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MEROMORPHIC L-FUNCTIONS

Meromorphic continuation of L-functions of p-adic representations By DAQING WAN*

1. Introduction

Let Fq be the nite eld of q elements of characteristic p. Let p be the completion of an algebraic closure of the p-adic rational numbers Qp . Let Rp be the ring of integers in a nite extension of Qp contained in p. Suppose that B (X ) is an r  r matrix (aij (X )) whose entries aij (X ) are elements of the convergent power series ring over Rp :

RpffX gg = Rp ffX1 ;


; Xn gg = f

X

u2Nn

auX u jau 2 Rp ; julim a = 0g; j!1 u

where N is the set of non-negative integers, X u = X1u


Xnun and juj = u1 +
P

+ un. Alternatively, the matrix B (X ) can be written as a power series bu X u with coecients in the (non-commutative) ring of r  r matrices Mrr (Rp ). We can then de ne the L-function of B (X ) on the ane n-space An=Fq as follow: (1:1) Y 1 L(B=An ; T ) = ; dx? d ( x ) q )


B (xq )B (x)) closed points x2An =Fq det(I ? T B (x where x is the Teichmuller lifting of x and d(x) denotes the degree of x over Fq . This in nite product is a well de ned power series with coecients in Rp because there are only nitely many closed points of a given degree. If V=Fq is an ane variety contained in An , then one de nes L(B=V; T ) to be the product in (1.1) except that we now restrict x to run only over the closed points of V=Fq . Using a standard reduction argument and Dwork's analytic construction of additive characters, one sees that the more general L-function L(B=V; T ) can be written as an L-function on the ane space An as de ned in (1.1) for a larger n and a dierent matrix B (X ). Thus we will be interested 1

( ) 1

*This work was partially supported by NSF and the UNLV Faculty Development Leave. The author wishes to thank P. Deligne for emphasizing the importance of uniform variation of L-functions, N. Katz for explaining the concrete meaning of F-crystals, and the Institute For Advanced Study for its hospitality. The author would also like to thank B. Dwork for helpful comments on an earlier version of the manuscript and for explaining to me two new shorter proofs of Theorem 1.2.

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only in L(B=An ; T ) in this paper, except for a brief discussion of the more general case at the end of section 2. The general L-function in (1.1) was rst investigated by Dwork [9-10]. If B (X ) is invertible in Mrr (RpffX gg Qp ), the analytic matrix B de nes (the Frobenius matrix of) an F-crystal on the ane space An =Fq (though not all F-crystals arise this way), see Katz [18]. The L-function in (1.1) is then the Katz L-function of the corresponding F-crystal on An . In the case that B is invertible in Mrr (Rp ffX gg) (the so-called unit root F-crystal), the functorial equivalence between unit root F-crystals and p-adic representations of 1 shows that the analytic matrix B (X ) de nes a continuous p-adic representation of the arithmetic fundamental group:

 : 1 (An =Fq ) ?! GLr (Rp): Equivalently, the representation  de nes a p-adic etale sheaf on An =Fq . In algebraic term, the fundamental group 1 (An =Fq ) is the Galois group of a separable closure of the function eld Fq (t1 ;


; tn ) of An =Fq modulo the inertia groups at the closed points of An =Fq . In the case that B is invertible in Mrr (Rp ffX gg), the L-function in (1.1) is the Artin L-function L(; T ) of the corresponding p-adic representation or the Grothendieck L-function of the corresponding p-adic etale sheaf. These L-functions are extremely general and

include all classical zeta functions and complex L-functions of algebraic varieties over nite elds as a very special case [8][17]. They naturally arise from Dwork's study [9-10] of p-adic analytic variation of a family of zeta functions over nite elds and Katz's [18] more general work on F-crystals. Based on the work of Dwork, Reich [23], Monsky-Washnitzer [20-21] on such L-functions, Katz made the following conjecture (Conjecture 6.1.1 in Katz [18], see also Crew [6] and Dwork [10, p53]): Katz Conjecture. The L-function L(B=An ; T ) of an F-crystal is a p-adic meromorphic function. Essentially, the conjecture was only known to be true in the so-called overconvergent case or more general for those B (X ) which decays faster than the logarithm function. Dwork's proof in several non-trivial cases [9-11] reduces again to the overconvergent case by using a dierent lifting of the Frobenius map (the so-called excellent lifting or canonical lifting). Unfortunately, excellent liftings rarely exist for general F-crystals. According to Dwork-Ogus [12, p113], the rst such example seems to be given by Mumford in a letter to Dwork (1972). Later, Sperber [27] showed that in the case of higher dimensional Kloosterman sums there does not exist any excellent lifting for the unit root F-crystal. Dwork-Ogus [12] and Oort-Sekiguchi [22] independently proved that under a mild assumption, a generic ordinary curve has no excellent

