lets: (6n K _ '(,n?, i: (6n',," (Pnf be natural morphisms.

The second vanishing cycle theorem describes the sheaves i *Rjs*9n K. In order to formulate it we shall introduce some notations. Denote by T the superposition (Yn/A -* An +_ V,,+1 X X - X, by D C Sym'X X X the incidence correspondence, by (n,A the restriction of -an to (Yn,A The inverse image of D under the morphism (a,nA, T): 6nA' Sym'X X X is denoted by JCn,A. Denote by ynA the restriction of ?nA to 3CCnA (this restriction is considered as a sheaf on WnA). Denote by 5n' the kernel of the homomorphism gn Sin . The isomorphism Vn +1 X X -* An induced by O 1 n: Vn +1 X X;C. Vn can be lifted canonically to an embedding 4 (n+I X X -9 6nj (if u E Vn+1 (Fq), z EX(Fq), u' = ,ln (U, Z), ( is the invertible subsheaf of ? corresponding to u, then (irn + )I1(u) is the set of one-dimensional vector subspaces of Ho(X0( Fq, -2 (0 ( det .) and irnI (u ') is the set of one- dimensional vector subspaces of H0 (X O Fq, (-2 0 Q(2z) ? det J) so that (ir+ 1)Y'(u) C 7rn'-(u ')). Put (6n = 44*(9n '+l (-1) 1 det ?). The inverse images of 3nA and 61n under the morphism Wni,K ,(,nA will be denoted by i3n and (6n1.

The action of Gal(K n /Kn) on 6(nK can be extended canonically to an action of this group on 5zn K and 61,nK By means of the homomorphism Gal(Kn /Kn) -) Gal(kn /kn ) we obtain an action of Gal(Kn /Kn ) on 6, K, In K and (Rn K. The action of Gal(Kn /Kn ) on (n K can be extended canonically to its action on the sheaves i*Rs* I:n

VANISHING CYCLE THEOREM II.

1) i*RJs*g { ZK = 0 if] > 1I 2) There exist isomorphisms i*s 3,nK :nK, i*R 's*y ,K f (R6K com-

patible with the action of Gal(Kn /Kn,)-

Remark. In particular, the theorem asserts that the inertia subgroup of Gal(Kn, /Kn) acts trivially on the sheaves i*Rjs*5inK.

We shall need the following theorem.


DELIGNE'S THEOREM. Suppose that r > 4g - 4. Denote by jacr the canonical morphism Symr X Pier X. Then RI(jacr) ?(r) = Ofor allj.

The proof of this theorem will be given in the appendix of the paper.

Remark. The assumption of the absolute irreducibility of the representation of ii (X? Fq ) corresponding to ? is essential to Deligne's theorem but not to the vanishing cycle theorems.

PROPOSITION 2.5. There exist Gal(Kn /Kn )-equivariant isomorphisms Hj((Y, , 3inK) Hj-i ((S nK (Rn K)

Proof. Consider the Leray spectral sequence for the morphism S: (pK ( 0 and the sheaf inK. By the base change theorem, H*((Qn1 P RJs* 7iK ) = H*Q((PK, i*Rjs 4FK ). According to the second vanishing cycle theorem, i *R Js * S nK = Oforj > I,i i*R ls K = is (Rn*s K = Kn . It remains for us to prove that H*((YnPK, 5,n K) = 0.

LEMMA. Let {: Symm- I X X X --+Symm X, a: Symm-2 X X X 4 Symm-1 X X Xbegiven by b(D, z) = D + z, a(D ', z) = (D' + z, z), where z eX(Fq), D E Symml- X(Fq) and D' E Symm2X(Fq) are considered as divisors on X 0 Fq. Then there is an exact sequence

(8) 0 -Xp(m) ? g(m-l) X ? det ?) -O 0

Proof. For a positive integer i c m denote by Yi the following sub- manifold of Xm X X: Yi(Fq )d{(x, ...,x,z)eXm+(Fq )Ixi=z}. Put Y = Umi=1 Yi. Denote by ir the projection Y -+ Xm. Define h:Y Symm-1 X X Xbyh(xl, ... .xm Z) =(mi=1x i-z, z). Then the diagram

y 7~~r xm

h

Symm-X X X - Sym 'X

is a Cartesian square. Therefore *8?*(m) = (h*ir* Mm 8)sm. For any l-adic sheaf l on Y there is a canonical resolution 0 -+ -+

98 v. G. DRINFELD

Picm-lX X X. Therefore it suffices to prove thatRJt* *8(") = 0. In view of the lemma it is enough to show that RJt*(8("-2) X ?) = 0 and R Jt ** ( (m -l 1) X det ?) = 0 for all j. This follows from Deligne's theorem since m -2 4g -2 > 4g -4. C

It follows from Proposition 2.5 that the inertia subgroup of Gal(Kn /Kn) acts trivially on Hj(W KK 5ZnK). Hence Rj(irn )* 5n, which is a locally constant sheaf on Vn by the first vanishing cycle theorem, can be extended to a locally constant sheaf on V, (we have used the following theorem due to M. Nagata and 0. Zariski: if M is a smooth variety, Z is a closed subset of M such that codim Z 2 2, then any etale covering of M\Z can be extended to an etale covering of M). As the geometric fibers of the morphism Vn PicnX are connected and simply connected (being projective spaces), RJ(rx )* 5Yn is the inverse image of a unique locally constant sheaf ORnJ on Picn X. It follows from Proposition 2.4 that if (a C ? is a maximal invertible subsheaf of degree n and u is the corresponding Fq - point of PicnX, then

(9) f(4C, (a) = ql+deg at (det ?)- 1 't((a) 1 (-1YTr(Fru 9 TnL)

