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ZETA FUNCTIONS OF TOTALLY RAMIFIED p-COVERS OF THE PROJECTIVE LINE 5 0 0 HANFENGLIANDHUIJUNEZHU 2 n a Abstract. InthispaperweprovethatthereexistsaZariskidenseopensubset J U defined over the rationals Q in the space of all one-variable rational func- tionswitharbitraryℓpolesofprescribedorders,suchthatforeverygeometric 0 pointf inU(Q),the L-functionoftheexponential sumoff ataprimephas 3 Newtonpolygonapproaching theHodgepolygonaspapproaches infinity. As an application to algebraic geometry, we prove that the p-adic Newton poly- ] T gon of the zeta function of a p-cover of the projective linetotally ramifiedat N arbitraryℓpointsofprescribedordershasanasymptoticgenericlowerbound. . h t a m 1. Introduction [ This paper investigates the asymptotics of the zeta functions of p-covers of the projective line which are totally (wildly) ramified at arbitrary ℓ points. Our ap- 2 v proach is via Dwork’s method on one-variable exponential sums. 3 Throughoutthispaperwefixpositiveintegersℓ,d ,...,d ,andletd:= ℓ d + 1 ℓ j=1 j 2 ℓ−2. For simplicity we assume d≥2 if ℓ=1. Let P =∞, P =0, P ,...,P be 4 1 2 3 P ℓ fixed poles in the projective line over Q of orders d ,...,d , respectively. Let f be 2 1 ℓ 1 a one-variablefunction overQ with these prescribedℓ poles. Itcan be written in a 3 unique form of partial fractions [13, Introduction]): 0 / d1 ℓ dj h (1) f = a xi+ a (x−P )−i 1,i ji j t a i=1 j=2i=1 X XX m with a ∈ Q. (Remark: we have assumed that f has a vanishing constant term ji : because this does not affect the p-adic Newton polygons of f.) Let A be the space v i of a ’s with ℓ a 6=0. It is an affine ( ℓ d )-space over Q. Let the Hodge X ji j=1 j,dj j=1 j polygonofA, denotedbyHP(A), be the lowerconvexgraphofthe piecewise-linear r Q P a function defined on the interval [0,d] passing through the two endpoints (0,0) and (d,d/2) and assuming every slope in the list below of (horizontal) length 1: d1−1 d2−1 dℓ−1 ℓ−1 ℓ−1 1 d −1 1 d −1 1 d −1 1 2 ℓ 0,...,0;1,...,1; ,··· , ; ,··· , ;......; ,··· , . d d d d d d z1 }| 1 { z2 }| 2 { zℓ }| ℓ { z }| { z }| { A non-smooth point on a polygon (as the graph of a piece-wise linear function) is called a vertex. We remark that the classical and geometrical ‘Hodge polygon’ for any curve (including Artin-Schreier curve as a special case) is the one with Date:February1,2008. 2000 Mathematics Subject Classification. 11,14. Key words and phrases. exponential sums, rational functions, Artin-Schreier covers, totally ramifiedcovers,L-functionofexponential sums,Newtonpolygon,Dworktheory. 1 2 HANFENGLIANDHUIJUNEZHU end points (0,0) and (d,d/2) and one vertex at (d/2,0). So the Hodge polygon in our paper is different from the classical Hodge polygon. We anticipate a p-adic arithmetic interpretation of our Hodge polygon, but it remains an open question. In [13] it is shown that in the case ℓ = 1 there is a Zariski dense open subset U defined over Q such that every geometric closed point f in U(Q) has p-adic Newton polygon approaching the Hodge polygon as p approaches ∞. Wan has proposedconjecturesregardingmultivariableexponentialsums,includingtheabove as a special case (see [10, Conjecture 1.15]). This series of study traces back at least to Katz [4, Introduction], where Katz proposed to study exponential