MEROMORPHIC L-FUNCTIONS

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lifting. As Dwork pointed out, excellent liftings may exist if one introduces more parameters. In fact, Dwork [10, p56] conjectured the existence of excellent liftings in this broader sense for a unit root F-crystal B (X ) which arises as the unit root part of certain overconvergent F-crystals. However, very little is known about Dwork's conjecture. Even if one assumes Dwork's conjecture, one gets the meromorphic continuation of the L-function only for those unit root F-crystals which come from certain overconvergent F-crystals. Most unit root F-crystals do not come from any overconvergent F-crystals. According to Dwork, Adolphson (around 1975, unpublished) found an example with the property that the trace of the related Dwork operator is not well de ned (more precisely, it is de ned but depends on the basis chosen) and thus the related Dwork operator is not compact. This suggests some evidence that Katz's conjecture might be false in general. However, even if an operator is not compact, it is conceivable that the Fredholm determinant (de ned with respect to a xed basis) is still meromorphic (not necessarily entire). Thus, as far as we can see, Adolphson's work does not disprove Katz's conjecture. Recently, Dwork-Sperber [13] proved the meromorphic continuation in a non-trivial disk for those B (X ) with logarithm decay. Our purpose of this paper is to use the Dwork trace formula to study the p-adic meromorphic continuation of the function L(B=An ; T ). We show that a weaker form of the Katz conjecture is true but the full form is false. To describe the results, let us write X u (1:2) B (X ) = buX ; bu 2 Mrr (Rp ): u2Nn De ne q (bu ) ; (1:3) h(B ) = julim inf ord logq juj j!1 where ordq (bu ) is the least rational number s such that qs divides all entries of bu . Since ordq (bu ) is non-negative, it is clear that 0  h(B )  1. We have the following results. THEOREM 1.1. The L-function L(B=An ; T ) is a p-adic meromorphic function in the open disk ordq (T ) > ?h(B ). In particular, if h(B ) = 1, then the L-function L(B=An; T ) is a p-adic meromorphic function in the entire p-adic plane. This result is not new. It is in fact a consequence of a more general theorem of Dwork-Sperber [13] on the compliment of a hypersurface (with a mild assumption on the hypersurface), except that in [13] a step function is used instead of the much simpler logarithm function. We shall also prove a strengthened form of Theorem 1.1 for a family of L-functions, see De nition 3.7 and Theorem 5.2 for the precise formulation. Roughly speaking, if we have

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DAQING WAN

P

a family of power series B (X; y) = bu (y)X u parameterized by y varying in some metric space Y and if each bu (y) is continuous in y such that q (bu (y))  ; (1:4) lim inf inf y ord logq juj juj!1 for some real number , then the family of L-functions L(B (X; y)=An ; T ) is uniformly meromorphic in y on the closed disk ordq (T ) > ?( ? ) for any  > 0. Furthermore, the zeroes and poles of the family L(B (X; y)=An ; T )

ow continuously in y on the closed disk ordq (T ) > ?( ? ) for any  > 0. This type of uniform results is the basis in [28] to prove the full strength of some conjectures of Goss [15] on global L-series of Drinfeld modules. It is expected that uniform results would also play an important role in the author's future investigation on Dwork's unit root zeta function [10]. The importance of uniform variation was rst pointed out to me by P. Deligne. The main new result is the following counter-example which shows that Theorem 1.1 is best possible in some sense. THEOREM 1.2. Let r = n = 1. For a positive integer h, let X B (X ) = 1 + qhu+1 cuX (qu ?1) ; cu 2 Rp: u>0

P Assume that the power series u>0 cu T u modulo the maximal ideal of Rp is not

a rational function over its residue eld (i.e., the coecients cu do not satisfy any linear recurrence relation in the residue eld), then the L-function L(B=A; T ) is meromorphic in the open disk ordq (T ) > ?h but not meromorphic on the closed disk ordq (T )  ?h .

To describe the nal result, recall that the degree of a rational function is the degree of its numerator minus the degree of its denominator while the total degree of a rational function is the degree of its numerator plus the degree of its denominator. P THEOREM 1.3. If B (X ) = u bu X u is a polynomial in X with coecients in Mrr (Rp ) of degree d, then the L-function L(B=An ; T ) is a rational function in

T whose degree is the negative of the degree of det(I ? Tb0 ) and whose total degree is at most

r

 n   X n n ? i + [d=(q ? 1)] i=0

i

n

:

Remark. In his p-adic theory, Dwork originally treated the overconvergent case, namely, those matrix B (X ) such that limjuj!1 inf ordjqu(jbu ) > 0. Dwork's n

overconvergent result on the ane space A was generalized to more general varieties (still overconvergent matrix) by Reich [23], Monsky [21] and Boyarsky

MEROMORPHIC L-FUNCTIONS

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[5]. Note that the hypothesis h(B ) = 1 in Theorem 1.1 is much weaker than the overconvergent hypothesis. If B (X ) with h(B ) = 1 is not a polynomial, then L(B=An ; T ) is meromorphic but not rational in general. If, in addition, the Euler factors at all closed points x 2 An =Fq have coecients which can be embedded in the ring of integers of a xed number eld, then the L-function L(B=An ; T ) can be viewed as a complex function. If this complex function has non-trivial radius of convergence, then the Borel lemma as used in Dwork's [7] rationality proof shows that in fact L(B=An ; T ) is a rational function over the complex numbers. In this case, the degree and total degree of the rational function L(B=An; T ) can be estimated by Bombieri's methods [3-4]. The bound for the total degree in Theorem 1.3 is best possible in general. However, if one takes into account which terms actually occur in the polynomial expansion of B (X ), then Theorem 1.3 can be improved in some cases using Adolphson-Sperber's [1] improvement of Bombieri's methods. The rationality result in Theorem 1.3 is natural and very useful in function eld case [28]. However, in our present p-adic case, we note that the rationality result in Theorem 1.3 does not include the rationality of the classical L-function of a nite character of 1 (An =Fq). It would be interesting to have a more general rationality theorem which includes both the nite character case and the polynomial case treated in Theorem 1.3. Even though Theorem 1.1 can be derived from a more general theorem of Dwork-Sperber [13], we shall include a self-contained treatment here for several reasons. First, the special ane space case treated here can be proved in a signi cantly simpler way. Second, this special ane space case is sucient for applications to function elds even in the most general case [28]. Third, since we also need a stronger family version of Theorem 1.1, it is by going through the known explicit proof of Theorem 1.1 that one sees the uniform variation of L-functions. The proof of Theorem 1.2 is a little more subtle. It requires a delicate precise estimate for the coecients of the L-function. Two shorter variations of the proof of Theorem 1.2 were kindly communicated to me by Dwork. However, we decide to use our original detailed proof. For simplicity, as in [29] we shall directly work with (in nite) matrices instead of operators on p-adic Banach spaces. This is sucient for meromorphic continuation, which is our main purpose here. However, to study deeper properties such as cohomological formula for L-functions (see Conjecture 6.1.2 in Katz [18]), one needs to develop a p-adic spectral theory as in Serre [26]. It is well known that Serre's theory applies in the case h(B ) = 1. Thus it is conceivable that the Dwork-Monsky-Washnitzer overconvergent theory extends to the much broader case h(B ) = 1. For most general B , our counterexample shows that the best one can hope for is a cohomological formula which accounts for the zeros and poles of the L-function in the open disk ordq (T ) > ?h(B ).