PROPOSITION 2.6. Suppose thatn A -n Vn+1 X X Pic 'XXX

The geometric fibers of c are connected. Therefore it is sufficient to prove that 036*c-*a*9Tn6j/ i36*5**(01n+1J-2(-1) M det 6).ThesheafT n*t corresponds to the representation of Gal(Kn /Kn) on Hi(Qn1Kg 5:nK ) which is isomorphic to Hi- (P,nK, 61nK,) (see Proposition 2.5). The representation of Gal(Kn/Kn) on Hj-l(@1PnK, 6,K) is in fact a representation of Gal(K-n /Kn). This representation of Gal(K-n /Kn) corresponds to the sheaf

r*y*9TJ i3* *c*a9nt Thus i3 * 9 Oj Rj-1(7 A)*(R


It remains for us to prove that Ri-1('rnt)* 61n - 6* ln+j2 2(-1) 1 det 6). By the definition of 61n we have Rji1(7rn")*6(Rn *Rj-lh*( 'n+l (-l) X det6), whereh: Yn+l XX- Vn+j XXisob-

tained by base change from 7rn+1 (Pn+ nl' Vn+1. It is clear that Rj-lh ('n+l (-l) 1 det 6) = (Rj-2(7rn+1)*9'n+1(-1)) 1 det 6. On the other hand, c-*(On + 1j-2 1 det 6) = (Rj-2(7r n+l)*:n +l) 1 det 6. It remains for us to prove that Ri 1 (7rn+i )*9'n +i Ri Orn+ )*I:n+1

Recall that 9'n + is the kernel of the epimorphism gn + 1 Y, :n+ 1 Therefore it is sufficient to show that Rj(rxn+l )*n+l = 0 for all j. This follows from Deligne's theorem since the morphism 7rn+1 (Pn + 1 Vn +i is obtained by base change from jacn_2: Symm-2X + Pic,-2X and gn+1 is obtained by the same base change from -(m2) (note that m - 2 2 4g - 2 > 4g - 4, so Deligne's theorem is applicable). D

Put C = det 6. Denote by anr the subtraction morphism PicnX X SymrX - PiC n-rX.

PROPOSITION 2.7. Suppose that n < h($) - 2g. Then (anr)*0Rt'n-r - OR nJ-2r( -r) N e(r).

Proof. Denote by r the natural morphism PicnX X Xr -+ PicnX X SymrX. After applying Proposition 2.6 r times we obtain an isomorphism r*(anr)*1tjn-r rT*(91ntj2r (-r) 1 Cv). It is easy to show that this isomorphism is compatible with the action of Sr (consider the restriction of our sheaves to PicnX X T, where T C Xr is the diagonal). Therefore our isomorphism induces an isomorphism (anr)*1 tn- r f fLn 2( -r) X e (r) F

PROPOSITION 2.8. Let ( and (' be maximal invertible subsheaves of ? such that deg (a < h(C) - 2g, deg a' < h() - 2g. Thenf(G$, (t) =

f(G$, a'). Proof In view of formula (9) it is sufficient to prove that if m, n <

h( ) - 2g, u E PicmX, v E PicnX, then

(10) qm ,u(u)Tr(Fru, Ytmj-2m) = q n p(v)Tr(Fr,, ,nj-2n)

It is enough to consider the case when v - u is the class of an effective divisor D. Put r = n - m. Then, according to Proposition 2.7, (anr)* OE6m1 2m - Mtnj-2n (-r) X C(r). Put w = (v, D) e PicnX X SymrX)(Fq). Then Tr(Fru, TOm1 2m) = Tr(Frw, (anr)*1m j-2m) = Tr(Fr,w nj-2n(_ r) 1 C(r)) = qrpt(D)Tr(FrV, nnj-2n), which is equivalent to (10). D

100 V. G. DRINFELD

The main theorem follows from Proposition 1.1 and 2.8.

3. Proof of the first vanishing cycle theorem. The theorem is formulated in Section 2. Here we shall slightly change our notations. In this section the number n is fixed (n < h(43) - 2g), therefore we shall write -r instead of 7rn , I' instead of 5:n and so on. Because of the geometric nature of the theorem we may replace the varieties (P and V occurring in it by (P 0 Fq and V(O Fq (5: is replaced by its inverse image on Y (P Fq). While proving the theorem we deal only with schemes over Fq. Therefore we shall write X, (S, V, etc. instead of X (0 Fq, (Pn (0 Fq, Vn (0 Fq etc. It is clear that, while proving the theorem, we may suppose that the sheaf 6 , which is implicit in its formulation, is not an l-adic sheaf but a locally constant A-module sheaf whose stalks are equal to A2, where A = OEvI' OE being the ring of integers of E, I a nonzero ideal of OE.

We shall prove more than the local constancy of R 7rx*5:. Namely, it will be proved that -r is locally acyclic with respect to 5: (for the definition of local acyclicity see [4],' p. 242). While proving the local acyclicity we may suppose 6 to be a constant sheaf. In this case E(m) = (O",=o (ta)*A, where t' is the natural morphism Sym'X X Symm-X - SymmX. For an integer a e [0, m] denote by 3C' the fiber product of JC and Sym'X X Symm-X over SymmX, and by r' the natural morphism 3C' -+ (P. Put 7r' = 7r o r'. As E>(m) = (?m0U=o (t')*A we have 5: = ?m0=o (#r)*A. Hence, to prove the local acyclicity of -r with respect to 5: it will be enough to show that for every a e [0, m] the morphism 7r': 3C' -+ V is locally acyclic (with respect to the constant sheaf). Because of the symmetry between a and m - a we may suppose that a 2 1/2 m 2 2g.

It is easy to see that the fibers of the natural morphism -y: 3C' -+ V X Symm-X are projective spaces. To be precise: let u be a closed point of V, D an effective divisor on X of degree m - a; a the invertible subsheaf of ? def def correspondingtou, i3 = ?C/ W(u D) = H0(X -' 063 Q(-D))* = Ext(6(*-D), (i), s(u, D) e W(u, D) the image of the canonical element of Ext(63, (i), CU,D the quotient of W(u, D) by the subspace generated by s(u,D); then y 1'(u , D) = P(CU,D). As deg( 6 - 1 X 0 s Q(-D)) = a 2 2g , W(u, D) is the fiber at (u , D) of a naturally defined locally free sheaf eg on V X Symm -X of dimension a - g + 1. It is clear that s(u, D) comes from a section s of this sheaf. Denote by C( the quotient of eg by the subsheaf generated by s. Then JC' and P(C) are isomorphic as schemes over V X Symm'X.

