Generic T-adic exponential sums in one variable

Journal of Number Theory 166 (2016) 276–297

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Journal of Number Theory www.elsevier.com/locate/jnt

Generic T -adic exponential sums in one variable ✩ Chunlei Liu a , Wenxin Liu b,∗ , Chuanze Niu c a

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, PR China b Department of Mathematics, Tianjin Chengjian University, Tianjin 300384, PR China c School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, PR China

a r t i c l e

i n f o

Article history: Received 18 November 2015 Received in revised form 20 February 2016 Accepted 21 February 2016 Available online 1 April 2016 Communicated by D. Wan

a b s t r a c t The classical and T -adic exponential sums associated to a Laurent polynomial are studied. An explicit polygon is proved to be the generic Newton polygon of the C-functions of the T -adic exponential sums. It gives the generic Newton polygons of the L-functions of p-power order exponential sums. © 2016 Elsevier Inc. All rights reserved.

MSC: 11L07 14F30 Keywords: T -adic Exponential sums C-functions Arithmetic polygon

✩

Supported by National Natural Science Foundation of China (Grant No. 11401285).

* Corresponding author. E-mail addresses: [email protected] (C. Liu), [email protected] (W. Liu), [email protected] (C. Niu). http://dx.doi.org/10.1016/j.jnt.2016.02.023 0022-314X/© 2016 Elsevier Inc. All rights reserved.

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1. Introduction 1.1. Classical exponential sums In this subsection, we recall the definition of the classical exponential sums of Laurent polynomials over finite fields. We fix a prime number p. Let Zp be the ring of p-adic integers, Qp the field of p-adic numbers, and Qp a fixed algebraic closure of Qp . Let Fq be a finite extension of Fp of degree b so that q = pb , Qq the unramified extension of Qp with residue field Fq , and Zq the integer ring of Qq .  Let Δ be an integral interval in R containing the origin, f (x) = au xu ∈ Fq [x±1 ] u∈Δ

be a Laurent polynomial. Let ψ be a character of Zp of order pm and write πψ = ψ(1) −1. Let x ˆ denote the Teichmüller lift of x ∈ Fq . Definition 1.1. The sum Sf,ψ (k) =



ψ(Tr Qqk /Qp (fˆ(ˆ x))) ∈ Zp [πψ ]

x∈F×k q

is called a pm -order exponential sum. And the function Lf,ψ (s) = exp(

∞ 

k=1

Sf,ψ (k)

sk ) ∈ 1 + sZp [πψ ][[s]] k

is called an L-function of a pm -order exponential sum. In general, the above L-function Lf,ψ (s) is rational in s. In particular, when f is nondegenerate, Lf,ψ (s) is a polynomial of degree pm−1 Vol(Δ), as was shown by Adolphson– Sperber [2] for m = 1, and by Liu–Wei [12] for all m ≥ 1. For a non-degenerate f , the location of the zeros of Lf,ψ (s) becomes an important issue. The p-adic theory of such L-functions was developed by Dwork, Bombieri [5,6], Adolphson–Sperber [1,2], Wan [14,15], Blache [3] for ψ of order p and by Liu–Wei [12] for all ψ. The reciprocal zeros of Lf,ψ (s) are algebraic integers, and one can talk about their archimedean and p-adic absolute values (their l-adic absolute values are trivial for prime l different from p by results of Deligne [8]). The archimedean absolute values for ψ of order p were determined by Denef–Loeser [9]. The p-adic absolute values were investigated by Adolphson–Sperber [1,2], Wan [14,15], and Liu–Wei [12]. 1.2. T -adic exponential sums In order to study the classical pm -order exponential sums for all m simultaneously, Liu–Wan [11] introduced the T -adic exponential sums, where T is a variable.

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Definition 1.2. The sum Sf (k, T ) =



Tr Q

(1 + T )

qk

ˆ

x)) /Qp (f (ˆ

∈ Zp [[T ]]

x∈F×k q

is called a T -adic exponential sum. And the function Lf (s, T ) = exp(

∞ 

Sf (k, T )

k=1

sk ) ∈ 1 + sZp [[T ]][[s]] k

is called a T -adic L-function of a T -adic exponential sum. The T -adic exponential sums interpolate the classical exponential sums. In fact, we have Sf,ψ (k) = Sf (k, T |πψ ) and Lf,ψ (s) = Lf (s, T |πψ ). As can be seen in Liu–Wan [11], instead of the usual way of extending the methods for ψ of order p to the case of pm -order, the T -adic exponential sum has the novel feature that it can sometimes transfer a known result for one non-trivial ψ to all. Definition 1.3. The function Cf (s, T ) = exp(

∞ 

−(q k − 1)−1 Sf (k, T )

k=1

sk ) k

is called a C-function of a T -adic exponential sum. The L-function and C-function determine each other: −1

Lf (s, T ) = Cf (s, T )Cf (qs, T )

and Cf (s, T ) =

∞ 

Lf (q j s, T ).

