Chunlei Liu
Generic Newton polygons for L-functions of exponential sums associated to Laurent polynomials in one variable are determined. The corresponding Hasse polynomials are also determined.
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Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol-Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent polynomial reciprocity laws. Our construction is based on Machide's result on his elliptic Dedekind-Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain …
Sums of Laurent series with bounded partial quotients
本文研究洛朗级数中具有有界部分商的实数表示问题，提出新的结果以扩展Hall和Cusick的经典结论。
Arithmetic hypergeometric $\mathcal {D}$-modules and exponential sums on reductive groups
本文定义约化群有限表示族关联的超几何指数和，引入带Frobenius结构的算术超几何$\boldsymbol{\textrm{D}}$-模研究该指数和，得到其估计并构造相关$F$-等晶体。
The $T$-adic exponential sum associated to a Laurent polynomial in one variable is studied. An explicit arithmetic polygon is proved to be the generic Newton polygon of the $C$-function of the T-adic exponential sum. It gives the generic Newton polygon of $L$-functions of $p$-power order exponential sums.
Let P(x) be a one-variable Laurent polynomial of degree (d_1,d_2) over a finite field of characteristic p. For any fixed positive integer s not divisible by p, we prove that the (normalized) p-adic Newton polygon of the L-functions of exponential sums of P(x^s) has a tight lower bound which we call `Hodge-Stickelberger polygon', depending only on d_1,d_2,s, and (p mod s). This Hodge-Stickelberger polygon is a weighted convolution of a `Hodge polygon' for L-function of exponential sum of P(x) and the `Newton polygon' for L-function of exponential sum of x^s (given by the classical Stickelberger…
$T$-adic exponential sums associated to a Laurent polynomial $f$ are introduced. They interpolate all classical $p^m$-power order exponential sums associated to $f$. The Hodge bound for the Newton polygon of $L$-functions of $T$-adic exponential sums is established. This bound enables us to determine, for all $m$, the Newton polygons of $L$-functions of $p^m$-power order exponential sums associated to an $f$ which is ordinary for $m=1$. Deeper properties of $L$-functions of $T$-adic exponential sums are also studied. Along the way, new open problems about the $T$-adic exponential sum itself ar…
Let $k$ be a finite field of characteristic $p$, $l$ a prime number distinct to $p$, $ψ:k\to \bar {\bf Q}_l^\ast$ a nontrivial additive character, and $χ:{k^\ast}^n\to \bar{\bf Q}_l^\ast$ a character on ${k^\ast}^n$. Then $ψ$ defines an Artin-Schreier sheaf ${\cal L}_ψ$ on the affine line ${\bf A}_k^1$, and $χ$ defines a Kummer sheaf ${\cal K}_χ$ on the $n$-dimensional torus ${\bf T}_k^n$. Let $f\in k[X_1,X_1^{-1},..., X_n,X_n^{-1}]$ be a Laurent polynomial. It defines a $k$-morphism $f:{\bf T}_k^n\to {\bf A}_k^1$. In this paper, we calculate the dimensions and weights of $H_c^i({\bf T}_{\bar …
Twistec T-adic exponential sums are studied. As an application, the behavior of the L-function under diagonal base chang is explicitely given.
The $C$-function of $T$-adic exponential sums is studeid. An explicit arithmetic bound is established for the Newton polygon of the $C$-function. This polygon lies above the Hodge polygon. It gives a sup-Hodge bound of the $C$-function of $p$-power order exponential sums.
We discuss exponential sums on affine space from the point of view of Dwork's p-adic cohomology theory
We prove a vanishing theorem for the p-adic cohomology of exponential sums on affine space. In particular, we obtain new classes of exponential sums on affine space that have a single nonvanishing p-adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.
We give two applications of our earlier work "Exponential sums on A^n, II" (math.AG/9909009). We compute the p-adic cohomology of certain exponential sums on A^n involving a polynomial whose homogeneous component of highest degree defines a projective hypersurface with at worst weighted homogeneous isolated singularities. This study was motivated by recent work of Garcia (Exponential sums and singular hypersurfaces, Manuscripta Math., v. 97 (1998), pp. 45-58). We also compute the p-adic cohomology of certain exponential sums on A^n whose degree is divisible by the characteristic.