Denote by Ythe subscheme of V X Symm-X defined by the equation


s = 0. If Y = 0, the morphism -r' would have been smooth and, hence, locally acyclic. In fact Y * 0 (for some a), but the following proposition shows that Y is a trivial bundle over V. This saves the situation.

PROPOSITION 3.1. Yis V-isomorphic to V X V1, wherej = n + a - 2g + 2.

Proof. We shall only construct a bijection Y(Fq) f V(Fq ) X Vj(Fq). The construction of the bijection Y(A) z+ V(A) X Vj(A) for any Fq-algebra A is quite similar.

Let u e V(Fq), (a be the corresponding invertible subsheaf of i, 6? - C/(i, c the canonical element of Ext(63, (i). The fiber of Y(Fq) over u is the set of effective divisors D on X of degree m - a such that the image of c in Ext(63(-D), 69) is equal to 0, i.e. the embedding 63(-D) -+(33 lifts to a morphism 63(-D) -+ ?. The lifting is unique since deg 63(-D) = deg ?- n - m + a = n + a -2g + 2 > n = deg (i. The image of 63(-D) in J is an invertible subsheaf of degree n + a - 2g + 2 = j. Thus we have constructed a mapping Y(Fq) V(Fq) X Vj(Fq). Its bijectiveness is clear. D

Remark. For every u E V(Fq) the embedding V X V1 z+ Y C- V X Symm-X induces an embedding iu: V, - Symm-'X. Note that iu does depend on u.

Fix Fq-points u e V, z e Symm-X. Put x = (u, z) e V X Symm`X. Denote by pru the composition V X Symm-X -+ Symm-X z+ u X Symm-XC4 V X Symm-X. Put C' = (pru )*C. Denote by 0 the henselization of the local ring of x. The inverse images of C, and C' on Spec 0 will be denoted by Co and %o'.

PROPOSITION 3.2. There exists an automorphism a of Spec 0 over V such that ou*Co C 0 '.

Proof. If x E Y then C and C' are locally free at x, so we may put or= id.

Now suppose that x e Y. Put Y' = (pru)-1(Y). Denote by YO and Yo' the inverse images of Y and Y' under the morphism Spec 0 -O V X Sym' X.

LEMMA. There exists an automorphism upf Spec 0 over Vsuch that r l(yo) = Y0'.

Proof. Denote by S the fiber of Spec 0 over u. Then S n Y0 = s n YO'. It follows from Proposition 3.1 that the identity automorphism of S n

102 v. G. DRINFELD

YO' can be extended to a V-isomorphism b: YO' YO Y0. Since 0 is the henselization of a local ring of a scheme which is smooth over V, b may be extended to a V-morphism a: Spec 0 -O Spec 0 such that the restriction of a to S is equal to the identity automorphism. a is an isomorphism because rls = id. D

Let a be the automorphism of Spec 0 constructed in the lemma. Put r = a - g + 1, M = H?(Spec 0, 0CO), M' = H?(Spec 0, C0'). Then M - Or/Oh, M' - Or/lOh/ where h = (hl, ... .hr) E or, h ' =(hl', .... hr /) E or, the equationsh1 = 0, . . ., hr = 0 define the subscheme ao-'(YO) C Spec 0 and the equations h1' = 0, ..., hr' = 0 define YO'. Since a-1(Yo) = Yo' the elements h1, .., hr generate the same ideal of 0 as hl . ... . hr'. It is easy to deduce from this fact that there exists an automorphism -y of or such that h' = 7y(h). Therefore M - M', whence it follows that a*Co - CO'. D

Since P(C') is V-isomorphic to V X Ufor some scheme Uover Fq, the morphism P(C(0') -+ V is locally acyclic ([4], p. 243, Corollary 2.16). It follows from this fact and Proposition 3.2 that the morphism P(C) -+ V is locally acyclic at every point of P(C) lying over u. As u may be arbitrary the morphism 7r': 3C' = P() - V is locally acyclic. We have shown at the beginning of the paragraph that the first vanishing cycle theorem follows from the local acyclicity of -rt.

4. Proof of the second vanishing cycle theorem. The theorem is formulated in Section 2. The notations used in this paragraph are the same as in Section 2 (and differ from those of Section 3).

PROPOSITION 4.1. There is a closed subscheme 3Cn C (Sn such that 1) the morphism 3C,n - Vln is a locally trivial bundle with Pm -g-1 as a

fiber; 2) JCn n (Pn = 3C, J Cn n pA = JCA Proof. Denote by ? the inverse image of ? on X X Vn,. Let (a C ? be

the universal invertible subsheaf. Put 33 = (i-1 (0 det i, C = ?/(i. The superposition (a ? tE ?0 () - det ? is trivial on (a ? (a and, therefore, induces a morphism (? (8 - det i, i.e. a morphism a: ?- (33. Note that a is an isomorphism over the open subsetX X Vn C X X Vn . If x E An (Fq ), (3X and Cx are the restrictions of (33 and C to X X x, ?ax: ex -+ c3X is induced by a, then Ker ax and Coker ax are sky-scraper sheaves concentrated in one


point, the stalks of Ker ax and Coker ax at this point being one-dimensional.

