(03) So when the order of { divides no N j at all then Z { (s) is holomorphic on C Now the N j are not intrinsically associated to f 1 f0g; but the or

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1 HOLOMORPHY OF LOCAL ZETA FUNCTIONS FOR CURVES Willem Veys Introduction (01) Let K be a nite extension of the eld Q p of p{adic numbers, R the valuation ring of K, P the maximal ideal of R, and K = R=P the residue eld with cardinality q For z 2 K we denote by ord z 2 Z [ f+1g its valuation, jzj = q ord z its absolute value, and ac(z) = z ord z its angular component, where is a xed uniformizing parameter for R Let f(x) 2 K[x]; x = (x 1 ; : : : ; x n ); be a non{constant polynomial over K, and { : R! C a character of R, the group of units of R (We formally put {(0) = 0) To these data one associates Igusa's local zeta function, which is the meromorphic continuation to C of Z { (s) = Z R n {(ac f(x))jf(x)j s jdxj for Re s > 0, where jdxj denotes the Haar measure on K n, normalized such that R n has measure 1 Igusa [I] showed that it is a rational function of q s (02) There is a formula for Z { (s) in terms of an embedded resolution (X; h) of f 1 f0g in A n, see Theorem 131; in particular we can describe its poles as follows Let E j ; j 2 T; be the (reduced) irreducible components of h 1 (f 1 f0g), and let N j and j 1 be the multiplicities of E j in the divisor of respectively f h and h (dx 1 ^ ^ dx n ) on X Then [I, (II)Theorem 1] all poles of Z { (s) are among the values s = j N j + 2k N j log q i with k 2 Z and j 2 T such that the order of { divides N j 1

2 (03) So when the order of { divides no N j at all then Z { (s) is holomorphic on C Now the N j are not intrinsically associated to f 1 f0g; but the order (as root of unity) of any eigenvalue of the local monodromy on f 1 f0g divides some N j, and those eigenvalues are intrinsic invariants of f 1 f0g (see 14) This observation inspired Denef [D2, Conjecture 442] to propose the following 04 Conjecture If f(x) is dened over a number eld F C, then for almost all completions K of F (ie for all except a nite number) we have the following : if the order of { doesn't divide the order of any eigenvalue of the (complex) local monodromy of f at any complex point of f 1 f0g, the Z { (s) is holomorphic on C (05) Denef showed that this conjecture is true for the relative invariants of a few prehomogeneous vector spaces In this paper we actually prove the conjecture for curves, ie for n = 2 and any f 2 F [x 1 ; x 2 ] 1 Explicit formulas (11) From now on we suppose n = 2 and we x f(x; y) 2 F [x; y] n F for some number eld F We will state the case n = 2 of general formulas for Z { (s) and for eigenvalues of monodromy on f 1 f0g in terms of an embedded resolution (12) Let (X; h) be the canonical embedded resolution (with normal crossings) of f 1 f0g in A 2 (C ); so in particular h : X! A 2 is a nite succession of (point{centered) blowing-ups, and no `unnecessary' blowing{ups occur We denote by E i ; i 2 T = T e [ T s ; the (reduced) irreducible components of h 1 (f 1 f0g), where E i is an exceptional curve for i 2 T e and an irreducible component of the strict transform of f 1 f0g for i 2 T s For each i 2 T let N i and i 1 be the multiplicities of E i in the divisor of respectively f h and h (dx ^ dy) on X The (N i ; i ) are called the numerical data of the resolution (X; h) We have N i ; i > 1 and if f 1 f0g is reduced, then (N i ; i ) = (1; 1) for i 2 T s Let also E i := E i n [ j6=i E j for i 2 T (13) Let K be the completion of F with respect to some maximal ideal of its ring of integers In order to state Theorem 131 we also consider the scheme{theoretical canonical embedded resolution (X K ; h K ) of f 1 f0g in A 2 (K), which is entirely dened over K Let Ei K; i 2 T K ; denote the (reduced) K{irreducible components of (h K ) 1 (f 1 f0g), and dene Te K ; Ts K ; E K i as above We have in particular that X K K C = X and that each Ei K K C = [ j2si E j, where all E j ; j 2 S i ; are mutually disjoint and have the same numerical data which we can therefore associate to Ei K and denote by (N i ; i ) (If the 2