Let Delta be an integral convex polytope containing the origin of dimension n in the n-dim real space and it is simplicial at all origin-less facets. Let A(Delta) be the space of all Laurent polynomials f parametered by its coefficients supported on the interior of Delta with with prescribed vertices. In this paper we prove that, if the interior points of Delta and its vertices generate almost all integral points in the cone of Delta, then there is a Zariski dense open subset U in A(Delta) defined over the rationals such that for every f in U(bar{Q}) and for p large enough the Newton polygon o…
We prove the intersection of Igusa's Conjecture of [Igusa, J., "Lectures on forms of higher degree", Lect. math. phys., Springer-Verlag, 59 (1978)] and the Denef - Sperber Conjecture of [Denef, J. and Sperber, S., "Exponential sums mod p^n and Newton polyhedra", Bull. Belg. Math. Soc., suppl. (2001) 55-63] on nondegenerate exponential sums modulo p^m.
The twisted $T$-adic exponential sum associated to $x^{d}+λx$ is studied. If $λ\neq0,$ then an explicit arithmetic polygon is proved to be the Newton polygon of the twisted $C$-function of the T-adic exponential sum. It gives the Newton polygons of the $L$-functions of twisted $p$-power order exponential sums.
The twisted $T$-adic exponential sum associated to a polynomial in one variable is studied. An explicit arithmetic polygon is proved to be the generic Newton polygon of the twisted $C$-function of the T-adic exponential sum. It gives the generic Newton polygon of the twisted $L$-functions of $p^m$-power order exponential sums, for all $m$.
We give bounds for exponential sums over curves defined over Galois rings. We first define summation subsets as the images of lifts of points from affine opens of the reduced curve, and we give bounds for the degrees of their coordinate functions. Then we get bounds for exponential sums, extending results of Kumar et al., Winnie Li over the projective line, and Voloch Walker over elliptic and $C_{ab}$ curves.
The $T$-adic exponential sum of a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the $C$-function of the T-adic exponential sum. This bound gives lower bounds for the Newton polygon of the $L$-function of exponential sums of $p$-power order.
The twisted $T$-adic exponential sum associated to a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the $C$-function of the twisted T-adic exponential sum. This bound gives lower bounds for the Newton polygon of the $L$-function of twisted $p$-power order exponential sums.
Given a prime $p$ and an integer $d>1$, we give a numerical criterion to decide whether the $\ell$-adic sheaf associated to the one-parameter exponential sums $t\mapsto \sum_xψ(x^d+tx)$ over ${\mathbb F}_p$ has finite monodromy or not, and work out some explicit cases where this is computable.
Our previous theorems on exponential sums often did not apply or did not give sharp results when certain powers of a variable appearing in the polynomial were divisible by p. We remedy that defect in this paper by systematically applying "p-power reduction," making it possible to strengthen and extend our earlier results.
In this article, we use Robba's method to give an estimate of the Newton polygon for the L function on torus and we can draw the Newton polygon in some special cases.
A translation of N.M. Katz's "Exponential Sums"
本文翻译并介绍Katz关于有限域上指数和的理论，涵盖数论背景、L-函数与Weil猜想、同调解释，并在后两章给出一般情形下的一致性定理及特定类的精确同调分析。
Laurent series expansions for the Barnes multiple zeta function
本文研究Barnes多重zeta函数的Laurent系数，在最高极点给出显式极限公式，在较低极点建立与Taylor系数的全阶关系，并利用整函数性质分析大阶行为，推广了常数项的有限部分表示。
We consider generalized Dedekind sums in dimension $n$, for fixed $n$-tuple of natural numbers, defined as sum of products of values of periodic Bernoulli functions. This includes the higher dimensional Dedekind sums of Zagier and Apostol-Carlitz' generalized Dedekind sums as well as the original Dedekind sums. These are realized as coefficients of Todd series of lattice cones and satisfy reciprocity law from the cocycle property of Todd series. Using iterated residue formula, we compute the coefficient of the decomposition of of the Todd series corresponding to a nonsingular decomposition of …
We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa's conjecture on exponential sums, with the log-canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for non-rational singularities, by comparing the log canonical threshold with a local notion of the motivic oscillation index.
