On the degree of the <span class="mathjax-formula">$L$</span>-function associated with an exponential sum

C ^OMPOSITIO M ^ATHEMATICA

A ^LAN A ^DOLPHSON S ^TEVEN S ^PERBER

On the degree of the L-function associated with an exponential sum

Compositio Mathematica, tome 68, n

^o

2 (1988), p. 125-159

<http://www.numdam.org/item?id=CM_1988__68_2_125_0>

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125

On the degree ^{of the} L-function associated with

an

exponential

^sum

ALAN ADOLPHSON* &#x26; STEVEN SPERBER**

*Department of Mathematics, Oklahoma State University, Stillwater, ^OK 74078, ^USA

**School of Mathematics, University of Minnesota, Minneapolis, ^{MN 55455, USA} Received 13 February 1986; accepted ^{in revised form 14} April ¹⁹⁸⁸

Compositio (1988)

© Kluwer Academic Publishers, Dordrecht - Printed in the Netherlands

0. Introduction

The purpose of the

present

paper is ^to derive estimates for the

degree

^{of the}

L-function associated with a certain

type

^of

exponential

^{sum defined over a}

finite field

Fq

^of characteristic _p. Let V be an affine

variety

^defined

by

^the

vanishing of polynomials {hi(X)}ri=1 ~ _Fq[X1,

^{... ,}

1; let f and {gi}si=1

be

regular

^{functions on V induced}

by polynomials f (X), {gi(X)}si=1 ~

Fq [X1 ,

^{... ,}

Xn]. Let 03C8

^{be a} non-trivial additive character of

Fq,

^and

^{Xi}si=1

a collection of non-trivial

multiplicative

characters

of F*,

^{extended to func-}

tions on

Fq by setting xl (0) =

^{0. Then the}

exponential

^sums of interest in this paper ^are

where

V(Fqm)

^{dénotes the}

Fqm-rational ^{points on V,}

^and

03C8(m) = 03C8° TrFqm/Fq, Xi(m)

⁼

~i03BFNFqm/Fq·

Associated to this collection of

exponential

^{sums is an}

L-function

known from the work of Dwork and Grothendieck to be a rational function of T with coefficients in the field

Q (03BEp, (q-l) ^(where

^the

^{symbol (m}

^denotes

an

arbitrary

^{choice of}

primitive

^{mth root} of 1 for all m

1).

* Partially supported by NSF Grant No. DMS-8401723.

** Partially supported by NSF Grant No. DMS-8301453.

It has been known for some time that the

pre-cohomological part

^of Dwork’s

theory

^{can be used to estimate the}

degree of L(T)

^{as rational func-}

tion

( =

Euler characteristic of L =

degree

numerator -

degree

^denomi-

nator)

and the "total"

degree

^of

L(T) (= degree

^{numerator +}

degree denominator).

^{The Euler} characteristic appears in the functional

equation

for

L,

^{and when L is a}

polynomial gives

the actual

degree

^{of L. The total}

degree

^of

L,

^{when combined with} information

concerning

the archimedean size of the

reciprocal

^{zeros and}

poles

^{of L}

gives

estimates for the absolute value of the

exponential

^sum

Km ,

^{often an}

important ingredient

in calcula- tions in

analytic

^number

theory.

^These invariants also appear in the ^recent work of Fried and Jarden

[9, 10].

^{The basic work in}

estimating

^{these two}

types

^of

degrees

of L-functions is due to Bombieri

[4, 5],

^{in the case of an}

exponential

^sum

involving only

^an additive character.

(Dwork

^{has an alter-}

native

approach involving

^{the use of}

cohomology

ⁱⁿ

simple non-singular

cases, and then

"deforming"

^to handle the

general case.)

^{In a series}

of papers [1, 2, 13],

the authors have

exploited

Bombieri’s

approach

^to

give improved

estimates in _{many cases,} and to extend the

applicability

of the results to

more

general

^{character sums, in}

particular, allowing multiplicative

^charac-

ters. In the

present

paper, ^we continue this

approach, utilizing

^{an idea of}

Dwork on how to reduce character ^sums such as

Km

^{above to additive}

character sums

(see (1.2) below).

^{We then}

study

^the

particular

^{character sum}

that so arises first

by using Adolphson’s

^{trace formula}

[3]

^to obtain estimates for the Frobenius

matrix;

^then

using

Bombieri’s

approach

^{in this}

particular

case, ^we derive the desired estimates.

One of the main results in this _paper is Theorem

(5.21 )

^which

gives

^a

sharp

estimate for the

degree

^as rational function of the L-function associated with

a certain

type

of character sum. From this

result,

^{we can} extract at once the

following

estimate for

degree

^{of L as} rational function

(Theorem (5.23)):

where

Ho

is the coordinate

hypersurface

^defined

by equation Xl X2 ... Xn = ^0, where do = deg f(X), di

⁼

deg gi (X)

^{and where}

Dn(d0, dl ,

^{... ,}

ds )

^denotes

the sum of all monomials in

do, dl , ... , ds

^of

degree

^n.

Another result that follows from

(5.23)

^{is the more}

general

^estimate

(Theorem 5.27):

where here v is defined over

Fq by

the simultaneous

vanishing

^of

{hj (X)}tJ =1 ~

Fq [X1,

^{... ,}

^{X,, ]}

^of

respective degrees given by {deg hj (X ) = dJ}tJ=1,

^and

where as

before f

^and

{gi}ki=t+

^{1 are}

^regular

^{functions on V induced}

^by polynomials f(X)

^and

gi(X)

^of

respective degrees do

^and

{di)ki=t+1.