j=0

By the last identity, one sees that Cf (s, T ) ∈ 1 + sZp [[T ]][[s]]. We view Lf (s, T ) and Cf (s, T ) as power series in the single variable s with coefficients in the T -adic complete field Qp ((T )). The C-function Cf (s, T ) was shown to be T -adic entire in s by Liu–Wan [11], so it behaves better than the L-functions. Therefore in the T -adic sense, the T -adic C-function behaves better than the T -adic L-function. Let C(Δ) be the cone generated by Δ, and M (Δ) = C(Δ) ∩ Z. There is a degree function deg on C(Δ) which is R≥0 -linear and takes the values 1 on the nonzero end points of Δ. That is, the degree function is defined by the formula  deg(a) =

0, a d(sgn(a)) ,

a = 0, a = 0,

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where d(ε) is the nonzero endpoint of Δ with sign ε = ±1. For a ∈ / C(Δ), we define deg(a) = +∞. Definition 1.4. A convex function on R≥0 which is linear between consecutive integers with initial value 0 is called the infinite Hodge polygon of Δ if its slopes between con∞ secutive integers are the numbers deg(a), a ∈ M (Δ). We denote this polygon by HΔ . Let NP T (Cf (s, T )) denote the T -adic Newton polygon of the C-function Cf (s, T ). ∞ Liu–Wan [11] proved the following Hodge bound NP T (Cf (s, T )) ≥ ordp (q)(p − 1)HΔ . 1.3. Main results  In this subsection, we state our main results. Let f (x) = au xu ∈ Fq [x±1 ] be u∈Δ  a Laurent polynomial with au =  0, where I is the set of the non-zero endpoints u∈I

of Δ. When f is a polynomial and ψ is of order p, Zhu [16–18] and Blache–Férard [4] studied the Newton polygon of the L-function Lf (s, π) by using Dwork’s p-adic analytic methods. Definition 1.5. Let 0 = a ∈ M (Δ). We define  δ∈ (a) =

1, {deg(a)} = {deg(pi)} for some i with ia > 0, deg(i) < {deg(a)}, 0, otherwise,

where {·} is the fractional part of a real number. We also define δ∈ (0) = 0. Definition 1.6. A convex function on R≥0 which is linear between consecutive integers with initial value 0 is called the arithmetic polygon of Δ if its slopes between consecutive integers are the numbers Δ (a) = (p − 1) deg(a) − δ∈ (a), a ∈ M (Δ), where · is the least integer equal or greater than a real number. We denote it by pΔ . We have the following position relation between the arithmetic and Hodge polygons. Theorem 1.7. We have ∞ pΔ ≥ (p − 1)HΔ .

Moreover, they coincide at the point Vol(Δ).

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Let D be the least common multiple of the nonzero endpoint(s) of Δ, and NP πψ (Cf (s, πψ )) denote the πψ -adic Newton polygon of the C-function Cf (s, πψ ). The main results are the following theorems. Theorem 1.8. If p > 3D, then NP T (Cf (s, T )) ≥ ordp (q)pΔ . Theorem 1.9. If p > 3D, then there is a non-zero polynomial H(y) ∈ Fq [yu | u ∈ Δ] such that NP T (Cf (s, T )) = ordp (q)pΔ if and only if H((au )u∈Δ ) = 0. Theorem 1.10. If p > 3D, then for all m ≥ 1 we have NP πψ (Cf (s, πψ )) = ordp (q)pΔ if and only if H((au )u∈Δ ) = 0. The polynomial H(y) in the above theorem is called the Hasse polynomial of Δ. Note that, if p  D, then L(s, πψ ) is a polynomial of degree pm−1 Vol(Δ) for all m ≥ 1 by results of Adolphson–Sperber [2] and Liu–Wei [12]. We shall deduce the following. Theorem 1.11. If p > 3D, then for all m ≥ 1 we have NP πψ (Lf (s, πψ )) ≥ ordp (q)pΔ on [0, pm−1 Vol(Δ)] with equality holding if and only if H((au )u∈Δ ) = 0. 2

If Δ = [0, d], when m ≥ m0 = 1+logp ( b(d−1) ), the slopes of the q-adic Newton poly8d gon of Lf (s, πψ ) have some stable properties as was proved by Davis–Wan–Xiao [7], i.e., the q-adic Newton slopes of Lf (s, πψ ) form a union of dpm0 −1 arithmetic progressions, with increment pm0 −m . As immediate corollaries of Theorem 1.11, the q-adic Newton slopes of Lf (s, πψ ) form a union of Vol(Δ) arithmetic progressions generically. Corollary 1.12. When Δ = [0, d], if p > 3d, then the q-adic Newton polygon of Lf (s, πψ ) lies above the convex polygon with initial point 0 and whose slopes between consecutive integers are d−1 

{

a=0

i Δ (a) + m | i = 0, 1, · · · , pm−1 − 1}. pm−1 p − pm−1

Moreover, the two polygons coincide if and only if H((au )u∈Δ ) = 0.

