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# On $L$-functions of certain exponential sums

## Zusammenfassung

Let denote the finite field of order (a power of a prime ). We study the -adic valuations for zeros of -functions associated with exponential sums of the following family of Laurent polynomials

 f(x1,x2,⋯,xn+1)=a1xn+1(x1+1x1)+⋯+anxn+1(xn+1xn)+an+1xn+1+1xn+1

where . When , the estimate of the associated exponential sum appears in Iwaniec’s work, and Adolphson and Sperber gave complex absolute values for zeros of the corresponding -function. Using the decomposition theory of Wan, we determine the generic Newton polygon (-adic values of the reciprocal zeros) of the -function. Working on the chain level version of Dwork’s trace formula and using Wan’s decomposition theory, we are able to give an explicit Hasse polynomial for the generic Newton polygon in low dimensions, i.e., .

MSC2010 Codes: 11S40, 11T23, 11L07

-func. of cert. exp. sums]On -functions of certain exponential sums ZhangTianjin FengShanghai \contactChern Institute of Mathematics, Nankai University, Tianjin 300071, P.R. Chinazhangjun04@mail.nankai.edu.cn \contactDepartment of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P.R. Chinad.d.feng@163.com \researchsupportedThe research of Jun Zhang is supported by the National Key Basic Research Program of China (Grant No. 2013CB834204), and the National Natural Science Foundation of China (Nos. 61171082, 10990011 and 60872025).

## 1 Introduction

-functions have been a powerful tool to investigate exponential sums in number theory. To estimate an exponential sum, people are interested in the zeros and poles of the corresponding -function. Mathematicians study the number of these zeros and poles, the complex absolute values, -adic absolute values of them for primes , and -adic absolute values of them, especially for some interesting varieties and exponential sums [6, 7, 13, 14, 15]. Deligne’s theorem gives the general information for complex absolute values of the zeros and poles of -function. For -adic absolute values, it is well-known that if then all the zeros and poles have -adic absolute value . However, for -adic absolute values, it is still very mysterious [9, 10, 11, 12, 19, 20], especially in higher dimensions.

In this paper, we consider the following family of Laurent polynomials

 f(x1,x2,⋯,xn+1)=a1xn+1(x1+1x1)+⋯+anxn+1(xn+1xn)+an+1xn+1+1xn+1 (1.1)

where . The exponential sum associated to is defined to be

 S∗k(f)=∑x1,…,xn∈F∗qkζTrkf(x1,…,xn)p,

where is a fixed primitive -th root of unity in the complex numbers and denotes the trace map from the -th extended field to the prime field . When , the estimate of exponential sum is vital in analytic number theory. It appears in Iwaniec’s work on small eigenvalues of the Laplace-Beltrami operator acting on automorphic functions with respect to the group .

To understand the sequence of algebraic integers, we study the -function associated to

 L∗(f,T)=exp(∞∑k=1S∗k(f)Tkk).

By the theorem of Adolphson and Sperber [1], the -function for non-degenerate is a polynomial of degree , where is the Newton polyhedron of defined explicitly later. As the origin is an interior point of the Newton polyhedron , by the theorem of Denef and Loeser [5], we have

###### Theorem 1.1.

Assume that is non-degenerate, that is,

 ∏(c1,c2,⋯,cn)∈{±1}n(2c1a1+2c2a2+⋯+2cnan+an+1)≠0.

Then, the -function associated to the exponential sum is pure of weight , i.e.,

 L∗(f,T)(−1)n=(n+1)!Vol(Δ(f))∏i=1(1−αiT)

with the complex absolute value .

On the other hand, for each -adic absolute value with prime , the reciprocal zeros are -adic units: So we next study the -adic slopes of the reciprocal zeros of of such a family of exponential sums for the remaining prime . When , such non-degenerate Laurent polynomials are always ordinary and the Hodge polygon is very clear. So, we only consider . One of our main results in this paper is

###### Theorem 1.2.

Assume that of the form (1.1) is non-degenerate.

(i) The polynomial has degree .

(ii) There exists a non-zero polynomial such that if has coefficients verifying , then for each , the number of reciprocal zeros of with -adic slope is

 (n+1k),

and for any rational number , there is no reciprocal zero of with -adic slope .

We identify a Laurent polynomial of the form (1.1) with the vector of its coefficients , as they are one-to-one correspondent to each other. Let be the open subset consisting all non-degenerate Laurent polynomials with Newton polyhedron , explicitly

 Mp(Δ)={a∈An+1|a1⋯an+1∏(±2a1+±2a2+⋯+±2an+an+1)≠0}.