^{In terms}

of total

degree,

^{we derive}

(Theorem (6.12)) (where

^the

given multiplicative characters xl

have the form

~i(0) 03BF NFq/Fq,

^{for suitable}

multiplicative

^characters

~i(0)

^of

F*p)

where D =

max {di}ki=0.

In the case when the

multiplicative

characters have

exponent pa’ 2013 ^1,

^the

same estimate holds with D

replaced by

^{a’D. In the case, where no}

multipli-

cative characters are

present

^at

all,

^{we derive} the estimate

(6.13):

This result _may be

compared

^with

[5,

^Theorem

2]

^{where the}

exponent

^that

appears

(when V is

^a closed subset of

An

^{which is not}

^equal

^to

anFq ^itself)

^is

2n + 1 rather than n.

We thank B. Dwork for his

encouragement

^and

helpful

^comments.

1. Definitions

Let _p be ^a

prime number, q . pa,

^{and let}

F q,

denote the finite field of

qm

elements. Let

Up

^denote

the p-adic

^number

field,

^{and let Q be the}

completion

of an

algebraic

^closure

of p .

^Let

^Ka

denote the

unique

unramified extension of

Op

^{in Q of}

^{degree a}

^over

Qp.

The residue class field of

Ka

^is

Fq.

^The

Frobenius

automorphism x - xP,

the canonical generator ^{of Gal}

(Fq/Fp),

lifts to a

generator

^{i of Gal}

(Ka/Qp).

^If

03BEq-1

^{is a}

(q - 1)st

^{root of 1 in}

Ka,

^{a so-called} Teichmuller

unit,

^then

!(q-l

^{1 =}

03BEpq-1.

^Let

00

⁼

Ka(03BEp), 03A91

⁼

Qp(03BEp),

^{and let}

^(Do

^and

^(Dl

^{be their}

respective ring of integers.

^Denote

by

"ord" the additive valuation on Q normalized so that ord _{p - 1} and denote

by "ordq"

the additive valuation normalized so that

ord,, q =

^1.

Let V be an

algebraic variety

^{defined over}

Fq.

^Let

{~i}si=1

^{be a collection}

of non-trivial

multiplicative

characters of

Fg

with values in Q

(all

such in fact have values in

Ka).

^We

extend Xi

^{to all of}

Fq by setting x;(0) =

^{0. For}

the trivial

multiplicative

^character xo , ^we define

~0(0) =

^1.

Let t/1

^be

a non-trivial additive character of

Fq. ^{Let f ’,} {gi}si=1

^be

^regular

^functions

on V. Associated with this data is the

family

^{of mixed or twisted}

exponential

sums

where

V(Fqm)

^{denotes the}

Fqm

^rational

^{points of V,}

the characters

t/1(m) (resp

~i(m)

^of

F, ^(resp F*)

^{are obtained}

from 03C8 (resp Xi) by composition

^{with the}

trace

(resp norm)

^{so that}

03C8(m) = 03C8 · TrFqm/Fq, ^~i(m)=

^{x.. 1}

NFqm/Fq.

Our

aproach

^{to the}

study

^of

(1.1)

^{will involve}

exponential

^{sums of the}

following type.

^{Let 03C9:}

F*q~

^{Q be the} Teichmüller

character,

^{the canonical}

generator of the cyclic group F:. ^{Let F(X ,} {Hi(X)}bi=1 ~ Fq [X1,

^{... ,}

^Xn].

Let {ji}bi=1 ~ Z,0 ji q -

^{2. Set}

We now assume V is affine and defined

by

^the simultaneous

vanishing

^of

polynomials {hi(X)}ti=1 ~ _{Fq [X1,}

^{... ,}

Xn].

^{We also assume that}

f

^and

{gi}si=1

^{are induced}

by polynomials f(X), {gi(X)}si=1 ~ Fq[X1,

^{1 ...}

^Xn].

Finally

^let

xl

^{1 =}

^03C9ji.

Then the relation of

(1.1)

^and

(1.2)

^is

given by

^the

following

^lemma.

(1.3)

^LEMMA.

If V

^is

affine

^as above and the notation above is used then

Here the notation

G,(X, ^03C8)

denotes the

negative of

^{the Gauss sum deter-}

mined

by

^a non-trivial

multiplicative

^and additive character

of Fq (x and t/1

respectively):

Proof. By (1.2),

^we may write

and since

~i(m)-1 (0) =

^{0 for all i =}

1,

..., s ^we obtain same result from the

sum on the

right

^{if we sum}

only

^over

(x, , z)

^E

Fnqm

^x

(F*qm)s

^x

Ftqm.

^Then

^by

orthogonality

^{of characters}

Note that if x E

V(Fqm)

^{such that}

gi(x)

^{= 0 for}

^{some i,}

^then

since

~i(m)-1

^{is a} non-trivial character of

F*qm.

^{On the other}

^hand,

^{if x e}

V(Fqm)

is fixed such that

gi(x)

^~

^0,

^{then the}

^change

^of

variables i = wigi(x)-1

yields

from which the desired conclusion now follows

using

^the

Hasse-Davenport relation;

Associated with the collections of elements in

Qo given by 03BBm

⁼

Km(V; f, ’, 1/1; {Xi, gi}si=1)

^{as in}

^(1.1)

^or

^{by 03BBm}

⁼

Sm (F, 03C8; {ji, Hi}bi=1)

^{as in}

^(1.2)

^{is an}

L-function

which is known to be a rational function of T with coefficients in

a «,, 03BEq-1).