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Corollary 1.13. When Δ = [−e, d], if p > 3D, then the q-adic Newton polygon of Lf (s, πψ ) lies above the convex polygon with initial point 0 and whose slopes between consecutive integers are the union (counting multiplicities) of {

i Δ (a) + m | i = 0, 1, · · · , pm−1 − 1}, a = −e, · · · , −1, 0, 1, · · · , d − 1. pm−1 p − pm−1

Moreover, the two polygons coincide if and only if H((au )u∈Δ ) = 0. 2. The T -adic Dwork theory In this section we review the T -adic analogue of Dwork theory on exponential sums. Let ∞ pi +∞   t E(t) = exp( ) = λi ti ∈ 1 + tZp [[t]] i p i=0 i=0

be the p-adic Artin–Hasse exponential series. Define a new T -adic uniformizer π of Qp ((T )) by the formula E(π) = 1 + T . Let π 1/D be a fixed D-th root of π. Let 

L(Δ) = {

ci π deg(i) xi | ci ∈ Zq [[π 1/D ]]}.

i∈M (Δ)

If u ∈ Δ, it is clear that E(πxu ) ∈ L(Δ). Write Ef (x) :=



E(πˆ au xu ),

au =0

then it is an element of L(Δ) since L(Δ) is a ring. Note that the Galois group of Qq over Qp acts on L(Δ) but keeping π 1/D as well as the variable x fixed. Let σ be the Frobenius element in the Galois group such that σ(ζ) = ζ p if ζ is a (q − 1)-th root of unity. Let Ψp be the operator on L(Δ) defined by the formula Ψp (



i∈M (Δ)

ci xi ) =



cpi xi .

i∈M (Δ)

Then the operator Ψ = σ −1 ◦ Ψp ◦ Ef acts on the T -adic Banach module B={



ci π deg(i) xi ∈ L(Δ) | ordT (ci ) → +∞ if deg(i) → +∞}.

i∈M (Δ) 1

We call it a Dwork’s T -adic semi-linear operator because it is semi-linear over Zq [[π D ]]. The b-iterate Ψb is linear over Zq [[π 1/D ]], since

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Ψb = Ψbp ◦

b−1 

i

i

Efσ (xp ).

i=0

One can show that Ψ is completely continuous in the sense of Serre [13]. So the deter1 1 minants det(1 − Ψb s | B/Zq [[π D ]]) and det(1 − Ψs | B/Zp [[π D ]]) are well-defined. We now state the T -adic Dwork trace formula given by Liu–Wan [11]. Theorem 2.1. We have 1

Cf (s, T ) = det(1 − Ψb s | B/Zq [[π D ]]). 1

The relationship between the T -adic Newton polygons of det(1 − Ψb s | B/Zq [[π D ]]) 1 and det(1 − Ψs | B/Zp [[π D ]]) is established by Liu–Niu [10]. Write 1

det(1 − Ψs | B/Zp [[π D ]]) =

+∞ 

(−1)i ci si .

i=0

Theorem 2.2. The T -adic Newton polygon of Cf (s, T ) is the lower convex closure of the points (m, ordT (cbm )), m = 0, 1, · · · . 3. Arithmetic estimate In this section, A is a finite subset of M (Δ) × Z/(b), and τ is a permutation of A. We shall estimate 

deg(pa − τ (a)),

a∈A

where deg(i, u) = deg(i). However, except in the ending paragraph, we assume that Δ = [0, d], A is a finite subset of {0, 1, 2, · · · } × Z/(b). Write  {x}, if {x} = 0,  {x} = 1 + x − x = 1, if {x} = 0. Lemma 3.1. We have m 

(δ< − δ∈ )(a) = #{1 ≤ a ≤ d{

a=0

where

m  m pa } | { } < { } }, d d d

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 δ< (a) =

1, 0,

283

 if { ad } < { pa d } otherwise.

Proof. Note that both δ∈ and δ< have a period d and have initial value 0. So we may assume that m < d. We have m 

δ∈ (a) =

a=0

a−1 m   a=1

1=

m−1  i=1

i=1

pi≡a(d)

m 

1 = #{1 ≤ i < m | i < d{

a=i+1

pi } ≤ m}. d

pi≡a(d)

And, by definition, m 

δ< (a) = #{1 ≤ a ≤ m | a < d{

a=0

pa }}. d

The lemma now follows. 2 Lemma 3.2. Let A, B, C be sets with A finite, and τ a permutation of A. Then #{a ∈ A | τ (a) ∈ B, a ∈ C} ≥ #{a ∈ A | a ∈ B, a ∈ C} − #{a ∈ A | a ∈ / B, a ∈ / C}. Proof. We have #{a ∈ A | a ∈ B, τ (a) ∈ B, a ∈ C} ≥ #{a ∈ A | a ∈ B, a ∈ C} − #{a ∈ A | a ∈ B, τ (a) ∈ / B}, and #{a ∈ A | a ∈ / B, τ (a) ∈ B, a ∈ C} ≥ #{a ∈ A | a ∈ / B, τ (a) ∈ B} − #{a ∈ A | a ∈ / B, a ∈ / C}. Note that #{a ∈ A ∩ B | τ (a) ∈ / B} = #{a ∈ A \ B | τ (a) ∈ B}. So #{a ∈ A | τ (a) ∈ B, a ∈ C} ≥ #{a ∈ A | a ∈ B, a ∈ C} − #{a ∈ A | a ∈ / B, a ∈ / C}. The lemma is proved. 2 For a ∈ A, we define  τ δ< (a)

=

1, if {deg(τ (a))} < {p deg(a)} , 0, otherwise.