sums in families instead of examining one at a time. He systematically studied families of multivariable Kloostermansum in [4]. Let Q be the extension field of Q generated by coefficients a ’s and poles f ji P ,...,P of f. For every prime number p we fix an embedding Q֒→Q once and 1 ℓ p forall. This fixes a placeP in Q lyingoverp ofresiduedegree a forsome positive f integer a. As usual, we let E(x) = exp( ∞ xpi/pi) be the p-adic Artin-Hasse i=0 exponential function. Let γ be a root of the p-adic logE(x) with ord (γ) = 1 . P p p−1 Then E(γ) is a primitive p-th root of unity and we set ζ := E(γ). Let F be the p p prime field of p elements. Let F be a finite field of pa elements. For k ≥ 1, let q ψ : F → Q(ζ )× be a nontrivial additive character of F . Henceforth we fix k qk p qk ψ (·)=ζTrFqk/Fp(·). Let ℓ d , andallpoles andleadingcoefficients a off be k p j=1 j j,dj p-adicunits. Letallcoefficientsa off arep-adicallyintegral. (Thesearesatisfied j,i Q when p is large enough.) Let S (f modP) = ψ (f(x)modP) where the sum k x k rangesoverall x in F \{P ,...,P } (where P arereductions of P modP). The qk 1 ℓ Pj j L-function of f at p is defined as ∞ L(f modP;T) = exp( S (f modP)Tk/k). k k=1 X This function lies inZ[ζ ][T]of degreed. Itis independent of the choiceof P (that p is, the embedding of Q ֒→Q ) for p large enough, but we remark that its Newton p polygon is independent of the choice of P for all p (see [15, Section 1]). One notes immediately that for every prime p (coprime to the leading coefficients, the poles and their orders) we have a map NP (·) which sends every p-adic integral point f p ofA(Z )totheNewtonpolygonNP (f)oftheL-functionofexponentialsumsoff p p atp. Givenanyf ∈A(Q),wehaveforplargeenoughthatf ∈A(Z )andhencewe p obtain the Newton polygon NP (f) of f at p. Presently it is known that NP (f) p p lies over HP(A) for every p. These two polygons do not always coincide. (See [15, Introduction].) Some investigation on first slopes suggests the behavior is excep- tional if p is small (see [7, Introduction]). There has been intensive investigation on how the (Archimedean) distance between NP (f) and HP(A) on the real plane p R2 varieswhenpapproachesinfinity. InspiredbyWan’sconjecture[10,Conjecture 1.15](provedin [13] for the one-variablepolynomial case), we believe that “almost all” points f in A(Q) satisfies lim NP (f) = HP(A). Our main result is the p→∞ p following. Theorem 1.1. Let A be the coefficients space {a } of the f’s as in (1). There is ji a Zariski dense open subset U defined over Q in A such that for every geometric closed point f in U(Q) one has f ∈ U(Z ) for p large enough (only depending on p ZETA FUNCTIONS OF TOTALLY RAMIFIED p-COVERS 3 f), and lim NP (f)=HP(A). p p→∞ The two polygons NP (f) and HP(A) coincide if and only if p≡1modlcm(d ) p j (see [15, Theorem 1.1]). The case ℓ = 1 is known from [13, Theorem 1.1]. For p 6≡ 1modlcm(dj), the point f = xd1 + ℓj=1(x−Pj)−dj does not lie in U. This means U is always a proper subset of A. P For any f ∈ A(F ) and the (generalized) Artin-Schreier curve C : yp−y = f, q f let NP(C ;F ) be the usual p-adic Newton polygon of the numerator of the zeta f q functionofC /F . Ifitisshrunkbyafactorof1/(p−1)verticallyandhorizontally, f q we denote it by NP(Cf;Fq). p−1 Corollary 1.2. Let notations be as in Theorem 