Zeta functions of totally ramified p-covers of the projective line

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Zeta functions of totally ramified p-covers of the projective line.

HANFENGLI(*) - HUIJUNEZHU(**)

ABSTRACT- In this paper we prove that there exists a Zariski dense open subset U defined over the rationals Q in the space of all one-variable rational functions with prescribed ` poles with fixed orders, such that for every geometric point f inU(Q), the L-function of the exponential sum of f at a prime p has Newton polygon approaching the Hodge polygon as p ap- proaches infinity. As an application to algebraic geometry, we prove that the p-adic Newton polygon of the zeta function of ap-cover of the projective line totally ramified at arbitrary`points with prescribed orders has an asymp- totic generic lower bound.

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`points. Our approach is via Dwork's method on one-variable exponential sums.

Throughout this paper we fix positive integers `;d₁;. . .;d_`, and let d:P_`

j1d_j` 2. For simplicity we assumed2 if`1. LetP₁ 1, P20, P3;. . .;P` be fixed poles inQof orders d1;. . .;d`, respectively.

Letf be a one-variable function overQwith these prescribed`poles. It can be written in a unique form of partial fractions: f P_d₁

i1a_1;ixⁱ

P_`

j2

P_d_j

i1a_j;i(x P_j) ⁱwitha_j;i2Q. (We assume that f has a vanish- ing constant term because this does not affect thep-adic Newton poly-

(*) Indirizzo dell'A.: Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada.

E-mail: hli@fields.toronto.edu

(**) Indirizzo dell'A.: Department of Maths and Stats, McMaster University, Hamilton, ON L8S 4K1, Canada.

E-mail: zhu@cal.berkeley.edu

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gons of f [13, Introduction]). Let A be the space of aj;i's with Q_`

j1aj;dj 60. It is an affine (P_`

j1dj)-space over Q. Let the Hodge polygon ofA, denoted by HP(A), be the lower convex graph of the pie- cewise-linear 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:

0;. . .;0 z}|{^` ¹

;1;z}|{. . .^` ¹;1

; 1

d1;. . .;d₁ 1 d1

z}|{^d¹ ¹

; 1

d2;. . .;d₂ 1 d2

z}|{^d² ¹

; . . . ;1

d`;. . .;d_` 1 d`

z}|{^d^` ¹ : 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 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`1 there is a Zariski dense open subsetU defined overQsuch that every geometric closed pointf inU(Q) has p-adic Newton polygon approaching the Hodge polygon as p ap- proaches 1. Wan has proposed conjectures regarding multivariable exponential sums, including the above 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 ex- amining one at a time. He systematically studied families of multivariable Kloosterman sum in [4].

LetQf be the extension field ofQgenerated by coefficients and poles P1;. . .;P` off. For every prime numberp we fix an embeddingQ,!Qp

once and for all. This fixes a placePinQ_f lying overpof residue degreea for some positive integera. As usual, we letE(x)exp (P₁

i0x^pⁱ=pⁱ) be thep-adic Artin-Hasse exponential function. Letgbe a root of the p-adic logE(x) with ordp(g)_p¹₁. ThenE(g) is a primitivep-th root of unity and we setz_p:E(g). LetFpbe the prime field ofpelements. LetFqbe a finite field of p^a elements. Fork1, letc_k:F_q^k!Q(z_p) be a nontrivial ad- ditive character ofF_q^k. Henceforth we fix c_k()z^Trp ^F^qk^=Fp⁽⁾. LetQ_`

j1dj, and all poles and leading coefficients a_j;d_j of f be p-adic units. Let all coefficients a_j;i off be p-adically integral. (These are satisfied when pis large enough.) LetS_k(fmodP)P

xc_k(f(x)modP) where the sum ranges

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over allxinF_q^knfP1;. . .;P`g(wherePjare reductions ofPjmodP). TheL- function off atpis defined as

L(fmodP;T)exp X¹

k1

S_k(fmodP)T^k=k

! :

This function lies inZ[z_p][T] of degreed. It is independent of the choice ofP (that is, the embedding ofQ,!Q_p) forplarge enough, but we remark that its Newton polygon is independent of the choice of P for all p (see [15, Section 1]). One notes immediately that for every primep(coprime to the leading coefficients, the poles and their orders) we have a map NPp() which sends everyp-adic integral pointf ofA(Z_p) to the Newton polygon NP_p(f) of theL-function of exponential sums offatp. Given anyf 2A(Q), we have forplarge enough thatf 2A(Z_p) and hence we obtain the Newton polygon NPp(f) offatp. Presently it is known that NPp(f) lies over HP(A) for everyp. These two polygons do not always coincide. (See [15, Intro- duction].) Some investigation on first slopes suggests the behavior is ex- ceptional ifpis small (see [7, Introduction]). There has been intensive investigation on how the (Archimedean) distance between NP_p(f) and HP(A) on the real planeR²varies whenpapproaches infinity. Inspired by Wan's conjecture [10, Conjecture 1.15] (proved in [13] for the one-variable polynomial case), we believe that «almost all» points f in A(Q) satisfy lim_p!1NP_p(f)HP(A). Our main result is the following.