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Lemma 3.3. If p > 3d, 0 ≤ l < b and A = ({0, 1, · · · , m − 1} × Z/(b)) ∪ {(m, i0 ), · · · , (m, il−1 )}, then 

τ δ< (a) ≥ b

m−1 

(δ< − δ∈ )(a) + l(δ< − δ∈ )(m).

a=0

a∈A

Proof. Firstly, we assume that l = 0. We have 

τ δ< (a) ≥ #{a ∈ A | {deg(τ (a))} ≤ {

a∈A

m−1  } < {p deg(a)} }. d

Applying the last lemma with B = {a ∈ A | {deg(a)} ≤ {

m−1  }} d

and C = {a ∈ A | {

m−1  } < {p deg(a)} }, d

we get 

τ δ< (a) ≥ #{a ∈ A | {deg(a)} ≤ {

a∈A

m−1  } < {p deg(a)} } d

− #{a ∈ A | {deg(a)} > {

m−1  } ≥ {p deg(a)} }. d

Since we have m−1  } < {p deg(a)} } d m−1 m−1  m−1  pa = b([ ] + 1)#{1 ≤ a ≤ d{ } |{ } < { } }, d d d d #{a ∈ A | {deg(a)} ≤ {

and m−1  } ≥ {p deg(a)} } d m−1 m−1  m−1  pa = b[ ]#{d ≥ a > d{ } |{ } ≥ { } } d d d d m−1 m−1  m−1  pa  = b[ ]#{1 ≤ a ≤ d{ } |{ } < { } }, d d d d #{a ∈ A | {deg(a)} > {

it follows that

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τ δ< (a) ≥ b#{1 ≤ a ≤ d{

a∈A

285

m−1  m−1  m−1  pa } |{ } < { } } ≥ b (δ< − δ∈ )(a). d d d a=0

Secondly, we assume that δ∈ (m) = 0. Extending the action of τ trivially from A to {0, 1, · · · , m} × Z/(b). By what we just proved, 

τ δ< (a) ≥ b

m 

(δ< − δ∈ )(a).

a=0

a∈{0,1,··· ,m}×Z/(b)

It follows that 

τ δ< (a) ≥ b

m 

(δ< − δ∈ )(a) − (b − l)δ< (m)

a=0

a∈A

≥b

m−1 

(δ< − δ∈ )(a) + l(δ< − δ∈ )(m).

a=0

Finally, we assume that δ∈ (m) = 1. We have 

τ δ< (a) ≥ #{a ∈ A | {deg(τ (a))} ≤ {

a∈A

m−1  } < {p deg(a)} }. d

Applying the last lemma with B = {a ∈ A | {deg(a)} ≤ {

m−1  }} d

and C = {a ∈ A | {

m−1  } < {p deg(a)} }, d

we get 

τ δ< (a) ≥ #{a ∈ A | {deg(a)} ≤ {

a∈A

m−1  } < {p deg(a)} } d

− #{a ∈ A | {deg(a)} > {

m−1  } ≥ {p deg(a)} }. d

Since we have m−1  } < {p deg(a)} } d m−1 m−1  m−1  pa = b([ ] + 1)#{1 ≤ a ≤ d{ } |{ } < { } }, d d d d #{a ∈ A | {deg(a)} ≤ {

and

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286

m−1  } ≥ {p deg(a)} } d m−1 m−1  m−1  pa = b[ ]#{d ≥ a > d{ } |{ } ≥ { } } + l1{ m } ≥{ m−1 } ≥{ pm } d d d d d d d m−1  m−1 m−1  pa  ]#{1 ≤ a ≤ d{ } |{ } < { } } + l1{ m } ≥{ m−1 } ≥{ pm } , = b[ d d d d d d d #{a ∈ A | {deg(a)} > {

it follows that  m−1  m−1  pa τ  } |{ } < { } } − l1{ md } ≥{ pm δ< (a) ≥ b#{1 ≤ a ≤ d{ d } d d d a∈A

=b

m−1 

(δ< − δ∈ )(a) + l(δ< − δ∈ )(m).

a=0

The proof of the lemma is completed. 2 Proposition 3.4. If p > 3d, A is of cardinality bm + l with 0 ≤ l < b, then 

τ (p deg(a) − deg(a) + δ< (a)) ≥ bpΔ (m) + lΔ (m).

a∈A

Moreover, the strict inequality holds if A is not of the form ({0, 1, · · · , m − 1} × Z/(b)) ∪ {(m, i0 ), · · · , (m, il−1 )}. Proof. Let A ⊂ M (Δ) × Z/(b) be an arbitrary set of the form ({0, 1, · · · , m − 1} × Z/(b)) ∪ {(m, i0 ), · · · , (m, il−1 )}. Define a permutation τ0 of A which agrees with τ on A ∩ A and acts trivially on A \ A, then 