1.1 and the above. For any f ∈ A(F ) we have NP(f;Fq) lies over HP(A) with the same endpoints, and they coincide q p−1 if and only if p≡1mod(lcm(d )). Moreover, there is a Zariski dense open subset j U defined over Q in A such that for every geometric closed point f in U(Q) one has f ∈U(Z ) for p large enough (only depending on f), and p NP(C ;F ) lim f q =HP(A). p→∞ p−1 Proof. This followsfromthe theoremaboveanda similarargumentasthe proofof Corollary 1.3 in [15], which we shall omit here. (cid:3) Remark 1.3. (1) The result in Theorem 1.1 and Corollary 1.2 does not depend on where those ℓ poles are (as long as they are distinct). (2)ByDeuring-Shafarevicformula(seeforinstance[3,Corollary1.5]),oneknows that NP (f) always has slope-0 segment precisely of horizontal length ℓ−1. By p symmetry it also has slope-1 segment of the same length. See Remark 1.4 of [15]. Plan of the paper is as follows: section 2 introduces sheaves of (infinite dimen- sional) ϕ-modules over some affinoid algebra arising from one-variable exponential sums. We consider two Frobenius maps α and α . Section 4 is the main technical 1 a part, where major combinatorics of this paper is done. After working out several combinatorial observations we are able to reduce our problem to an analog of the one-variablepolynomialcaseasthatin[13]. NowbacktoSection3weimprovethe key lemma 3.5 of [13] to make the generic Fredholm polynomial straightforwardto compute. Section 5 uses p-adic Banach theory to give a new transformation theo- remfromα toα foranya≥1. Thisapproachisverydifferentfrom[9]or[14]. It 1 a shredssomenewlightonp-adicapproximationsofL-functionsofexponentialsums and we believe that it will find more application in the future. Finally at the end of section 5 we prove our main result Theorem 1.1. Acknowledgments. Zhu’sresearchwaspartiallysupportedbyanNSERCDiscovery grant and the Harvard University. She thanks Laurent Berger and the Harvard mathematics department for hospitality during her visit in 2003. The authors also thanks the referee for comments. 4 HANFENGLIANDHUIJUNEZHU 2. Sheaves of ϕ-modules over affinoid algebra The purpose of this section is to generalize the trace formula (see [14, Section 2])for anexponentialsum to that for families of exponentialsums. See [2] for fun- damentals in rigidgeometryand see [1] for an excellentsetup for rigidcohomology related to p-adic Dwork theory. Let O := Z [ζ ] and Ω := Q (ζ ). Fix a positive integer a. Let Ω be the 1 p p 1 p p a unramified extension of Ω of degree a and O its ring of integers. Let Pˆ ,...,Pˆ 1 a 1 ℓ in O× be Teichmu¨ller lifts of P ,...,P in F . Similarly let A be that of a a 1 ℓ pa j,i j,i and let A~ denote the sequence of A (we remark that for most part of the paper j,i A~ will be treated as a variable). Let τ be the lift of Frobenius to Ω which fixes a Ω1. Then τ(Aji)= Apji. Let 1≤ j ≤ ℓ. Pick a root γ1/dj of γ in Qp (or in Zp, all the same) for the rest of the paper, and denote Ω′1 :=Ω1(γ1/d1,...,γ1/dj). Let O1′ be its ring of integers. Let Ω′ := Ω Ω′ and let O′ be its ring of integers. Then a a 1 a the affinoid algebra O′hA~i (with A~ as variables) forms a Banach algebra over O′ a a under the supremum norm. Let 0 < r < 1 and r ∈ |Ω′| . Let A be the affinoid with ℓ deleted discs a p r centering at Pˆ ,...,Pˆ each of radius r