Denote by pr the projection X X Vn Vn and by Q the inverse image of Q onX X Vn . The exact sequence 0 -+ a -+ C -+ 0 induces a morphism f: pr*(a0- C 0) ( - C ) Q) R'pr*Q = 0vn. Put 3C = Kerf. If x e Vn, (ix, Cx and Qx are the restrictions of (i, C and Q to X X x, then H1 (X X x, (ix ex (0 Qx) = 0 (because the quotient of (ix- 0 x () 0x by its torsion subsheaf has a degree not less than deg - 2n - 1 + 2g -2 = m - 1 2 4g - 1 > 2g - 2. Therefore pr*((iV- ' (0 C 0 l) is a locally free sheaf whose fiber at a point x is equal to H?(X X x, (ix-ly (0 Cx (8 0 Qfx) is surjective because in the opposite case the exact sequence 0 -+ ( -+ C, - 0 would have split over some point x e Vn, which is impossible since deg (ix = n < h(Gi). Asf is surjective and pr*((' - 1 ( 0 2 (g Q) is locally free, 3C is also locally free. Put OR = pr*((i - a ( 3 3D 0). OR is locally free and the fiber of OR at a point x e Vn is equal to H?(X x x, xy- ?63x s& Qlx). It is clear that (P- = P(M*). The morphism a: C( - (33 induces a morphism h: 3C -S OR. It is easy to see that h induces injective mappings of the fibers. Therefore h induces an isomorphism between P(3C*) and a closed subscheme 3Cn C (Sn. The subscheme 3C,n has the desired properties. D

Let us slightly change our notations. Recall that the support of 5:n iS equal to 3Cn. Henceforth we shall consider 5:n as a sheaf on JCn (not on (9n). Accordingly, 5:nK and (Rn7K will be considered as sheaves on JCn K , and 5FnK as that on C K . Denote by JC,/ the fiber product of JCn and Spec On over Vn . In this paragraph s will denote the morphism JC K -+ JC,/ (not (PYK (P0,,) and i will denote the morphism JC,nK -+J (not (pnK (n

We shall formulate and then prove a more precise statement than the second vanishing cycle theorem. To do this we need some notations. Denote byKn the field of fractions of O?. Denote by JCnK the fiber of JCno over Spec Kn and by s the embedding JCnK JCnj. Denote by :nyK and 5no the inverse images of gn under the morphisms JCnK -+( n , n -+ (Pn

PutZn = 4(JCn+1 X X), where b: (Pn + X X + (PAn is the embedding defined in Section 2 (before the formulation of the second vanishing cycle theorem). It is clear thatZn C JC,t. Denote byZnK the fiber of Zn over Spec Kn and by t the natural embedding 3Cn \ZnK \+ JC K. Denote by h the embedding of the generic point of JCnZK into the whole scheme JCnK.

Recall that in Section 2 JC A was defined as the inverse image of the incidence correspondence D C SymmX X X under a morphism

104 V. G. DRINFELD

(spn SymmX X X. In factD = Symm-lX X X, so we obtain a morphism 3iC Symm-lX X X. Denote byf the corresponding morphism JCn K SymmlX X X.

Let a, j3 denote the same objects as in the exact sequence (8) (recall that b is a morphism Symm-lX X X -+ SymmX, a is a morphism Symm-2X X XC_+ Symm-lX X X, j is a morphism i,b*6(m) 6(m-1) 1 It is clear that 5Y'f =K pil, (m), ii,K = t,t*p**(g(m-2)(-1) 1 det 6).

Note that if sp: 3C -+ OR is a morphism of sheaves on JCnK inducing an isomorphism of their generic stalks, then there is a natural morphism atr - h*h*3C. In particular, j: ib*6(m) X;(m-1) & 6 induces an isomorphism h*f*8,*(m) hh*f*((m-i) X 6) and, therefore, a morphism 3:f(6(ml) X 6) h*h*f* ,*(m) = h*h*YnK .

Besides, note that if 9l is an l-adic sheaf on CnK, and 9l are its inverse images on JCnK and iCnK, then there is a canonical exact sequence

O -- H' (Gal(Kn/Kn), i*s4,71)

-- i*R'ls*ft - H0(Gal(kn/Kn), i*R ls*0) - 0

In particular, if the action of Gal(K /Kn) on iRKs Z istrivialforj = 0, 1, then we obtain a canonical exact sequence

0 -+ i*s**Y(-1) -+ i*Rls*ft --+ i*Rls*0 -+ 0

Finally, note that the isomorphism s 5n n ifnK induces a morphism 5n? S*5:n. Thus we obtain a canonical morphism :nK -+ i*S*5:nK

VANISHING CYCLE THEOREM II (REFINED VERSION). 1. The canonical morphism 5Yn K i*s*5YnK is an isomorphism.

2) i*Rjs*5Y K = Oforj > 1. 3) The restriction of R ls*5nK to ZnK is equal to zero. 4) The action of Gal(Kn /Kn) on R 1 *5ynK is trivial. 5) The canonical morphism i*s *5nK (-1) + i*R ts *:V K induces an

isomorphism h*i*s*5n eK (-1) h+ h*i*R ls*5nK. The resulting morphism i*R 1S5 n- h *h *i*s* 5nK (-1) induces an isomorphism t*i*Rls* :n1 s t*h*h*i*s*5Y K (-1).

6) The morphism 3:f*(6(m'1) 1 6) h*h*5nK is an isomorphism. The second vanishing cycle theorem in its former formulation results

from the theorem just formulated. Indeed, from the commutative diagram


0 t*i*R s *()n t*i*R S*53n 0

t *h h*i*s* * (-1)

t*h *h* 5,(K 1)

O t i*n K(_) 1 t ** (8 m 1)-1 Fs) > t *f* ((m 1) 1 det 8) 0

having exact rows (the lower row being the inverse image of the exact sequence (8)) it follows that t*i*Rls*In nK t*f*((m-2)(-1) 1 det 6). By statement 3) this isomorphism induces an isomorphism i*RlS*g:nK t!t*f*(6 )H1) X det 6) = (R K n.