THEOREM1.1. LetAbe the coefficients spacefaj;igof the f⁰s as in the beginning of the paper. There is a Zariski dense open subsetUdefined over Q in A such that for every geometric closed point f in U(Q) one has f 2 U(Z_p)for p large enough (only depending on f ), and

p!1lim NPp(f)HP(A):

The two polygons NP_p(f) and HP(A) coincide if and only if p1mod lcm(d_j) (see [15, Theorem 1.1]). The case`1 is known from [13, Theorem 1.1]. For p61mod lcm(d_j), the point f x^d¹P_`

j1(x P_j) ^d^j does not lie inU. This meansU is always a proper subset ofA.

For any f 2A(Fq) and the (generalized) Artin-Schreier curve C_f :y^p yf, let NP(C_f;Fq) be the usualp-adic Newton polygon of the numerator of the zeta function of C_f=F_q. If it is shrunk by a factor of 1=(p 1) vertically and horizontally, we denote it by^NP(C_p ^f₁^;F^q⁾

COROLLARY1.2. Let notation be as in Theorem1:1and the above. For anyf 2A(F_q)we have^NP(f;F_p ₁^q⁾lies overHP(A)with the same endpoints, and

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they coincide if and only ifp1mod (lcm(dj)). Moreover, there is a Zariski dense open subsetUdefined overQinAsuch for every geometric closed point f inU(Q)one hasf2 U(Zp)for p large enough (only depending on f), and

p!1lim

NP(C_f;F_q)

p 1 HP(A).

PROOF. This follows from the theorem above and a similar argument as the proof of Corollary 1.3 in [15], which we shall omit here. p REMARK1.3. (1) The results in Theorem 1.1 and Corollary 1.2 do not depend on where those`poles are (as long as they are distinct).

(2) By Deuring-Shafarevic formula (see for instance [3, Corollary 1.5]), one knows that NP_p(f) always has slope-0 segment precisely of horizontal length` 1. By 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 dimensional)W-modules over some affinoid algebra arising from one-variable exponential sums. We consider two Frobenius mapsa1andaa. Section 4 is the main technical 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-variable polynomial case as that in [13]. Now back to Section 3 we improve the key lemma 3.5 of [13] to make the generic Fredholm polynomial straightforward to compute. Section 5 usesp-adic Banach theory to give a new transformation theorem froma1to a_afor anya1. This approach is very different from [9] or [14]. It shreds some new light onp-adic approximations ofL-functions of exponential sums 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's research was partially supported by an NSERC Discovery grant and the Harvard University. She thanks Laurent Berger and the Harvard mathematics department for hospitality during her visit in 2003. The authors also thank the referee for comments.

2. Sheaves ofW-modules over affinoid algebra.

The purpose of this section is to generalize the trace formula (see [14, Section 2]) for an exponential sum to that for families of exponential sums.

See [2] for fundamentals in rigid geometry and see [1] for an excellent setup for rigid cohomology related top-adic Dwork theory.

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LetO1:Zp[z_p] andV1:Qp(z_p). Fix a positive integera. LetVabe the unramified extension ofV1of degreeaandOaits ring of integers. Let P^₁;. . .;P^` inO_a be TeichmuÈller lifts ofP₁;. . .;P`inFp^a. Similarly letA_j;i be that ofa_j;iand let~Adenote the sequence ofA_j;i(we remark that for most part of the paper~Awill be treated as a variable). Lettbe the lift of Fro- benius toVa which fixesV1. Thent(Aji)A^p_ji. Let 1j`. Pick a root g^1=d^jofginQp(or inZp, all the same) for the rest of the paper, and denote V⁰₁:V1(g^1=d¹;. . .;g^1=d^j). Let O⁰₁ be its ring of integers. Let V⁰_a:VaV⁰₁ and letO⁰_abe its ring of integers. Then the affinoid algebraO⁰_ah~Ai(with~A as variables) forms a Banach algebra overO⁰_aunder the supremum norm.

