L-FUNCTIONS OF CERTAIN EXPONENTIAL SUMS OVER FINITE FIELDS II XIN LIN AND CHAO CHEN ABSTRACT. Inthis paper,we computethe q-adic slopes of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson-Sperber’s work on toric exponential sums and Wan’s decomposition theorems. 1. INTRODUCTION A fundamental problem in number theory is to estimate the reciprocal zeros and poles of L-functions of certain exponential sums over finite fields of characteristic p. Deligne’s theorem on Riemann hypothesis gives general information for the complex absolute values of the zeros and poles. The theorem also indicates that the reciprocal roots and poles are ℓ-adic units if ℓ is a prime and ℓ =6 p. So the remaining unknown part is the p-adic estimate of the L-function which can be identified with the p-adic version of Riemann hypothesis for the L-function. In this paper, we study the p-adic slopes of L-functions of a special family of exponential sums. Let Fq be the finite field of q elements with characteristic p. For each positive integer ∗ k, let Fqk denote the degree k finite extension of Fq and Fqk denote the set of non-zero ∗ elements in Fqk . Let ψ : Fp → C be a fixed nontrivial additive character over Fp and r Trk : Fqk → Fp be the trace map. Let n = i=1 ni be a partition of a positive integer n. We introduce n variables P xij, for 1 ≤ i ≤ r, 1 ≤ j ≤ ni, and n positive integral constants bij, for 1 ≤ i ≤ r, 1 ≤ j ≤ ni. ∗ In this paper, we are concerned with the following class of exponential sums: for ai ∈ Fq and bij ∈ Z>0, r ni Sk(~a)= ψ Trk xij , r ai arXiv:2107.13357v1 [math.NT] 28 Jul 2021 =1 i=1 j=1 !! i=1 n bij Xi x X X P Qj=1 ij ∗ where the sum is over all xij ∈ Fqk . Since the case of bij divisible by p can be reduced to p ∤ bij through a change of variables, in this paper we assume p ∤ bij for 1 ≤ i ≤ r, 1 ≤ j ≤ ni. The estimate of Sk(~a) plays an important role in analytic number theory when n = 4, ni =2 and bij =1. It is a crucial ingredient in Heath Brown’s work on the divisor function d3(n) in arithmetic progressions [HB86]. It also appears in Friedlander and Iwaniec’s work on estimating certain averages of incomplete Kloosterman sums in application to 2010 Mathematics Subject Classification. Primary 11S40, 11T23, 11L07. Key words and phrases. Exponential sums, L-function, Laurent polynomials, Newton polygon, Hodge polygon, Decomposition theory, Weight computation. 1 2 XIN LIN AND CHAO CHEN the divisor problem of d3(n) [FI85]. Relying on the estimate of Sk(~a) and Friedlander- Iwaniec’s result, Zhang gained a boundary of the error terms in his work on twin prime conjecture [Zha14]. For the complex absolute values of exponential sums, Katz [Kat87] proved that (n−1)k/2 |Sk(~a)| ≤ c1q , r ni where c1 = i=1 1+ j=1 bij − 1. To understand the p-adic absolute values of the exponential sums, we study p-adic slopes Q P of the reciprocal roots and poles of the associated L-function ∞ T k (1.1) L(~a, T ) = exp Sk(~a) . k ! Xk=1 When n =4, ni =2 and bij =1, our previous paper [CL20] showed that 6 L(~a, T )=(1 − T )(1 − qT ) (1 − αiT ), i=1 Y where the p-adic norms of the reciprocal roots are given by 1 if i =1. −1 q if i =2, 3. |αi|p = −2 q if i =4, 5. q−3 if i =6. In this paper, we obtain q-adic slopes for the above general class of exponential sums. Note that our main result is consistent with Katz’s complex estimate. Theorem 1.1. Let D = lcm{bij|1 ≤ i ≤ r, 1 ≤ j ≤ ni}. (1) The associated L-function is a polynomial given by d−r (−1)n r−1 L(~a, T ) = (1 − T ) (1 − αiT ), i=1 Y r ni where d = i=1 1+ j=1 bij . (2) If p ≡ 1(modD), for each m = 0, 1, . , nD, the number of reciprocal zeros of Qn P L(~a, T )(−1) with q-adic slope m/D is the coefficient of xm+D in the following generating function ni 1 r (1+ j=1 b )D n−r 1 − x ij G(x)= 1 − xD P , ni D/bij i=1 j=1 1 − x Y and for any rational number m∈ / {0, 