on the rigid projective line P1 over Ω′ 1 ℓ a (as defined in [15]). The topology on A is given by the fundamental system of r strict neighborhood A with r ≤ r′ < 1 and r′ ∈ |Ω′| . Let A be A for some r′ a p r unspecifiedrsufficientlycloseto1−(thepreciseboundonthesizeofrwasdiscussed in[15, Section2]). LetH(Ω′) be the ring ofrigidanalytic functions onA overΩ′. a a Then it is a p-adic Banach space over Ω′. It consists of functions in one variable a X of the form ξ = ∞ c Xi + ℓ ∞ c (X −Pˆ )−i where c ∈ Ω′ and i=0 1,i j=2 i=1 j,i j j,i a ∀j ≥1,lim |cj,i|p =0. Its normis defined as||ξ||=max (sup |cj,i|p). (See [15, i→∞ ri P P P j i ri Section2.1].) LetH(Ω′hA~i):=H(Ω′)⊗ˆ Ω′hA~iwhere⊗ˆ meansp-adiccompletion a a Ω′a a afterthetensorproduct. Itisap-adicBanachmodulesoverΩ′hA~iwiththenatural a norm on the tensor product of two Banach modules defined by the followings. For any v ⊗w ∈ H(Ω′)⊗Ω hA~i let || v ⊗w|| = inf(max (||v ||·||w ||)), where a a i i i the inf ranges over all representatives v ⊗ w with v ⊗ w = v ⊗ w . P P i i i i i i From the p-adic Mittag-Leffler decomposition theorem derived in [15, Section 2.1], P P P we can generalize it to the decomposition of Ω′hA~i as a Banach Ω′hA~i-module. a a Write X1 = X or Xj = (X −Pˆj)−1 for 2 ≤ j ≤ ℓ. Let Zj = γ1/djXj. Note that ~b = {1,Zi,...,Zi} is a formal basis of the Banach Ω′hA~i-module H(Ω′hA~i), w 1 ℓ i≥1 a a that is, every v in H(Ω′hA~i) can be written uniquely as an infinite sum of c′ Zi’s a j,i j with c′j,i ∈ Ω′ahA~i and |cr′j,′ii|p → 0 as i → ∞, where r′ = r p−(p−11)dj. The Banach module H(Ω′hA~i) is orthonormalizable (even though ~b is not its orthonormal a w basis). 1 In this paper we extend the τ-action so that it acts on γdJ trivially for any J. Below we begin to construct the Frobenius operator α on H(Ω′hA~i). Recall the 1 a p-adicArtin-HasseexponentialfunctionE(X). TakeexpansionofE(γX)atX one gets E(γX) = ∞ λ Xm for some λ ∈ O . Clearly ord λ ≥ m for all m=0 m m 1 p m p−1 m ≥ 0. In particular, for 0 ≤ m ≤ p−1 the equality holds and λ = γm. Let P m m! ZETA FUNCTIONS OF TOTALLY RAMIFIED p-COVERS 5 F (X ):= dj E(γA Xi). Then j j i=1 j,i j Q ∞ F (X )= F (A ,...,A )Xn, j j j,n j,1 j,dj j n=0 X where F :=0 for n<0 and for n≥0 j,n (2) F := λ ···λ Am1···Amdj, j,n m1 mdj j,1 j,dj wherethe sumrangesoverallmX,...,m ≥0and dj km =n. It isclearthat 1 dj k=1 k F liesinO [A ,...,A ]. OneobservesthatF (X )∈O hA ,...,A ihX i, j,n 1 j,1 j,dj j Pj 1 j,1 j,dj j the affinoid algebra in one variable X (actually it lies in O [A ,...,A ]hX i). j 1 j,1 j,dj j Taking product over j = 1,...,ℓ, we have that F(X) := ℓ F (X ) lies in j=1 j j H(O hA~i). Let τa−1 be the push-forward map of τa−1, that is, for any function ξ, a ∗ Q τa−1(ξ) = τa−1◦ξ◦τ. For example, τa−1(B/(X −Pˆp)) = τa−1(B)/(X −Pˆ) for ∗ ∗ anyB ∈O hA~i andPˆ a Teichmu¨ller lift of someP. LetU be the Dworkoperator 1 p andletF(X)denotethe multiplicationmapbyF(X),asdefinedin[15,Section2]. Let α :=τa−1◦U ◦F(X) denote the compositionmap. Then α is a τa−1-linear 1 ∗ p 1 endomorphism of H(Ω′hA~i) as a Banach Ω′hA~i-module. a a LetS betheaffinoidoverΩ′ withaffinoidalgebraΩ′hA~i. IfLisasheafofp-adic a a Banach Ω′hA~i-module (with formal basis) and α is the Frobenius map which is a 1 τa−1-linearwithrespecttoΩ′hA~i,thenwecallthepair(L,α )asheaf of ϕ-module a 1 of infinite rank. Note that