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We feel that such an optimal theory to be plausible in some sense and it would apparently have useful applications in Katz's [19] theory of overconvergent padic modular forms, see also Gouvea [16]. A preliminary evidence is obtained in [30], where it is shown that the related subring of the power series ring is Noetherian, generalizing the result of Fulton [14]. Most results in this paper remain valid if one replaces the p-adic ring Rp by any complete discrete valuation ring of characteristic zero or characteristic p (the same p). This generalization is described in the last section. The characteristic p version of our results will be used in a future paper [28] to prove some conjectures of Goss on L-functions of Drinfeld modules and t-motives. There is a striking analogue between Dwork's p-adic theory for L-functions of F-crystals over a non-archimedean eld and Ruelle's [25] theory for dynamic zeta functions over an archimedean eld. In fact, both theories essentially employ the same type of techniques, such as transfer operators, spectral theory and De Rham type cohomology. One can sometimes transport both results and proofs to each other. In view of the connection of the Ruelle zeta function with dynamic system and statistical mechanics [24], it might be natural and interesting to reformulate Dwork's theory as a theory of p-adic dynamic system. Such reformulation of Dwork's powerful results may have fruitful applications in statistical mechanics.

2. Reduction to torus Gnm For a subset S  f1; 2;


; ng (possibly empty), let BS (X ) be the analytic matrix obtained from B (X ) by setting xi = 0 for all i 62 S . The matrix B (X ) can also be viewed as an analytic matrix on the torus Gnm and we may follow

(1.1) to de ne the L-function (2:1) Y L(B=Gnm ; T ) =

1

: det(I ? T d(x) B (xqd(x)?1 )


B (xq )B (x))

closed points x2Gnm =Fq De ne L(BS =GjmS j ; T ) in a similar way (2:2) L(B=An ; T ) =

for each S . One checks easily that L(BS =GjmSj; T ):

Y

S f1;2;


;ng

Then Theorems 1.1-1.2 are reduced to the corresponding Theorems 2.1-2.2 for torus Gnm described below. Let X u (2:3) B (X ) = buX ; bu 2 Mrr (Rp ); n u2Z where u runs over all lattice points in Zn and juj = ju1 j +


+ jun j. Thus, the matrix B (X ) is now a Laurent series with coecients bu in Mrr (Rp ) and we

MEROMORPHIC L-FUNCTIONS

7

are in a slightly more general situation. De ne q (bu ) (2:4) h(B ) = julim inf ord logq juj : j!1 Since ordq (bu ) is non-negative, it is clear that 0  h(B )  1 and h(BS )  h(B ) for all subsets S of f1; 2;


; ng. THEOREM 2.1. The L-function L(B=Gnm ; T ) is a p-adic meromorphic function in the open disk ordq (T ) > ?h(B ). THEOREM 2.2. Let r = n = 1. For a positive integer h, let X B (X ) = 1 + qhu+1 cu X (qu ?1) ; cu 2 Rp: u>0 P Assume that the power series c T u modulo the maximal ideal of R

p is not a u>0 u rational function over its residue eld, then the L-function L(B=Gm ; T ) can not be extended to a p-adic meromorphic function on the closed disk ordq (T )  ?h.

P

THEOREM 2.3. If B (X ) = bu X u is a polynomial in X with coecients in Mrr (Rp ) of degree d, then the L-function L(B=Gnm ; T ) is a rational function in

T whose degree is zero and whose total degree is at most   r2n n + [d=n(q ? 1)] :

In the more general case case that B (X ) is a Laurent polynomial in X , the L-function L(B=Gnm ; T ) is still a rational function whose degree is zero and whose total degree can be estimated using the method of Adolphson-Sperber [1]. By (2.2), it is clear that Theorem 2.1 implies Theorem 1.1. For Theorem 1.2, one checks that L(B=A; T ) = det(I ? TB (0))(?1) L(B=Gm ; T ) and thus L(B=Gm ; T ) is meromorphic in a given disk if and only if L(B=A; T ) is meromorphic in the same disk. In the case of Theorem 1.3, for a nonempty subset S 2 f1; 2;


; ng the L-function L(BS =GjmS j; T ) has degree zero by Theorem 2.3; while for the empty set S = , the L-function L(BS =GjmS j; T ) is just the Euler factor det(I ? TB (0))(?1) = det(I ? Tb0 )(?1) . The bound for the total degree in Theorem 1.3 can not be reduced to the (larger) bound for the total degree in Theorem 2.3. It follows, instead, from a slightly more subtle trace formula (see the end of section 5 for the proof). Remark. If B (X ) is the rank one matrix corresponding to the overconvergent power series for the exponential sums associated to a polynomial of one variable, then the L-function is a rational function whose degree is not zero in general. Comparing with Theorem 2.3, one deduces the observation

