Gauss Sums, Kloosterman Sums, and Monodromy Groups. (AM-116), Volume 116 9781400882120

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Annals of M athem atics Studies Number 116

Gauss Sums, K loosterm an Sums, and M onodrom y Groups by Nicholas M. Katz

PRIN C ETO N UNIVERSITY PRESS

PRIN CETO N , NEW JER SEY

Copyright (c) 1988 by Princeton University Press ALL RIGHTS RESERVED

T he A nnals of M athem atics Studies are edited by W illiam Browder, Robert P. Langlands, John Milnor and Elias M. Stein Corresponding editors: Stefan Hildebrandt, H. Blaine Lawson, Louis Nirenberg, and D avid Vogan

C lothbound editions of Princeton University Press books are printed on acid-free paper, and binding m aterials are chosen for strength and durability. Paperbacks, while satisfactory for personal collec­ tions, are not usually suitable for library rebinding

P rinted in the U nited States of America by Princeton University Press, 41 W illiam Street Princeton, New Jersey

L ib r a r y o f C o n g r e s s C a t a lo g in g - in - P u b lic a t io n D a t a K atz, Nicholas M ., 1943G auss sums, K loosterm an sums, and monodrom y groups (The Annals of m athem atics studies ; 116) Bibliography: p. 1 . G aussian sums.

2 . K loosterm an sum s. 3. Homology

theory. 4. M onodrom y groups. I. T itle. II. Series: Annals of m athem atics studies ; no. 116. Q A 246.8.G 38K 37 1987

512’.7

ISBN 0-691-08432-7 (alk. paper) ISBN 0-691-08433-5 (pbk.)

87-45525

Contents I n t r o d u c ti o n C hapter 1. B re a k s a n d S w an C o n d u c to rs 1.0 The basic setting 1.1-1.10 Definitions and basic properties of breaks, breakdecompositions, and Swan conductors 1.11-1.17 Represent ation-theoretic consequences of particular arrays of breaks 1.18-1.20 Detailed analysis when Swan = 1 C hapter 2. C u rv e s a n d T h e ir C o h o m o lo g y 2.0 2.1 2.2 2.3

Generalities Cohomology of wild sheaves Canonical calculations of cohomology The Euler-Poincare and Lefschetz trace formulas

C hapter 3. E q u id is tr ib u tio n in E q u a l C h a r a c te r is tic 3.0-3.5 The basic setting 3.6 The equidistribution theorem 3.7 Remark on the integration of sufficiently sm ooth functions C hapter 4. G a u ss S u m s a n d K lo o s te r m a n S u m s: K lo o s te r m a n S h eav es 4.0 Kloosterm an sums as inverse Fourier transform s of mono­ mials in Gauss sums 4.1 The existence theorem for Kloosterm an sheaves 4.2 Signs of pairings 4.3 The existence theorem for n — 1, via the sheaves C 0 (cf. C hapters 5, 6 , 7, 8 ). It m ay seem strange a t first sight to devote so m uch attention to lisse sheaves on so particular a curve as G m; it would certainly be so if we were in characteristic zero. However, if X is any sm ooth geometrically curve over a perfect field k of characteristic p > 0, then there exists a non-em pty Zariski open set U in X and a finite etale fc-morphism U —►G m. This analogue of Belyi’s striking theorem [Be] about open sets of curves over Q being finite etale over P 1 — {0,1, oo} is easily proven (take a general function / : X —* P 1 to reduce to the case when X is open in P 1; shrinking X , we m ay assume X = A 1 —T, where T is finite etale additive subgroup of A 1—this is where we use characteristic p— the quotient group A 1/r is again A 1, via the explicit projection ir : t —+ H7 (t —7 ), and this same 1r is the required finite etale map of A 1 —V to G m). Its moral is th a t in characteristic p, whatever can happen for lisse sheaves on general curves already happens for lisse sheaves on G m. In the same chain o f ideas, it is natural to try to classify all lisse sheaves on G m with specified local m onodrom y a t both zero and infinity. It turns out (cf. 8.7) th a t the Kloosterm an sheaves can be specified among all lisse sheaves on G m exactly by specifying their local monodromy a t both zero and infinity. This intrinsic characterization suggests th a t Kloosterm an