Let 0<r<1 and r2 jV⁰_aj_p. Let A_r be the affinoid with ` deleted discs centering atP^1;. . .;P^`each of radiusron the rigid projective line P¹ over V⁰_a (as defined in [15]). The topology on Ar is given by the fundamental system of strict neighborhood A_r⁰ with rr⁰<1 and r⁰2 jV⁰_aj_p. LetAbeA_rfor some unspecifiedrsufficiently close to 1 (the precise bound on the size of r was discussed in [15, Section 2]). Let H(V⁰_a) be the ring of rigid analytic functions onAoverV⁰_a. Then it is ap- adic Banach space overV⁰_a. It consists of functions in one variableX of the form jP₁

i0c_1;iXⁱP_`

j2

P₁

i1c_j;i(X P^_j) ⁱ where c_j;i2V⁰_a and 8j1;limi!1jcj;ij_p

rⁱ 0. Its norm is defined as kjk maxj( sup_i^jc_r^j;ii^j^p).

(See [15, Section 2.1].) LetH(V⁰_ah~Ai): H(V⁰_a)^_V⁰_aV⁰_ah~Aiwhere^ means p-adic completion after tensoring. It is a p-adic Banach modules over V⁰_ah~Ai with the natural norm on the tensor product of two Banach modules defined by the followings. For any P

vw2 H(V⁰_a)Vah~Ai let kP

vwk inf ( max_i(kv_ik kw_ik)), where the inf ranges over all representatives P

iv_iw_i withP

vwP

iv_iw_i. From the p-adic Mittag-Leffler decomposition theorem derived in [15, Section 2.1], we can generalize it to the decomposition of V⁰_ah~Ai as a Banach V⁰_ah~Ai- module. WriteX1XorXj(X P^j) ¹ for 2j`. LetZj g^1=d^jXj. Note that~bw f1;Zⁱ₁; ;Zⁱ_`g_i1is a formal basis of the BanachV⁰_ah~Ai- moduleH(V⁰_ah~Ai), that is, everyvinH(V⁰_ah~Ai) can be written uniquely as an infinite sum of cj;iZⁱ_j's with cj;i2V⁰_ah~Ai and ^jc^j;i_ri^j^p!0 as i! 1. The Banach moduleH(V⁰_ah~Ai) is orthonormalizable (even though~bwis not its orthonormal basis).

In this paper we extend thet-action so that it acts ong^dJ¹ trivially for any J. Below we begin to construct the Frobenius operator a₁ onH(V⁰_ah~Ai).

Recall thep-adic Artin-Hasse exponential functionE(X). Take expansion of E(gX) at X one gets E(gX)P₁

m0lmX^m for some lm2 O1. Clearly ordplm_p^m₁ for allm0. In particular, for 0mp 1 the equality

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holds andl_m_m!^g^m. LetF_j(X_j):Q_d_j

i1E(gA_j;iX_jⁱ). Then F_j(X_j)X¹

n0

F_j;n(A_j;1;. . .;A_j;d_j)Xⁿ_j; whereFj;n:0 forn<0 and forn0

F_j;n:X

l_m₁ l_m_djA^m_j;1¹ A^m_j;d^dj_j; 1

where the sum ranges over all m₁;. . .;m_d_j 0 andP_d_j

k1km_kn. It is clear that F_j;n lies in O₁[A_j;1;. . .;A_j;d_j]. One observes that F_j(X_j)2 2 O1hAj;1;. . .;Aj;djihXji, the affinoid algebra in one variableXj(actually it lies inO1[A_j;1;. . .;A_j;d_j]hX_ji). Taking product overj1;. . .; `, we have that F(X):Q_`

j1F_j(X_j) lies inH(O_ahAi). Let~ tâ ¹be the push-forward map of tâ ¹, that is, for any function j, tâ ¹(j)tâ ¹jt. For example, tâ ¹(B=(X P^^p))tâ ¹(B)=(X P) for any^ B2 O₁h~Ai and P^ a Teich- muÈller lift of someP. LetUpbe the Dwork operator and letF(X) denote the multiplication map by F(X), as defined in [15, Section 2]. Let a₁:tâ ¹U_pF(X) denote the composition map. Thena₁is atâ ¹-linear endomorphism ofH(V⁰_ah~Ai) as a BanachV⁰_ahAi-module.~

LetSbe the affinoid overV⁰_awith affinoid algebraV⁰_ah~Ai. IfLis a sheaf ofp-adic BanachV⁰_ah~Ai-module (with formal basis) anda1is the Frobenius map which ist^a ¹-linear with respect toV⁰_ah~Ai, then we call the pair (L;a1) asheaf ofW-module of infinite rank. Note that the pair (H(V⁰_ah~Ai);a1) can be considered as sections of a sheaf ofV⁰_ah~Ai-module of infinite rank onA.

This is intimately related to Wan'snuclears-module of infinite rank(see [11]) if replacing hissby ourt^a ¹. Wan has definedL-functions of nuclear s-modules and he also showed that it isp-adic meromorphic on the closed unit disc (see Wan's papers [11, 12] which proved Dwork's conjecture).