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that Dwork's analytic formula for additive characters can not be replaced by a polynomial. Now, we brie y discuss the more general L-function L(B=V; T ), where B (X ) is the matrix given in (1.2) and V is an ane (or quasi-ane) subvariety in An =Fq . We emphasis that our L(B=V; T ) is not the most general form. We assumed that B (X ) is a global convergent power series as in (1.2). One can de ne L(B=V; T ) for a more general Krasner analytic matrix B (X ) on V , which in general can not be written as a global convergent power series in (1.2). The study of such more general L-functions needs the more general Reich-Monsky trace formula. We have essentially restricted ourself to the simplest torus and ane space cases in this paper. THEOREM 2.4. Assume that B (X ) is a convergent power series as in (1.2). Then, the L-function L(B=V; T ) is a p-adic meromorphic function in the open disk ordq (T ) > ?h(B ). Clearly, we may assume that V is ane. Let be a non-trivial additive character of Fq . Let V be de ned by the equations g1 (X ) =


= gk (X ) = 0. Dwork's analytic formula for an additive character shows that there is an overconvergent power series (X; Z1 ;


; Zk ) such that





L(B=V; T ) = L B (X )(X; Z1 ;


; Zk )=An+k ; T=qk : Applying Theorem 1.1, one gets the meromorphic continuation of L(B=V; T ) only in the smaller disk ordq (T ) > ?h(B ) + k because of the shifting factor 1=qk . This is weaker than the expected bound. If one uses the Dwork trace formula for the ane space An (see Lemma 4.3) which is a little bit more subtle than the simplest torus case, then it can be proved with a little more work that L(B=V; T ) is actually meromorphic in the expected disk ordq (T ) > ?h(B ).

3. Trace and Fredholm Determinant The trace and determinant of an in nite matrix can be de ned in usual way as long as everything is convergent. To be self-contained and rigorous, we recall precisely what we mean by trace and Fredholm determinant. De nition 3.1. Let M = (au;v ) be an in nite matrix with entries in the quotient eld of Rp, where the row number u and the column number v run over an ordered countable set which we identify as the set of positive integers. (i). If limu!1 au;u = 0, we de ne the trace of M to be the sum (3:1)

tr(M ) =

1 X u=1

au;u:

MEROMORPHIC L-FUNCTIONS

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(ii). If for each integer k  1, the following sum is convergent, (3:2)

k = (?1)k

X u;

sign()au ;(u )


auk ;(uk ) ; 1

1

where the summation is over all ui 's with 1  u1 <


< uk and all permutations  of the ui 's, then we de ne the Fredholm determinant of M to be the power series (3:3)

det(I ? TM ) = 1 +

1 X k=1

k T k :

Note that for a xed k, the sum k is convergent if and only if (3:4)

lim a


auk ;(uk ) = 0 uk !1 u1 ;(u1 )

uniformly for all 1  u1 < u2 <


< uk and all permutations  of fu1 ; : : : ; uk g. Since we are in non-archimedean situation, convergence of an in nite series is independent of the order of summation. Let Y = (Yu;v ) (1  u; v < +1) be an in nity matrix all of whose coecients are indeterminates. For each positive integer k, one can de ne the power Y k of Y with the usual multiplication rule. The coecients of Y k are formal power series in the variables Yu;v . De ne tr(Y ) and k (Y ) as in De nition 3.1 with au;v replaced by Yu;v . Then tr(Y ) and k (Y ) are also formal power series in the variables Yu;v . It is clear that tr(M ) (resp. k (M )) is de ned if and only if tr(Y ) (resp. k (Y ) ) is convergent at the specialization Yu;v = au;v . We rst prove a well known simple lemma, which is essentially a universal version of the classical Newton formula. LEMMA 3.2. Let Y = (Yu;v ). Then we have the following identity of formal power series

(3:5)

det(I ? TY ) = exp(?

1 tr(Y k ) X k=1

k k T ):

Proof. It is well-known that (3.5) is true for all nite matrices. Let Y (m) be

the specialization of Y such that Yu;v = 0 for all u and v satisfying max(u; v) > m. Then (3.5) holds for all Y (m). Let k (Y ) be the coecient of T k in the expansion of the right side of (3.5) as a formal power series. We want to show that (3:6)

k (Y ) = k (Y ) (k = 1; 2; : : :):

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The terms of k (Y ) and k (Y ) are monomials of degree k in the Yu;v 's. Since (3.6) holds for all specializations Y = Y (m), it follows that (3.6) holds in general. COROLLARY 3.3. Let Sk be tr(Y k ) and k (Y ) be de ned as above. Then for all positive integers m, the expression m (Y ) is a polynomial of the Si 's (1  i  m) with rational coecients. Similarly, for all positive integers m, the expression Sm is a polynomial of the i (Y ) (1  i  m) with rational coecients. More precisely, we have

X

i1

(?1)i +


+im i !


iS1 !1
i

2Si m


mim ; 1 m i +2i +


+mim =m    i


 im X i +


+ i S 1 m m m : i +


+ i 1 m (?1) (3:8) m = i ; : : : ; i i +


+ im 1 m 1 i +2i +


+mim =m (3:7) m =

1

im

1

1

2

2

1

1

1

2

Proof. By (3.5), we have 1 X

(3:9)

1+

k=1

k

Tk =

1 (?1)m X 1 S X k km m! ( k T ) :

m=0

k=1

The rst formula follows from the power series expansion of (3.9). To prove (3.8), we take the logarithm on both sides of (3.5) and deduce that 1 S X k k=1

(3:10)

k

X T k = (?1) log(1 + m T m) m=1 1 (?1)d X 1 X = ( m tm )d : d m=1 d=1 1

The second formula follows from the power series expansion of (3.10). COROLLARY 3.4. Let M = (au;v ) be an in nite matrix with entries in the quotient eld of Rp . Then det(I ? TM ) is de ned if and only if tr(M k ) is de ned for all positive integers k. In which case, we have 1 tr(M k ) X

(3:11)

det(I ? TM ) = exp(?

k=1

k k T ):

Proof. det(I ? TM ) is de ned if and only if k (M ) is de ned for all k  0. By Corollary 3.3, this is true if and only if all Sk (M ) = tr(M k ) are de ned. Equation (3.11) is then a consequence of (3.5). The corollary is proved. We now study the meromorphic continuation of the Fredholm determinant.