Introduction

5

sheaves are more intrinsic th an their construction in term s of exponen­ tial sums m ight suggest. The proof of this characterization relies heavily upon the /-adic Fourier transform on A 1 as developed by Brylinski and Laumon, which we discuss a t some length (cf. 8.2-8.5). After this brief survey of the ideas, techniques, and results to be applied, we now tu rn to the actual concrete problems to which they will be applied. We will begin by explaining the original m otivating problem. Given a prime num ber p and an integer a which is prime to p, the Kloosterm an sum is defined as the complex num ber Kl(p, a) =

e x p f — (* + j/)V x y = a m od p x,y£Z/pZ

This sum is real (replace ( x , y ) by (—# , —?/)), and its absolute value is at m ost 2y/p, this last estim ate due to Weil as a consequence of the “Riem ann Hypothesis” for curves over finite fields. Therefore, there exists a unique angle 0(p, a) E [0, w] for which — Kl(p, a) = 2y/p cos 0(p, a). T he problem is to understand how the angles 0(p, a) are distributed in [0,7r] for a fixed non-zero integer a, as p varies over all the primes not di­ viding a . Although there is no compelling conceptual justification, and not a great deal of “com puter evidence,” it is nonetheless tem pting to believe th a t for fixed a ^ 0 in Z, the angles {0(p,a)}p are equidistributed in [0, tt] with respect to the “S ato -T ate measure” (2/7r) sin2 0 d0. Unfortunately, we have absolutely no contribution to make to this question. O ur results are concerned rather with the distribution of the angles 0(p, a) for fixed p as a varies over F * . We prove (cf. C hapter 13) th a t as p f oo, the p — 1 angles {0(p, a )}aep x “become” equidistributed in [0,7r] for the S ato -T ate measure, and we give bounds for the error term . In a given characteristic p > 0, once we pick a prime num ber / ^ p and an /-adic place A of the field Q(Cp) = E , there is a natural lisse, rank two jE^-sheaf, “K l^(2)” , on the m ultiplicative group G m (g) F p in characteristic p, whose local “traces of Frobenius” are (minus) the Kloosterm an sums in question. The equidistribution property of the angles {0(p, a)}a€p x then results (via Weil II) from the fact the geometric monodrom y group of the rank-two sheaf Kl^, (2) is a Zariski-dense subgroup of Sl(2), cf. C hapter 11.

Introduction

6

For any integer n > 2, one m ay consider the more general Kloosterm an sums Kl(f>, n, a) —

^2

exp ( — " (*i H--------1"***)]

x i ...xn = a m od p X \, . .. , xn m od p

^

and their analogues over finite extension fields Y q of F p; one again con­ structs a n atu ral lisse rank n E \-shea f “K l^(n )” on G m F p (for n — 2 this is the param etrization

-c ;

a

)

of conjugacy classes in K = SU(2) by [0, 7r]). It follows from Weil II th a t the conjugacy classes {0(p, ra, a)}aeF x have an approxim ate equidistribution, with explicit error estim ate, for the measure on — {conjugacy classes in K } which is the direct image of H aar measure on K (for K = SU( 2 ), this measure on « [O', 7r] is S ato -T ate measure (2/tt) sin 2 0 dO). The problem is to com pute the Zariski closure G g e 0 m of the geometric monodromy group of the lisse jE^-sheaf Kln (^>) on G m ® F p. A priori, the group G g e 0 m “depends” on three param eters (p, n, A ) . In fact, it is independent of A , and very nearly independent of p as well. For n > 2 even, the sheaf K l^(n) carries an alternating autoduality respected by Trf001X1, and we prove th at, in fact G ge0m

= Sp(n)

for n even, p arbitrary.

For n > 3 odd, and p odd, we prove th a t (jtgeom — SL(n)

if pTl IS odd.

For n > 3 odd, and p = 2 , the sheaf K l^(n) carries a symmetric auto­ duality respected by 7 r f eom, and we prove th a t Ggeom = SO(n)

if p = 2 , n > 3 odd, n ^ 7.

For n = 7 and p = .2, a general argum ent based on classification shows th a t either G geom = SO(7), or G geom = the subgroup G 2 of SO(7), where G 2 c—►SO (7) is the seven-dimensional irreducible representation of G 2.

Introduction

7

We initially expected to find 5 0 (7 ) as the answer. To our surprise, we found instead th a t O'geom — O 2

for p — 2, 71 — 7.