Similarly if Àm

⁼

Sm (F, 03C8; {ji, H}bi=1)

denotes the

right-side of (1.2)

^but

where the sum runs

only

^over

(x, ) ~ (F*qm)n ^+b,

^then

^(1.4)

^{defines an asso-}

ciated L-function which

again belongs

^to

0 «,, (q-l) (T). Finally,

^{if V is}

affine/Fq

^with coordinate

ring Fq [X1,

^{... ,}

Xn ]/I (where

^{I is an ideal in}

Fq[X1,

^{... ,}

^Xn]),

^{we can define}

^V*,

^the

complement

ⁱⁿ V of the coordinate

hypersurface Ho having equation X1X2

^...

Xn =

^{0. Thus the collection}

03BBm = Km (V*;f, 03C8; {gi, ^~i}bi=1)

^{also defines via}

^(1.4)

^{an element}

of Q(03BEp, 03BEq-1)(T),

^the associated L-function.

The

proof of

^the

following

remark is identical with the

proof

^{of Lemma}

(1.3).

REMARK

(1.5).

^{Let the}

hypotheses

^{and notations be the same as those in}

Lemma

(1.3).

^Then

Finally, (1.3)

^and

(1.5)

^{admit the}

following

L-function formulations:

COROLLARY

(1.6).

^{Let the}

hypotheses

^{and notations be the same as those in}

Lemma

(1.3).

^Then

and

The

importance

of this result for the

present study

^{is that}

properties

^{of the}

L-functions associated with

exponential

^sums

^(1.1)

^{may be} established

by

studying

^the L-functions associated with ^sums of the

type (1.2).

2. Trace formula

Our

approach

will utilize the basic framework of Dwork’s

theory;

ⁱⁿ par- ticular we will need

Adolphson’s

form of the trace formula

[3].

^{We review}

here the

major

features of the

theory

^that will be used in

subsequent

^sections.

Let 7 = (7i,

^...,

jk)

^{be an ordered}

k-tuple

^of

integers satisfying 0 ji q -

2. We define a

p-adic

Banach space of formal series in

integral

powers of ^{one set} of variables

{Xi}ni=1

and fractional _powers of another set of

variables {Yi}ki=1.

^More

precisely

^let

We will write

XR

YM to denote

03A0ni=1 XRii. 03A0kj=1 YjMj.

^In the notation

of (1.2),

let

do = deg F(X), di

⁼

deg Hi(X).

^{In terms of these}

quantities,

^{we can}

define a

weight

^{function on}

0393(j)

where we

employ

^for

brevity

the usual dot

product

and the notation

1 R |

⁼

03A3ni=1 Ri

for ordered

tuples.

^This

weight

function takes values in

(q - 1)-1 Z0

^{and satisfies the}

following properties:

(iii)

^{there exists}

positive constants 81

^and 82 such that

We define an

O0 -module

of formal series for

b,

^{c E}

R, b

&#x3E; 0.

and an

S2o-vector

^space

In fact

L(j) (b)

^{is a}

^p-adic

^{Banach space of}

b(r(j))-type

ⁱⁿ Serre’s notation

[12].

We now define a

completely

continuous

03A90 -linear operator

^a

acting

^on

L(j)(b). Let 03C8q

^act

^by

Observe that

(R, M)

^E

0393(j)

^{if and}

^only

^if

^{(qR, qM)}

^E

0393(j). ^Clearly, 03C8q(L(j)(b, ^{c» 9} L(j)(qb, ^c).

Let

E(X) =

exp

(03A3~i=0 Xpl/pi)

be the Artin-Hasse

exponential series;

let _y E Q be a root of

03A3~j=0 Xpl/pi

^{= 0 with ord} y =

1/(p - 1).

^Then

0,,,(x) = E(03B3x)

^{is a}

splitting

function in the

terminology

^of

[7];

if we write

03B8~(X) = 03A3~m=0 BmXm,

^{it is a} consequence of the

p-adic integrality

^{of the}

coefficients of the Artin-Hasse series that

Suppose

^{now that we write}

where

A (i)

is a finite set of ordered

n-tuples

^of

non-negative integers

^{for each}

i e

{0, ^1,

^{... ,}

k}

^{and where ~}

Fq.

^Let

A(iw(i)

^{denote the}

Teichmüller

lifting of (i)w(i),

^{so that}

(A(i)w(i))q

⁼

A(i)w(i), and 03C4(A(i)w(i))

⁼

(A(i) )p

Consider

If we write

F(X, Y) = 03A3(R,M)~0393(0) ^{F(R, M)X’ YM}

^{then from}

^(2.6)

where the sum runs over ordered

1-tuples (1

⁼

03A3ki=0

^card

(i)), ({mw(0) }w(0)~(0),

...,

{mw(k)}(k)~A(k)),

^of

non-negative integers satisfying

Thus

using (2.5)

where the infimum runs over

all 1-tuples

^of

non-negative integers satisfying (2.7).

Hence

where the infimum is taken over the same set.

Combining

^the

equalities

ⁱⁿ

(2.7)

^{with the}

inequalities |w(i)| d; ,

^{we obtain}

Thus

which

yields

^{the estimate}

so that

Let

then

In fact

multiplication by Fo(X, Y)

^{defines via}

(2.3)

^an

endomorphism

^of

L(j))(P/q(P- 1)).