τ δ< (a) ≥

a∈A



τ0 δ< (a) − #(A \ A).

a∈A

By Lemma 3.3, we have  a∈A

τ δ< (a) ≥ b

m−1 

(δ< − δ∈ )(a) + l(δ< − δ∈ )(m) − #(A \ A).

a=0

Since p > 3d, if deg(a) > deg(b), we have p deg(a) − deg(a) ≥ 2 + p deg(b) − deg(b). Note that #(A \ A) = #(A \ A), and A is the set of cardinality bm + l consisting of elements in M (Δ) × Z/(b) with the least possible degrees, therefore,

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(p deg(a) − deg(a))

a∈A

=



(p deg(a) − deg(a)) + (

a∈A

≥



287





−

a∈A\A

)(p deg(a) − deg(a))

a∈A\A

(p deg(a) − deg(a)) + 2#(A \ A)

a∈A

=b

m−1 

(p deg(a) − deg(a)) + l(p deg(m) − deg(m)) + 2#(A \ A).

a=0

The proposition now follows from the definitions of pΔ (m) and Δ (m). 2 We now assume that Δ ⊂ Z, and A is a finite subset of M (Δ) × Z/(b) of cardinality bm. For an integer m ≥ 1, we define Am = {a ∈ M (Δ) | Δ (a) ≤ pΔ (m) − pΔ (m − 1)}. Proposition 3.5. If p > 3D, then 

deg(pa − τ (a)) ≥ bpΔ (m).

a∈A

Moreover, the strict inequality holds if m is a turning point of pΔ , and A = Am × Z/(b). Proof. For the permutation τ , if we define ⎧ ⎪ ⎨ 0, τ0 (a) = τ (0), ⎪ ⎩ τ (a),

a = 0, a = τ −1 (0), a = 0, τ −1 (0),

then we have 

deg(pa − τ (a)) ≥

a∈A



deg(pa − τ0 (a)).

a∈A

We define sgn((i, u)) = sgn(i). Write Bi = {a ∈ A | sgn(a) = (−1)i } and Bij = {a ∈ A | sgn(a) = (−1)i , sgn(τ0 (a)) = (−1)j }. Define a new permutation τ1 as follows:

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288

• τ1 = τ0 on B11 and B22 ; • τ1 maps B1 \ B11 to B1 \ τ0 (B1 ); • τ1 maps B2 \ B22 to B2 \ τ0 (B2 ). We have 

deg(pa − τ (a)) ≥

a∈A

≥





deg(pa − τ0 (a)) ≥

a∈A



deg(pa − τ1 (a))

a∈A

(p deg(a) − deg(τ1 (a)) + 1{deg(τ1 (a))} <{p deg(a)} ).

a∈A

The proposition now follows the last one. 2 4. The Dwork semi-linear operator Let O(π α ) denote the term whose π-adic order is not less than α. In this section we prove the following theorem. Theorem 4.1. Let p > 3D. Then ordπ (cbm ) ≥ bpΔ (m). Moreover, if m < Vol(Δ) is a turning point of pΔ , then cbm = ±Norm(det(γpi−j )i,j∈Am ) + O(π bpΔ (m)+1/D ), where Norm is the norm map from Qq (π 1/D ) to Qp (π 1/D ). Write 

Ef (x) =

γi xi .

i∈M (Δ)

Lemma 4.2. We have 

γi = π deg(i) 

jnj =i

 j∈Δ nj =deg(i)



λn j a ˆj j + O(π deg(i) +1 ). n

j∈Δ

j∈Δ

Proof. This follows from the fact that γi =

  j∈Δ

jnj =i,nj ≥0



π j∈Δ

nj

 j∈Δ

n

λn j a ˆj j ,

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289

and the fact that 

nj ≥ deg(i) if

j∈Δ



2

jnj = i.

j∈Δ

Fix a normal basis {ξ¯u }u∈Z/(b) of Fq over Fp . Let ξu be their Teichmüller lift of ξ¯u . Then {ξu }u∈Z/(b) is a normal basis of Qq over Qp , and σ acts on the basis {ξu }u∈Z/(b) as a permutation. Let (γ(i,u),(j,ω) )i,j∈M (Δ),1≤u,ω≤b denote the matrix of Ψ on B ⊗Zp Qp (π 1/D ) with respect to the basis {ξu xi }i∈M (Δ),1≤u≤b . Lemma 4.3. We have γ(r,w),(l,u) = O(π deg(pr−l) ). Proof. This follows from Lemma 4.2 and the fact that (ξu γpr−l )σ

−1

=

b 

2

γ(r,w),(l,u) ξω .