Perhaps exceptional Lie groups aren’t so exceptional after all. All of these results on G geom are proven in C hapter 11 , as consequences of some general classification theorem s (11.6, 11.7) which are of independent interest. In C hapter 12 , we show th a t for given p and n, the actual A-adic image of 7rpom e ? rath er th an the Zariski closure) is “as big as possible,” cf. 12.1, 12.2, 12.5.2, and 12.6.2. The ideas and argum ents of this chapter are due entirely to Ofer G abber. In C hapter 13, we apply the results of C hapter 11 to the equidistribution of K loosterm an angles. The results of this chapter make it reasonable to ask whether for n > 2 a fixed integer, and a ^ O a fixed non-zero integer, the conjugacy classes {0 (p, n ,a )} p, as p runs over odd primes which are prime to a, are equidistributed for H aar measure in for K a com pact form of Ggeom ( = Sp(n) if n is even, SL(ra) if odd). For n = 2, this is the m otivating problem w ith which we began. A second problem (cf. C hapter 9) which we consider is the following. Let F q be a finite field, a non-trivial C-valued additive character of F^, and X a generator of the group of all C-valued m ultiplicative characters of F * . For each integer 1 < a < q — 2, the gauss sum g(i’, x a) = ^ 2 has absolute value y/q. Let us tem porarily denote by 0(a) » („ )

S' V?

the corresponding angle, viewed as a point on the unit circle. It is well known (cf. [Ka-1]) th a t as q | oo, the q — 2 points 0(a) E S 1 “become” equidistributed in 5 1 with respect to usual H aar measure; this equidis­ tribution results from the fact th a t the Kloosterm an sheaves K l^(n) on G m 0 F p are lisse sheaves of rank n which are pure of weight n — 1 . Moti­ vated by an early paper of D avenport [Dav], we ask about the distribution, for a given integer r > 1 , of the r-tuples of “successive” angles (0(a + 1), 0(a -h 2 ) , . . . , 0(a + r)) E ( 5 1)1*,

Introduction

8

as a runs over the interval 0 < a < q — 2 — r. We prove th a t as q | oo, these q — l — r points in (S 1)1, “become” equidis­ trib u ted w ith respect to H aar measure. The proof is based on the fact th a t monomials in gauss sums (which arise in applying the classical Weyl criterion) are the m ultiplicative Fourier transform of certain “K loosterm an sums w ith m ultiplicative characters” of the form i>(xi + -----l-a;»)xi(^i)---X n(!C »), 371...3?fi— CL

xieFf where Xi >*• • >X n are m ultiplicative characters of F *. For fixed ip and Xi>---»Xn, there is a natural lisse, rank n E \-sh eaf K1W>; X u • • • j X n ) on G m y, then G (x+ ) acts trivially on H om z(M (x), AT(y)), whereas G(x) acts trivially on N ( y ), so a G (z)-in variant in Hom is a G(a?)-equivariant map ip : M (x ) —►N (y), which necessarily factors through the covariants B ut for any Z[l/p][G (y)] module, the natural map from in­ variant to covariants ( M ( x ))g ^

— M (x)

(M (x))a w

15

Breaks and Swan Conductors

is an isomorphism, the inverse being the averaging operator tt( x ). As ( M ( x ) ) G(x) = 0 , we have ( M x )) g (x) — 0 , so V* = 0 . ■Therefore x is the unique break of H om z(M (x), N (y )) when x > y. In the x = y case, G (# + ) acts trivially on both N ( x ) and M ( x ), so on both M ( x ) 0 N ( x ) and on Hom z(M (ai), N (x ) ). Therefore both of these have all breaks < x. | 1.4. L e m m a . I f A is a Z [1 /p]-algebra, and M a left A-module on which P acts A-linearly through a finite discrete quotient, then in the breakdecomposition M = 0 M (a;) ) X> 0 each M ( x ) is an A-submodule o f M . For any A-algebra B, the breakdecomposition o f B 0, M ( x ) is an A-module direct factor of M by the preceding corollary. For A noetherian local and M free of finite rank, we find th a t M ( x ) is projective of finite type, so free of finite type. Because M v is a subm odule of H om z(M , A), the second assertion follows from (1.3). The last assertion is obvious from the change of rings formula (1.4) (M 0 a B )(x ) = M ( x ) ® a B.

|

16

C hapter 1

1 . 6 . D e fin itio n . For M as in 1.5 above, we define its Swan conductor to be the non-negative real num ber (it is in fact a rational num ber, cf. [Ka-5]) x rk a ( M ( x )).