By Theorem 1.2(ii), we have determined -adic slopes of all reciprocal zeros of for “almost all” ’s in except a Zariski closed subset defined by the polynomial . The polynomial is called a Hasse polynomial of the Newton polyhedron . Next, we try to give an explicit Hasse polynomial. We will see that it is already quite complicated to explicitly determine the Hasse polynomial for we consider, a priori for more general polyhedrons. For low dimensions, we obtain the following explicit formulae of Hasse polynomials of the Newton polyhedron of the Laurent polynomials of the form (1.1). Working on the chain level version of Dwork’s trace formula, we prove

###### Theorem 1.3.

When , a Hasse polynomial can be taken to be

 hp(Δ)(a1,a2,a3)=∑0≤u+v≤p−12u,v∈Z1(u!v!(p−1−2u−2v)!)2a2v1a2u2ap−1−2u−2v3.

For , it has already been a bit more complicated than the case . Using the chain level version of Dwork’s trace formula, we can easily give the condition when Newton polygon and Hodge polygon coincide at the first break point just as we treat in the case . However, to find out when they meet at the second break point with the same method above, it needs us to compute the determinant of a matrix of size whose entries are all polynomials. In fact, it even requires a while to write down the matrix, let alone to compute the determinant. To deal with this problem, we use the boundary decomposition theorem of Wan to divide the “characteristic power series” det into “interior pieces” and then handle them piece by piece.

###### Theorem 1.4.

When , a Hasse polynomial can be taken to be

 hp(Δ)(a)=T(a)∑0≤u+v+w≤p−12u,v,w∈Z1(u!v!w!(p−1−2u−2v−2w)!)2a2u1a2v2a2w3ap−1−2u−2v−2w4,

where and is explicitly presented by Formula (3.5) which is essentially the determinant of a matrix whose entries are all polynomials.

The rest of this paper is organized as follows. To make this paper self-contained, we first recall some baic concepts and results about -functions, Newton polygon and Hodge polygon of -functions. Then we review some powerful tools to study -functions, such as Dwork’s -adic method and Wan’s decomposition theory. Finally, we use these methods to investigate -functions of the family of Laurent polynomials we consider in (1.1).

## 2 Preliminaries

### 2.1 Exponential sums and L-functions

Let be the finite field of elements with characteristic . For each positive integer , let be the finite extension of of degree . Let be a fixed primitive -th root of unity in the complex numbers. For any Laurent polynomial , we form the exponential sum

 S∗k(f)=∑x1,…,xn∈F∗qkζTrkf(x1,…,xn)p,

where denotes the multiplicative group of non-zero elements in and denotes the trace map from to the prime field . To understand the sequence of algebraic integers, we form the generating function of

 L∗(f,T)=exp(∞∑k=1S∗k(f)Tkk),

which is called the -function of the exponential sum . The study of has fundamental importance in number theory. For example, it connects with the zeta functions over finite fields. Consider

 Uf(Fq)={x1,…,xn∈F∗q∣f(x1,…,xn)=0}

the affine toric hypersurface defined by a Laurent polynomial

 f(x1,…,xn)∈Fq[x1,x−11,…,xn,x−1n].

Let denote the number of solutions of in . Its zeta function is given by

 Z(Uf,T)=exp(∞∑k=1#Uf(Fqk)Tkk).

It can be easily shown that

 qk#Uf(Fqk)=(qk−1)n+S∗k(x0f), (2.1)

and we have

 Z(Uf,qT)=Z(Gnm,T)L∗(x0f,T).

Thus we see that in order to study the zeta function, it suffices to study the -function . Also the study of exponential sums and the associated -functions has important applications in analytic number theory, and some applied mathematics such as coding theory, cryptography, etc.

By a theorem of Dwork-Bombieri-Grothendieck, the following generating -function is a rational function:

 L∗(f,T)=exp(∞∑k=1S∗k(f)Tkk)=∏d1i=1(1−αiT)∏d2j=1(1−βjT), (2.2)

where zeros and poles are non-zero algebraic integers. Equivalently, for each positive integer , we have the formula

 S∗k(f)=d2∑j=1βkj−d1∑i=1αki. (2.3)

Thus, our fundamental question about the sums is reduced to understanding the reciprocal zeros and .