ω=1

Lemma 4.4. If A is a finite subset of M (Δ), then det((γi,j )i,j∈A×Z/(b) ) = ±Norm(det((γpi−j )i,j∈A )), where Norm is the norm map from Qq (π 1/D ) to Qp (π 1/D ). Proof. Let V =



Qq (π 1/D )ei be a vector space over Qq (π 1/D ), and F be the linear

i∈A

operator on V whose matrix with respect to the basis {ei } is (γpi−j )i,j∈A . Let σ act on V coordinate-wise. It is easy to see that (γi,j )i,j∈A×Z/(b) is the matrix of σ −1 ◦ F over Qp (π 1/D ) with respect to the basis {ξu ei }i∈A,u∈Z/(b) . Therefore det((γi,j )i,j∈A×Z/(b) ) = det(σ −1 ◦ F ) = ± det(F | Qp (π 1/D )) = ±Norm(det(F )). The lemma is proved. 2 Proof of Theorem 4.1. We have cbm =



det((γi,j )i,j∈A ),

A

where A runs over all subsets of M (Δ) × Z/(b) with cardinality bm. By Proposition 3.5 and Lemma 4.3, if p > 3D, A is a subset of M (Δ) × Z/(b) with cardinality bm, and SA is the permutation group of A, then ordπ det((γi,j )i,j∈A ) ≥ max

τ ∈SA

 a∈A

ordπ (γa,τ (a) ) ≥ max

τ ∈SA

 a∈A

deg(pa − τ (a)) ≥ bpΔ (m).

290

C. Liu et al. / Journal of Number Theory 166 (2016) 276–297

Moreover, if m is a turning point of pΔ , and A = Am × Z/(b), then the strict inequality holds. By Lemma 4.4, we have det(γi,j )i,j∈Am ×Z/(b) = ±Norm(det((γpi−j )i,j∈Am )). The theorem is proved. 2 5. The Hasse polynomial In this section we study det(γpi−j )i,j∈Am . 0 Definition 5.1. For each positive integer m, we define Sm to be the set of permutations τ of Am satisfying τ (0) = 0, and

τ (a) ≥ deg(pa) − deg(pa) − deg(n), a = 0, d(sgn(a)) where n is the element of maximal degree in Am ∩ (sgn(a)N). Lemma 5.2. Let p > 3D, m < Vol(Δ) a turning point of pΔ , and τ a permutation of Am . Then 

deg(pa − τ (a)) ≥ pΔ (m),

a∈Am 0 with equality holding if and only if τ ∈ Sm .

Proof. We assume that M (Δ) = N. The other cases can be proved similarly. In this case, sgn(a) = +1 if a = 0, and the element n of maximal degree in Am ∩ sgn(a)N is m − 1. Let d be the nonzero endpoint of Δ. For a = 0, we have deg(pa − τ (a)) ≥ 0 with equality holding if and only if τ (a) = 0. For a = 0, we have 

pa − n pa − τ (a) ≥  d d

with equality holding if and only if pa pa − n τ (a) ≥ − . d d d It follows that

C. Liu et al. / Journal of Number Theory 166 (2016) 276–297



deg(pa − τ (a)) ≥

n 

291

deg(pa − n) = pΔ (m)

a=1

a∈Am

0 with equality holding if and only if τ ∈ Sm . The theorem is proved. 2

Definition 5.3. For each positive integer m, we define Hm (y) =



sgn(τ )

0 τ ∈Sm



 

i∈Am



j∈Δ



n

λnj yj j ∈ Zp [yj | j ∈ Δ].

j∈Δ

jnj =pi−τ (i)

nj =deg(pi−τ (i))

j∈Δ

Lemma 5.4. Let p > 3D, and m < Vol(Δ) a turning point of pΔ . Then det(γpi−j )i,j∈Am = Hm ((ˆ aj )j∈Δ )π pΔ (m) + O(π pΔ (m)+1/D ). Proof. Let Sm be the set of permutations of Am . We have det(γpi−j )i,j∈Am =

 

γpi−τ (i) .

τ ∈Sm i∈Am

2

The lemma now follows from Lemma 4.2 and Lemma 5.2.

Definition 5.5. The reduction of Hm modulo p is denoted as H m , and is called the Hasse polynomial of Δ at m. Theorem 5.6. If p > 3D, and m < Vol(Δ) is a turning point of pΔ , then H m is non-zero. Proof. Define deg(yj ) = |j|. Then  i∈Am

 



j∈Δ



n

λnj yj j

j∈Δ

jnj =pi−τ (i)

nj =deg(pi−τ (i))

j∈Δ

has degree  i∈Am

|pi − τ (i)| =



sgn(i)(pi − τ (i))

i∈Am

=p



sgn(i)i −

i∈Am

≥p





sgn(i)τ (i)

i∈Am

sgn(i)i −

i∈Am

≥ (p − 1)

 i∈Am



i∈Am

sgn(i)i,

sgn(τ (i))τ (i)

292

C. Liu et al. / Journal of Number Theory 166 (2016) 276–297

with equality holding if and only if τ preserves the sign. Therefore it suffices to show that the reduction of 1 Hm (y) =