3 resolution (X; h) is entirely dened over K there is a natural bijection E K i! E i = E K i KC and there is no need to consider X K For example we can choose F `big enough' such that (X; h) is already dened over F itself) 131 Theorem [D1, Theorems 21 and 22] For almost all completions K of F (ie for all except a nite number) we have the following (i) If { is not trivial on 1 + P, then Z { (s) is constant on C (ii) If { is trivial on 1 + P, then q 2 Z { (s) = c ; + X i2t K djn i c fig q 1 q i+sn i 1 + X fi;jgt K djn i ;djn j c fi;jg (q 1) 2 (q i+sn i 1)(q j +sn j 1) ; where d is the order of { and the c I ; I T K ; are constants depending on { and on the K{rational points on the reduction mod P of \i2i E K i In particular c fi;jg = 0 if E K i \ E K j = ; For a complete denition and a cohomological interpretation of those numbers c I we refer to [D1, xx2-3] or [D2, Theorem 34 and 35] We will only need the following vanishing property of some c fig We denote by () the complex Euler{Poincare characteristic 132 Theorem [D1, Theorem 11] Let d denote the order of { For almost all completions K of F we have the following Fix one i 2 T K with djn i and such that Ei K intersects no other E K j with djn j If ( E K i K C ) = 0 then c fig = 0 in Theorem 131(ii) (14) We now remind the denition of local monodromy [M] Fix b 2 C 2 with f(b) = 0 Let B C 2 be a small enough ball with center b; the restriction fj B is a locally trivial C 1 bration over a small enough pointed disc D C n f0g with center 0 Hence the dieomorphism type of the Milnor ber F b := f 1 ftg \ B of f around b does not depend on t 2 D, and the counterclockwise generator of the fundamental group of D induces an automorphism of H (F b ; C ) which is called the local monodromy of f at b 141 Theorem [A, Theorem 3] For b 2 f 1 f0g let P i (t) denote the characteristic polynomial of the monodromy action on H i (F b ; C ) for i = 0; 1 Then P 0 (t) Y P 1 (t) = (t N i 1) ( Ei \h 1fbg) : i2t (142) In particular if b is a smooth point of f 1 f0g red, thus h 1 (b) 2 E i0 for some i 0 2 T s, then H 1 (F b ; C ) = 0 and P 0 (t) = t N i 0 1 3

4 And if b is a singular point of f 1 f0g red, then Y P 1 (t) = P 0 (t) (t N i 1) ( Ei ) ; i2t 0 e where T 0 e T e ranges over all exceptional curves E i occurring in the local resolution of b, ie such that h(e i ) = fbg 143 Remark If f 1 f0g is reduced then P 0 (t) = t 1 for all b 2 f 1 f0g 2 Concrete eigenvalues An important point in our proof of Conjecture 04 for curves is the knowledge of concrete eigenvalues of monodromy, determined in Proposition 24 We rst state some standard facts about the (local) canonical embedded resolution of one singular point, and a congruence theorem between numerical data, which we frequently use in its proof (21) Let b be a singular point of f 1 f0g red and denote T 0 e := fi 2 T e jb 2 h(e i )g (i) We have that [ i2t 0 e E i is connex (ii) For intersecting E i and E j ; fi; jg T 0 e; there is an unambiguous chronology between E i and E j with respect to creation in the resolution process; we denote E i E j if E i is created before E j in the resolution process Clearly E i E j, N i < N j (22) For any i 2 T let k i denote the number of intersection points of E i with the other components of h 1 (f 1 f0g) Remark that ( E i ) = 2 k i Theorem [L, Lemme II2] Fix one exceptional curve E i, intersecting k i components E 1 ; : : : ; E ki Then times other Xk i j=1 N j 0 mod N i : (For a short conceptual proof and generalizations, see [V, Proposition 21] or [D2, Lemma 611]) 4