Swan(M ) = x>0

1.7. R e m a rk s . Clearly we have Swan(M ) = 0 if and only if M = M ( 0 ) = M p is trivial as a representation of P . For any ring homomorphism A —►B w ith B noetherian local, we have Sw an(M ) = Swan(M 0 ^ B ). The m ost useful cases of this are B = residue field of A , and, if A is a do­ main, B = fraction field of A. Notice also th a t M and M v = Hom a (M , A) have the same Swan conductor. We now consider the situation where the given action of P extends to an action of 7, or of D.

1 . 8 . L e m m a . I f M is a Z[1 / p]-module on which I (resp. D ) acts} such that the action o f P factors through a finite discrete quotient o f P, then in the break-decomposition M = 0 M ( i) , each M ( x ) is I-stable (resp. D-stable). Proof. The groups P and /(*) for x > 0 are all normal in I (resp. D ), so the elements of I (resp. D) com m ute with all of the idem potents tt( x ) and 7r(a;+) used to define the break-decomposition. | Now let A be a complete noetherian local ring with finite residue field F a of characteristic / ^ p, and let M be a free A-module of finite rank on which D (resp. I) acts continuously. Because an open subgroup of finite index in A u t^ (M ) is pro-/ (e.g., the subgroup of elements which induce the identity on M ® a F a ), while P is pro-p, the action of P on M autom atically factors through a finite discrete quotient of P. T hus we m ay speak of the breakdecomposition of M ; it provides a canonical direct sum decomposition as A[D] (resp. A[/])-m odule M = 0 M (i) a?>0

in which each M ( x ) is a free A-module of finite rank, whose formation com m utes w ith arbitrary extension of scalars A —►B. 1.9. P r o p o s itio n . Let A be a complete noetherian local ring with finite residue field F \ of characteristic I ^ p, and M a free A-module o f finite rank on which I acts continuously. Then for every x > 0, the product

Breaks and Swan C onductors

17

x i k a ( M ( x )) is an integer > 0. In particular, the Swan conductor Sw an(M ) is an integer > 0, and Swan(M ) = 0 i f and only i f M is tame in the sense that M = M p . Proof. In view of the break-decomposition M = 0 M ( # ) , it suffices to prove universally th a t Swan(M ) is an integer. Because Swan(M ) = Sw an(M 0 F a ), we are reduced to the case when A is a finite field F a of characteristic / ^ p. In this case, M is itself finite, so the representation of I factors through a finite quotient G of I. In this case, the com pati­ bility ([Se-1], pp. 80-82) between upper and lower num bering shows th a t Sw an(M ) coincides with the integer “6(M )” of ([Se-2], 19.3). | 1.10. R e m a r k . Suppose now th a t E \ is a finite extension of Q j, / ^ p, with integer ring G \ and residue field Fa- Let M be a finite-dimensional E \ - vector space on which D (resp. I) operates continuously and J^A-linearly. By compactness, there exists an 0A -lattice M in M , (i.e., a free ^A-m odule M of finite rank w ith M 0 E \ = M ) which is D -stable (resp. /-stable). Ox Therefore P acts on M , and hence on M , through a finite quotient. The break-decomposition of M is obtained from th a t of M by the extension of scalars G \ E \ . Therefore the E \ resp. G \ y resp. Fa-representations M 0 E \ = M, Ox

M,

M 0 Fa

all have the same breaks with the same m ultiplicities. In particular, Swan(M ) = Swan(M ) = Swan(M 0 F a) for any C>A-form M of M . 1.11. L e m m a . Let I be a prime number I p, E \ a finite extension of Q i, D \ its integer ring, ¥ \ its residue field. Let M be a non-zero finite di­ mensional E \ (resp. F a )-vector space which is a continuous representation o f I. Suppose that M p = 0 and Sw an(M ) = 1 . Then ( 1) The unique break o f M is x = 1/dim (A f), and its multiplicity is dim (M ). (2) representation o f I, M is absolutely irreducible. (3) I f M is a quasi-unipotent representation o f I (e.g., i f M is the re­ striction to I o f a continuous representation o f D, and i f no finite extension o f the residue field of K contains all l-power roots o f unity) then the image o f I in A ut(M ) is finite.

18

C hapter 1 (4) I f M as in (3) above is the restriction to I o f a continuous rep­ resentation o f D, then an open subgroup o f D acts by scalars. In particular, the image o f D in A ut(M ) is finite i f and only i f d e t ( M) is a character o f finite order o f D.