MEROMORPHIC L-FUNCTIONS

11

De nition 3.5. Let h be a xed real number. We de ne Mh to be the set

of in nite matrices M = (au;v ) such that the entries of M are bounded in the quotient eld of Rp and almost all rows of M are divisible by qh? for every  > 0. More precisely, the matrix M is in Mh if ordq (M ) = inf u;v ordq (au;v ) is bounded from below and for any given  > 0, there is an integer N > 0 such that for all u > N and all v, the following inequality holds: (3:12) ordq (au;v )  h ? : The next result gives an estimate for the radius of convergence of the Fredholm determinant det(I ? TM ). PROPOSITION 3.6. Let M = (au;v ) 2 Mh . Assume that the Fredholm determinant det(I ? TM ) is well de ned. Then det(I ? TM ) is p-adic analytic in the open disk ordq (T ) > ?h. Proof. For any given  > 0, there is an integer N > 0 such that (3.12) holds. By de nition, det(I ? TM ) = 1 + where

m = (?1)m

X u;

1 X

m=1

m T m;

sign()au ;(u )


aum ;(um ) : 1

1

Since the ui are distinct, it follows that for m > N , ordq (m )  inf fordq (au ;(u ) ) +


+ ordq (aum ;(um ) )g u;  Nordq (M ) + (m ? N)(h ? ): This inequality implies that lim inf ordq (m )  h ? : 1

m!1

1

m

We have proved that det(I ? TM ) is p-adic analytic in the open disk ordq (T ) > ?h + . Since  can be taken to be arbitrarily small, the proposition is proved. Next, we consider the variation of a family of Fredholm determinants parameterized by a parameter y in some metric space Y , not necessarily compact. In all applications we have in mind, the space Y will be a subset of the ring of p-adic integers Zp with its p-adic topology. Thus, we shall assume that Y is a subset of Zp with its induced p-adic topology. De nition 3.7. (i). Let h be a xed real number. A family of in nite matrices M (y) = (au;v (y)) is called uniformly in Mh if the entries of M (y) are uniformly bounded in the quotient eld of Rp and almost all rows of M (y) are

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uniformly divisible by qh? for every  > 0. More precisely, the family M (y) is uniformly in Mh if ordq (M (y)) = inf u;v;y ordq (au;v (y)) is bounded from below and for any given  > 0, there is an integer N > 0 such that for all u > N , all v and all y, the following inequality holds: (3:13) ordq (au;v (y))  h ? : P (ii). Let f (y; T ) = m0 fm (y)T m be a family of power series, where each fm(y) is a function from Y to Rp . The family f (y; T ) is called continuous if each coecient fm (y) is a continuous function. For a real number , the family of functions f (y; T ) is called uniformly analytic (or entire) on the closed disk ordq (T )   if (3:14) lim inf(inf ordq (fm (y)) + m) = 1: m!1 y In terms of the Gauss norm jjf (y; T )jj = supm jfm (y)jq , this means that the maximum value jjf (y; q T )jj of f (y; T ) on the closed disk ordq (T )   is uniformly bounded in y, where j?jq is the absolute value on Rp normalized such that j1=qjq = q. The family f (y; T ) is called uniformly meromorphic on the closed disk ordq (T )   if f (y; T ) can be written as a quotient f1 (y; T )=f2 (y; T ), where both f1 (y; T ) and f2 (y; T ) are uniformly analytic on the closed disk ordq (T )  . It is clear that the product and quotient of two uniformly meromorphic families (parameterized by the same parameter y) are still uniformly meromorphic. (iii). Let f (y; T ) be parameterized by some metric space Y and let  be a real number. Following a terminology used by Goss, we say that the zeroes of the family f (y; T ) ow continuously in y on the closed disk ordq (T )   if the following two conditions hold. (1). For each y, the function f (y; T ) is analytic on the closed disk ordq (T )  . (2). For any  > 0, there is a real number  > 0 such that whenever jy1 ? y2 jq < , we have the inequality jjf (y1; q T ) ? f (y2; q T )jj < : In terms of the Gauss norm, this means that jjf (y; q T )jj is uniformly continuous in y. These two conditions implies that for any given  > 0, there is a real number  > 0, such that whenever jy1 ? y2 jq < , the two functions f (y1; T ) and f (y2 ; T ) have the same number s of zeroes on the closed disk ordq (T )  , which can be arranged as f1 (i);


; s (i)g for 1  i  2 such that jj (1) ? j (2)jq <  for all 1  j  s. We say that the zeroes and poles of the family f (y; T ) ow continuously in y on the closed disk ordq (T )   if f (y; T ) can be written as a quotient f1 (y; T )=f2 (y; T ), where the zeroes of f1(y; T ) and the zeroes of f2 (y; T ) ow continuously in y on the closed disk ordq (T )  . The following result is immediate.

MEROMORPHIC L-FUNCTIONS

13

LEMMA 3.8. Let f (y; T ) be a continuous family of power series. If the family f (y; T ) is uniformly analytic on the closed disk ordq (T )  , then the zeroes of f (y; T ) ow continuously in y on the closed disk ordq (T )  . The converse is also true if Y is compact. The proof of Proposition 3.6 yields the following version of Proposition 3.6 for a family. PROPOSITION 3.9. Let M (y) = (au;v (y)) be a family of in nite matrices which are uniformly in Mh . Assume that the Fredholm determinant det(I ? TM (y)) is well de ned for each y. Then for any  > 0, the family det(I ? TM (y)) is uniformly analytic on the closed disk ordq (T ) > ?(h ? ).