5 23 Lemma Let E 0 be an exceptional curve with k 0 = 1 Then for some r > 1 there exists a unique path : : : E 0 E 1 E 2 E r in the resolution graph consisting entirely of exceptional curves, such that (1) k j = 2 for j = 1; : : : ; r 1; (2) k r > 3; (3) N 0 jn j for all j = 1; : : : ; r; (4) N 0 < N 1 < < N r Remark In a (dual) resolution graph the E i ; i 2 T; are denoted by dots and intersections between them by lines, connecting the associated dots Proof By the algorithm of canonical embedded resolution one easily sees that E 1 must be an exceptional curve and that E 0 E 1 So N 0 < N 1 and furthermore N 0 jn 1 by Theorem 22 Clearly k 1 6= 1, and if k 1 > 3 we are done So suppose k 1 = 2 Then there is exactly one E 2 continuing the path E 0 E 1 E 2 Again by the algorithm of resolution E 2 must be an exceptional curve and E 1 E 2, implying that N 1 < N 2 Now N 1 j(n 0 + N 2 ) by Theorem 22 and we derive that N 0 jn 2 Again we have k 2 6= 1 and if k 2 > 3 we are done By niteness of T the same argument abuts to some E r with k r > 3 24 Proposition Let E j be an exceptional curve with k j > 3 Then e 2i N j of the local monodromy of f at the (singular) point h(e i ) of f 1 f0g is an eigenvalue Proof We use the notations of (14) We claim that X i2t 0 e N j jn i ( E i ) > 0 which implies by 142 that e 2i N j monodromy action at b = h(e j ) is a zero of the characteristic polynomial P 1 (t) of the Since ( E j ) > 1(, k j > 3) this statement is true, except perhaps when at least one exceptional curve E 0 with ( E o ) = 1(, k 0 = 1) and N j jn 0 occurs in h 1 fbg Now by Lemma 23 we can associate to each such curve E 0 a curve E r with ( E r ) > 1 and N j jn r which is dierent from E j since N 0 < N r So our claim could only be false if there 5

6 exist at least two dierent exceptional curves E 0 and E 0 0 with k 0 = k 0 0 = 1, N j jn 0 and N j jn0, 0 for which the respectively associated E r and E 0 r of Lemma 23 are equal In fact 0 by the resolution algorithm this is impossible for more than two exceptional curves E 0 and E0, 0 hence we only have to prove that this situation cannot occur if k r (= k 0 r0) = 3 Now in this case the resolution graph, at the stage of the resolution process when E r (= E 0 r0) is just created, looks like E r = : : : E0 r 0 : : : E 0 E 1 E 2 E r 1 E 0 r 0 1 E 0 1 E 0 0 ~E where ~ E is the (possibly reducible) strict transform of f 1 f0g Since [ i2t 0 e E i is connex, no other exceptional curves E`; ` 2 T 0 e; can occur at that stage (ie ~ E intersects only Er ); so in particular E j is created after this stage and N j > N r (> N 0 ), contradicting that N j jn 0 3 Holomorphy conjecture for curves 31 Theorem Let f(x; y) 2 F [x; y] for some number eld F For almost all completions K of F we have the following If the order of doesn't divide the order of any eigenvalue of the (complex) local monodromy of f at any complex point of f 1 f0g, then Z { (s) is holomorphic on C Proof Let d denote the order of {, which we may suppose trivial on 1 + P by Theorem 131(i) We will prove that Z { (s) = q 2 c ; in the expression of 131(ii) By Proposition 24 and Lemma 23 it is clear that d cannot divide any N i ; i 2 T e ; associated to E i with k i > 3 or k i = 1 We have furthermore that d - N i ; i 2 T s ; since by (142) those N i are orders of monodromy eigenvalues We conclude that (eventually) djn i only for exceptional curves E i with k i = 2 We x such an exceptional curve E i with djn i, intersecting say E 1 and E 0 1 Suppose that djn 1 ; then E 1 too is an exceptional curve with k 1 = 2, intersecting also say E 2 Now by Theorem 22, applied to E 1, we have that djn 2 Continuing the same argument contradicts the niteness of the resolution graph So d - N 1 and, analogously or by Theorem 22, also d - N 0 1 6