Proof. We have Sw an(M ) = ^ z d i m M ( x ) — 1 and M (0) = 0. Since each term x dim M ( x ) w ith x > 0 is a non-negative integer, there is exactly one such term which is non-zero, say xodim M (xo) = 1. Because M (0) = 0 by hypothesis, the decomposition of M as 0 M ( a r ) shows M = M ( x o), whence dimM(a?o) = dim M , xq — 1/ dim M . If M ' C M is a non-zero 7-sub-representation, then M' ( x ) C M{ x ) for every x > 0, so the only possible break of M 9 is 1 /d im M , whence Sw an(M ') = dim M ' / dim M ; as Sw an(M ') is an integer, we m ust have M = M *. As this argum ent is equally valid after extending scalars to any finite extension of E \ (resp. F a), we get the absolute irreducibility. For (3) and (4), only the E \ case is not obvious. For (3), the condition on the residue field guarantees th a t the local m onodrom y theorem applies: there exists an open subgroup of I on which the representation is unipotent. The associated nilpotent endomorphism N is 7-equivariant (cf. 7.0.5), hence a nilpotent scalar (7 acts absolutely irreducibly), hence zero, whence the representation is trivial on an open subgroup of 7. For (4), every element of D normalizes the finite (by (3)) image of 7 in A ut(M ). Therefore an open subgroup of D com m utes with the image of 7. By the absolute irreducibility of 7, this open subgroup of D acts by scalars in A ut(M ). | 1.12. L e m m a . Let E \ be a finite extension of Qi , I ^ p, with residue field Fa* Let M be a non-zero finite dimensional E \ (resp. F a )-vector space on which I acts continuously and irreducibly. Then the unique break o f M is x = Swan( M ) / dim (M ), and its multiplicity is dim (M ). Proof. This is obvious from the fact th a t in the break-decomposition M = 0 M ( x ) , each M ( x ) is an 7-submodule; by irreducibility M m ust be a single M( x ) . | 1.13. Before continuing, we m ust recall th a t for every integer N > 1 prime to p, the inertia group 7 has a unique open subgroup I ( N ) of index N . In term s of the short exact sequence

i - p - / - n z,(i)-i. l± r >

Breaks and Swan C onductors

19

I ( N ) is the kernel of the projection of I onto the/f/v quotient of If we think of I as being the absolute galois group of the m aximal unramified extension K nr of K inside K sep, then I ( N ) corresponds to the /^ -e x te n sio n of K nr obtained by adjoining the AT-th root of any uniformizing param eter 7T of K nr. The wild inertia subgroup P of I is also the p-Sylow subgroup of /(AT), b ut the upper num bering filtration on it changes; if we think of /(AT) as the absolute galois group of K nr(w1/ 1*), its upper num bering filtration is related to th a t of I by the simple change of scale I(JV)W =

for all x > 0 .

From this it follows th a t if E \ is a finite extension of Q/, I p, with residue field F a , then we have the following behavior of breaks and m ulti­ plicities under the operations of restriction and induction of E \ (resp. F a)representations. 1.13.1. If M is a finite-dimensional continuous E \ (resp. F a ^representa­ tion of I with breaks Xi of m ultiplicity n*, then its restriction to I ( N ), which we denote by [Af)*(M), has breaks N x i with the same m ultiplicity ni\ more precisely, we have ([AT]*(M))(x) = M ( x / N ) for each x > 0 , and consequently Swan([Af|*(M)) = iVSwan(M ). 1.13.2. If M is induced from a finite dimensional E \ (resp. F a ^representa­ tion V of I ( N ), w ritten M = [AT]*(T^), we have M * (A f) = ®

7V

(for 7 running over a set of coset representatives of ///(A T )), whence M ( x / N ) = ([iV]*(M))(a:) = ®

j ( V ( x ) ) = , ® ( 7 ^ )( * ).

Thus if V has breaks Xi of m ultiplicity n», its induction [A/r]*(F) has breaks X {/N of m ultiplicity ATn*, and consequently Swan([AT]*(V0) = Sw an(F). 1.14. P r o p o s itio n . Let E \ be a finite extension o f Q i , l ^ p, with residue field F a. Let M be a non-zero finite dimensional E \ (resp. F \/-ve cto r space on which I acts continuously. Suppose that Sw an(M ) = a,