^If

denotes

inclusion,

^{then i and a =}

03C8q 03BF F0(X, ^{Y) 03BF}

^{i are}

completely

^con-

tinuous

endomorphisms of L(j)(P/(P - ¹⁾⁾

^{in the sense of Serre}

[12].

^Fur-

thermore the trace formula of

Adolphson [3] yields

For a

completely

continuous

endomorphism

^{a, Tr}

(ocm)

^{and the Fredholm}

determinant det

(I - Ta)

^are

well-defined, independent

^{of choice of ortho-}

normal

basis,

^{and are related}

by

Let £5 denote the

operator

Then

(2.9)

^is

equivalent by (2.10)

^to

3. Reduction

step

Our method

gives significantly

better estimates for the Newton

polygon

^of

det

(I - Ta)

^{in the case when all}

the x;

^have

exponent p - 1,

^{i.e. all}

the j;

are divisible

by (q - 1)/(p - 1)

^{= 1 + p + ... +}

pa-1.

^If

with

0 vi

p -

2,

^we may define

an

03A91 (= Qp(03BEp))-linear completely

continuous

endomorphism

^of

L(j)(p/(p - ^1)).

^{We have the}

relationship

and the

relationship

of Fredholm determinants

given by

where the

product

^{is taken over all}

roots 03BE

^of

(a =

^1. This establishes the

following

^result.

THEOREM

(3.4).

^{Under the}

hypothesis (3.1),

^a

point (x, y)

^E

^R2

^{is a vertex}

of the

^Newton

polygon of det (I - T03B1) computed

^{with respect to} the valuation

"ordq" ^if

^and

only if (ax, ay)

^{is a vertex}

of

^{the Newton}

polygon of detol (I - Tao) computed

^with respect ^to the valuation "ord".

We are thus reduced in the case

(3.1)

^to

estimating

the Newton

polygon

of

detol (1 - Tao).

^{In the other cases we} estimate the Newton

polygon

^of

det

(I - T03B1) directly

^{and use} the somewhat weaker results.

4. Estimates for

Frobenius;

^Newton

polygon

Let {03BE1,... , 03BEa}

^{be an}

integral

^{basis for}

Qo

^over

QI

that has the

property

of

p-adic

directness

[7, §3c], i.e.,

^for any

{03B21,..., Pal

⁹

f2l,

An orthonormal basis for

L(j)(p/p - ¹⁾

^{as an}

03A91-linear

space ^can then be obtained from the set

by multiplying

^{each i E I}

by

^{a suitable}

constant y,

^e

S2o .

^We obtain first estimates for the Frobenius matrix with

respect

^to

I(j).

^{In order to do so, it}

is convenient to rewrite

F(X, Y )

^{in terms}

of I(j);

ⁱⁿ

particular,

^if

then ord

(F(R, M, l)) w(R, M)/(p - 1) d0,

^and

If we write

then ord

u(l, ; l’)

^0.

Furthermore,

^{if i =}

03BE1 XRYM,

^{i’ =}

ç/,XR’ ^YM’

belong

^to

I(j)

^{and we write}

then

and with the

given

^{notation we obtain}

using

^the

properties (2.3).

If we write

then

(-1)m cm

^{is the sum of all m x m}

principal

^{minors of the matrix of} ao with

respect

^{to an} orthonormal basis. Since

lU)

differs from an orthonormal basis

only by

^scalar

multiples,

^{it is} easy ^{to see} that

(-1)m cm

is also the sum

of the m x m

principal

minors of the matrix

(A(i, i’))i,i’~I(J).

^Thus

137 where the infimum is taken over all

collections {i(r)

⁼

(R(r),M(r),l(r)}r=1,2, ...m

of m distinct elements in

I(j),

^{and over all u E}

permutations

^{on m letters. As}

a consequence of

(4.1),

^we obtain from

(4.2)

in which the infimum is taken over all collections of m distinct elements from

I(j) -

For each K

~ Z0,

^{we define}

Summarizing

^our above result

(4.3)

^{we have}

THEOREM

(4.5). Suppose

^the

integers {ji 1 i k= ,

^are all divisible

by (q - 1)/(p - 1).

Then the Newton

polygon of det (I - Ta)

^{is contained in the convex closure}

in

R2 of

^the

points (0, 0)

^and

It remains to

compute W(K).

^{This will} be done in the next theorem. We

employ

^the

following

^notation:

THEOREM

(4.6). If K |v| d0,

^then

W(K) =

^0.

Assume K a |v| d0.

^Let

K0 = K - |v| d0

⁼

Qdo(p - 1)

^{+ r}

^{with 0 ,} Q, 0

^r

do(p - 1).

Case

(i).

^{r ~ 0:}

If, in ^addition, Ko

^{+ v}

d ~

^{0 mod}

( p - 1),

^then

W(K) =

^0.

If Ko

^{+ v d =}

Q’ (p - 1)

^then

where N ⁼

(NI,

^{... ,}

Nk )

^E

(Z0)k

^{and the sum is taken over all such N}

satisfying 0 |N| 1 Q.

Proof.

^Let

Then

and all terms involved are rational

integers. Clearly

^if

K0

^{is not a}

multiple

of

do(p - 1)

^{a solution}

(R, M)

^of

(4.7)

^{must have}

positive

^{max term on}

the left. But then

K0

^{+ v - d - 0 mod}

(p - 1).