When proving the theorem we may suppose 6 to be not a locally constant l-adic sheaf but the constant sheaf A2, where A denotes the quotient of the ring of integers of E by a nonzero ideal. For any integer r e [0, m] we denote by JCn,r the fiber product of JCn and SymrX X Symm-rX over SymmX and by JC,r the fiber product of JCn,r and Spec On over Vn. The fibers of JC n,r over Spec Kn Spec Kn and Spec Kn will be denoted by n r' JcK rCK L thJCK C4 j)0 r I JC yr. r K rac 0 n,r, n,r Let ir n,r n,rs S *

106 V. G. DRINFELD

f*(6 (m-l) ) = 0m 0(rr)((yr) A (? (6)* A) . Thus we have reduced the second vanishing cycle theorem to the following one.

THEOREM A. Let rEiZ, 0 c r c m. 1) The canonical isomorphism A (ir)*(Sr)*A is an isomorphism. 2) (ir )*RJ (Sr )*A = O for j > 1. 3) The restriction of R' (sr)*A to Z'n,r is equal to zero. 4) The action of Gal(Kn /Kn) on R' (Sr )A is trivial. 5) There is an isomorphism (tr)*(ir)*R l(sr )*A ( tr )*(_Yr )* A(-)1) (*

(tr )*(br )* A(-1) such that the following diagram is commutative:

(tr )*(jr )* (sr )*A )*(tr )*A(-1) (? (tr )*(br )*A(- 1)

1t1 (tr )*?(ir )*?(Sr )*?A(-1l) (tr )*?A(- 1)

6) The natural morphism (_,r)*A ((5r)* A_- (hr) A is a isomorphism.

Proof of statement 6). It suffices to show that a) rr(Bf n Pr) * 3C,n", b) Br and Pr are smooth varieties, whose irreducible components have the same dimension as JC,n. Since 7r(Br n Pr) is the set of divisors from Pn, containing v with multiplicity greater than one, we have rr(Br n Pr) * Jc K. It is clear that Br is the fiber of the natural morphism so: Symr-lx x Symm-rX + Picm-1X over a Kn-point of Picm-lX. As m 2 4g we have r - 1 > 2g -2 orm - r > 2g -2. Therefore either jacr1: Symr-lX Picr- lX, orjacm-r: Symm-rX PicCm-rX or both morphisms are smooth. Hence so is smooth. So Br is a smooth variety whose irreducible components have dimensionm - 1 - g = dim CiKn. Pr has the same properties because Pr - Bm-r.

Statements 1)-5) of Theorem A will be proved in the remaining part of the section. Due to the symmetry between r and m - r we may suppose that r c 1/2 * m.

To calculate (ir)*Rj(Sr)*A and (tr)*(ir)*Rj(Sr)*A we shall construct a kind of singularity resolution of RCn,r. Let e denote the same sheaf as in the proof of Proposition 4.1. Denote by Quotr (C) the scheme over Vn representing the following functor: if S is a scheme over Vn, pr:X X S -+ S the projection, Cs the inverse image of C, onX X S, then MorV (S, Quotr(C)) is


defined to be the set of quotient sheaves 9 of CS such that the support of 9 is finite over S and pr* 9 is a locally free sheaf of rank r.

We shall define a canonical morphism Quotr(C) ,+ SyMrX. To do this we need the following construction. Let 9 be a coherent sheaf on X X S such that the support of 9 is finite over S and pr* 9 is a locally free Os -module sheaf of rank r. Every point of X X S has a neighborhood U such that the restriction of 9 to U may be represented as the cokernel of a monomor- phismf: (uk * (uk. Denote by Du the closed subscheme of U defined by deff = 0. It is easy to show that 1) Du does not depend on the choice off, 2) there is a closed subscheme D C X X S such that Du = D n U for every U, 3) the morphism D -+ S is flat and finite of degree r. D will be called the determinant of G.

Now we proceed to the construction of Quotr(C) SymrX. Let S be a scheme, so a morphism S -+ Quotr(C(), 9 the corresponding quotient sheaf of CS. Then the determinant of 9 defines a morphism S -+ SymrX. Thus we have constructed a functor morphism Mor(?, Quotr(C)) -+ Mor(?, SymrX) and therefore a morphism Quotr(C) -+ SymrX.

Denote by a, C, Q, ? the inverse images of a, C, Q, ? on X X Quotr(C() and by pr the projectionX X Quotr(C() + Quotr(C(). Let 9l be the subsheaf of C such that C/9l is the universal quotient sheaf of C. Put 3 = pr*(a' 01 X X Q). If x E Quotr (C), (ix 9x DX. are the restrictions of a, Z9, Q toX X x, thenH1 (X X x, '0(ix-1 ( 9x (Qx) = 0 (since the quotient of

axy 0 (ix 0 Ox by its torsion subsheaf has a degree not less than deg ? - 2n -1 -r + 2g -2 m -r -1 (m/2) -1 2g - 1). Therefore 3 is locally free and its fiber over x is equal to H?(X X x, a(x-1' 0 x (0 Qx). Consider the superposition 3 = pr(- 10( I (8 0 Q) - pr*(a - 1( 8 Q) -) R 1 pr* Q, where the second morphism is defined via the exact sequence 0 a eC -+ C, - 0. As R 'pr* Q is equal to the structure sheaf, the morphism 3 Rlpr*Q defines a section a E H0(Quotr(C), 3*). Denote by 61 the quotient of 3* by the subsheaf generated by a and put JCnr = P(1).

Now we shall define a morphism Vr: Cn,r Cn,r * Put R = pr*(a -0 C, ( Q). The embedding 3 -+ OR induces an epimorphism so: Mt -+ . so in its turn induces an epimorphism JC* -+ 61, where JC is the inverse image under the morphism Quotr( ) -+ Vn of the sheaf JC from the proof of Prop- osition 4.1. Denote by , the superposition Cn, r = P(01) P(3CM*)C P(MC)

kn and byf the superposition JCnmr Quotr( ) + SymrX. JCn,r may be considered as a closed subscheme of JCn X SymrX. It is easy to show that the image of (l,, f ): Sn,r - n X SymrX is contained in JCn,r Thus we obtain a morphism JCn, r nJCl, r which will be denoted by Vr

108 V. G. DRINFELD

The morphism 'r: JCn, r 3Cn,r is the desired "resolution of singulari- ties" of JCn, r. In fact, fCn,r is not smooth in general. However the singulari- ties of the morphism JCn r -+ Vln are rather simple (see Proposition 4.3).