dim (M ) = n,

(a, n ) = 1,

20

C hapter 1

and that the unique break o f M is a / n with multiplicity n. Then: ( 1) M is absolutely irreducible. (2) Write n = nopv with no prime to p. Then over a finite extension of E \ (resp. E \ ) , M is induced from a pv-dimensional representation V o f /(n o ), and the restriction o f V to P is absolutely irreducible. A s I (no)-representation, all breaks o f V are a /p v . (3) Over a finite extension o f E \ (resp. T?\), the restriction o f M to P is the direct sum o f no pairwise-in equivalent absolutely irreducible pv-dimensional representations o f P , whose isomorphism classes are fixed by /(n o ) and cyclically permuted by / / /( n o ) . Proof. If M ' is a non-zero sub-representation of M , then its unique break is a /n . Therefore Sw an(M ') ■= d im (M ')(a /n ). Because Sw an(M ') is an integer, b u t (a, n) = 1, we find th a t n divides dim (M '). As n = dim (M ), we infer th a t dim (M ') = dim (M ), whence M ' = M . Repeating the argum ent over a finite extension field, we see th a t M is absolutely irreducible. To prove (2) and (3), we argue as follows. Because P acts on M through a finite p-group quotient G, M is P - semisimple. Extending scalars, we m ay and will assume th a t every irreducible representation of P which factors through G is absolutely irreducible. Let M = 0 a M a be the P-isotypical decomposition of M . Because P is norm al in / , / perm utes the non-zero M a ; because I operates irreducibly it perm utes them transitively. The stabilizer of one of these blocks is open in / , and it contains P . Therefore it has some index N prime to p, so is equal to I ( N ) by unicity. By unicity I ( N ) is norm al in / , so I ( N ) is the stabilizer of each non-zero P-isotypical block M a . If we pick one of them , say M ao, we have an /(iV )-direct sum decomposition

M = 0

7Mao

over a set of representatives 7 of I / I ( N ) . This means th a t M is induced from the representation M ao of I ( N ) . Because M as /-representation has unique break a /n , it follows th a t M ao as /(^ -re p re s e n ta tio n has unique break N a / n = a ( N / n ), with m ultiplicity n / N , and it has Swan = a. In particular, n / N is an integer, and it is prime to a (because (a, n) = 1 by hypothesis). T hus we m ay replace I by I ( N ) and M by M ao, and begin all over again w ith the additional hypothesis M is isotypical as a representation of P. Let V be a non-zero P-irreducible subspace of M . By our prelim inary extension of scalars, we have insured th a t V is absolutely irreducible as a

Breaks and Swan Conductors

21

P-representation. Therefore its dimension is a power of p : dim 17 = pv for some v > 0. T hus it suffers to show th a t in fact V — M . For this we argue as follows. Because M is P-isotypical, every transform j V of V by an element 7 € 7 is isomorphic to V as a P-representation. If we denote by p :P - * A u t( T 7 ) the action of P on V , this means th a t for any 7 € 7, the representation p7 : P —►Aut(V)- defined by

is equivalent to p. We will show th a t p m ay be extended to a continuous representation of 7 on V , from which it follows th a t Sw an(F) is an integer. G ranting this for a m om ent, we notice th a t as V is a P-subm odule of M , its unique break is a /n , with m ultiplicity dim(V'). Therefore Swan(T7) is given by S w an(y) = d im (F )(a /n ). As S w an(F ) is an integer, but (a, n) = 1 , we have dim V = 0 mod n, whence n = d i m V , whence V = M , as required. It remains to show th a t p extends to 7. For this, we choose an element 7 G 7 which m aps onto a topological generator of Y\j^p Z/( 1) in the short exact sequence Wp Taking a cluster point in 7 of the sequence of {7 pn}n>o, we m ay further suppose 7 has its pro-finite order prime to p. Then 7 defines an isomorphism P K Z notp ^ 7 . Let V be an G \-form of the P -m odule V (in the case of E \ ) . Recall th a t because P acts irreducibly on V through a finite p-group, the F,\[P]-m odule V 0 F a is irreducible. Now consider the representations p and p7 of P on V (resp. on V’). They are isomorphic over E \ (resp. F x), so we can certainly write down an E \ (resp. Fx)-equivalence A between them , which by suitable scaling m ay be assumed in the E \-case to m ap V to V and to induce a non-zero m ap of V 0 Fx to itself. Because V 0 F a (resp. V') is P-irreducible, the non-zero map A 0 F a (resp. A) m ust be an isomorphism. Therefore we m ay choose an isomorphism A between p and p7 on V (resp. on V): = A p (g)A ~ 1 for all g e P,

22

C hapter 1

with A E A u to x(V ) (resp. A u tp x(V )). Replacing A by a pn-th power of A, and 7 by the same pn-th power of 7 , we m ay further assume th a t A has pro-finite order prime to p, and th a t 7 still defines an isomorphism

P * Znot P ^ I . Now we can write down an explicit continuous extension p of p on all of 7, by defining, for g E P and n E Z not p, p(97n ) = p(a)An .