On the other

hand,

^if

Ko

^{+ v - d =}

Q’(p - 1),

the solutions

(R, M) of (4.7)

^are those for which

which

completes

^{the case}

(i).

If r ⁼ 0 and

(R, M)

^{is a} solution of

(4.7) then |M| Q

^and

If v. d ^~ 0 mod _{p -}

1,

^{then the}

only

^{solutions of}

(4.8)

^{arise when}

Q

⁼

1 MI |

^and

On the other hand if v - d ⁼

E(p - 1)

^one obtains other solutions to

(4.8)

^{in addition to the solutions of the}

inequality (4.9) when 1 MI |

⁼

Q.

^In

particular 1 M 1 Q

^and

yield

the additional solutions.

This

completes

^the

proof

^{of case}

(ii)

and the theorem.

5.

Degree

of the L-function

It is known that

L({S*m(F, 03C8, tji Hi}ki=1)}, ^T)

^{is a} rational function of T with coefficients in

Q(03BEp, 03BEq-1). ^(It

^follows

^again

^from

^(2.11)

^{and the Dwork}

rationality

^criterion

[6].)

^{We write}

so that the

degree

^of

L({S*m}, T)(-1)n+k+1

^{as a} rational function

is r1

^- r2.

Inverting (2.11)

^and

solving

^for the Fredholm determinant of a

yields

where

LEMMA

(5.2) [4, Corollary

^{to Lemma}

3]. If L({S*m (F, 03C8, {ji,Hi)ki=1)}, T)(-1)n+k

is written as in

(5.1),

^then

where the sums X’ are over all m such that the summands are

positive.

^D

By

^a well-known formula

(Knopp,

^Infinite

Series,

^Ch.

14),

^{for 1 1}

This

yields

^{at once that}

as x - +00. Since

ordq(qmQ)

^{= m +}

ord,(Q), (5.5) implies

that the left side of

(5.3) equals

It is our intention next to estimate the

asymptotic

behavior of the

right-

side of

(5.3).

It will be useful to establish a lemma first. Let

{di}ki=0

^{and r}

be for the

present arbitrary positive integers.

^Let

Dr(d0,

^{... ,}

dk)

^{denote the}

sum of all monomials in the

{di}ki=0

^of

degree

^r

Let

where the outer sum runs over all k + 1

tuples (No,..., Nk )

^{of non-}

negative integers satisfying 03A3ki=0 N =

^03BB.

LEMMA

(5.9).

^Assume

{di}ki=0

^{and r are}

^arbitrary positive integers.

^Then

Proof.

^We

proceed by

^{induction on} k. The case k ⁼ 0 is immediate. We

assume the result now for k and consider

(using

the induction

hypothesis)

x

(dr-1k+1Dl(d0,d1,..., ^dk))

⁺

O(03BBr+k) (summing over j and using (5.4)).

(5.10)

Now we assert

for the left side is

equal

^to

by

^the usual beta function evaluation.

Substituting (5.11)

into the last

expression

ⁱⁿ

(5.10) yields finally

Of _course,

which

completes

the induction

proof

of Lemma 5.9. D Note that in the notation of Theorem

(4.6),

^{if we view}

Q, Q’,

^{and E as}

functions of

K,

^then

LEMMA

(5.13).

^{Consider K such that}

do(p - 1)

^{does not divide}

^{K -} |v do (case (i)

^{in the}

language of

^Theorem

(4.6)).

^Then

(i) W(K)

⁼

0, if p -

^{1 does not divide}

K - 1 v do

^{+ v d.}

Proof of (ii). Clearly (5.12) implies

If we set

No = Q - |N|,

^then

so that

where the ^sum runs over ordered k +

1-tuples

^of

non-negative integers (No,

^{... ,}

Nk) satisfying L7=0 N¡

⁼

Q. Using (5.4)

^{and lemma}

(5.9)

^with

r ⁼ n -

1, Â

⁼

Q

^{we obtain}

immediately

^the desired conclusion. 1:1 In

precisely

^{the same} way ^we prove the

following

^result.

LEMMA

(5.14).

^{Consider K such that}

do( p - 1)

^{does divide}

K - 1 v do (case (ii)

^{in the}

language of

^Theorem

(4.6)).

(i) If

p - 1 does ^not divide v -

d,

^then

THEOREM

(5.15).

^The

right

^side

of (5.3) equals

Proof.

The calculation will

depend

^{on whether or not v · d ~ 0}

mod

( p - 1).

Assume first that v ·

d ~

^{0 mod}

( p - 1).

^{Consider the}

contribution to the

right-side

^of

(5.3) coming

^{from those K for which}

Ko =1=

^{0 mod}

(do(p - 1);

^more

particularly,

those K for which

Ko

⁺

v - d ~ 0 mod

(p - 1). Using

the notation of

(4.6)

^we may write

K = Q’ ( p - 1) + |v|d0 - v · d

and re-express the

right-side

^of

(5.3)

ⁱⁿ

terms of

Q’, namely

where _co is a fixed constant

(co

⁼

(v · d - |v|d0)/(p - 1)) independent

^of

Q’,

^{and where we} have used the estimates of

(5.13).