Before studying the morphism Cn, r + VVn we shall investigate the canonical morphism QUOtr(C() -+ Vn X SymrX. Denote byz the superposition An z+ Vn+1 X X -+ X. Let M C An X SymrX be the inverse image of the incidence correspondence under the morphism (z, id): An X SymrX + X X SymrX. It is clear that M is a smooth subvariety of Vn X SymrX having codimension 2.

PROPOSITION 4.2. Quotr(C) istheblow-upof Vn X SymrX along M. Proof. Denote by r the graph of z and by Jr the sheaf of ideals of r C

X X Vn.

LEMMA 1. C( - Jr (8 (5p

Proof. The canonical morphism a: C( -+ 6 is injective since C( and (63 are flat over VVn and a is an isomorphism over X X Vn. The cokernel of a is the restriction of (63 to a closed subscheme rF C X X Vn. It is clear that r' and r are equal as sets. We must prove that r' and r are equal as schemes.

Since the non-empty geometric fibers of the morphism r' -+ Vn consist of a single point (without nilpotents), the projection X X Vn -+ Vn maps rF isomorphically onto a closed subscheme An ' C VVn. Denote by (' and i' the restrictions of (a and ? to X X An'. It is clear that (' is contained in an invertible subsheaf of ? ' of degree n + 1. Therefore An' lies in the image of the morphism Vn+1 X X - Vn, i.e. An' C An - It is clear that An C An'- Thus Ai' A= , and hence r' = r. E

It follows from Lemma 1 that Quotr(C) = QUOtr(30) Consider the following functor F from the category of schemes over Vn

X SymrX to the category of sets: let S be a scheme over 1n X SymrX, 35 the inverse image Of Ir under the morphismX X S -+ X X Vn, D C X X S the relative divisor defined by the morphism S -+ SymrX, 3D the restriction of 35s to D, pr :X X S -+ S the projection; then F(S) is defined to be the set of those quotient sheaves of pr* 3D which are locally free and r-dimensional. We have an obvious embedding Mor(S, QUOtr(3r)) C> F(S) (here "Mor" denotes the set of morphisms over Vn X SymrX). Put Y = X X Vn, Z = X X Symr-lX X Vn. Let a:Z Y, :Z SymrX X Vn be the natural morphisms.

LEMMA 2. F is represented by P(A rf* a*3P).

TWO-DIMENSIONAL l-ADIC REPRESENTATIONS 109

Proof. Let S be a scheme over SymrX X Vn, 9 the inverse image of B*oa*3r under the morphism S -+ SymrX X VV,. Then F(S) is the set of r- dimensional locally free quotient sheaves of 9. Therefore F is represented by a closed subscheme T C P(ArI3*a*30). To prove that T = P(Aro*a*3) it suffices to show that the dimensions of the geometric fibers of 13*a*3r do not exceed r + 1.

Apply the functor 3*a* to the exact sequence

(11) 0 3r Or pOr ?

where Or is considered as a sheaf on Y. Note that i* and x'* are exact, a*O y = Oz, ,B*a(*Or = OM (here OM is considered as a sheaf on SymrX X Vn). Thus we obtain an exact sequence

(12) ? Ò*a*3r Ò*Oz ÒM ?

It follows from (12) that the dimensions of the geometric fibers of f*a'*3r do not exceed r + 1 (indeed, the restriction of 0* ax*3r to the complement of M is a locally free sheaf of rank r, and the restriction of 3*?a*3r to M is a locally free sheaf of rank r + 1). 0

So Quotr(35) is a closed subscheme of P(Arf* a*3p).

LEMMA 3. Let 9 be a coherent sheaf on X X Sflat over S and such that the support of 9 isfinite over S. Let D CX X S be the determinant of 9. Denote by pr the projection X X S -- S. Then everyfunction a E H?(D, OD) acts on the sheaf det pr*9 as multiplication by N(a), where N:H?(D, OD) -4 H?(S, Os) is the norm mapping.

Proof. a) Suppose S is the spectrum of a field. If the statement to be proved holds for a subsheaf 9' C 9 and for 9/9', then it holds for 9. Therefore, it suffices to consider the case when there are no non-trivial subsheaves in 9. In this case the statement is obvious.

b) Now let us consider the general case. We may assume S to be affine. In this case g can be represented as a quotient of 4k XS (for some k). There- fore it is enough to consider the case S = Quotr(Oxk). A standard argument shows that QUotr((9xk) is smooth. Therefore a function on QUotr(Oxk) is uniquely defined by its values at the points of QUotr(o xk). Thus, we have reduced the general case to the already studied case when S is the spectrum of a field. I

110 V. G. DRINFELD

Denote by H the group scheme over SymrX X Vn representing the following functor: for a scheme S over SymrX X V Mor(S, H)-H?(S, (9*), where S is obtained from S by the base change 3:Z - SymrX X Vn. Let v: H -f Gm be the norm homomorphism. For every coherent sheaf OR on Z, H acts naturally on j* OR. In particular, H acts on 3* a*3r and hence on Ari3* a*3r. Denote by 9l the maximal quotient sheaf of Ar3* ay*3r on which H acts via v. It follows from Lemma 3 that QUotr(0r) C P(0Z) and that the embedding QUOtr( 0) -- P(0Z) induces a bijection of the sets of Fq -points. It is easy to show that 1) the fiber of QUotrG5r) over every point of M is isomorphic to P' (at least up to nilpotents), 2) the fiber of Quotr(0r) over every point of (SymrX X Vn )\M consists of a single point. Therefore the dimension of a geometric fiber of 9l is equal to 1 or to 2.

