STEVEN SPERBEROn the degree of the L-function associatedwith an exponential sumCompositio Mathematica, tome 68, no 2 (1988), p. 125-159<http://www.numdam.org/item?id=CM_1988__68_2_125_0>
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Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiques
On the degree of the L-function associated with anexponential sum
ALAN ADOLPHSON* & 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
The purpose of the present paper is to derive estimates for the degree of theL-function associated with a certain type of exponential sum defined over afinite field Fq of characteristic p. Let V be an affine variety defined by thevanishing of polynomials {hi(X)}ri=1 ~ Fq[X1, ... , 1; let f and {gi}si=1be 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=1a 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 thispaper 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 anL-function
known from the work of Dwork and Grothendieck to be a rational functionof T with coefficients in the field Q (03BEp, (q-l) (where the symbol (m denotesan 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.
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It has been known for some time that the pre-cohomological part ofDwork’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 + degreedenominator). The Euler characteristic appears in the functional equationfor L, and when L is a polynomial gives the actual degree of L. The totaldegree of L, when combined with information concerning the archimedeansize of the reciprocal zeros and poles of L gives estimates for the absolutevalue of the exponential sum Km , often an important ingredient in calcula-tions in analytic number theory. These invariants also appear in the recentwork of Fried and Jarden [9, 10]. The basic work in estimating these twotypes of degrees of L-functions is due to Bombieri [4, 5], in the case of anexponential sum involving only an additive character. (Dwork has an alter-native approach involving the use of cohomology in simple non-singularcases, 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 improvedestimates in many cases, and to extend the applicability of the results tomore general character sums, in particular, allowing multiplicative charac-ters. In the present paper, we continue this approach, utilizing an idea ofDwork on how to reduce character sums such as Km above to additivecharacter sums (see (1.2) below). We then study the particular character sumthat so arises first by using Adolphson’s trace formula [3] to obtain estimatesfor the Frobenius matrix; then using Bombieri’s approach in this particularcase, 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 witha certain type of character sum. From this result, we can extract at once thefollowing 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 ) denotesthe 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):
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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, andwhere as before f and {gi}ki=t+ 1 are regular functions on V induced bypolynomials f(X) and gi(X) of respective degrees do and {di)ki=t+1. In termsof total degree, we derive (Theorem (6.12)) (where the given multiplicativecharacters 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 thatappears (when V is a closed subset of An which is not equal to anFq itself) is2n + 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 qmelements. Let Up denote the p-adic number field, and let Q be the completionof an algebraic closure of p . Let Ka denote the unique unramified extensionof Op in Q of degree a over Qp. The residue class field of Ka is Fq. TheFrobenius 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. Denoteby "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 collectionof non-trivial multiplicative characters of Fg with values in Q (all such in facthave values in Ka). We extend Xi to all of Fq by setting x;(0) = 0. For
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the trivial multiplicative character xo , we define ~0(0) = 1. Let t/1 bea non-trivial additive character of Fq. Let f ’, {gi}si=1 be regular functionson V. Associated with this data is the family of mixed or twisted exponentialsums
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 thetrace (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 canonicalgenerator 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 ofpolynomials {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 thefollowing 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/1respectively):
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Proof. By (1.2), we may write
and since ~i(m)-1 (0) = 0 for all i = 1, ..., s we obtain same result from thesum on the right if we sum only over (x, , z) E Fnqm x (F*qm)s x Ftqm. Then byorthogonality 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)-1yields
from which the desired conclusion now follows using the Hasse-Davenportrelation;
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 anL-function
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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) butwhere 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 isaffine/Fq with coordinate ring Fq [X1, ... , Xn ]/I (where I is an ideal inFq[X1, ... , Xn]), we can define V*, the complement in V of the coordinatehypersurface Ho having equation X1X2 ... Xn = 0. Thus the collection03BBm = 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 inLemma (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 inLemma (1.3). Then
and
The importance of this result for the present study is that properties of theL-functions associated with exponential sums (1.1) may be established bystudying the L-functions associated with sums of the type (1.2).
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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 reviewhere the major features of the theory that will be used in subsequent sections.
Let 7 = (7i, ..., jk) be an ordered k-tuple of integers satisfying0 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 ofanother 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 candefine a weight function on 0393(j)
where we employ for brevity the usual dot product and the notation1 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 > 0.
and an S2o-vector space
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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). Then0,,,(x) = E(03B3x) is a splitting function in the terminology of [7]; if we write03B8~(X) = 03A3~m=0 BmXm, it is a consequence of the p-adic integrality of thecoefficients 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 eachi e {0, 1, ... , k} and where ~ Fq. Let A(iw(i) denote theTeichmüller lifting of (i)w(i), so that (A(i)w(i))q = A(i)w(i), and 03C4(A(i)w(i)) = (A(i) )pConsider
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
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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
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In fact multiplication by Fo(X, Y) defines via (2.3) an endomorphism ofL(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 Fredholmdeterminant 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 ofdet (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
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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 ofdetol (I - Tao) computed with respect to the valuation "ord".