^This

yields

^{a contribu-}

tion to the

right

^{side of}

(5.3) equal

^to

The contribution from those K for which

do( p - 1)

^divides

Ko

is obtained in similar fashion

writing K

⁼

Qd0 (p - 1) + |v| d0

^and

re-expressing

^the

right-side of (5.3)

^{in terms of}

Q.

^This

yields

^a contribution to the

right

^side

of

(5.3) equal

^to

Since

this

completes

^the

proof

^of the theorem when v

d ~

^{0 mod}

(p - 1).

On the other

hand,

^{if v d ~ 0 mod}

(p - 1),

^{then the two cases}

Ko

⁺

v · d ~ 0 mod p - 1 (i.e. K0

^~

0 mod p - 1) and K0 ~ 0 mod (do(p - 1))

145

are not

independent.

^When

Ko -

⁰

(mod

p -

1)

^{in all cases}

(whether

^or

not

K0~0

^mod

do( p - 1))

^{there is a} contribution to the

right

^side

of (5.3) equal

^to

When

Ko

^{~ 0 mod}

do(p - 1),

^{there is an} additional contribution to the

right-side

^of

(5.3) equal

^to

This

completes

^the

proof

of the theorem.

THEOREM (5.16). If

then

Proof.

^The

right-most inequality

^{follows from a}

comparison

^of

(5.6)

^and

(5.15)

^{as x ~ + oo. The}

inequality

^on the left is a consequence via

(2.11)

^of

the

following

^lemma

provided by

the referee.

LEMMA

(5.11). Let f be a

^rational

function, f(0)

^{= 1} such that

for

^some

m E

N, f(1-~)-m

^{is an}

entire function (where f (t)~

⁼

f(qt)

^{so that b = 1 -}

~).

Then

degree f

^0.

Proof. Write f

⁼

03A0si=1 (1 - ÀitYB

ei ⁼ + ^{1. We let the}

cyclic multiplica-

tive group

~q~

^{act on Q}

by multiplication.

^The intersection of the orbits under this action with

{03BB1,

^{... ,}

03BBs} partitions

^{this set. For each}

equivalence class,

^{we choose so that =}

qm03C4 (for

^{some m}

0)

^for

every

in the class.

Then

f

⁼

II)=1 ^{(1 -} 03C4jt)hJ(~)

where each

hj(~) ~ Z [~] and J =1= J’ implies 03C4j/03C4j, ~ ~q~.

^Thus

⁽¹

^-

03C4jt)hJ(~)(1-~)-m

^{has no} factor in common with

(1

^{As a} consequence,

(1 -

Li

t)hj(~)(1 -

^{q»-m is}

entire,

^{and we}

may ^assume

f

⁼

(1 - rt)1(9).

^Now

so we may write

h(x) = (1 - x)’ 03A3n0 anxn.

^Thus

^h(x)

^{0 for x in the}

interval

[0, 1),

^{so that}

h(1) a

^{0. But}

h(1)

⁼

degree f.

REMARK 1. We also wish to treat the case in which

F(x)

^{= 0}

identically. (Or

what is almost the same

thing,

^{for our}

purposes, do

⁼

0).

^{We revise the}

discussion in

preceding

^{sections as follows. Let}

In terms of

these,

^{we define}

where we have

preserved

the notation for

Hi

from Section 2.

Furthermore,

let

so that

and

If j

⁼

(q - 1)v/( p - 1), we define as in §3, 03B1’n = 03C8p03BF03C4-103BFF’(X ^{Y) and}

obtain ^the

analogous

^{result to Theorem}

(3.4)

^for

03B1’0

^{and a’.}

The

argument

^{is now clear: The Newton}

polygon

^{of det}

(1 - Ta’)

may be estimated as follows:

THEOREM

(4.5)/. Suppose F(x)

^{= 0}

identically. Suppose j

⁼

(q - 1)v/

(p - 1).

^{Then the Newton}

polygon of

^det

(1 - Ta’) is

contained in the

convex closure in

1R2 of

^the

points (0, 0)

^and

where

We _prove the

following

^{in the same} way ^as

(4.6).

where the sum on the

right

^{is taken over}

k-tuples

^{N =}

(NI,

^{... ,}

Nk) of non-negative integers satisfying 1 N 1

⁼

Q’.

This result leads to the

following

theorem in the same way ^as

(5.16)

^was

proved.

THEOREM

(5.16)’. If j

⁼

(q - 1)v/( p - 1)

^{where v =}

(VI

^{... ,}

03BDk)

^E

(Z0)k,

^then

REMARK 2. In the

case ji #

0 for all i ⁼

1, ... , k

^{then as we have shown}

in Section 1

where the latter uses the notation of

[2]. ^However,

^since

Dn (d1,

^{... ,}

dk) (L7=1 di)n (and

^{this is a strict}

inequality

^{unless n = 1 or k =}

1)

^the upper estimate

of (5.16)’ is

^an

improvement

^{on the upper estimate}

of [2,

^Theorem

4],

which we believed then to be best

possible

^and

generically

^{attained. The}

argument

^{we used in}

[2]

^involved

estimating degree Z(X, T)

where X is the

complement

^{in An of the}

hypersurface

^defined

by

^the

vanishing

^of

the

polynomial Xl X2

^...

Xn Hl H2 ... Hk

^{and then}

utilizing

^{the known}

relationship

However our method of

estimating degree Z(X, T)

^treated

^H

⁼

H1 H2 ... Hk

^{as it would a}

generic polynomial

^of

degree 03A3ki=1 di

^{and did not}

make use of the

special

^feature

^{of H} namely

^its

reducibility

^to

improve

^the

estimate to

Dn (dl ,

^{... ,}

dk).