7 E K i Now the complex decomposition E K i K C = [ j2si E j for i 2 T K e induces in fact that K C = [ j2siej and moreover all E j ; j 2 S i ; are mutually isomorphic So we can associate k i := k j ; j 2 S i ; to Ei K and formula X 131(ii) for Z { (s) reduces to q 1 q 2 Z { (s) = c ; + c fig q i+sn i 1 : i2t K e k i =2;djN i Since ( E K i K C ) = P j2s i ( E j ) = 0 for all appearing i 2 T K e, Theorem 132 yields the announced result 32 Example Let F = Q and f(x; y) = y 5 (y x) 2 + x 9 Its resolution (X; h) is entirely dened over Q with associated resolution graph and numerical data E 1 E 5 E 6 E 4 E 3 i E 2 E 0 N i where E 0 is the strict transform of f 1 f0g and E 1 ; : : : ; E 6 are the exceptional curves By (14) the characteristic polynomials of monodromy are t 1 and (t 1)(t 45 1); so its eigenvalues have orders 1,3,5,9,15 and 45 In particular when the order of { is 2,7,13,18 or 26 we have that Z { (s) is constant on C (for almost all completions Q p of Q) This example illustrates the fact that we really need Theorem 132 in our proof of the holomorphy conjecture for curves, for there can exist exceptional curves E i such that N i doesn't divide the order of any eigenvalue of monodromy (33) One can (for arbitrary n) extend the denition of Z { (s) as follows Let : K n! C be a Schwartz-Bruhat function, ie a locally constant function with compact support Then Z Z ;{(s) = (x){(ac f(x))jf(x)j s jdxj ; K n so taking for the characteristic function of R n yields the classical denition When is residual, meaning that Supp R n and (x) only depends on x mod P, Theorems 131 and 132 remain valid with analogous constants c I ; I T K [D2, Theorem 34] So our proof of the holomorphy conjecture for curves generalizes to Z ;{(s) for residual 34 Remark Denef and Loeser [DL, x3] associate to f 2 C [x 1 ; : : : ; x n ] and r 2 N n f0g the so{called topological zeta function Z (r) top(s) For n = 2 we have that Z (r) top(s) = (A 2 n f 1 f0g) + X i2t rjn i ( E i ) i + sn i + 7 X fi;jgt rjn i ;rjn j (E i \ E j ) ( i + sn i )( j + sn j ) :

8 The holomorphy conjecture for Z (r) top(s) can be formulated as follows If d 2 N nf0g doesn't divide the order of any eigenvalue of the local monodromy of f at any point of f 1 f0g, then Z (r) top(s) is holomorphic on C for all r such that djr Our proof of 31 remains valid for Z (r) top(s) since for i 2 T e we have that k i = 2, ( E i ) = 0 References [A] N A'Campo, La fonction zeta d'une monodromie, Comment Math Helv 50 (1975), 233{248 [D1] J Denef, Local zeta functions and Euler characteristics, Duke Math J 63 (1991), 713{721 [D2] J Denef, Report on Igusa's local zeta function, Sem Bourbaki 741, Asterisque 201/202/203 (1991), 359{386 [DL] J Denef and F Loeser, Caracteristiques d'euler{poincare, Fonctions Zeta locales, et modications analytiques, J Amer Math Soc (to appear) [I] J Igusa, Complex powers and asymptotic expansions I, J Reine Angew Math 268/269 (1974), 110{130; II, ibid 278/279 (1975), 307{321 [L] F Loeser, Fonctions d'igusa p{adiques et polyn^omes de Bernstein, Amer J Math 110 (1988), 1{21 [M] J Milnor, Singular points of complex hypersurfaces, Princeton Univ Press, 1968 [V] W Veys, Congruences for numerical data of an embedded resolution, Compositio Math 80 (1991), 151{169 KULeuven, Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium Current address: IHES, 35 Route de Chartres, F Bures-sur-Yvette, France address: veys@frihes61bitnet 8