We are thus reduced in the case (3.1) to estimating the Newton polygonof detol (1 - Tao). In the other cases we estimate the Newton polygon ofdet (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 propertyof 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 beobtained from the set
by multiplying each i E I by a suitable constant y, e S2o . We obtain firstestimates for the Frobenius matrix with respect to I(j). In order to do so, itis convenient to rewrite F(X, Y ) in terms of I(j); in particular, if
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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 aowith respect to an orthonormal basis. Since lU) differs from an orthonormalbasis only by scalar multiples, it is easy to see that (-1)m cm is also the sumof the m x m principal minors of the matrix (A(i, i’))i,i’~I(J). Thus
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where the infimum is taken over all collections {i(r) = (R(r),M(r),l(r)}r=1,2, ...mof m distinct elements in I(j), and over all u E permutations on m letters. Asa 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 closurein R2 of the points (0, 0) and
It remains to compute W(K). This will be done in the next theorem. Weemploy the following notation:
THEOREM (4.6). If K |v| d0, then W(K) = 0. Assume K a |v| d0. LetK0 = 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.
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Proof. Let
Then
and all terms involved are rational integers. Clearly if K0 is not a multipleof do(p - 1) a solution (R, M) of (4.7) must have positive max term onthe 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
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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. Inparticular 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 Twith coefficients in Q(03BEp, 03BEq-1). (It follows again from (2.11) and the Dworkrationality 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
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LEMMA (5.2) [4, Corollary to Lemma 3]. If L({S*m (F, 03C8, {ji,Hi)ki=1)}, T)(-1)n+kis 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 leftside 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 rbe for the present arbitrary positive integers. Let Dr(d0, ... , dk) denote thesum 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.
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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. Weassume 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
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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 asfunctions 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.
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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) withr = 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
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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 thecontribution to the right-side of (5.3) coming from those K for whichKo =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) interms of Q’, namely
where co is a fixed constant (co = (v · d - |v|d0)/(p - 1)) independent ofQ’, 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 obtainedin similar fashion writing K = Qd0 (p - 1) + |v| d0 and re-expressing theright-side of (5.3) in terms of Q. This yields a contribution to the right sideof (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))
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are not independent. When Ko - 0 (mod p - 1) in all cases (whether ornot 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 theright-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) ofthe following lemma provided by the referee.
LEMMA (5.11). Let f be a rational function, f(0) = 1 such that for somem 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 orbitsunder this action with {03BB1, ... , 03BBs} partitions this set. For each equivalenceclass, 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’ implies03C4j/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 wemay assume f = (1 - rt)1(9). Now
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so we may write h(x) = (1 - x)’ 03A3n0 anxn. Thus h(x) 0 for x in theinterval [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. (Orwhat is almost the same thing, for our purposes, do = 0). We revise thediscussion 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
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and
If j = (q - 1)v/( p - 1), we define as in §3, 03B1’n = 03C8p03BF03C4-103BFF’(X Y) andobtain 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 theconvex 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) ofnon-negative integers satisfying 1 N 1 = Q’.
This result leads to the following theorem in the same way as (5.16) wasproved.
THEOREM (5.16)’. If j = (q - 1)v/( p - 1) where v = (VI ... , 03BDk) E(Z0)k, then
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REMARK 2. In the case ji # 0 for all i = 1, ... , k then as we have shownin 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 upperestimate 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. Theargument we used in [2] involved estimating degree Z(X, T) where X is thecomplement in An of the hypersurface defined by the vanishing ofthe polynomial Xl X2 ... Xn Hl H2 ... Hk and then utilizing the knownrelationship
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 notmake use of the special feature of H namely its reducibility to improve theestimate to Dn (dl , ... , dk).
REMARK 3. We believe the estimates of (5.16) and (5.16)’ are genericallyattained. 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 bythe vanishing of the polynomial X1X2 II7=ql Hi. Using (5.18), (5.19) and
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(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 linearhypersurfaces in An Fq when n > 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) anddegree 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’sare 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 andji ~ 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 defineB’ = B u 0/1, a subset of 1/. Let A ~ S, B ~ S denote arbitrary subsets.We define
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where if f (X) E Fq[S] then fA (X) E Fq[A] is the result of specializing thevariables Xi, i ~ A, to equal zero, and where if W is a collection indexed byV, LB, is the subcollection indexed by B’. A similar definition is immediatefor 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 thefollowing 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 +1be regular functions on V induced by polynomials F(X), 1 ~
Fq[X1, Xn] of respective degrees d0, {di}ki=t+1. Then
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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 boundfor the Newton polygon of det (I - Ta) (Theorems (4.5) and (4.6)). Recallthe 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 occuramong 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 totallength of the projections on the x-axis of the sides of slope n + k - mof 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
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det (I - Ta) such that
It is then clear that the total length of the projections on the x-axis of thesides of slope 5 r of the Newton polygon of det (I - Ta) is x(r). HenceNm = x(n + k - m) and by (6.2)
Theorem (4.5) gives us a sequence of points lying on or below the boundaryof the Newton polygon of det (I - Ta). The next step is to determine whichof 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 positivereal 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 theinequality holds with N - 1 in place of N:
Adding NW(N) to both sides:
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But
which is 0 by hypothesis.