Finally we definea_a:a^a₁.

Recall thata₁ is at^a ¹-linear (with respect toV⁰_ah~Ai) completely continuous endomorphism on thep-adic Banach moduleH(V⁰_ah~Ai) overV⁰_ahAi.~ Let 1J1;J`. Write (a1Zⁱ_J)P^J1 P₁

n0(t^a ¹C_J^n;i₁_;J)Zⁿ_J₁ for someC^n;i_J₁_;J in V⁰_ahAi. The matrix of~ a₁, consisting of all these t^a ¹C^?;?_J₁_;J's, is a nuclear matrix (see section 5). This 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 P_d₁

i1a_1;ixⁱP_`

j2P_d_i

i1a_j;i(x P_j) ⁱ2A(Fq) and let ^f be its TeichmuÈller lift with coefficient a_jibeing lifted to A_ji. Let H(V⁰_ah~Ai)_r be the Banach module H(V⁰_ah~Ai) for some suitably chosen

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0<r<1with r2 jV⁰_aj_pclose enough to1 . Then L(f=Fq;T)det (1 TaajH(V⁰_ahAi))~

det (1 Tqa_ajH(V⁰_ah~Ai))

lies inO_ah~Ai[T]as a polynomial of degree d. Its TeichmuÈller specializa- tion of~A inO_alies inZ[z_p][T].

PROOF. The proof is similar to that of [14, Lemma 2.7]. Let H^y:S

0<r<1H(V⁰_ah~Ai)_r. Then it is the Monsky-Washnitzer dagger space.

Then aa is a completely continuous endomorphism on H^y and the determinant det (1 TaajH^y)det (1 TaajH(V⁰_ahAi)~ _r) for any r within suitable range in (0;1) is independent of r. Finally one knows that the coefficients are all integral so lie inO_aand coefficient ofT^mvanishes for all

md. We omit details of the proof. p

3. Explicit approximation of the Frobenius matrix.

This section uses some standard techniques in p-adic approximation and it is very 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 f(1)0. Let f(Zⁿ_j)_dⁿ_jforj2 orⁿ_d_j¹forj3. Order the elements in~b_w ase₁;e₂; such thatf(e1)f(e2) . Consider the infinite matrix representing the endomorphisma1of theV⁰_ah~Ai-moduleH(V⁰_ahAi) with respect to the basis~

~b_w. This matrix can be written ast^a ¹M, where each entry ist^a ¹C_J^n;i₁_;Jfor 1J₁;J`.

Our goal of this section is to collect delicate information about entries of the matrixM. Recall the polynomialFJ;nJ inO1[~A] as in (1), which we have already built up some satisfying knowledge. Below we will expressC^?;?_J₁_;Jas a polynomial expression in these F_J₁_;n_J₁'s. In this paper the formal expansion of C_J^?;?₁_;J will always mean the formal sum in O⁰_a[A] by the com-~ position of (2) and the formula in Lemma 3.1.

Forn;i1, and ifJ1 orJ11 then fori0 or forn0 respectively one has

C_J^n;i₁_;J g^dJⁱ ^dJⁿ¹H^np;i_J₁_;J J11;2 g^dJⁱ ^dJⁿ¹P_np

mnC^n;mP^^{np m}_J₁ H^m;i_J₁_;J J₁3 8<

2 :

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where C^?;?2Zp is defined in [15, Lemma 3.1] and H^?;?_J₁_;J2 Oah~Aiis for- mulated in Lemma 3.1 below. Indeed, we recall that C^n;m is actually a rational integer and it only depends onn,mandp.

LEMMA3.1. Let~n:(n₁;. . .;n_`)2Z^`₀. (1)For i;n0, then H^n;i_1;Jis equal to X F_1;n₁ X

0mJnJ J61

F_J;m_J n_Ji 1 m_Ji 1

P^ⁿ_J^J ^m^J 0

B@

1 CA 0

B@

Y

j61;J

Xⁿ^j

mj0

F_j;m_j n_j 1 mj 1

P^ⁿ_j^j ^m^j 0

@

1 A

1 A; where the sum ranges over all~n2Z^`₀such that nn1i P_`

j2n_jand theor depends on J1or J61, respectively.

(2)For J₁;J61, one has that H^n;i_J₁_;Jis equal to X F_J₁_;n_J₁ X

m^J6JJ0¹

F_J;m_J( 1)^m^Jⁱ n_Jm_Ji 1 m_Ji 1

(P^_J P^_J₁) ⁽ⁿ^J^m^Jⁱ⁾ 0

BB

@

1 CC A 0

BB

@

X¹

m1n1

F1;m1

m1

n₁ P^^m_J₁¹ ⁿ¹

!