Without any smoothness condition of , one does not even know exactly the number of zeros and the number of poles, although good upper bounds are available, see [3]. On the other hand, Deligne’s theorem on the Riemann hypothesis [4] gives the following general information about the nature of the zeros and poles. For the complex absolute value , it says

 |αi|=qui/2, |βj|=qvj/2, ui∈Z∩[0,2n], vj∈Z∩[0,2n]

where denotes the set of integers in the interval . Furthermore, each (resp. each ) and its Galois conjugates over have the same complex absolute value. For each -adic absolute value with prime , the and are -adic units:

 |αi|l=|βj|l=1.

For the remaining prime , Deligne’s integrality theorem implies that

 |αi|p=q−ri, |βj|p=q−sj, ri∈Q∩[0,n],sj∈Q∩[0,n],

where the -adic absolute value is normalized such that . Strictly speaking, in defining the -adic absolute value, we have tacitly chosen an embedding of the field of algebraic numbers into an algebraic closure of the -adic number field . Note that each (resp. each ) and its Galois conjugates over may have different -adic absolute values. The precise version of various types of Riemann hypothesis for the -function in (2.2) is then to determine the important arithmetic invariants . The integer (resp. ) is called the weight of the algebraic integer (resp. ). The rational number (resp. ) is called the slope of the algebraic integer (resp. ) defined with respect to . Without any smoothness condition on , not much more is known about these weights and the slopes, since one does not even know exactly the number of zeros and the number of poles. Under a suitable smoothness condition, a great deal more is known about the weights and the slopes , see Adolphson-Sperber [1], Denef-Loesser [5] and Wan [16, 18].

To investigate the slopes , Newton polygon was introduced.

### 2.2 Newton polygon and Hodge polygon

Let

 f(x1,…,xn)=J∑j=1ajxVj, aj≠0,

be a Laurent polynomial in . Each is a lattice point in and the power means the product . Let be the convex closure in generated by the origin and the lattice points . This is called the Newton polyhedron of . If is a subset of , we define the restriction of to to be the Laurent polynomial

 fδ=∑Vj∈δajxVj
###### Definition 2.1.

The Laurent polynomial is called non-degenerate if for each closed face of of arbitrary dimension which does not contain the origin, the partial derivatives

 {∂fδ∂x1,…,∂fδ∂xn}

has no common zeros with over the algebraic closure of .

Assume now that is non-degenerate, then the -function is a polynomial of degree by a theorem of Adolphson-Sperber [1] proved using -adic methods, where denotes the volume of . The complex absolute values (or the weights) of the zeros can be determined explicitly by a theorem of Denef-Loeser [5] proved using -adic methods. They depend only on , not on the specific and as long as is non-degenerate with . Hence, the weights have no variation as and varies. As indicated above, the -adic absolute values of the zeros are always 1 for each prime . Thus, there remains the intriguing question of determining the -adic absolute values (or the slopes) of the zeros. This is the -adic Riemann hypothesis for the -function . Equivalently, the question is to determine the Newton polygon of the polynomial

 L∗(f,T)(−1)n−1=n!V(f)∑i=0Ai(f)Ti, Ai(f)∈Z[ζp].

The Newton polygon of , denoted by NP, is defined to be the lower convex closure in of the following points

 (k,ordqAk(f)), k=0,1,…,n!V(f).

And a point in is called a break point of the Newton polygon if the left segment and the right segment have different slopes

Let be the parameter space of over with fixed . Let be the set of non-degenerate over with fixed . It is a Zariski open smooth affine subset of . It is non-empty if is large enough, say . Thus is again a smooth affine variety defined over . The Grothendieck specialization theorem [17] implies that as varies, the lowest Newton polygon

 GNP(Δ,p)=inff∈Mp(Δ)NP(f)

exits and is attained for all in some Zariski open dense subset of . The lowest polygon can then be called the generic Newton polygon, denoted by GNP.

A general property is that the Newton polygon lies on or above a certain topological or combinatorial lower bound, called the Hodge polygon HP which we describe bellow.

Let denote the -dimensional integral polyhedron in containing the origin. Let be the cone in generated by . Then is the union of all rays emanating from the origin and passing through . If is a real number, we define . For a point , the weight is defined to be the smallest non-negative real number such that . If such does not exist, we define .

It is clear that is finite if and only if . If is not the origin, the ray emanating from the origin and passing through intersects in a face of codimension that does not contain the origin. The choice of the desired -codimensional face is in general not unique unless the intersection point is in the interior of . Let be the equation of the hyperplane in , where the coefficients are uniquely determined rational numbers not all zero. Then, by standard arguments in linear programming, one finds that the weight function can be computed using the formula:

 ω(u)=n∑i=1eiui (2.4)

where denotes the coordinates of .