1Q, . , nD}, there is no reciprocal zero of L(~a, T )(−1)n with q-adic slope m/D. Corollary 1.2. Assume bij =1 (1 ≤ i ≤ r, 1 ≤ j ≤ ni). (−1)n r (1) The polynomial L(~a, T ) has degree i=1 (1 + ni) − 1. r (2) Let d = (1 + ni). If ni is even for 1 ≤ i ≤ n, we have i=1 Q r2+3r+2 d− 2 Q 2 r (−1)n r−1 r −r L(~a, T ) = (1 − T ) (1 − γ0T )(1 − qT ) 2 (1 − γiT ) (1 − αjT ), i=1 j=1 Y Y L-FUNCTIONS OF CERTAIN EXPONENTIAL SUMS OVER FINITE FIELDS II 3 n−1 where ordq γ0 =0, ordq γi = 1(1 ≤ i ≤ r), |γi| = q 2 (0 ≤ i ≤ r), ordq αj > 1 n−1 and |αj| ≤ q 2 . n (3) For each m = 0, 1, 2,...,n, the number of reciprocal zeros of L(~a, T )(−1) with q-adic slope m is the coefficient of xm+1 in the following generating function r ni G1(x)= (1 + x + ··· + x ) , i=1 Y and for any rational number m∈ / {0, 1,...,n}, there is no reciprocal zero of L(~a, T )(−1)n with q-adic slope m. Our approach is to reduce this theorem to the L-function of toric exponential sums and i then apply the systematic results available for such toric L-functions. Let Ni = l=1 nl for 1 ≤ i ≤ r and N0 =0. Let xij = xNi−1+j and bij = bNi−1+j (1 ≤ i ≤ r, 1 ≤ j ≤ ni). ±1 ±1 P Consider a Laurent polynomial f ∈ Fq[x1 ,...,xn+1] defined by n r ai f(x1, x2,...,xn+1)= xi + xn+1 − 1 , Ni bl i=1 i=1 l=N 1+1 xl ! X X i− ∗ where ai ∈ Fq and bl ∈ Z>0. For any positive integer k,Q let ∗ Sk(f)= ψ(Trk(f)) ∗ xi∈F Xqk be the associated exponential sum. Its generating L-function is defined to be ∞ k ∗ ∗ T L (f, T ) = exp Sk(f) . k ! Xk=1 ∗ The following equation describes the relationship between Sk(~a) and Sk(f), (−1)n 1 S (~a)= + S∗(f). k qk qk k Based on the relationship, it’s easy to check that n 1 n (1.2) L(~a, T )(−1) = L∗ (f,T/q)(−1) . 1 − T/q So it suffices to evaluate the reciprocal roots or poles of L∗(f, T ). By Adolphson and Sper- ∗ (−1)n ber’s theorem [AS89], L (f, T ) is a polynomial if p ∤ lcm{bij}. Under this assump- n tion, we compute the q-adic Newton polygon of L∗(f, T )(−1) , which indicates the q-adic slope information of the L-function. Furthermore, using Wan’s decomposition theorem, we show that the L-function has the following form d−r ∗ (−1)n r−1 L (f, T ) = (1 − T )(1 − qT ) (1 − βiT ), i=1 Y r ni where d = i=1 1+ j=1 bij . This leads to our main theorem. This paper is organized as follows. In section 2, we review some technical methods and Q P theorems including Adolphson-Sperber’s theorems and Wan’s decomposition theorems. In section 3, we prove the main results. 4 XIN LIN AND CHAO CHEN 2. PRELIMINARIES ±1 ±1 2.1. Rationality of the generating L-function. Let f ∈ Fq x1 ,...,xn be a Laurent polynomial and its associated exponential sum is defined to be ∗ (2.1) Sk(f)= ψ(Trk(f)), ∗ xi∈F Xqk ∗ where Trk : Fqk → Fp is the trace map and ψ : Fp → C is a fixed nontrivial additive ∗ character. It’s a classical problem to give a good estimate for the valuations of Sk (f) in analytic number theory. In order to compute the absolute values of the exponential sums, ∗ we usually study the generating L-function of Sk(f) given by ∞ k ∗ ∗ T L (f, T ) = exp Sk(f) ∈ Q(ζp)[[T ]]. k ! Xk=1 By a theorem of Dwork-Bombieri-Grothendieck [Dwo62,Gro66], the generating L-function is a rational function, d1 (1 − α T ) L∗(f, T )= i=1 i , d2 Qj=1(1 − βjT ) where all the reciprocal zeros and poles areQ non-zero algebraic integers. After taking loga- rithmic derivatives, we have the formula d2 d1 ∗ k k (2.2) Sk(f)= βj − αi , k ∈ Z≥1, j=1 i=1 X X which implies that the zeros and poles of the generating L-function contain critical infor- mation about the exponential sums. From Deligne’s theorem on Riemann hypothesis [Del80], the complex absolute values of reciprocal zeros and poles are bounded as follows ui/2 vj /2 |αi| = q , |βj| = q ,ui ∈ Z ∩ [0, 2n], vj ∈ Z ∩ [0, 2n]. For non-archimedean absolute values, Deligne [Del80] proved that |αi|ℓ = |βj|ℓ =1 when ℓ is a prime and ℓ =6 p.