Y

j61;J1;J

X¹

mj0

Fj;mj( 1)^m^j njmj 1 m_j 1

(P^j P^J1) ⁽ⁿ^j^m^j⁾ 0

@

1 A

1 A where the sum ranges over all~n2Z^`₀such that nn_J₁i P

j6J1n_jif JJ₁and nn_J₁ P

j6J1n_j if J6J₁.

(3)For J161and J1we have that H^n;i_J₁_;J is equal to X FJ1;nJ1 X

m1n1 i

F1;m1

m1i n₁

P^^m_J₁¹^{i n}¹

!

Y

j6J1;1

X¹

mj0

Fj;mj( 1)^m^j n_jm_j 1 m_j 1

(P^j P^J1) ⁽ⁿ^j^m^j⁾ 0

@

1 A

1 A where the sum ranges over all~n2Z^`₀such that nn_J₁ P

j6J1n_j.

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PROOF. We shall use «^P^{^}^j» to mean expansion atP^j. Clearly for anyJ1

one hasFJ1(XJ1)Xⁱ_J₁ ^P^{^}^J¹P₁

n0FJ1;nXⁿⁱ.

ForJ2 one has the expansion atP^1 1:

F_J(X_J)X_Jⁱ X¹

m0

F_J;m(X ¹(1 P^_JX ¹) ¹)^mi

^{^}

P1X¹

m0

F_J;m X¹

kmi

k 1 mi 1

P^^k_J ^(mi)X ^k

X¹

n0

(Xⁿ

m0

F_J;m ni 1 mi 1

P^^{n m}_J )X ^{n i}:

ForJ₁61 andJ61;J₁, its expansion atP^_J₁ is:

F_J(X_J)X_Jⁱ X¹

m0

F_J;m(X_J₁¹ (P^_J P^_J₁)) ^(mi)

^

PJ1X¹

m0

F_J;m( 1)^miX¹

n0

nmi 1 mi 1

(P^_J P^_J₁) ^(nmi)X_J₁ⁿ

X¹

n0

X¹

m0

F_J;m( 1)^mi nmi 1 mi 1

(P^_J P^_J₁) ^(nmi)

! X_J₁ⁿ:

ForJ161 andJ1 then one has FJ(X)X_Jⁱ ^P^{^}^J¹X¹

n0

X¹

mn i

FJ;m mi n

P^^{mi n}_J₁

! X_J₁ⁿ: ByF(X)X_Jⁱ (FJ(XJ)X_Jⁱ)Q

j6JFj(Xj), and Key Computational Lemma of [15], one can compute and obtain (F(X)X_Jⁱ)P^J1 for the case J11 or J₁61 presented respectively in the two formulas in our assertion. This

proves the lemma. p

REMARK3.2. If we are dealing with the case of unique pole at1then one sees easily thatC_1;1^?;?lies inO⁰₁[~A]. This greatly reduces the complexity of situation.

The following results were presented in [15]. See Section 3 and in particular, Theorem 3.7 of [15] for a proof. We shall uset_J₁ to denote the lower bound in Lemma 3.3 c).

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LEMMA3.3. Let notation be as above.

(a) For all J and n_Jwe haveordp(F_J;n_J)^d_p^nJ^dJ^e₁_d_J_(pⁿ^J₁₎: (b) For all J₁;J, and all n;i we haveord_p(H^n;i_J₁_;J)_d_J^{n i}

1(p 1): (c) For any J1and any n we haveordp(C^n;?_J₁_;?)_dⁿ_J

1 or ⁿ_d_J¹

1 depending on J11;2 or 3J1`. Moreover, ordp(C_J^n;i₁_;J₁) d^{np i}_d_J

1e=(p 1) or

d⁽ⁿ ^1)p_d_J ⁽ⁱ ¹⁾

1 e=(p 1)depending on J₁1;2or3J₁`, respectively.

3.2 ±Approximation by truncation.

From the previous subsection one has noticed an unpleasant feature of C^?;?_J₁_;Jfor the purpose of approximation byF_J₁_;n_J₁'s. First, in the sum for (2) whenJ₁3, the range ofmis too «large». Second,H^?;?_J₁_;Jof Lemma 3.1 is generally an infinite sum of F_J₁_;n_J₁'s. In this subsection we will define an approximation in terms of truncated finite sum ofF_J₁_;n_J₁'s. Below we prove two lemmas which will be used for approximation in Lemma 4.3.

For any integer 0<tp, let ^tC^n;i_J₁_;J be the same as C_J^n;i₁_;J except for J₁3 its sum ranges over all m in the sub-interval [(n 1)p1;

(n 1)pt].