I

1.14.1. R em ark. In 1.14, the hypothesis on the breaks serves only to guar­ antee th a t M is absolutely irreducible: 1.14.2. V ariant. Let E \ be a finite extension o fQ i, I ^ p, with residue field F \ . Let M be a non-zero finite dimensional E \ (resp. F \)-vector space on which I acts continuously and absolutely irreducibly. Write dim (M ) = tiqPv, with no prime to p. Then over a finite extension o f E \ (resp. F \ ) , M is induced from a pv-dimensional representation V o f /(n o ), and the restriction o f V to P is absolutely irreducible. Furthermore, the restriction o f M to P is the direct sum of no pair­ wise inequivalent absolutely irreducible pv-dimensional representations of P , whose isomorphism classes are fixed by /(n o ) and cyclically permuted by / / /( n o ) . Proof. Exactly as in the proof of 1.14, we reduce easily to the case when M is isotypical as a P-representation. If V is an absolutely irreducible P-subm odule of M , the argum ent of 1.14 shows th a t V extends to a repre­ sentation V i of I. If M \ P ^ nV , then H om p(V i,M ) is an n-dimensional representation of I / P . Extending scalars if necessary, this representation of I / P contains a one-dimensional character, say x*”1? of I / P j simply because I / P is abelian. Therefore we have H om /(V i, M ® x) i 1 0Because both Vi and M 0 % are /-irreducible, this non-zero /-m orphism m ust be an isomorphism of V\ with M ® x- Restricting to P , we obtain V ~ M |P , as required. | 1.15. C o ro lla ry . Hypotheses as in 1.14 above, «/dim (M ) is a power of p, then M is absolutely irreducible as a P-representation. Proof. We have no = 1, whence over an extension field M = V is absolutely irreducible as P-representation. |

23

Breaks and Swan Conductors

1 .16. C o ro lla ry . Hypotheses and notations being as in Proposition 1.14, let x : I E * (resp. F * ) be a continuous character of order dividing no (recall no — the “prime-to-p part” o f d im (M )). Then M is isomorphic to M 2 , M p = 0 and Swan(M ) = 1, and we write dim (M ) — n — nopv with no prime to p. We denote by G the Zariski closure of the subgroup p (I) in A ut(M ), which we view as a linear algebraic group over E \ (resp. F a), given with a faithful representation on M . (Recall (1.11) th a t if p is quasi-unipotent, then p(7) is finite, so in this case G is ju st the finite group p (/), viewed as an algebraic group. For example, this is autom atic over F a.) 1.19. L e m m a . Hypotheses and notations as in 1.18 above, we have ( 1) The given representation of G on M is absolutely irreducible; in particular, as dim M > 2 , G is not abelian. (2) Over an finite extension of E \ (resp. o f ¥ \ ) , any linear representa­ tion o f G o f dimension < n = dim M is abelian. In particular, G has no faithful representations of dimension < n, and every absolutely irreducible representation o f G of dimension < n is o f dimension one. (3) A n y linear representation of G of dimension < n is trivial on the finite subgroup p(P ). (4) I f p(I) is finite, then the index of p(P) in p(I) is prime to p and > n, and the quotient p (J)/p (P ) is cyclic. Proof. Assertion (1) holds because the subgroup p(I) of the rational points of G already acts absolutely irreducibly on M (cf. 1 .1 1.(2)). For ( 2 ), let xp : G —> G L {V ) be a linear representation of G of dimension < n, and consider the composite I —> ■p(I) C G - * G L (V ). Because p has all breaks 1 /n , i.e. is trivial on for x > 1 /n , we see th a t this composite is also trivial on /(*) for x > 1/n , so all the breaks of x/>op are < 1/n . B ut d im (F ) < n, so Sw an(F) < 1 , whence Sw an(F) = 0 . Therefore ip o p is trivial on P , thus proving assertion (3); I acts on V through its abelian quotient J tame. So for any a, /3 £ I, we have [^(pO*))) ^(p(/?))] = 0 in E n d (F ). Fixing /?, we see by Zariski density th a t ['ip(g), ^ ( p(/3))] = 0 for all g G G. Now fixing g £ G, we see by Zariski density th a t [tl>(g)\ V,(p/)] = 0 for all g, g* £ G, whence ip is abelian. For (4), the quotient p ( /)/p (P ) is a finite quotient of /tame5 so m ust be cyclic of some order N > 1 prime to p. Consider the restriction of p to