REMARK 3. We believe the estimates of

(5.16)

^and

(5.16)’

^are

generically

attained. We note that in the case of

(5.16)’

^if

and if

are in

general position

^{in the sense that no three} of them intersect ⁱⁿ

Pi,

^then

where here

again X

^{is the}

complement

ⁱⁿ

A2Fq

^{of the}

hypersurface

^defined

by

the

vanishing

^{of the}

polynomial X1X2 II7=ql Hi. Using (5.18), (5.19)

^and

(5.20),

^{we obtain}

which is the upper estimate in

(5.16)’ when

^{n = 2 and}

d1 = ...

⁼

dk

^{= 1.}

It does not seem difficult to extend this

example

^{to the case of linear}

hypersurfaces

ⁱⁿ

An Fq

^{when n} &#x3E; 2.

REMARK 4.

(cf.

^Remark

following

Theorem 4 in

[2].) By

the result

of Deligne [11] on

Euler-Poincaré

characteristics, degree L ({Sm(F, 03C8;{ji, Ai 1)1, ^T)

^and

degree L({S*m(F, 03C8; {ji, Hi})}, T)

^are

independent

of the choice of the

{ji}ki=1.

^In

particular,

^we may conclude the

following.

THEOREM

(5.22).

^For

arbitrary

^choice

of

Using (1.5)

^and

(5.22)

^we find in the

particular

^{case when none of the}

ji’s

are divisible

by

q - 1 the

following.

COROLLARY

(5.23).

^Let

Ho

^{be the union}

of

the coordinate

hyperplanes.

^Then

where do = degf(x),di= deg gi/f(x)

^and

{gi(x)}si=1 ~ Fq[x1,

^{... ,}

^Xn]

inducing f and {gi}si=1 respectively

^{on V.}

bMARK 5. Let us renumber if necessary so that ji = 0, if i = 1,..., t and ji ~ 0,

^{if t i k. Let F =}

{X1,

^...,

Xn} F = {1, ... , t}, U = {t

⁺

^1,

^{... ,}

k},

V = F ~ 0/1. For each subset B of

f,

^{we define}

B’ = B u

0/1,

^a subset of 1/. Let A ~

S, B ~ S

^denote

arbitrary

^subsets.

We define

where

if f (X)

^E

Fq[S] then fA (X)

^E

Fq[A]

is the result of

specializing

^the

variables

Xi,

ⁱ

~ A,

^to

equal

^{zero, and where} if W is a collection indexed

by V, LB,

^{is the} subcollection indexed

by

^{B’. A} similar definition is immediate for

S*m,A~B.

^Then

In terms of L-functions

From

(5.16)

^{we obtain}

where |A| 1 (respectively 1 B 1)

denotes the

cardinality

^{of the}

given

^{set. Utiliz-}

ing

the weaker estimate

we obtain via

(1.3)

^the

following

^result

THEOREM

(5.26). If

^{V is}

defined

^over

Fq ^by

the simultaneous

vanishing of

{Hi(X)}ti=1 ~ Fq[X1,..., Xn] then

Greater

precision

may of course be obtained if the upper estimate in

(5.25)

is attained in this case for all A ~ S B ~L.

Finally

^{we note that the}

following

estimate follows

directly

^from

(5.26).

THEOREM

(5.27).

^{Assume V is}

defined

^over

Fq by

the simultaneous

vanishing of

{hi(X)}ti=1 ~ Fq[X1, ^{X ,, ] where} degree hi (X) = di.

^Let

F

^and

{gi}ki= t +

¹

be

regular functions

^{on V induced}

by polynomials F(X),

1 ~

Fq[X1, Xn] of respective degrees d0, {di}ki=t+1.

^Then

Proof. Fixing

^{A and}

using

we

get immediately

^from

(5.24)

^{that the} upper estimate is

majorized by

A calculation with binonical coefficients then

yields

which concludes the

proof

^of

(5.27).

6. Total

degree

of the L-function

In this section we estimate the total

degree

^of

L({S*m(F, ^1/1; {ji, Ail)l, ^T).

Suppose

first that the

integers {ji}ki=1

^are all divisible

by (q - 1)/(p - 1).

We follow the method of

[5]

^{to deduce} this estimate from the lower bound for the Newton

polygon

^{of det}

(I - Ta) (Theorems (4.5)

^and

(4.6)).

^Recall

the basic idea: From

(5.1), (2.11),

and the definition of à we have

But

[15, Exp. ^XXI,

^Cor.

5.5.3(iii)]

says that 0

ord,Qi, ordq ~j

^{n + k.}

Hence the

reciprocal

^{zeros and}

poles

^on the left hand side of

(6.1 )

^{all occur}

among the

reciprocal

^zeros

of 03A0n+km=0 det (I - qmT03B1)(n+km) of ordq n

^{+ k.}

Let

Nm

be the number of

reciprocal

^{zeros of det}

(I - qm T03B1) of ordq n

^{+ k}

(i.e.,

the number of

reciprocal

^{zeros of det}

(I - Ta) of ordq n

^{+ k -}

^m).