 1 τ ∈Sm





sgn(τ )

i∈Am





j∈Δ



n

λnj yj j

j∈Δ

jnj =pi−τ (i)

nj =deg(pi−τ (i))

j∈Δ

1 0 is nonzero, where Sm consists of the sign-preserving permutations of Sm . The proof is due to the maximal-monomial-locating technique of Zhu [16], as was used by Blache– Férard [4]. For Δ = [0, d], a general twisted Hasse polynomial was shown to be nonzero in 1 Liu–Niu [10]. We shall prove that the reduction of Hm (y) is non-zero when Δ = [−e, d]. 1 Firstly, we show the set Sm is nonempty. Let n1 and −n2 be the integers of the maximal degree in Am ∩ N and Am ∩ (−N) respectively, then n1 ≤ d − 1 and n2 ≤ e − 1. For 1 ≤ k ≤ n1 , write pk = qk d + rk with 1 ≤ rk ≤ d − 1. For −n2 ≤ l ≤ −1, write rk rk −n1 1 pl = −ql e − rl with 1 ≤ rl ≤ e − 1. Then τ ∈ Sm is equivalent to τ (k)  d ≥ d − d

for 1 ≤ k ≤ n1 , and

τ (l) −e

≥

rl e

−

rl −n2 e 

for −n2 ≤ l ≤ −1. Assume we have the order

rk1 < · · · < rks ≤ n1 < rks+1 < · · · < rkn1 , and rl1 < · · · < rlt ≤ n2 < rlt+1 < · · · < rln2 , 1 then τ ∈ Sm if and only if τ (ki ) ≥ rki for 1 ≤ i ≤ s and τ (lj ) ≤ −rlj for 1 ≤ j ≤ t. 1 Therefore, such τ exists and Sm is nonempty. 1 (y) appear in As a polynomial of yd and y−e , the leading terms of Hm



sgn(τ )

1 τ ∈Sm

×

n1 

(k) [ pk−τ ] δτ (k) d

λ[ pk−τ (k) ] λδτ (k) yd

k=1 −1 

y

(k) } d{ pk−τ d

d

[ pl−τ (l) ] δτ (l)

λ[ pl−τ (l) ] λδτ (l) y−e −e

k=−n2

−e

y

(l) −e{ pl−τ } −e

,

where δτ (k) = 1d(pk−τ (k)) and δτ (l) = 1e(pl−τ (l)) . The coefficients are p-adic units since 1 λi = i!1 for 0 ≤ i ≤ p − 1. To show the reduction of Hm (y) is nonzero, it suffices to show 1 among the leading terms of Hm (y), there is exactly one maximal monomial with respect to the lexicographic order. 1 If τ ∈ Sm , then n1 −1   pk − τ (k) pl − τ (l) ]+ ] [ [ d −e

k=1

=

n1  k=1

l=−n2

qk +

−1  l=−n2

ql +

n1 −1   rk − τ (k) r + τ (l) ]+ ] [ [ l d e

k=1

l=−n2

C. Liu et al. / Journal of Number Theory 166 (2016) 276–297

=

n1 

qk +

k=1

≤

n1 

−1 

ql +

l=−n2

qk +

k=1

−1 

293

s t   rl + τ (lj ) rk − τ (ki ) ]+ ] [ i [ j d e i=1 j=1

ql ,

l=−n2

with the equality holding if and only if τ (ki ) = rki for 1 ≤ i ≤ s and τ (lj ) = −rlj for 1 ≤ j ≤ t. Therefore the leading terms appear for such τ that τ (ki ) = rki for 1 ≤ i ≤ s and τ (lj ) = −rlj for 1 ≤ j ≤ t. For s < i ≤ n1 and t < j ≤ n2 , we have pl −τ (l )

(ki ) δτ (ki ) = δτ (lj ) = 1, d{ pki −τ } = rki −τ (ki ) and −e{ j −e j } = −rlj −τ (lj ). Therefore d among the leading terms, the maximal monomial with respect to the lexicographic order appears exactly when

τ (ki ) = rki , 1 ≤ i ≤ s; τ (lj ) = rlj , 1 ≤ j ≤ t; τ (ki ) = max{{1, 2, · · · , n1 } \ {τ (k1 ), · · · , τ (ki−1 )}}, i = s + 1, · · · , n1 ; τ (lj ) = min{{−n2 , · · · , −1} \ {τ (l1 ), · · · , τ (lj−1 )}}, i = t + 1, · · · , n2 . The theorem is proved. 2 Definition 5.7. We define H =



H m , where the product is over all turning points

m

m < Vol(Δ) of pΔ . Theorem 5.8. If p > 3D, then H is non-zero. Proof. This follows from the last theorem. 2 6. Proof of the main theorems We shall prove Theorems 1.7, 1.8, 1.9, 1.10 and 1.11. Firstly we prove Theorem 1.7, which says that ∞ pΔ ≥ (p − 1)HΔ ,

with equality holding at the point Vol(Δ). Proof of Theorem 1.7. We assume that Δ = [0, d]. The other cases can be proved similarly. It suffices to show that ∞ pΔ ≥ (p − 1)HΔ on [0, d],

with equality holding at the point d. Let 0 < m ≤ d. We have

C. Liu et al. / Journal of Number Theory 166 (2016) 276–297

294



pΔ (m) =

(

(p − 1)a  − δ∈ (a)) d

(

a pa  −   + δ< (a) − δ∈ (a)) d d

(

 pa  − 1) + (δ< − δ∈ )(a) d

0≤a


=

0≤a


=

1≤a


=

1≤a
=p

[

pa ]+ d

1≤a


(δ< − δ∈ )(a)

1≤a
 pa  a − { }+ d d

1≤a
1≤a
∞ = (p − 1)HΔ (m) +



1

1≤a
 pa  a pa − { } + #{1 ≤ a < m | d{ } ≥ m}. d d d

1≤a
1≤a
In particular, we have ∞ pΔ (d) = (p − 1)HΔ (d).