Let be the least common denominator of the rational numbers . It follows from (2.4) that for a lattice point in , we have

 ω(u)∈1D(δ)Z≥0, (2.5)

where denotes the set of non-negative integers. It is easy to show that there are lattice points such that the denominator of is exactly . Let be the least common multiple of all the ’s:

 D(Δ)=lcmδD(δ),

where runs over all the -codimensional faces of which do not contain the origin. Then by (2.5), we deduce

 ω(Zn)∈1D(Δ)Z≥0∪{∞}. (2.6)

The integer is called the denominator of . It is the smallest positive integer for which (2.6) holds. But be careful that there may not have a lattice point such that the denominator of is exactly .

For an integer , let

 WΔ(k)=♯{u∈Zn∣ω(u)=kD}

be the number of lattice points in with weight . This is a finite number for each . The Hodge numbers are defined to be

 HΔ(k)=n∑i=0(−1)i(ni)WΔ(k−iD),k∈Z≥0.

Hodge number is the number of lattice points of weight in a certain fundamental domain corresponding to a basis of the -adic cohomology space used to compute the -function. Thus, is a non-negative integer for each . Furthermore, by cohomology theory,

 HΔ(k)=0, for k>nD

and

 nD∑k=0HΔ(k)=n!V(Δ).
###### Definition 2.2.

The Hodge polygon HP of is defined to be the lower convex polygon in with vertices

That is, the polygon HP is the polygon starting from the origin and has a side of slope with horizontal length for each integer . For , the point

 (k∑m=0HΔ(k),1Dk∑m=0kHΔ(k))

is called a break point of the Hodge polygon if .

The lower bound of Adolphson and Sperber [1] says that if , then NP HP and they have the same endpoint. The Laurent polynomial is called ordinary if NP HP. Combining with the definition of the generic Newton polygon, we deduce

###### Proposition 2.1.

For every prime and every , we have the inequalities

 NP(f)≥GNP(Δ,p)≥HP(Δ).

Let be the moduli space of those such that , is nondegenerate with respect to and NPHP. In Dwork’s terminology, is called the Hasse domain of the generic Laurent polynomial defined over with contained in , and it is a Zariski-open subset of (possibly empty). Moreover the complement of in is an affine variety defined by a polynomial in the variables (coefficients of ), called the Hasse polynomial and denoted by . Very little about Hasse polynomials is known. It is very difficult to compute Hasse polynomial in general. In this paper, we study a family of Laurent polynomials and determine the Hasse polynomials in low dimensions.

### 2.3 Diagonal local theory

A Laurent polynomial is called diagonal if has exactly non-constant terms and is -dimensional (i.e., a simplex). In this case, the -function can be computed explicitly using Gauss sums. Let

 f(x)=n∑j=1ajxVj, aj∈F∗q, (2.7)

where are the vertices of an -dimensional integral simplex in . Let its vertex matrix be the non-singular matrix

 M=(V1,…,Vn),

where each is written as a column vector.

###### Proposition 2.2.

For in (2.7), is non-degenerate if and only if is relatively prime to det.

###### Proof.

Note that has only one face of dimension not containing the origin. For this face, let , We have

 xi∂f∂xi=n∑j=1Vji(ajxVj)=n∑j=1Vjiyj, (1≤i≤n). (2.8)

The partial derivatives have no common zeros with is equivalent to the linear equations of () have no common zeros in (2.8), which is equivalent to that is relatively prime to det.

For any other face of dimension , by a orthogonal transformation, we can assume is on the hyperplane , which reduce to the above situation.

Consider the solutions of the following linear system

 M⎛⎜ ⎜⎝r1⋮rn⎞⎟ ⎟⎠≡0 (mod 1), ri rational, 0≤ri<1. (2.9)

Let be the set of solutions of (2.9). It is easy to see that is a finite abelian group and its order is precisely given by

 |det(M)|=n!V(Δ).

Let be the prime to part of . It is an abelian subgroup of order equal to the prime to factor of det. In particular, if is relatively prime to det , i.e., is non-degenerate.

Let be an integer relatively prime to the order of , then multiplication by induces an automorphism of the finite abelian group . The map is called the -map of denoted by , where

 {mr}=({mr1},…,{mrn})

and denotes the fractional part of the real number . For each element , let be the smallest positive integer such that multiplication by acts trivially on , i.e.

 (md(m,r)−1)r∈Zn.

Let be the set of such that , We have the disjoint -degree decomposition

 Sp(Δ)=⋃d≥1Sp(m,d).