the pair (H(Ω′hA~i),α ) can be considered as sections a 1 of a sheaf of Ω′hA~i-module of infinite rank on A. This is intimately related to a Wan’s nuclear σ-module of infinite rank (see [11]) if replacing his σ by our τa−1. Wan has defined L-functions of nuclear σ-modules and he also showed that it is p-adicmeromorphicontheclosedunitdisc(seeWan’spapers[11,12]whichproved Dwork’s conjecture). Finally we define α :=αa. a 1 Recall that α is a τa−1-linear (with respect to Ω′hA~i) completely continuous 1 a endomorphismonthep-adicBanachmoduleH(Ω′hA~i)overΩ′hA~i. Let1≤J ,J ≤ a a 1 ℓ. Write (α Zi) = ∞ (τa−1Cn,i )Zn for some Cn,i in Ω′hA~i. The matrix of α , consi1stinJgPôJ1f all thne=se0τa−1C⋆J,1⋆,J’s, Jis1a nuclear mJa1t,Jrix (seae section 5). This 1 P J1,J matrix is the subject of the next section. Below we extend Dwork, Monsky and Reich’s trace formula to families of one-variable exponential sums. Theorem 2.1. Let f = d1 a xi+ ℓ di a (x−P )−i ∈A(F ) and let fˆ i=1 1,i j=2 i=1 j,i j q be its Teichmu¨ller lift with coefficient a being lifted to A . Let H(Ω′hA~i) be the P Pji P ji a r Banach module H(Ω′hA~i) for some suitably chosen 0<r <1 with r ∈|Ω′| close a a p enough to 1−. Then det(1−Tα |H(Ω′hA~i)) L(f/F ;T) = a a q det(1−Tqα |H(Ω′hA~i)) a a lies in O hA~i[T] as a polynomial of degree d in T. Its Teichmu¨ller specialization of a A~ in O lies in Z[ζ ][T]. a p Proof. Theproofissimilartothatof[14,Lemma2.7]. LetH† := H(Ω′hA~i) . 0<r<1 a r Then it is the Monsky-Washnitzer dagger space. Then α is a completely con- a tinuous endomorphism on H† and the determinant det(1 − TαS|H†) = det(1 − a 6 HANFENGLIANDHUIJUNEZHU Tα |H(Ω′hA~i) ) for any r within suitable range in (0,1) is independent of r. Fi- a a r nally one knows that the coefficients are all integralso lies in O and coefficient of a Tm vanishes for all m>d. We omit details of the proof. (cid:3) 3. Explicit approximation of the Frobenius matrix Thissectionusessomestandardtechniquesinp-adicapproximationanditisvery technical. The readers are recommended to skip it at first reading and continue at the next section. 3.1. The nuclear matrix. Let notation be as in the previous section. Assign φ(1) = 0. Let φ(Zn) = n for j ≤ 2 or n−1 for j ≥ 3. Order the elements j dj dj in ~b as e ,e ,··· such that φ(e ) ≤ φ(e ) ≤ ···. Consider the infinite matrix w 1 2 1 2 representingthe endomorphismα of the Ω′hA~i-module H(Ω′hA~i) withrespect to 1 a a the basis~b . This matrix canbe writtenas τa−1M,where eachentry is τa−1Cn,i w J1,J for 1≤J ,J ≤ℓ. 1 Our goal of this section is to collect delicate information about entries of the matrixM. RecalltheispolynomialF inO [A~]asin(2),whichwehavealready J,nJ 1 built up some satisfying knowledge. Below we will express C⋆,⋆ as a polynomial J1,J expressionintheseF ’s. InthispapertheformalexpansionofC⋆,⋆ willalways J1,nJ1 J1,J meantheformalsuminO′[A~]bythecompositionof(3)andtheformulainLemma a 3.1. For n,i≥1, and if J =1 or J =1 then for i≥0 or for n≥0 respectively one 1 has (3) Cn,i = γdiJ−dJn1HJn1p,,Ji J1 =1,2 J1,J  γdiJ−dJn1 nmp=nCn,mPˆJn1p−mHJm1,,iJ J1 ≥3 P where C⋆,⋆ ∈Z is defined in [15, Lemma 3.1] and H⋆,⋆ ∈O hA~i is formulated in p J1,J a Lemma 3.1 below. Indeed, we recallthat Cn,m is actually a rational integer and it only depends on n,m and p. Lemma 3.1. Let ~n:=(n ,...,n )∈Zℓ . 