LEMMA3.4. Let3J₁`,1J`. Let nd_J₁and id_J. (1)For p large enough, one has

ord_p(C_J^n;i₁_;J ^pC^n;i_J₁_;J)>n 1 d_J₁ d

p 1: 3

(2)There is a constantb>0depending only on d such that for tb one has

ord_p(^pC^n;i_J₁_;J ^tC^n;i_J₁_;J)>n 1 dJ1

d p 1: 4

PROOF. (1) By [15, Lemma 3.1], one knows that for anym(n 1)p one has ordp(C^n;m)1 and hence ordp(C^n;i_J₁_;J)1(_dⁱ_J _dⁿ_J

1)_p¹₁. For nd_J₁ and for p large enough one has 1(_dⁱ_J _dⁿ_J

1)_p¹₁>ⁿ_d_J¹

1 _p^d₁. Combining these two inequalities, one concludes.

(2) We may assumeJ13. Then for any 1vp, by Lemma 3.3, ord_p(H⁽ⁿ_J₁_;J^1)pv;i)(n 1)pv i

dJ1(p 1) > n dJ1

i dJ

1

p 1n 1 dJ1

d p 1;

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ifvbfor someb>0 only depending ond. Therefore, ordp(^pC^n;i_J₁_;J ^tC^n;i_J₁_;J)>n 1

d_J₁ d p 1:

This finishes our proof. p

Fixbfor the rest of the paper. We will truncate the infinite expansion of H^?;?_J₁_;J. Letw>0 be any integer. ForJ₁1;2 let^wH^np;i_J₁_;Jbe the sub-sum in H^np;i_J₁_;Jwhere~n(n1;. . .;n`) are such thatnJ1 npandnj lie the interval [ w;w] forj6J1. Similarly, forJ13 let^wH^m;i_J₁_;Jbe the sub-sum ofH^m;i_J₁_;J where ~n ranges over the finite set of vectors (n1;. . .;n`) such that n_J₁ (n 1)pandn_jlie in the interval [ w;w] forj6J₁. Consider^bC^n;i_J₁_;J as a polynomial expression inH^?;?_J₁_;J's, then we set^wK^n;i_J₁_;J:^bC^n;i_J₁_;J(^wH^n;i_J₁_;J):

LEMMA3.5. There is a constantadepending only on d such that ordp(^bC^n;i_J₁_;J ^aK^n;i_J₁_;J)>n 1

d_J₁ d p 1: 5

PROOF. This part is similar to Lemma 3.4 2), so we omit its proof. p 3.3 ±Minimal weight terms.

Theweightof a monomial (with nonzero coefficient) (Q_`

j1

Q_d_j

i1A^k_j;i^j;i) in O⁰_a[~A] is defined asP_`

j1P_d_j

i1ikj;i:For example, the weight ofA^a_1;2A^b_1;3is equal to 2a3b. We will later utilize the simple observation that every monomial inF_J;n_J is of weightn_J.

We call those entries withJ₁Jthe diagonal one (or blocks). As we have seen in Lemma 3.1, the off-diagonal entries are less manageable while the diagonal entries behave well in principle. For any integer 0<tp, let ^tM:(^tC^n;i_J₁_;J) with respect to the basis arranged in the same order as that for M. Consider the diagonal blocks, consisting of

pC^?;?_J;J's. Despite ^pC^?;?_J;J lives in O⁰_ah~Ai, its minimal weight terms live in P^^Z_Jg^{i n}^dJO₁[~A_J].

LEMMA3.6. Let p>d_jfor all j. The minimal weight monomials of^pC^n;i_J;J (with J1;2) live in the termg^{i n}^d¹F1;np iwhere d1>n;i0unless n0 and i>0. For J3and n2, the minimal weight monomials of^pC^n;i_J;J

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live in the term

g^{i n}^dJC^n;(n ^1)p1P^^p_J ¹FJ;(n 1)p (i 1)

where d_J>n;i1.

PROOF. This follows from Lemma 3.1. We omit its proof. p Given akkmatrixM:(m_ij)_1i;jkwith a given formal expansion of m_ij2 O⁰_ah~Ai, theformal expansionof detM means the formal expansion asP

s2Sksgn(s)Q_k

n1m_ij where the product is expanded according to the given formal expansion mij. For example, if mijC_J^?;?₁_;J then its formal expansion is given by composition of (2) and formulas in Lemma 3.1.

LEMMA3.7. Let notation be as above and let p>djfor all j. Then in the formal expansion ofdet (^pM)^[k] in O⁰_ah~Ai, all minimal weight terms are fromQ_`

J1det^pC^n;i_J;J(with n;i1in a suitable range for J1;2and with n;i2for J3) of the diagonal blocks.