In this paper we define the $p$-density of a finite subset $D\subset\ma{N}^r$, and show that it gives a good lower bound for the $p$-adic valuation of exponential sums over finite fields of characteristic $p$. We also give an application: when $r=1$, the $p$-density is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in $D$.
We propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new predicted uniform behavior. The exponential sums have summation sets consisting of integers modulo $p^m$ lying $p$-adically close to $y$, and the proposed bounds are uniform in $p$, $y$, and $m$. We give evidence for the conjecture, by showing uniform bounds in $p$, $y$, and in some values for $m$. On the way, we prove new bounds for log-canonical thresholds which a…
In this article, we consider the estimation of exponential sums along the points of the reduction mod $p^{m}$ of a $p$-adic analytic submanifold of $ \mathbb{Z}_{p}^{n}$. More precisely, we extend Igusa's stationary phase method to this type of exponential sums. We also study the number of solutions of a polynomial congruence along the points of the reduction mod $% p^{m}$ of a $p$-adic analytic submanifold of $\mathbb{Z}_{p}^{n}$. In addition, we attach a Poincare series to these numbers, and establish its rationality. In this way, we obtain geometric bounds for the number of solutions of the…
We deduce Katz's theorems for $(A,B)$-exponential sums over finite fields using $\ell$-adic cohomology and a theorem of Denef-Loeser, removing the hypothesis that $A+B$ is relatively prime to the characteristic $p$. In some degenerate cases, the Betti number estimate is improved using toric decomposition and Adolphson-Sperber's bound for the degree of $L$-functions. Applying the facial decomposition theorem in \cite{W1}, we prove that the universal family of $(A,B)$-polynomials is generically ordinary for its $L$-function when $p$ is in certain arithmetic progression.
We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power $q$. For fixed $d$, we restrict to moduli $q$ so that there is a unique subgroup of invertible classes modulo $q$ of order $d$. We study distribution properties of these families of sums as $q$ grows and we establish equidistribution results in some regions of the complex plane which are described as the image of a multi-dimensional torus via an explicit Laurent polynomial. In some cases, the region of equidistribution can be interpreted as the one delimited by a hypocycloid, or as a Mi…
On the Laurent series expansions of the Barnes double zeta function
本文研究Barnes双ζ函数在s=1和s=2处的Laurent级数展开，推导出系数的极限表达式，得到类似于欧拉-斯蒂尔杰斯常数的类比，并分析其渐进行为，揭示其与经典理论的区别。
This thesis contains some new results on the uniform rationality of Poincaré series of p-adic equivalence relations and Igusa's conjecture on exponential sums
On partial T-adic exponential sums and partial exponential sums with p-power conductor
本文推广了Liu和Wan的T进指数和理论到部分情形，证明了相关L函数的T进亚纯性，从而给出p幂导子特征部分L函数有理性的p进证明，并给出Newton-over-Hodge估计。
Recent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of exponential sums are described first, concentrating on developments involving smooth Weyl sums. Next, arithmetic variants of classical inequalities of Bessel and Cauchy-Schwarz are discussed. Finally, some emerging connections between the circle method and arithmetic geometry are mentioned.
We give more evidence for Patterson's conjecture on sums of exponential sums, by getting an asymptotic for a sum of quartic exponential sums over $\Q[i].$ Previously, the strongest evidence of Patterson's conjecture over a number field is the paper of Livné and Patterson \cite{LP} on sums of cubic exponential sums over $\Q[ω], ω^3=1.$ The key ideas in getting such an asymptotic are a Kuznetsov-like trace formula for metaplectic forms over a quartic cover of $GL_2,$ and an identity on exponential sums relating Kloosterman sums and quartic exponential sums. To synthesize the spectral theory and …
We improve an existing result on exponential quadrilinear sums in the case of sums over multiplicative subgroups of a finite field and use it to give a new bound on exponential sums with quadrinomials.
In this paper, we focus on computing the higher slope Hasse polynomials of L-functions of certain exponential sums associated to the following family of Laurent polynomials $f(x_1,\ldots ,x_{n+1})=\sum_{i=1}^na_i x_{n+1}\left(x_i+\frac{1}{x_i}\right)+a_{n+1} x_{n+1}+\frac{1}{x_{n+1}}$, where $a_i \in \F^*_{q}$, $i=1,2, \ldots, n+1$. We find a simple formula for the Hasse polynomial of the slope one side and study the irreducibility of these Hasse polynomials. We will also provide a simple form of all the higher slope Hasse polynomials for $n=3$, answering an open question of Zhang and Feng.