1 ℓ ≥0 (1) For i,n≥0, then Hn,i is equal to 1,J n +i−1 F · F J PˆnJ−mJ  1,n1  J,mJ m +i−1 J  X 0≤mXJ6=J≤1nJ (cid:18) J (cid:19)     nj n −1 · F j Pˆnj−mj ,  j,mj m −1 j  j6=Y1,J mXj=0 (cid:18) j (cid:19)   where the sum ranges over all ~n∈Zℓ such that n=n ±i− ℓ n and the + ≥0 1 j=2 j or − depends on J =1 or J 6=1, respectively. P ZETA FUNCTIONS OF TOTALLY RAMIFIED p-COVERS 7 (2) For J ,J 6=1, one has that Hn,i is equal to 1 J1,J n +m +i−1 XFJ1,nJ1 ·mJX6=JJ≥10FJ,mJ(−1)mJ+i(cid:18) JmJ +Ji−1 (cid:19)(PˆJ −PˆJ1)−(nJ+mJ+i)   ∞  m · F 1 Pˆm1−n1 mX1=n1 1,m1(cid:18)n1(cid:19) J1 ! ∞ n +m −1 F (−1)mj j j (Pˆ −Pˆ )−(nj+mj)  j,mj m −1 j J1  j6=Y1,J1,J mXj=0 (cid:18) j (cid:19)   where the sum ranges over all ~n∈Zℓ such that n=n +i− n if J =J ≥0 J1 j6=J1 j 1 and n=n − n if J 6=J . J1 j6=J1 j 1 P (3) For J 6=1 and J =1 we have that Hn,i is equal to 1 P J1,J m +i F · F 1 Pˆm1+i−n1 X J1,nJ1 m1X=n1−i 1,m1(cid:18) n1 (cid:19) J1 ! ∞ n +m −1 · F (−1)mj j j (Pˆ −Pˆ )−(nj+mj)  j,mj m −1 j J1  j6=YJ1,1 mXj=0 (cid:18) j (cid:19)   where the sum ranges over all ~n∈Zℓ such that n=n − n . ≥0 J1 j6=J1 j Proof. We shall use “P=ˆj” to mean expansion at Pˆ . ClearPly for any J one has j 1 F (X )Xi Pˆ=J1 ∞ F Xn+i. J1 J1 J1 n=0 J1,n For J ≥2 one has the expansion at Pˆ =∞: P 1 ∞ F (X )Xi = F (X−1(1−Pˆ X−1)−1)m+i J J J J,m J m=0 X ∞ ∞ P=ˆ1 F k−1 Pˆk−(m+i)X−k J,m m+i−1 J m=0 k=m+i(cid:18) (cid:19) X X ∞ n n+i−1 = ( F Pˆn−m)X−n−i. J,m m+i−1 J n=0 m=0 (cid:18) (cid:19) X X For J 6=1 and J 6=1,J , its expansion at Pˆ is: 1 1 J1 ∞ F (X )Xi = F (X−1−(Pˆ −Pˆ ))−(m+i) J J J J,m J1 J J1 m=0 X ∞ ∞ Pˆ=J1 F (−1)m+i n+m+i−1 (Pˆ −Pˆ )−(n+m+i)X−n J,m m+i−1 J J1 J1 m=0 n=0(cid:18) (cid:19) X X ∞ ∞ n+m+i−1 = ( F (−1)m+i (Pˆ −Pˆ )−(n+m+i))X−n. J,m m+i−1 J J1 J1 n=0 m=0 (cid:18) (cid:19) X X 8 HANFENGLIANDHUIJUNEZHU For J 6=1 and J =1 then one has 1 ∞ ∞ F (X)Xi Pˆ=J1 F m+i Pˆm+i−n X−n. J J n=0 m=n−i J,m(cid:18) n (cid:19) J1 ! J1 X X By F(X)Xi = (F (X )Xi) · F (X ), and Key Computational Lemma J J J J j6=J j j of [15], one can compute and obtain (F(X)Xi) for the case J = 1 or J 6= Q J PˆJ1 1 1 1 presented respectively in the two formulas in our assertion. This proves the lemma. (cid:3) Remark 3.2. If we are dealing with the case of unique pole at ∞ then one sees easily that C⋆,⋆ lies in O′[A~]. This greatly reduces the complexity of situation. 1,1 1 The following results were presented in [15]. See Section 3 and in particular, Theorem 3.7 of [15] for a proof. We shall use t to denote the lower bound in J1 Lemma 3.3 c). Lemma 3.3. Let notation be as above. ⌈nJ⌉ (a) For all J and n we have ord (F )≥ dJ ≥ nJ . J p J,nJ p−1 dJ(p−1) (b) For all J ,J, and all n,i we have ord (Hn,i )≥ n−i . 1 p J1,J dJ1(p−1) (c) For any J and any n we have ord (Cn,⋆ )≥ n or n−1 depending on J = 1 p J1,⋆ dJ1 dJ1 1 1,2or 3≤J ≤ℓ. Moreover, ord (Cn,i )≥⌈np−i⌉/(p−1) or ⌈(n−1)p−(i−1)⌉/(p− 1 p J1,J1 dJ1 dJ1 1) depending on J =1,2 or 3≤J ≤ℓ, respectively. 1 1 3.2. Approximation by truncation. From the previous subsection one has noticed anunpleasantfeature of C⋆,⋆ for the purpose ofapproximationbyF ’s. J1,J J1,nJ1 First, in the sum for (3) when J ≥3, the range of m is too ‘large’. Second, H⋆,⋆ 1 J1,J of Lemma 3.1 is generally an infinite sum of F ’s. In this subsection we will J1,nJ1 define an approximation in terms of truncated finite sum of F ’s. Below we J1,nJ1 prove two lemmas which will be used for approximationin Lemma 4.3. For any integer 0<t≤p, let tCn,i be the same as Cn,i except for J ≥3 its J1,J J1,J 1 sum ranges over all m in the sub-interval [(n−1)p+1,(n−1)p+t]. Lemma 3.4. Let 3≤J ≤ℓ, 1≤J ≤ℓ. Let n≤d and i≤d . 