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_Jstand for the vector (A_J;1;. . .;A_J;d_J). As we have noticed earlier the polynomialF_J;n_JinO₁[~A_J] has every monomial of equal weightn_Jfor any J. For simplicity we assumen;i1 here. Using data from Lemma 3.1, we find all minimal weight monomials inH^?;?_J₁_;J's illustrated below by an arrow:

H^np;i_1;1 !F_{1;np i};H^np;i_1;J2!F_1;npi;H^np;i_2;2 !F_{2;np i};H^np;i_2;J62!F_2;np. One also notes that forJ13 one has thatH⁽ⁿ_J₁_3;J^1)p1;i!FJ1;(n 1)p (i 1)ifJ1J, and H⁽ⁿ_J₁_3;J^1)p1;i!F_J₁_;(n _1)p1ifJ₁6J. One notices from (2) and the above that the minimal weight monomials of ^pC^n;i_J₁_;J live in H^np;i_J₁_;J if J11;2 and in H⁽ⁿ_J₁_;J^1)p1;iif 3J1`:

Recall that forJ1 the range foriisi0. In all other cases the range isi1. From the above we conclude our claim in the beginning of the proof. Consequently, all minimal weight monomials in the formal expansion of the determinant detM^[k]come from the diagonal blocks. By Lemma 3.6,C_1;1^0;iandC^1;i_J;J(withJ3) both have their minimal weight equal to 0 if i0 and >0 if i>0. Then it is not hard to conclude that the minimal weight monomials of det (C^n;i_1;1)_n;i0 (resp. det (C_J;J^n;i)_n;i1) are from det (C_1;1^n;i)_n;i1(resp. det (C_J;J^n;i)_n;i2). p

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For 1J`, letD^[k]_J :det (FJ;ip j)_1i;jk2 O1[A1;. . .;Ad].

PROPOSITION3.8. Letp > djfor allj. The minimal weight monomials ofdet ((^pCî;j_J;J)_1i;jk)forJ 1,2(resp.det ((^pCî;j_J;J)_2i;jk)forJ 3) lie in D^[k]_J (resp.D^[k_J ^1]). Every monomial ofD^[k]_J (resp.D^[k_J ^1]) corresponds to a monomial in the formal expansion ofdet ((^pCî;j_J;J)_1i;jk)forJ 1,2(resp.

det ((^pC^i;j_J;J)_2i;jk) for J 3) by the same permutation s2Sk in the natural way.

PROOF. It follows from Lemmas 3.6 and 3.7 above. p

3.4 ±Local at each pole.

For ease of notation, we drop the subindexJ for the rest of this subsection. One should understand that d;A_i;F_{ip j};Dn stand for d_J;A_J;i;F_{J;ip j};D^[n]_J , respectively. Let 1nd 1 and let S_n be the permutation group. LetD_n:det (F_{ip j})_1i;jn2 O₁[A₁;. . .;A_d]. Then we have the formal expansion ofD^[n]:

D^[n]X

s2Sn

sgn(s)X Yⁿ

i1

g_s;i;

where the second P

runs over all termsgs;i of the polynomialFip s(i) in O₁[A₁;. . .;A_d].

PROPOSITION3.9. Let1nd. Then there is a unique monomial in the above formal expansion of D^[n] with highest lexicographic order (according to Ad;. . .; A1). Moreover, the p-adic order of this monomial (with coefficient) is minimal among thep-adic orders of all monomials in the above formal expansion.

REMARK3.10. We shall fix the uniques₀found in the proposition for the rest of the paper. The minimal p-adic order of this monomial (with coefficient) is equal to (7) while every row achieves its minimal order in Lemma 3.3c). We shall use this fact later.

PROOF. Denote byrthe least non-negative residue ofpmodd. Recall the n by n matrix rn: frijg_1i;jn where rij:dd^{ri j}_d e (ri j). The properties of this matrix can be found in [13, Lemma 3.1]. LetQ_n

i1h_s;ibe a highest-lexicographic-order-monomial in the formal expansion of D^[n].

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Then hs;i must be the highest-lexicographic-order-monomial in Fip s(i), which is easily seen to becip s(i)A^ip_d^d^s(i)orcip s(i)A^b_d^ip^d^s(i)^cAd ri;s(i)depending on r_i;s(i)0 or not, wherec_ip _s(i)2V1. We show first that

s(i)kfor any 1i;knwithr_ik0:

6

Suppose that (6) does not hold. Pick a pair (i;k) among the pairs failing (6) such thatjs(i) kjis minimal. Says(j)k. Define another permutation s⁰2Snbys⁰(i)kands⁰(j)s(i) whiles⁰(s)s(s) for all others. Denote by hs⁰;t the highest-lexicographic-order-monomial in F_tp _s⁰_(t). Then it is easy to see that the lexicographic order ofQ_n

t1h_s⁰_;t is strictly higher than that ofQ_n

t1h_s;t, which is a contradiction. Therefore (6) holds.