4. Dwork trace formula The L-function L(B; T ) can be de ned in two ways. One is via the Euler product expansion as in (2.1). The other is to use character sums via the exponential function. The second way can be obtained from (2.1) by taking the logarithmic derivative (or just using (3.5)): 1 Tk X n L(B=Gm ; T ) = exp( k Sk (B )); k=1

where Sk (B ) is the character sum X (4:1) Sk (B ) =

x(qk ?1) =1; x2 np

tr(B (xqk? )


B (xq )B (x)); 1

the notation tr denotes the trace of a matrix and xq = (xq1 ;


; xqn ). Similarly, the Dwork trace formula has two forms. One is via the trace of an in nite matrix. The other is via the Fredholm determinant of the same in nite matrix. This in nite matrix is simply FB = (bqu?v )u;v2Zn , called the in nite Frobenius matrix associated with B (X ). Thus, the (u; v) block entry of FB is the r  r matrix bqu?v 2 Mrr (Rp ) from (2.3). Since limjuj!1 bu = 0, one checks immediately that the power FBk is well de ned for each positive integer k. The Dwork trace formula says that the trace of FBk is also de ned and related to the character sum Sk (B ). LEMMA 4.1 (additive form). For each integer k  1, we have (4:2) Sk (B ) = (qk ? 1)n tr(FBk ): Proof. For k = 1, by de nition,

S1 (B ) =

X

x(q?1) =1

tr(

X

u2Zn

bu xu)

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=

X

u2Zn

x q? X n (

= (q ? 1)

(4:3)

X

tr(bu )

1)

=1

xu

tr(b(q?1)u )

u2Zn n = (q ? 1) tr(FB ):

In deriving (4.3), we used the relation X ui  q ? 1; if (q ? 1) divides ui : (4:4) xi = 0; otherwise q? xi(

1)

=1

For k > 1, the proof is similar. Again, by de nition, X Sk (B ) = tr(B (xqk? )


B (xq )B (x)) 1

=

(4:5)

x(qk ?1) =1

X

u(k?1) ;


;u(0) 2Zn = (qk ? 1)n

tr(bu k?


bu ) (

(0)

1)

X

X

xqk? u k? +


+qu 1 (

x(qk ?1) =1

qk?1 u(k?1) +


+qu(1) +u(0) 0 (mod (qk ?1))

1)

(1)

+u(0)

tr(bu k?


bu ); (

(0)

1)

where the i in u(i) is a superscript. Note that all sums above are convergent. To show that (4.5) is actually equal to the number (qk ? 1)n tr(FBk ), one simply writes all lattice point solutions of the equation (4:6) qk?1u(k?1) +


+ qu(1) + u(0) = (qk ? 1)v(k?1) in the form u(k?1) = qv(k?1) ? v(k?2) ; u(k?2) = qv(k?2) ? v(k?3) ; (4:7) .. . u(0) = qv(0) ? v(k?1) : Then X Sk (B ) = (qk ? 1)n tr(bqv k? ?v k?


bqv ?v k? ) k ? n v ;


;v 2Z (4:8) k n k = (q ? 1) tr(FB ) is well de ned and the lemma P is proved. Remark. If B (X ) = u2Nn bu X u , where u runs only over the set of nonnegative integers, then we can simply take the in nite Frobenius matrix FB to be (bqu?v )u;v2Nn . The point is that in this case, for u(k?1) ;


; u(0) ; v(k?1) 2 Nn, equations (4.6) and (4.7) guarantee that all v(i) 2 Nn. Alternatively, if (

(

1)

(0)

1)

(

2)

(0)

(

1)

15

MEROMORPHIC L-FUNCTIONS

one of the lattice points v(i) is not in Nn, then the corresponding term in (4.8) is easily seen to be zero (simply looking at the most \negative" point v(i) ). The above lemma and Corollary 3.4 show that the Fredholm determinant of FB is well de ned. Expanding (qk ? 1)n in (4.2) and using the identity (3.5), one gets the following multiplicative form of the Dwork trace formula. LEMMA 4.2 (multiplicative form). We have (4:9)

L(B=Gnm ; T )(?1)n?1

Equivalently, we have (4:10)

det(I ? TFB ) =

1 Y i=0

=

Yn i=0

n

det(I ? qi TFB )(?1)i ( i ) :

n (L(B=Gnm ; qi T )(?1)n? )( 1

i?1 i ):

+

Note that (4.10) follows from (4.2) by expanding (qk ? 1)?n . These formulas show that the L-function is completely determined by the Fredholm determinant det(I ? TFB ) of the in nite Frobenius matrix FB . If the matrix FB de nes in some sense a compact operator (completely continuous in the sense of Serre [26]) of a certain p-adic Banach space, then the Fredholm determinant is p-adic entire and Lemma 4.2 gives the meromorphic continuation of the L-function to the entire p-adic plane. However, the matrix FB does not in general de ne a compact operator. In the case when h(B ) = 1, it is well known that FB does de ne a compact operator. The Dwork trace formula for the ane space An can be derived from its torus version in (4.9) via a combinatorial argument using the torus decomposition (2.2). Since the ane space version will be used in section 7, we now carry out the deduction. P Let B (X ) = u2Nn bu X u be a power series with coecients in Mrr (Rp ). The in nite Frobenius matrix is then given by (bqu?v )u;v2Nn . For a subset U of f1;


; ng, de ne FBU to be the submatrix (bqu?v ) with u; v 2 Nn such that ui  1 and vi  1 for all i 2 U . De ne FB (U ) to be the submatrix (bqu?v ) of FBU satisfying the further condition that ui = vi = 0 for all i 62 U . If we order the m = 2n subsets of f1;


; ng as U1 ;


; Um in such a way that jUj j  jUj+1j, then it is easy to check that the in nite matrix FB has the following triangular form 0 F (U ) 


 1 BB B 0 1 FB (U2 )


 C FB = B C: .. .. C @ ...