1 J1 J (1) For p large enough, one has n−1 d (4) ord (Cn,i −pCn,i ) > + . p J1,J J1,J dJ1 p−1 (2) There is a constant β >0 depending only on d such that for t≥β one has n−1 d (5) ord (pCn,i −tCn,i ) > + . p J1,J J1,J dJ1 p−1 Proof. (1) By [15, Lemma 3.1], one knows that for any m ≤ (n − 1)p one has ord (Cn,m) ≥ 1 and hence ord (Cn,i ) ≥ 1+( i − n ) 1 . For n ≤ d and p p J1,J dJ dJ1 p−1 J1 for p large enough one has 1+( i − n ) 1 > n−1 + d . Combining these two dJ dJ1 p−1 dJ1 p−1 inequalities, one concludes. (2) We may assume J ≥3. Then for any 1≤v ≤p, by Lemma 3.3, 1 (n−1)p+v−i n i 1 n−1 d ord (H(n−1)p+v,i) ≥ >( − ) + + , p J1,J dJ1(p−1) dJ1 dJ p−1 dJ1 p−1 ZETA FUNCTIONS OF TOTALLY RAMIFIED p-COVERS 9 if v ≥β for some β >0 only depending on d. Therefore, n−1 d ord (pCn,i −tCn,i ) > + . p J1,J J1,J dJ1 p−1 This finishes our proof. (cid:3) Fix β forthe restofthe paper. We willtruncate the infinite expansionofH⋆,⋆ . J1,J Let w >0 be any integer. For J =1,2 let wHnp,i be the sub-sum in Hnp,i where 1 J1,J J1,J ~n= (n ,...,n ) are such that n −np and n lie the interval [−w,w] for j 6= J . 1 ℓ J1 j 1 Similarly, for J ≥ 3 let wHm,i be the sub-sum of Hm,i where ~n ranges over the 1 J1,J J1,J finite set of vectors (n ,...,n ) such that n −(n−1)p and n lie in the interval 1 ℓ J1 j [−w,w] for j 6= J . Consider βCn,i as a polynomial expression in H⋆,⋆ ’s, then 1 J1,J J1,J we set wKn,i :=βCn,i (wHn,i ). J1,J J1,J J1,J Lemma 3.5. There is a constant α depending only on d such that (6) ord (βCn,i −αKn,i ) > n−1 + d . p J1,J J1,J dJ1 p−1 Proof. This part is similar to Lemma 3.4 2), so we omit its proof. (cid:3) 3.3. Minimalweightterms. Theweightofamonomial(withnonzerocoefficient) ( ℓ dj Akj,i) in O′[A~] is defined as ℓ dj ik . For example, the weight j=1 i=1 j,i a j=1 i=1 j,i of Aa Ab is equal to 2a+3b. We will later utilize the simple observation that Q 1,2Q1,3 P P every monomial in F is of weight n . J,nJ J We call those entries with J = J the diagonal one (or blocks). As we have 1 seen in Lemma 3.1, the off-diagonalentries are less manageable while the diagonal entries behave well in principle. For any integer 0 < t ≤ p, let tM := (tCn,i ) J1,J with respect to the basis arranged in the same order as that for M. Consider the diagonal blocks, consisting of pC⋆,⋆’s. Despite pC⋆,⋆ lives in O′hA~i, its minimal J,J J,J a weight terms live in PˆJZγid−JnO1[A~J]. Lemma 3.6. Let p >d for all j. The minimal weight monomials of pCn,i (with j J,J i−n J =1,2) live in the term γ d1 F1,np−i where d1 >n,i≥0 unless n=0 and i>0. For J ≥3 and n≥2, the minimal weight monomials of pCn,i live in the term J,J γid−JnCn,(n−1)p+1PˆJp−1FJ,(n−1)p−(i−1) where d >n,i≥1. J Proof. This follows from Lemma 3.1. We omit its proof. (cid:3) Given a k × k matrix M := (m ) with a given formal expansion of ij 1≤i,j≤k m ∈ O′hA~i, the formal expansion of detM means the formal expansion as ij a sgn(σ) k m where the product is expandedaccordingto the givenfor- maσl∈eSxkpansion mn=.1Foirjexample, if m =C⋆,⋆ then its formal expansion is given P Qij ij J1,J by composition of (3) and formulas in Lemma 3.1. Lemma 3.7. Let notation be as above and let p>d for all j. Then in the formal j expansionofdet(pM)[k] inO′hA~i,allminimalweighttermsarefrom ℓ detpCn,i a J=1 J,J (with n,i ≥ 1 in a suitable range for J = 1,2 and with n,i ≥ 2 for J ≥ 3) of the Q diagonal blocks. 