We study averages over squarefree moduli of the size of exponential sums with polynomial phases. We prove upper bounds on various moments of such sums, and obtain evidence of un-correlation of exponential sums associated to different suitably unrelated and generic polynomials. The proofs combine analytic arguments with the algebraic interpretation of exponential sums and their monodromy groups.
We establish bounds of triple exponential sums with mixed exponential and linear function. The method we use is by Shparlinski together with a bound of additive energy from Roche-Newton, Rudnev and Shkredov.
This paper provides a technique for evaluating some nonlinear Gaussian sums in closed forms. The evaluation is obtained from the known values of simpler exponential sums.
We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2πiag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning additive properties of multiplicative subgroups imply new bounds for the sums under consideration.
In this paper, we completely determine the slopes and weights 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. One consequence of our main result is a sharp estimate of these exponential sums. Another consequence is to obtain an explicit counterexample of Adolphson-Sperber's conjecture on weights of toric exponential sums.
This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.
In this paper we introduce two nonlinear functionals, the H and J functional, in the theory of complete exponential sums.
In this paper, we compute the $q$-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.
We study the p-adic valuations of roots of L-functions associated with certain families of exponential sums of Laurent polynomials in n variables over a finite field. The families we consider are reflection and Kloosterman variants of diagonal polynomials. Using decomposition theorems of Wan, we determine the Newton and Hodge polygons of a non-degenerate Laurent polynomial in one of these families.
In this paper we expose our main results about rank problems concerning persymmetric matrices over F_2 associated to some exponential sums.
We study upper bounds for sums of Dirichlet characters. We prove a uniform upper bound of the character sum over all proper generalized arithmetic progressions, which generalizes the classical Polya and Vinogradov inequality. Our argument is based on getting an upper bound for the l1 norm of the Fourier coefficients of a generalized arithmetic progression. Our method also applies to give upper bounds for polynomial exponential sums.
In this paper we obtain some new estimates for multilinear exponential sums in prime fields with a more general class of weights than previously considered. Our techniques are based on some recent progress of Shkredov on multilinear sums which has roots in Rudnev's point plane incidence bound. We apply our estimates to obtain new results concerning exponential sums with sparse polynomials and Weyl sums over small generalized arithmetic progressions.
We use Bourgain's recent bound for short exponential sums to prove certain independence results related to the distribution of squarefree numbers in arithmetic progressions.
Barban-Davenport-Halberstam type theorems for exponential sums and Piatetski-Shapiro primes
该研究在数论领域建立了三类Barban-Davenport-Halberstam型定理，分别适用于素数指数和、Piatetski-Shapiro素数及该类素数上的指数和，拓展了相关定理的应用范围。
Two Kinds Hybrid Power Mean Involving Two-Term Exponential Sums and Dedekind Sums
本文利用解析方法、二次剩余变换及 Dedekind 和性质，研究两类含两项指数和与 Dedekind 和的混合幂平均计算问题，并给出两个渐近公式，推广了已有结论。
We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on elliptic curves.
In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}<γ< 1$.
We obtain several estimates for bilinear form with exponential sums with binomials $mx^k + nx^\ell$. In particular we show the existence of nontrivial cancellations between such sums when the coefficients $m$ and $n$ vary over rather sparse sets of general nature
Several asymptotic expansions and formulas for cubic exponential sums are derived. The expansions are most useful when the cubic coefficient is in a restricted range. This generalizes previous results in the quadratic case and helps to clarify how to numerically approximate cubic exponential sums and how to obtain upper bounds for them in some cases.
The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on Shparlinski's bound for exponential sums attached to certain recurrence sequences over finite fields.
Let $γ<1<c$ and $19(c-1)+171(1-γ)<9$. In this paper, we establish an asymptotic formula for exponential sums over Piatetski-Shapiro primes $p=[n^{1/γ}]$ in arithmetic progressions.
In this paper we construct a Dwork theory for general exponential sums over affinoids in Witt towers. Using this, we compute the degree of the $L$-function, its Hodge polygon and examine when the Hodge and Newton polygons coincide.
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