. . A 0 0


FB (Um )

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Thus, (4:11)

det(I ? TFB ) =

Y U 2f1;


;ng

det(I ? TFB (U )):

This decomposition is a special case of the trivial boundary decomposition from Theorem 5.1 in [29]. More general, for a subset V 2 f1;


; ng, we have Y (4:12) det(I ? TFBV ) = det(I ? TFB (U )): V U

If V is the empty set, then (4.12) reduces to (4.11). LEMMA 4.3 (trace formula for the ane space, see (8.4) in [29]). We have (4:13)

L(B=An; T )(?1)n?1

=

Yn Y

i=0 jU j=i

det(I ? qn?i TFBU )(?1)n?i ;

where U denotes a subset of f1;


; ng.

Proof. By the torus decomposition (2.2) and Lemma 4.2, one derives that Y L(B=An; T )(?1)n?1 = L(BS =GjmSj ; T )(?1)n?1 S f1;2;


;ng Y Y det(I ? qjV j TFBS )(?1)n+jSj+jV j : = S V S

Applying (4.11) with FB replaced by FBS , we compute that YY Y det(I ? qjV j TFB (U ))(?1)n L(B=An ; T )(?1)n? =

jS j+jV j

+

1

S V S U S

=

YY V U

det(I ? qjV j TFB (U ))(?1)

n+jV j

P

U;V S (?1)

jS j

:

The above sum of signs over S is (?1)n or zero depending on whether U contains the compliment V of V or not. Applying (4.12), we then conclude that Y L(B=An ; T )(?1)n? = det(I ? qjV j TFBV )(?1)jV j : 1

This proves (4.13).

V

5. Meromorphic continuation and rationality We prove Theorem 2.1, its family version and Theorem 2.3 in this section. We rst prove Theorem 2.1. By Lemma 4.2, it suces to prove the meromorphic continuation of det(I ? TFB ) in the open disk ordq (T ) > ?h(B ). Recall that q (bu ) ; F = (b h = h(B ) = julim inf ord qu?v )u;v2Zn : log juj B j!1 q

MEROMORPHIC L-FUNCTIONS

17

We can assume that h is nite. For a given  > 0, there is an integer N > 0 such that for all juj > N, (5:1)

ordq (bu )  (h ? 2 ) logq juj:

De ne a non-negative weight function w(u) by (5:2)

 (h ?  ) log juj; q 2 w(u) = 0;

if juj > 0 if u = 0.

Let au;v be the twisted matrix qw(v)?w(u) bqu?v in Mrr (Rp ). Let FB be the twisted in nite Frobenius matrix (au;v )u;v2Zn . Clearly, the power (FB )k is well de ned and has the same trace as FBk for each positive integer k. Thus det(I ? TFB ) is well de ned and equal to det(I ? TFB ). To nish the proof, by Proposition 3.6 we need to prove that FB 2 Mh? for any  > 0. Take a positive integer N to be so large that N > N and (5:3)

N )  h ? : (h ? 2 ) logq (q ? N  

If juj  N , we have the trivial inequality ordq (au;v )  ?w(u)  ?w(N ). To prove the assertion that FB 2 Mh , it suces to prove the claim that ordq (au;v )  h ?  for all juj > N and all v. Assume juj > N . If v = 0 or qu, one checks that ordq (au;v )  w(qu) ? w(u) = (h ? =2) > h ?  and the claim is true. We now assume that juj > N and v is dierent from 0 and qu. There are two cases. If jqu ? vj  N , then by (5.1) and (5.2) we deduce that ordq (au;v )  (h ? 2 ) logq jjuvjj

= (h ? 2 ) logq qjuj + (jjuvjj ? qjuj)

 (h ? 2 ) logq (q ? NN )   h ? :

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DAQING WAN

If jqu ? vj > N, we use the inequality ab  a + b ? 1 (i.e., (a ? 1)(b ? 1)  0) for a; b  1 and deduce that

? vj ordq (au;v )  (h ? 2 ) logq jvjjqu juj  (h ? 2 ) logq jvj + jquju?j vj ? 1  (h ? 2 ) logq ( qjujju?j 1 )  (h ? 2 ) logq (q ? N1 )   h ? :

The claim is true and the proof of Theorem 2.1 is complete. The proof shows that the meromorphic continuation is uniform if we have a uniform family B (X; y). We make this precise. De nition 5.1. Let B (X; y) be a family of power series parameterized by y in some metric space Y : X B (X; y) = bu(y)X u ; u2Nn where bu (y) is in Mrr (Rp ) for each y and bu (y) goes to zero uniformly in u. Let  be a real number. The family B (X; y) is called uniformly  log-convergent if we have q (bu (y))  : lim inf inf y ord log juj juj!1

q

The family B (X; y) is called continuous in y if each coecient bu (y) is a continuous map. This continuity is automatically uniform in u since bu (y) goes to zero uniformly in y as juj goes to in nity. We have the following strengthened form of Theorems 1.1 and 2.1. THEOREM 5.2. Let B (X; y) be a family of power series parameterized by y in some metric space Y . Assume that the family B (X; y) is continuous in y and uniformly  log-convergent. Then for any  > 0, the zeroes and poles of the family of L-functions L(B (X; y)=Gnm ; T ) (resp. L(B (X; y)=An ; T )) ow continuously on the closed disk ordq (T )  ?( ? ). Proof. We prove only the assertion for the L-function L(B (X; y)=Gnm ; T ) on the torus as the proof for L(B (X; y)=An ; T ) is completely similar. Since B (X; y) is uniformly  log-convergent, the above proof of Theorem 2.1 shows that the twisted Frobenius matrix FB = (qw(v)?w(u) bqu?v (y)) is uniformly in M?=2 for any  > 0. By Proposition 3.9, the family of Fredholm determinants