LEMMA 4. 9T = SM 0 det f* Oz, where 3mis the sheaf of ideals of the subscheme M C SymrX X Vn.

Proof. The embedding 3*a ?*3r * Oz (see the exact sequence (12)) induces a morphism A ri* y*3, - det f* Oz9. We shall denote its kernel and image by 3C' and 9 ', respectively. Local calculations by means of (12) show that 1) 3C' is the direct image of a locally free sheaf on M of rank r - 1, 2) 9Yl' = M (8 det f* Oz, 3) the exact sequence

(13) 0 3C'- Ar3*,*3,p,+ 9+ '-0

splits locally. Denote by JC the subsheaf of A ri*,*3, such that 9l = (A r0*,,*3,)1

JC. Let us show that JC = 3C'. Indeed, if 3C ? 3C ' the fibers of 9l over some points of (SymrX X Vn )\M would have been equal to zero, and if 3C C 3C ', 3C * 3C', then the dimension of the fiber of 9l at some point of M would have been greater than 2. Therefore JC = 3C' and hence 9l = 9Y ' = 5M 0 det A* Oz.

It follows from Lemma 4 that P(9Z) = P(3M), i.e. P(9Z) is the blow-up of SymrX X VVn along M. QUotr(5r) is a subscheme of P(9Z) which is equal to P(9Z) as a set. Since P(0Z) is reduced we have QuotrG5r) = P(0Z). OI

Denote by Blowr the inverse image of M in Quotr (C) and by Propr the proper transform of An X SymrX, i.e. the irreducible component of the inverse image of An X SymrX in Quotr(C() which is not contained in Blowr. Denote by JCir the inverse image of An under the morphism $iCnr V, . Denote by Propr, Blowr the inverse images of Propr, Blowr under the

morhs det - morphism JCn,r -+Quotr(C). Put Singr. = ProPr nlBlowr.


PROPOSITION 4.3. 1) 3C,r is equal to Propr U BlOWr as a scheme. 2) The morphism JCn r\ Singr -+ V, is locally acyclic. 3) There is an open smooth subvariety U C C,n,r such that a) U D

Singr, b) U n Propr and U n Blowr are smooth subvarieties of U having codimension 1 and intersecting transversally.

4) The morphism Singr \An is smooth.

Proof. Recall that JCn,r P(61), where 61 is the quotient of a locally free sheaf P5 on Quotr(C) by a subsheaf generated by a certain section a E Ho(Quotr (C), 3 *). Denote by Wthe open subset of QUOtr (C) defined by the inequality a * 0.

LEMMA. W D Propr.

Proof. Let i, C, 9l denote the same objects as in the definition of 3. For a pointx E QUOtr (C) denote by ?., ex, 9DIX the restrictions of i, (C, 9 DI to X X x. If x E Quotr (C)\ W, the embedding TLx C+x lifts to an embedding TLx - Jx and therefore 9Tx is invertible. On the other hand, if x e Propr the torsion of 9Tx is nonzero. D

Since the morphism JCn,r '+ QUOtr(C() is smooth over W, statements 1), 3) and 4) result from the lemma. Statement 2) is proved in the same way as the analogous statement in Section 3. D

Denote by Zn,r the inverse image of Zn in 3FCn

PROPOSITION 4.4. 1) The morphism Vr: eCn,r r 3RCn,r induces an isomorphism Vr' (SCn,r\Zn,r) JCn,r\Zn,r

2) The geometric fiber of Vr over a point of Zn,r either consists of a single point, which does not belong to Singr, or is isomorphic to Pl. In the second case thefiber lies in Blowr and intersects Singr in exactly one point, the intersection being transversal.

Proof, Let x E Cn.(Fq), D be the image of x in Sym X(Fq), y be the image of x in 3rn (Fq ). Denote by ? and C2 the inverse images of ? and Q on X 0 Fq. The pointy corresponds to an invertible subsheaf (a C ? and an embedding a: (a ( ?-F1 -+ , where C, = ?/(i (aois defined up to an element of Fq*). Put 9Z = Coker a. Let D' e SymmX(Fq) be the determinant of MY. It is easy to see that D' 2 D. Denote byFx the fiber of Pr overx. Fx is the variety of quotient sheaves of Ml whose determinant is equal to D.

112 V. G. DRINFELD

If xgZn,r(Fq), then 9Y -OD'" (where /9D' iS the direct image of the structure sheaf of the subscheme D' C X 0 Fq). Therefore FX consists of a single point. A similar argument also works if x E (3Cn,r\Zn,r)(A), whereA is an arbitrary algebra over Fq. This proves statement 1).

Now let x E Zn, r (Fq). Then - OD'- ? ( , where t is an Fq -point of X is contained in D'. Denote by k and k' the multiplicities of t in D and D'. It is clear that 0 c k ' k'. It is easy to show that if k = 0 then FX consists of a single point lying in Propr\ Blowr, and if k = k ' thenF consists of a single point lying in Blowr\Propr. Now let 0 < k < k', x- be the image of x in Vn (Fq) X SymrX(Fq), F the fiber of the morphism QUOtr(eC) -_ VVn X SymrX over x-. It is easy to see that x- E M(Fq) and the natural morphism FX

F is an isomorphism. Thus statement 2) is proved. L

Now let us prove statements 1)-5) of Theorem A. Denote by 3CX'r, BlowrK, ProprK and SingrK the fibers of fCn,r, Blowr, Propr and Singr over Spec Kn. Denote by -P the morphism JCX,r JCK,r induced by Pr. By the first part of Proposition 4.4, -r maps Pr'(3CK,r\ZK,r) isomorphically onto JC r\Z K . It is easy to show that BlowrK n V r '(CK nr\ZK r) is mapped onto Br\ZKn,r and ProprK r irj1 (JCnr\Znr) is mapped ontoPr\Zknr. Therefore, using Proposition 4.3 we obtain statement 5) of Theorem A and the following statements, which are weaker than statements 1), 2), 4) of the theorem: 1 ') the canonical morphism (tr)*A -+ QrMir)*(Sr) *A is an isomorphism; 2') (tr)*(ir)*Rj(Sr)*A = 0 for j > 1; 4') the action of Gal(Kn /K) on (tr)*RJ(sr)*A is trivial. It remains for us to prove that if x E ZKnr(kun) and W J is the fiber of RJ(sr)*A over x, then W J = 0 forj > 0 and the natural homomorphism A_- W 0 is an isomorphism.