COROLLARY (6.6). Suppose in addition that W(K) is a non-decreasing functionof K. Then for N = 0, 1, 2, ... ,
Proof. The hypothesis that W(K) is non-decreasing implies that we maytake à = 1 in Lemma (6.5). D
The function W(K) that appears in Theorem (4.5) is not non-decreasing, asis easily seen -from Theorem (4.6). However, we shall see that on a certainsubsequence of the sequence of points given in Theorem (4.5) W(K) behaveson 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-decreasingfunctions of Q. We may reformulate Theorem (4.6) as follows. Put K0 =K - |v| d0 and write
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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 let03BB(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 that03BB(Nr) 2r. From the definition of 03BB, it is clear that it suffices to take
Thus the point (x(r), y(r)), where
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satisfies (6.3). So from (6.4)
Next, we estimate 03A3Nn+k-mK=0 W(K). We have from (6.7) and the definitionof N,
Set D = max {di}ki=0. We have from the definitions of W , W
Thus
Hence
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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 orequal to
Using (n + 1)n/n! en and (n + 1)n-1/(n - 1)! 2en-1, this is less thanor equal to
Weakening this estimate slightly to produce a more compact formula, wehave the following result.
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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 theL-function when the coordinate hyperplanes are included in the exponentialsum. 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/Fpwhere Xi(0) is a non-trivial multiplicative character of Fp*.
where f(X) and {gi(X)}ki=t+11 of respective degrees do = deg f(X),di = deg gi(X) induce the given regular functions f and {gi}ki=t+1 on V;
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and where V itself is defined by the simultaneous vanishing of {hj(X)}tj=1 ofrespective degrees, deg hj = d,.
Consider now the special case where all the multiplicative characters aretrivial, i. e., ji = 0, for i = 1, 2, ... , k. Then by Lemma (1.3),
total deg L({Sm(F, 03C8; {0, Hi})}, T) = total deg L({Km(V, f, 03C8)}, T),
where V is the affine variety defined by the vanishing of the I7i’s. It is known[8] and [14] that over some extension field of Fq, V can be defined by nequations. Furthermore, this extension of scalars does not change the totaldegree of L({Km(V, f, 03C8)}, T) (see [5]). Hence for this particular L-function,we may assume that k n in Theorem (6.12).
COROLLARY (6.13).
This may be compared with [5, Theorem 2], where the exponent thatappears is 2n + 1 rather than n.We can prove somewhat weaker results without the assumption that the
ji’s are divisible by (q - 1)/(p - 1). Let a’ be chosen so that the charactersxl , ... , Xs have orders dividingpa’ - 1. The conclusion of Theorem (4.5) isstill valid provided the Newton polygon in question is the Newton polygon ofdet (I - Ta) computed with respect to ordpa’, rather than ordq. The subse-quent arguments may then be repeated without change to establish thefollowing.
THEOREM (6.14). If in Theorem (6.12) the Xi’s have exponent pa’ - 1, then theconclusion of the Theorem is valid when D is replaced by a’D.
We do not know whether the conclusions of Theorem (6.12) are valid underthe weaker hypothesis of Theorem (6.14).
References
1. A. Adolphson and S. Sperber; Exponential sums on the complement of a hypersurface.American Journal of Math. 102 (1980) 461-487.
2. A. Adolphson and S. Sperber; Character sums in finite fields. Comp. Math. 52 (1984)325-354.
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3. A. Adolphson: On the Dwork trace formula. Pacific Journal of Math. 113 (1984) 257-268.4. E. Bombieri: On exponential sums in finite fields. American Journal of Math. 88 (1966)
71-105.
5. E. Bombieri: On exponential sums in finite fields, II. Inventiones Math. 47 (1978) 29-39.6. B. Dwork: On the rationality of the zeta function of an algebraic variety. American
Journal of Math. 82 (1960) 631-648.7. B. Dwork: On the zeta function of a hypersurface. Publ. Math. I.H.E.S. 12 (1962) 5-68.8. D. Eisenbud and E.G. Evans, Jr: Every algebraic set in n-space is the intersection of n
hypersurfaces. Inventiones Math. 19 (1973) 107-112.9. M. Fried and M. Jarden: Field Arithmetic, Springer-Verlag, Berlin, Heidelberg 1986.
10. M. Fried: L-series on a Galois Stratification, preprint.11. L. Illusie: Theorie de Brauer et caractéristique d’Euler-Poincaré. Astérisque 82-83 (1981)
161-172.
12. J.P. Serre: Endomorphismes completement continus des espaces de Banach p-adiques.Publ. Math. I.H.E.S. 12 (1962) 69-85.
13. S. Sperber: On the L-functions associated with certain exponential sums, Journal ofNumber Theory 12 (1980) 141-153.
14. U. Storch: Bemerkung zu einem Satz von M. Kneser. Arch Math. 23 (1972) 403-404.15. P. Deligne and N. Katz: Groupes de Monodromie en Géométrie Algébrique, Lecture
Notes in Math. No. 340. Berlin, Heidelberg, New York: Springer-Verlag (1973).