Notice that for permutationss⁰⁰2S_nsatisfying (6), the degree ofA_din Q_n

t1hs⁰⁰;t does not depend on the choice ofs⁰⁰, wherehs⁰⁰;t is the highest- lexicographic-order-monomial inF_tp _s⁰⁰_(t). Then the proof of [13, Lemma 3.2] shows that there exists a unique s₀2S_n such that Q_n

t1h_s₀_;t has highest lexicographic order among the corresponding monomials for all s⁰⁰2S_nsatisfying (6). In fact,s₀is exactly the permutation in [13, Lemma 3.2]. By the above discussion, this monomialQ_n

t1hs0;t also has the unique highest lexicographic order in the formal expansion ofD^[n].

Next we show that thep-adic order ofQ_n

i1h_s₀_;iis minimal (among the p-adic orders of the monomials in the formal expansion of D^[n]). Let Q_n

i1g_s;ibe an arbitrary monomial in the formal expansion of D^[n]. Then clearly ordp(gs;i) dîp_d^s(i)e ^pi ^s(i)r_d î;s(i)for all 1in. Sincehs0;iis the highest-lexicographic-order-monomial in F_ip _s₀_(i), one sees easily that ord_p(h_s₀_;i)^pi ^s⁰^(i)r_d î;s⁰⁽ⁱ⁾ for all 1in. From (6) it is easy to see that r_i;j r_i;s₀_(i)j s₀(i) for all 1i;jn. It follows that ord_p(Q_n

i1g_s;i) ord_p(Q_n

i1h_s₀_;i). p

REMARK3.11. In the proof of Proposition 3.9 we have noticed that the s₀is exactly the permutation in [13, Lemma 3.2]. Therefore, one can always taket00, that is,f_n⁰(~A)60 in [13, Lemma 3.5].

4. Newton polygon ofa1.

Recall that HP(A) lives on the real plane over the interval [0;d]. Be- cause of Remark 1.3, one only has to consider the part of NP_p(f) with slope

<1, that is, to consider the part of NPp(f) over the interval [0;d `]. This part is our focus of this section. Suppose for some 1kd `, the point

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(k;c0) is a vertex on HP(A). Then one notices that c₀X²

J1

X^k^J

i1

i=d_JX^`

J3

X^k^J

i1

(i 1)=d_J

for a sequence of nonnegative integers k₁;. . .;k_` such that k1. . .k`k. This sequence is unique because (k;c0) is a vertex.

From now on we fix such ak.

For our purpose we also fix the residue classes ofpmodd_Jfor allJ. Let r_J;ij be the least nonnegative residue of (ip j)modd_J. Let s₀ be the permutation in S_k which is the union of those permutations found in Proposition 3.9 locally at each pole PJ. Let sJ1 be the rational number defined by

sJ1 :(p 1)kJ1(kJ11)=2

d_J₁

PkJ1

i1r_J₁_;i;s₀_(i) d_J₁ 7

where and is taken according to J11;2 or J13. Let s₀:s₁ s`. Clearlys₀ c₀(p 1)<kd `.

Letaandbbe the integers chosen in Lemmas 3.4 and 3.5 (they depend only ond). LetQ⁰:Q(g^1=d¹;. . .;g^1=d^`).

LEMMA4.1. For any J₁1;2,1J`and for any n;i in their range, for p large enough, there is a polynomial G^n;i_J₁_;JinQ⁰(~P)[~A]such that

aK_J^n;i₁_;Jg^(p ^1)n=d^J¹U_J₁_;nG^n;i_J₁_;Jmodg^(p ^1)n=d^J¹^d1: For the case J₁3, one has a similar G^n;i_J₁_;Jsuch that

aK_J^n;i₁_;Jg^(p ¹⁾⁽ⁿ ^1)=d^J¹UJ1;nG^n;i_J₁_;Jmodg^(p ¹⁾⁽ⁿ ^1)=d^J¹^d1;

where U_J₁_;n is a p-adic unit depending only on the the row index(J₁;n).

PROOF. We use the same technique as [13], so we only outline our proof here for the case J1J1. Let n_j2[ a;a] for j6J1. Let n_J₁npP

j6J1n_j i. For any~n(n₁;. . .;n_`) in this range, we have Fj;nj g^nj^djQjmodg^nj^dj^d1

and

FJ1;nJ1 g^nJ^dJ¹¹VJ1QJ1modg^nJ^dJ¹¹^d1;