Denote by 3C?n,r the fiber product of JCnr and Spec On over Vn, by sr the natural morphism 3C'nr 'C?n r. Denote byF, the inverse image of x in SC r and by K, the restriction of R(s^r)*A to FX (R(s^ )*A is the direct image of A in the derived category). Then W i = Hi(FX, K,). It follows from Proposition 4.3 that 1) the restriction of (r)*A to 3Cnr is equal to A, 2) R 1 (sr)*A = a*A(-1) where oa is the natural embedding gingrK + 3C?n, r 3) RI(sr)*A = 0 forj > 0, 4) the restriction of R(sr)*A to BlowrK is equal to R(,)*A, Ar being the natural embedding BlowrK\Sing rK- BlowrK. It follows from these facts and the second part of Proposition 4.4 that either Fx consists of a single point and K. = A, or F. is a projective line, Fx f singrK consists of a single point a and K., R-y* A, where -y is the natural embedding Fx\{ a } Fx . In both cases H(F, K,) = O forj > O, H?(Fx, Kx) = A. Thus Theorem A is proved.


Appendix (proof of Deligne's theorem).

DELIGNE'S THEOREM. LetX be a smooth connected projective curve of genus g over an algebraically closed field F, jacr: SymrX -- PicrX the canonical morphism, p a continuous absolutely irreducible representation of r1 (X) in a vector space of dimension d over a finite extension E of Ql, 1 being a prime number differentfrom the characteristic of F. Denote by ; the locally constant sheaf on X corresponding to p and by 8(r) the corresponding sheaf on SymrX (see the beginning of Section 2). Assume that d > 1, r > d(2g - 2). Then Rj(jacr)* (r) = Ofor allj.

The proof given below is due to Deligne and is published by his con- sent. The author of this paper bears the responsibility for possible short- comings of the account.

Proof. First of all let us prove the following statement which is weaker than Deligne's theorem.

LEMMA. The sheaves Rj(jacr)* &(r) are locally constant.

Proof. It suffices to prove the analogous statement in the case when 8 is a locally constant sheaf of A-modules whose stalks are isomorphic to Ad, where A is a quotient of the ring of integers of E by a nonzero ideal. To this end we shall prove that jacr is locally acyclic with respect to 8(r). When proving the local acyclicity we may assume that & = Ad. In this case 8(r) =

3r + - +rd=r3(rj, . * *, *rd), where 93(rl, * . ., rd) is the direct image of the constant sheaf on SymrlX X ... X SymrdX. Therefore, it is enough to prove that the natural morphism SymrlX X ... X SymrdX -_ PicrX is smooth provided r1 + + rd-r > d(2g - 2). To do this it suffices to show that the morphism SymriX _ PicriX is smooth for some i. Indeed, ri > 2g -2 for some i, because r, + * * * + rd > d(2g -2). F-]

Now let us suppose that the theorem is wrong. Letj be the least number such that R i (jacr)* 8(r) * 0. According to the lemma Rj(jacr)* 8(r) is locally constant and, therefore, corresponds to a representation X of _ (piCrX). Since 71 (PicrX) is abelian, X contains a subrepresentation of dimension 1 (maybe after replacing Eby its finite extension). Therefore, having replaced E by its finite extension we obtain a one-dimensional locally constant sheaf X on PicrX such that Ho(PicrX, 31Z R'(jacr)* 8(r)) * 0. Then according to the Leray spectral sequence Hi(SymrX, &(r) (g (jacr)* 31Z) ? 0. Put (a = p* 31Z, where sp is the natural embedding X -+ PicrX ((i does not depend on the choice of p). It is well known that (jacr )*3

114 V. G. DRINFELD

= (r). Therefore &(r) (D ?acr)*M = &(r), where & = C?(. Hence H'(Sy- m X, & (r)) E 0.

On the other hand, let us calculate H*(SymrX, &(r)) using the Kunneth formula. As p is irreducible we have H?(X, 8) = H2(X, 8) = 0. Therefore H (SymrX, ;(r)) = 0 for i E r, Hr(Symrx, ;(r)) = ArHl(X, &). But dim H' (X, 8) = d(2g - 2) < r, so ArHi (X, 8) = 0. Thus H*(SymrX, g(r)) = 0.

So we have obtained a contradiction, which proves the theorem.

PHYSICO-TECHNICAL INSTITUTE OF LOW TEMPERATURE, KHARKOV, USSR.

REFERENCES

[1] V. G. Drinfeld, Langlands' conjecture for GL(2) over functional fields, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), vol. 2, pp. 565-574.

[2] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathe- matics N114, Springer-Verlag, Berlin (1970).

[3] P. Deligne, Les constantes des equations fonctionnelles des fonctions L, in Modular Functions of One Variable II, pp. 501-597 Lecture Notes in Mathematics N349, Springer-Verlag, Berlin (1973).

[4] Cohomologie Etale (SGA 41/2), Lecture Notes in Mathematics N569, Springer-Verlag, Berlin (1977).

[5] Theorie des Topos et Cohomologie Etale des Sch6mas (SGA 4), tome 3, Lecture Notes in Mathematics N305, Springer-Verlag, Berlin (1973).

[6] A. Weil, Dirichlet Series and Automorphic Forms, Lecture Notes in Mathematics N189, Springer-Verlag, Berlin (1971).

[7] R. P. Langlands, Base Change for GL(2), Princeton University Press and University of Tokyo Press (1980).

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