Then

To estimate

Nm ,

^{we use Theorems}

(4.5), (4.6),

^and the fact that

Nm

is the total

length

^{of the}

projections

^on the x-axis of the sides of

slope n

^{+ k - m}

of the Newton

polygon

^{of det}

(I - Ta). Suppose

^{we can find}

points

(x(r), y(r))

^E

R2, r = 0, 1, 2,

^...

lying

^{on or} below the Newton

polygon

^of

det

(I - Ta)

^{such that}

It is then clear that the

total length

^{of the}

projections

^on the x-axis of the sides of

slope 5 r

^{of the Newton}

polygon

^{of det}

(I - Ta)

^is

x(r).

^Hence

Nm = x(n

^{+ k -}

m)

^and

by (6.2)

Theorem

(4.5) gives

^{us a} sequence of

points lying

^{on or below the}

boundary

of the Newton

polygon

^{of det}

(I - Ta).

^{The next}

step

^{is to} determine which of these

points satisfy (6.3)

^{for a}

given

^{r. We}

begin

^{with a}

simple

^lemma.

LEMMA 6.5.

Suppose W(K)

^{is a}

real-valued function of

^{K and ô is a}

positive

real number such that

for

^{N =}

0, 1, 2,

^...

Then for

^{N =}

^{0, 1, 2,}

^{... ,}

Proof. By

^{induction on}

N,

^{the case N = 0}

being

^trivial.

Suppose

^the

inequality

holds with N - 1 in

place

^{of N:}

Adding NW(N)

^{to both sides:}

But

which is 0

by hypothesis.

COROLLARY

(6.6). Suppose

ⁱⁿ addition that

W(K)

^{is a}

non-decreasing function of

^K.

Then for

^{N =}

^{0, 1, 2,}

^{... ,}

Proof.

^The

hypothesis

^that

W(K)

^is

non-decreasing implies

^{that we} may

take à ⁼ 1 in Lemma

(6.5).

^D

The function

W(K)

^that appears in Theorem

(4.5)

^{is not}

non-decreasing,

^as

is

easily

^seen -from Theorem

(4.6). ^However,

^{we shall see that on a certain}

subsequence

^{of the} sequence of

points given

in Theorem

(4.5) W(K)

^behaves

on average ^as

though

^{it were}

non-decreasing.

Let E ⁼

[v· d/(p - 1)]

^{be as} in Theorem

(4.6)

^and

put

where N ⁼

(Nl ,

^...,

Nk )

^E

(Z0)k

^{and d =}

(dl,

^...,

dk).

^For

⁰

^r

do ( p - 1),

^{if r - - v}

d (mod

p -

1) put

where N - (No, N1,

^...,

Nk)

^e

(Z0)k+1 ^{and d =} (d0, d1,

^{... ,}

^dk),

^other-

wise set

Wr (Q) -

^{0. It is}

easily

^{seen that}

W and W

^are

non-decreasing

functions of

Q.

We may reformulate Theorem

(4.6)

^{as follows. Put}

K0 =

K - |v| d0

^{and write}

where

0 r d0(p - 1).

^Then

W(K)

^{= 0 if}

K lvld,

^{and for}

K 1 v do

^{we have}

Now consider N such that

N - |v| d0

^{= -}

1 (mod do ( p - 1))

^{and let}

03BB(N) = [(N - |v| d0)/(p - 1)]. Using (6.7)

^and the trivial estimate K

Qdo(p - 1)

^we

get

Now

apply Corollary (6.6)

^to

W and W

^{to conclude}

It follows that for such

N,

the line from the

origin through

has

slope 03BB(N)/2.

^{So to}

get slope

^{r, we must choose N =}

Nr

^{such that}

03BB(Nr)

^{2r. From} the definition of

03BB,

it is clear that it suffices to take

Thus the

point (x(r), y(r)),

^where

155 satisfies

(6.3).

^{So from}

(6.4)

Next,

^we

estimate 03A3Nn+k-mK=0 W(K).

^{We have from}

(6.7)

and the definition

of N,

Set D ⁼ max

{di}ki=0.

^We have from the definitions of

W , W

Thus

Hence

From

(6.8)

^{and the}

elementary

^estimate

we get

total

degree 03C8; {ji, Ri })), ^T)

Using

^the

inequalities

and

rs/s! er/(| r - s |

⁺

1)

^{one sees} that this last

expression

^{is less than or}

equal

^to

Using (n

⁺

1)n/n! en and (n

⁺

1)n-1/(n - 1)! 2en-1,

this is less than

or

equal

^to

Weakening

this estimate

slightly

^to

produce

^{a more}

compact formula,

^we have the

following

^result.

157 THEOREM

(6.11).

total

degree L({S*m(F, ^{03C8; {ji,} I7il)l, T)

From this we can also derive ^an estimate for the total

degree

^{of the}

L-function when the coordinate

hyperplanes

^{are included in the}

exponential

sum.

Apply

^Theorem

(6.11)

^to estimate total

degree

^{of each}

L({S*m,A~B}, T) (see (5.24)):

where a ⁼

cardinality

^{of A. From}

(5.24),

total

degree L({Sm(F, 03C8; {ji, Hi})}, ^T)

The estimate

leads

immediately

^to

(6.12)

THEOREM. Assume that each character _Xi has the

form

Xi =

iO) 0 NFq/Fp

where

Xi(0)

^{is a} non-trivial

multiplicative

^character

of Fp*.

where

f(X)

^and

{gi(X)}ki=t+1

¹

of respective degrees do = deg f(X),

di

⁼

deg gi(X)

^{induce the}

given regular functions f

^and

{gi}ki=t+

^{1 on}

^V;