Let be a multiplicative character and let

 Gk(q)=−∑a∈F∗qχ(a)−kζTr(a)p (0≤k≤q−2)

be the Gauss sums. Then we have

###### Theorem 2.1 ([18]).
 L∗(f/Fq,T)(−1)n−1=∏d≥1∏r∈Sp(q,d)(1−Tdn∏i=1χ(ai)ri(qd−1)Gri(qd−1)(qd))1d,

where .

The Stickelberger theorem for Gauss sums is

###### Theorem 2.2 ([8]).

Let . Let be the sum of the -digits of in its base expansion. That is, , , . Then,

 ordpGk(q)=σp(k)p−1.

By Theorems 2.1 and 2.2, with a calculation, we have the ordinary criterion for a diagonal Laurent polynomial .

###### Theorem 2.3 ([18]).

Let be the largest invariant factor of . Let be the largest invariant factor of . If , then the diagonal Laurent polynomial in (2.7) is ordinary at . In particular, if , then the diagonal Laurent polynomial in (2.7) is ordinary at .

In order to study the (generically) ordinary property of -functions and determine a Hasse polynomial , we are going to briefly review Dwork’s trace formula, Wan’s descent theorem and Wan’s decompostion theory for -function.

### 2.4 Dwork’s trace formula

Let be the field of -adic numbers. Let be the completion of an algebraic closure of . Let for some positive integer . Denote by ord the additive valuation on normalized by ord , and denote by ord the additive valuation on normalized by ord. Let denote the unramified extension of in of degree . Let , where is a primitive -th root of unity. Then is the totally ramified extension of of degree . Let be the compositum of and . Then is an unramified extension of of degree . The residue fields of rings of algebraic integers of and are both , and the residue fields of rings of algebraic integers of and are both . Let be a fixed element in satisfying

 ∞∑m=0πpmpm=0, ordpπ=1p−1.

Then, is a uniformizer of and we have

 Ω1=Qp(π).

The Frobenius automorphism of Gal lifts to a generator of Gal such that . If is a -st root of unity in , then .

Let be the Artin-Hasse exponential series:

 E(t)=exp(∞∑m=0tpmpm)=∏k≥1,(k,p)=1(1−tk)μ(k)/k

where is the Möbius function. The last product expansion shows that the power series has -adic integral coefficients. Thus, we can write

 E(t)=∞∑m=0λmtm, λm∈Zp.

For (what we need below), more precise information is given by

 λm=1m!, ordpλm=0, 0≤m≤p−1. (2.10)
 λm=1m!+1p(m−p)!, ordpλm≥0, p≤m≤2p−1. (2.11)

The shifted series

 θ(t)=E(πt)=∞∑m=0λmπmtm (2.12)

is a splitting function in Dwork’s terminology. The value is a primitive -th root of unity, which will be identified with the -th root of unit used in our definition of the exponential sums as given in the introduction.

For a Laurent polynomial

 f(x1,⋯,xn)∈Fq[x1,x−11,⋯,xn,x−1n],

we write . Let be the Teichmüller lifting of in . Thus, we have . Set

 F(f,x)=J∏j=1θ(ajxVj) (2.13)
 Fa(f,x)=a−1∏i=0Fτi(f,xpi). (2.14)

Note that (2.12) implies that and are well defined as formal Laurent series in with coefficients in .

To describe the growth conditions satisfied by , write

 F(f,x)=∑r∈ZnFr(f)xr.

Then from (2.12) and (2.13), one checks that

 Fr(f)=∑u(J∏j=1λujaujj)πu1+⋯+uJ, (2.15)

where the outer sum is over all solutions of the linear system

 J∑j=1ujVj=r, uj≥0, uj integral. (2.16)

Thus, if (2.16) has no solutions. Otherwise, (2.15) implies that

 ordFr(f)≥1p−1infu{J∑j=1uj},

where the inf is taken over all solutions of (2.15).

For a given point , recall that the weight is given by

 ω(r)=infu{J∑j=1uj|J∑j=1ujVj=r,uj≥0,uj∈R},

where the weight is defined to be if is not in the cone generated by and the origin. Thus,

 ordFr(f)≥ω(r)p−1, (2.17)

with the obvious convention that if . Let be the closed cone generated by the origin and . Let be the set of lattice points in . That is,

 L(Δ)=Zn∩C(Δ).

For real numbers and with , define the following two spaces of -adic functions:

 L(b,c)={∑r∈L(Δ)Crxr∣Cr∈Ωa, ordpCr≥bω(r)+c}
 L(b)=⋃c∈RL(b,c).

one checks from (2.17) that

 F(f,x)∈L(1p−1