MEROMORPHIC L-FUNCTIONS

19

det(I ? TFB(X;y) ) is uniformly analytic on the closed disk ordq (T ) > ?( ? ) for any  > 0. Since B (X; y) is continuous, the Euler product de nition of L-functions shows that the family of L-functions L(B (X; y)=Gnm ; T ) is continuous in y. The trace formula (4.10) shows that the family of Fredholm determinants det(I ? TFB(X;y) ) is also continuous in y. We have proved that det(I ? TFB(X;y) ) is a continuous family of uniformly analytic functions on the closed disk ordq (T ) > ?( ? ). By Lemma 3.8, the zeroes of the family det(I ? TFB(X;y) ) ow continuously on the closed disk ordq (T ) > ?( ? ). The theorem then follows by applying the trace formula (4.9). The proof is complete. Now, we turn to proving Theorem 2.3 and Theorem 1.3. Write X B (X ) = bu X u ; bu 2 Mrr (Rp ): jujd;u2Nn The in nite Frobenius matrix FB is (bqu?v )u;v2Nn . The block entry bqu?v is easily seen to be zero if jqu ? vj > d since B (X ) is a polynomial of degree d. We order the set Nn in such a way that juj > jvj implies that u > v. Then the ?
n + k rst k elements in Nn are just the lattice points u = (u1 ;


; un ) 2 Nn with u1 +


+ un  k. With this ordering, for u  v and juj > [d=(q ? 1)], we have jqu ? vj  (q ? 1)juj > d and thus bqu?v = 0. We have proved that the in nite matrix FB has the block form (5:4)

F   FB = 01 F ; 2

where F1 is the nite matrix (bqu?v ) with u and v run over the lattice points in Nn such that both juj  [d=(q ? 1)] and jvj  [d=(q ? 1)], and F2 is the in nite matrix (bqu?v ) with u and v run over the lattice points in Nn such that both juj > [d=(q ? 1)] and jvj > [d=(q ? 1)]. Furthermore, any block entry on or below the main diagonal of F2 is zero. This implies that det(I ? TFB ) = det(I ? TF1 ) is a polynomial of degree at most n + [d=(q ? 1)] n + [d=(q ? 1)] (5:5) r [d=(q ? 1)] = r : n This and Lemma 4.2 together give Theorem 2.3. If U is a subset of f1;


; ng, recall that the FBU is the submatrix (bqu?v ) such that ui  1 and vi  1 for all i 2 U . One computes that det(I ? TFBU ) is a polynomial of degree at most n + [d=(q ? 1)] ? jU j n ? jU j + [d=(q ? 1)] (5:6) r [d=(q ? 1)] ? jU j = r : n This and Lemma 4.3 together give the total degree bound in Theorem 1.3.

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Example. Let k be a positive integer and let B (X ) =

P

u jujk(q?1) bu X be

a polynomial in X = (X1 ;


; Xn ) of degree d = k(q ? 1) with coecients bu in Mrr (Rp ). Assume that k  q ? 2 and bu = 0 if u is not divisible by (q ? 1). Then for u 6= v and juj; jvj  k, one has bqu?v = 0 since qu ? v  u ? v 6 0 (mod (q ? 1)). Thus, the upper left matrix F1 in (5.4) is of the diagonal form whose block matrices on the main diagonal are the r  r matrices b(q?1)u with juj  k. It follows that (5:7)

det(I ? TFB ) = det(I ? TF1 ) =

Y

jujk

det(I ? Tb(q?1)u ):

If each b(q?1)u in (5.7) is invertible, then det(I ? TFB ) is a polynomial of degree exactly n + k n + [d=(q ? 1)] r n =r n and for each subset U of f1;


; ng, det(I ? TFBU ) is a polynomial of degree exactly n ? jU j + k n ? jU j + [d=(q ? 1)] r =r : n n

If each b(q?1)u is invertible modulo p, then there cannot have any cancellation in (4.9) and (4.13). Thus, the bounds for the total degree of L-functions in Theorems 1.3 and 2.3 are sharp in this case.

6. Counterexamples We now begin to prove Theorem 2.2. Let r = n = 1. In this case, a general B (X ) is a p-adic analytic function of one variable. Write

B (X ) =

1 X k=0

bk X k ; b0 = 1; bk 2 Rp ; klim b = 0: !1 k

To construct best possible examples for the meromorphic continuation theorem, we further suppose that (6:1) bk = 0 unless k is of the form qi ? 1: Let FB be the in nite Frobenius matrix associated with B , namely, FB = (bqu?v ), where u and v run over the non-negative integers. Let FB0 be the submatrix of FB consisting of those entries bqu?v with u  1 and v  1. Then, the Frobenius matrix FB has the following block form: 1   (6:2) FB = 0 F 0 : B

21

MEROMORPHIC L-FUNCTIONS

Now, (6:3)

?




det (I ? TFB ) = (1 ? T )det I ? TFB0 = (1 ? T )

1 X k=0

k T k ;

where the k are the coecients in the expansion of det (I ? TFB0 ). More precisely, 0 = 1 and for k  1, X X sign()bqu ?(u )


bquk ?(uk ) ; (6:4) k = (?1)k 1

1u1