Note that  a  pa  a pa − − { } + #{1 ≤ a < m | d{ } ≥ m} ≥ d d d d

1≤a
1≤a
1≤a
 1≤a
{

pa } ≥ 0. d

d{ pa d }
It follows that ∞ pΔ (m) ≥ (p − 1)HΔ (m).

2

We now prove Theorem 1.8, which says that NP T (Cf (s, T )) ≥ ordp (q)pΔ if p > 3D. Proof of Theorem 1.8. The theorem now follows from Theorem 2.2 and the estimate in Theorem 4.1. 2 Lemma 6.1. Let A(s, T ) be a T -adic entire series in s with unitary constant term. If 0 = |t|p < 1, then NP t (A(s, t)) ≥ NP T ((A(s, T )), where NP t (A(s, t)) denotes the t-adic Newton polygon of A(s, t). Moreover, the equality holds for one t if and only if it holds for all t.

C. Liu et al. / Journal of Number Theory 166 (2016) 276–297

295

Proof. Write A(s, T ) =

∞ 

ai (T )si .

i=0

The inequality follows from the fact that ai (T ) ∈ Zq [[T ]]. Moreover, NP t (A(s, t)) = NP T ((A(s, T )) if and only if ai (T ) ∈ T e Zq [[T ]]× for every turning point (i, e) of the T -adic Newton polygon of A(s, T ). It follows that the equality holds for one t if and only if it holds for all t. 2 Lemma 6.2. Let p > 3D and ψ be a character of Zp of order pm , if the equality NP πψ ((Cf (s, πψ )) = ordp (q)pΔ holds for one m ≥ 1, then it holds for all m ≥ 1, and we have NP T (Cf (s, T ) = ordp (q)pΔ . Proof. This follows from Theorem 4.1 and Lemma 6.1. 2 Lemma 6.3. Let ψ be a character of Zp of order pm , then NP πψ (Cf (s, πψ )) = ordp (q)pΔ if and only if NP πψ (Lf (s, πψ )) = ordp (q)pΔ on [0, pm−1 Vol(Δ)]. Proof. Assume that Lf (s, πψ ) =

pm−1 Vol(Δ)

(1 − βi s). Then

i=1

Cf (s, πψ ) =

∞  j=0

m−1

j

Lf (q s, πψ ) =

∞ p 

Vol(Δ) 

j=0

i=1

(1 − βi q j s).

Therefore the slopes of the q-adic Newton polygon of Cf (s, πψ ) are the numbers j + ordq (βi ), 1 ≤ i ≤ pm−1 Vol(Δ), j = 0, 1, · · · .

C. Liu et al. / Journal of Number Theory 166 (2016) 276–297

296

One can show that the slopes of pΔ are the numbers j(p − 1) + pΔ (i) − pΔ (i − 1), 1 ≤ i ≤ pm−1 Vol(Δ), j = 0, 1, · · · . It follows that NP πψ (Cf (s, πψ )) = ordp (q)pΔ if and only if NP πψ (Lf (s, πψ )) = ordp (q)pΔ on [0, pm−1 Vol(Δ)].

2

We now prove Theorems 1.9, 1.10 and 1.11. By the above lemmas, it suffices to prove the following. Theorem 6.4. Let p > 3D and ψ be a character of Zp of order p, then NP πψ (Lf (s, πψ )) = ordp (q)pΔ on [0, Vol(Δ)] if and only if H((au )u∈Δ ) = 0. ∞ at Proof. It is known that the q-adic Newton polygon of Lf (s, πψ ) coincides with HΔ ∞ the point Vol(Δ). By Theorem 1.7, pΔ coincides with (p − 1)HΔ at the point Vol(Δ). It follows that the πψ -adic Newton polygon of Lf (s, πψ ) coincides with ordp (q)pΔ at the point Vol(Δ). Therefore it suffices to show that

NP πψ (Lf (s, πψ )) = ordp (q)pΔ on [0, Vol(Δ) − 1] if and only if H((au )u∈Δ ) = 0. From the identity Cf (s, πψ ) =

∞ 

Lf (q j s, πψ ),

j=0

and the fact the q-adic orders of the reciprocal roots of Lf (s, πψ ) are no greater than 1, we infer that NP πψ (Lf (s, πψ )) = NP πψ (Cf (s, πψ )) on [0, Vol(Δ) − 1]. Therefore it suffices to show that NP πψ (Cf (s, πψ )) = ordp (q)pΔ on [0, Vol(Δ) − 1] if and only if H((au )u∈Δ ) = 0. The theorem now follows from the T -adic Dwork trace formula and Theorem 4.1 and Lemma 5.4. 2

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