10 HANFENGLIANDHUIJUNEZHU Proof. We will show that picking an arbitrary entry on the diagonal block, every off-diagonal entry on the same row has strictly higher minimal weight among its monomials. Let A~ stand for the vector (A ,...,A ). As we have noticed earlier the J J,1 J,dJ polynomial F in O [A~ ] has every monomial of equal weight n for any J. J,nJ 1 J J For simplicity we assume n,i ≥ 1 here. Using data from Lemma 3.1, we find all minimal weight monomials in H⋆,⋆ ’s illustrated below by an arrow: Hnp,i → J1,J 1,1 F ,Hnp,i → F ,Hnp,i → F ,Hnp,i → F . One also notes 1,np−i 1,J≥2 1,np+i 2,2 2,np−i 2,J6=2 2,np that for J ≥ 3 one has that H(n−1)p+1,i → F if J = J, and 1 J1≥3,J J1,(n−1)p−(i−1) 1 H(n−1)p+1,i →F ifJ 6=J. One noticesfrom(3)andthe abovethatthe J1≥3,J J1,(n−1)p+1 1 minimalweightmonomials ofpCn,i live in Hnp,i if J =1,2 andin H(n−1)p+1,i if J1,J J1,J 1 J1,J 3≤J ≤ℓ. 1 RecallthatforJ =1therangeforiisi≥0. Inallothercasestherangeisi≥1. Fromtheaboveweconcludeourclaiminthebeginningoftheproof. Consequently, allminimal weight monomials in the formalexpansionof the determinant detM[k] come from the diagonal blocks. By Lemma 3.6, C0,i and C1,i (with J ≥ 3) both 1,1 J,J have their minimal weight equal to 0 if i = 0 and > 0 if i > 0. Then it is not hard to conclude that the minimal weight monomials of det(Cn,i) (resp. 1,1 n,i≥0 det(Cn,i) ) are from det(Cn,i) (resp. det(Cn,i) ). (cid:3) J,J n,i≥1 1,1 n,i≥1 J,J n,i≥2 [k] For 1≤J ≤ℓ, let D :=det(F ) ∈O [A ,...,A ]. J J,ip−j 1≤i,j≤k 1 1 d Proposition3.8. Letp>d forallj. Theminimalweightmonomialsofdet((pCi,j ) j J,J 1≤i,j≤k for J =1,2 (resp. det((pCi,j ) for J ≥3 ) lie in D[k] (resp. D[k−1]). Every J,J 2≤i,j≤k J J monomial of D[k] (resp. D[k−1]) corresponds to a monomial in the formal expan- J J sion of det((pCi,j ) ) for J =1,2 (resp. det((pCi,j ) for J ≥3) by the J,J 1≤i,j≤k J,J 2≤i,j≤k same permutation σ ∈S in the natural way. k Proof. It follows from Lemmas 3.6 and 3.7 above. (cid:3) 3.4. Local at each pole. For ease of notation, we drop the subindex J for the rest of this subsection. One should understand that d,A ,F ,D stand for i ip−j n d ,A ,F ,D[n], respectively. Let 1 ≤ n ≤ d − 1 and let S be the per- J J,i J,ip−j J n mutation group. Let D :=det(F ) ∈O [A ,...,A ]. Then we have the n ip−j 1≤i,j≤n 1 1 d formal expansion of D[n]: n D[n] = sgn(σ) g , σ,i σX∈Sn XiY=1 wherethesecond runsoveralltermsg ofthepolynomialF inO [A ,...,A ]. σ,i ip−σ(i) 1 1 d Proposition 3.9P. Let 1 ≤ n ≤ d. Then there is a unique monomial in the above formal expansion of D[n] with highest lexicographic order (according toA ,...,A ). d 1 Moreover, the p-adic order of this monomial (with coefficient) is minimal among the p-adic orders of all monomials in the above formal expansion. Remark 3.10. We shall fix the unique σ found in the proposition for the rest of 0 the paper. The minimalp-adicorderofthis monomial(with coefficient)is equalto (8) while every row achieve its minimal order in Lemma 3.3c). We shall use this fact later.