Breaks and Swan Conductors

25

I ( N ) . All of its breaks a re N / n , and by construction p (I(N )) = p(P) is a finite p-group. We m ust show th a t N > n, i.e., th a t the unique break of p restricted to I ( N ) is > 1. This is a special case of the following lemma, applied to I = I ( N ) and to V = M viewed as I ( N ) -representation. 1 1.20. L e m m a . Let V be a finite-dimensional vector space over a field of characteristic ^ p, on which I acts linearly, and such that the action o f I factors through a finite discrete p-group quotient of I. Then every non-zero break o f V i s > 1. Proof. By the break-decomposition V = @ ^ ( # ) , we m ay reduce to the case when V has a single break, say xq . We may suppose xq > 0, otherwise there is nothing to prove. Let G be a finite p-group quotient of I through which the action factors, and let G — Go D G i D G 2 D • • •

be the lower-numbering filtration. In term s of lower num bering the Swan conductor is given by Sw an(F ) = £

f £ r dim (V yV ^)#G °

In our situation, G is a p-group, so G q = G i, but all breaks of V are > 0, so V Gl = 0, whence the term i = 1 of the formula gives the inequality Swan(V') > d im V . Because v has a single break, xq , we have Sw an(F) = #o dim (F ), whence xq > 1, as asserted. |

CHAPTER 2

Curves and Their Cohomology 2.0. G e n e ra litie s . Let k be a perfect field, C / k a proper sm ooth geomet­ rically connected curve over fc, K = k(C) the function field of C. For each closed point a? of C, denote by the fraction field of the henselization of the discrete valuation ring in K which “is” x. Choose a separable closure of K { x}, and a i£-linear field em bedding K 8ep ►K The local galois group Dx

G a l(A f/}/ A {r})

is naturally identified to the decomposition subgroup of G a\ ( K 8ep/ K ) a t the place x of K sep defined by the chosen em bedding of K 8ep into . Similarly, the inertia subgroup I x C D x is itself the inertia subgroup of G a\ ( K 8ep/ K k sep), k 8ep the separable closure of k in K 8ep, for the same place x of K sep. Let U C C be a non-em pty open set. If we view the chosen K sep as a geometric generic point fj of {7, we have a natural identification 2.0.1

7Ti(C/, fj) = the quotient of G a \(K 8ep/ K ) by the smallest closed norm al subgroup containing I x for all x £ U .

F ix a prime num ber / ^ char(ib), and denote by A an “/-adic coefficient ring,” i.e., A is either a finite extension E \ of Q/, or a complete noetherian local ring with finite residue field F \ of characteristic /. For T a lisse sheaf of finitely generated A-modules on (7, its fibre Tf\ is a continuous repre­ sentation of 7Ti(Ufj) on a finitely generated A-module, and the construction T \-+ Ffj defines an exact A-linear equivalence of abelian categories

.

2 0.2

/lisse sheaves of \ finitely generated ' A-modules on

/finitely generated \ A-modules given with a continuous / 1-linear action \ o f 7Ti(C/, 77) /

For a closed point x E C, the decomposition group D x C G a l(K 8ep/ K ) maps to 7T\(JJyfj) , so acts on In particular, the inertia groups I x a t all points of C —U act on (the I x for x E U operate trivially by hypothesis).

Curves and Their Cohomology

27

2.0.3. For x £ C — [/, we m ay speak of the break-decomposition of as Ptf-representation. The breaks which occur in it are called “the breaks of T a t x .” We say th a t T is tam e at x if Px acts trivially on i.e., if 0 is the only break of T at x. We say th a t T is totally wild at x if (Tf\)Px — 0, i.e., if 0 is not a break of T at x. 2.0.4. In view of the exactness properties of the break-decomposition (cf. 1.1), each of the conditions “tam e at x ,” “totally wild a t x ” defines a full sub-category, stable by sub-object, quotient, and extension, of the abelian category of all lisse sheaves of finitely generated A-modules on U . For T \ and T 2 tam e a t x, and Q totally wild a t x, the tensor product T\ < 8 >a P i is tam e a t x, while T \ Q is totally wild at x. For A an /-adic coefficient ring, and T a lisse sheaf of finitely generated A-modules on P , the com pact and ordinary cohomology groups P * (C /0 ib seP ,P )

H i {U