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 $H_c^i({\bf T}_{\bar k}^n, {\cal K}_χ\otimes f^\ast {\cal L}_ψ)$ under some non-degeneracy conditions on $f$. Our results can be used to estimate sums of the form $$\sum_{x_1,..., x_n\in k^\ast} χ_1(f_1(x_1,..., x_n))... χ_m(f_m(x_1,..., x_n))ψ(f(x_1,..., x_n)),$$ where $χ_1,..., χ_m:k^\ast\to {\bf C}^\ast$ are multiplicative characters, $ψ:k\to {\bf C}^\ast$ is a nontrivial additive character, and $f_1,..., f_m, f$ are Laurent polynomials.
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 $m$.
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.
We study the $p$-adic absolute value of the roots of the $L$-functions associated to certain twisted character sums, and additive character sums associated to polynomials $P(x^d)$, when $P$ varies among the space of polynomial of fixed degree $e$ over a finite field of characteristic $p$. For sufficiently large $p$, we determine in both cases generic Newton polygons for these $L$-functions, which is a lower bound for the Newton polygons, and the set of polynomials of degree $e$ for which this generic polygon is attained. In the case of twisted sums, we show that the lower polygon defined in \c…
In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.
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.
In their recent work, the authors (2016) have combined classical ideas of A. G. Postnikov (1956) and N. M. Korobov (1974) to derive improved bounds on short character sums for certain nonprincipal characters with powerful moduli. In the present paper, these results are used to bound sums of the Mobius function twisted by characters of the same type, complementing and improving some earlier work of B. Green (2012). To achieve this, we obtain a series of results about the size and zero-free region of $L$-functions with the same class of moduli.
We consider a generalized Gauss sum supported on matrices over a number field. We evaluate this Gauss sum and relate it to the number of totally isotropic subspaces of related quadratic spaces. Then we consider a further generalization of such a Gauss sum, realizing its value in terms of numbers of totally isotropic subspaces of related quadratic spaces.
The aim of this paper is to construct new Dedekind type sums. We construct generating functions of Barnes' type multiple Frobenius-Euler numbers and polynomials. By applying Mellin transformation to these functions, we define Barnes' type multiple l-functions, which interpolate Frobenius-Euler numbers at negative integers. By using generalizations of the Frobenius-Euler functions, we define generalized Dedekind type sums and prove corresponding reciprocity law. We also give twisted versions of the Frobenius-Euler polynomials and new Dedekind type sums and corresponding reciprocity law. Further…
We produce a new proof of the reciprocity law for the twisted second moment of Dirichlet L-functions that was recently proved by Conrey. Our method is to analyze certain two-variable sums where the variables satisfy a linear congruence. We show that these sums satisfy an elegant reciprocity formula. In the case that the modulus is prime, these sums are closely related to the twisted second moment, and the reciprocity formula for these sums implies Conrey's reciprocity formula. We also extend the range of uniformity of Conrey's formula.
$T$-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…
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.
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.
We obtain new bounds for short sums of isotypic trace functions associated to some sheaf modulo prime $p$ of bounded conductor, twisted by the Mobius function and also by the generalised divisor function. These trace functions include Kloosterman sums and several other classical number theoretic objects. Our bounds are nontrivial for intervals of length at least $p^{1/2+\varepsilon}$ with an arbitrary fixed $\varepsilon >0$, which is shorter than the length at least $p^{3/4+\varepsilon}$ in the case of the Mobius function and at least $p^{2/3+\varepsilon}$ in the case of the divisor function r…
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.
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.
We construct isomorphisms between spaces of vector-valued modular forms for the dual Weil representation and certain spaces of scalar-valued modular forms in the case that the underlying finite quadratic module $A$ has order $p$ or $2p$, where $p$ is an odd prime. The isomorphisms are given by twisted sums of the components of vector-valued modular forms. Our results generalize work of Bruinier and Bundschuh to the case that the components $F_γ$ of the vector-valued modular form are antisymmetric in the sense that $F_γ = -F_{-γ}$ for all $γ\in A$. As an application, we compute restrictions of …
In this paper, we study various twisted A-harmonic sums, named following the seminal log-algebraicity papers of G. Anderson. These objects are partial sums of new types of special zeta values introduced by the first author and linked to certain rank one Drinfeld modules over Tate algebras in positive characteristic by Anglès, Tavares Ribeiro and the first author. We prove, by using techniques introduced by the second author, that various infinite families of such sums may be interpolated by polynomials, and we deduce, among several other results, properties of analogues of finite zeta values b…
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.
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.
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.
Let $λ_g (n)$ be the Fourier coefficients of a holomorphic cusp modular form $g$ for $\mathrm{SL}_2 (\mathbb{Z})$. The aim of this article is to get non-trivial bound on non-linearly additively twisted sums of the Fourier coefficients $λ_g (n)$. Precisely, we prove for any $3/4 < β< 3/2$, $β\neq 1 $, the following non-trivial estimate $$ \sum_{n \leq N}λ_g(n)\,e(α\, n^β)\ll_{g, α, β, \varepsilon} N^{\frac{1}{2}+ \fracβ{3} +\varepsilon} + N^{\frac{3}{2}-\frac {2β}{3} + \varepsilon}, $$ for any $\varepsilon > 0$. This is the first time that non-trivial estimate for such sums is achieved for $1 <…
Let $$S(N) = \sum_{n \sim N}^{\text{smooth}} \, λ_{f}(n) \, χ(n),$$ where $λ_{f}(n)$'s are Fourier coefficients of Hecke-eigen form, and $χ$ is a primitive character of conductor $p^{r}$. In this article we prove a sub-Weyl strength bounds for $S(N)$. Indeed, we obtain $$S(N) \ll \, N^{\frac{5}{9}} \ p^{\frac{13r}{45}},$$ provided that $ p^{13r/20} \leq N \leq p^{4r/5}$. Note that the above bound for $S(N)$ is non-trivial if $N\geq \left(p^{r}\right)^{\frac{2}{3}-\frac{1}{60}}$.
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.
Let f be a Hecke-Maass or holomorphic primitive cusp form for $SL(2,\mathbb{Z})$ with Fourier coefficients $λ_{f}(n)$. Let $χ$ be a primitive Dirichlet character of modulus p, where p is a prime number. In this article we prove the following Weyl-type bound: for any $ε>0$, $$\sum_{|n| \ll N}λ_{f}(n)χ(n) \ll_{f,ε}N^{3/4 }p^{1/6}(pN)^ε.$$ We can see an improvement of the range $N > p^{3/4}$ to the range $N > p^{2/3}$ and we get a bound for $S_{f,χ}(N)$ without going into the $L$-function.
In this paper, we sharpen earlier work of the first author, Luca and Mulholland, showing that the Diophantine equation $$ A^3+B^3 = q^αC^p, \, \, ABC \neq 0, \, \, \gcd (A,B) =1, $$ has, for "most" primes $q$ and suitably large prime exponents $p$, no solutions. We handle a number of (presumably infinite) families where no such conclusion was hitherto known. Through further application of certain {\it symplectic criteria}, we are able to make some conditional statements about still more values of $q$, a sample such result is that, for all but $O(\sqrt{x}/\log x)$ primes $q$ up to $x$, the equa…
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
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.
We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of $q$, and we use the theory of Deligne and Katz to estimate certain complete exponential sums in several variables and prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of $q$. When at least one of the forms $f$ an…
We prove a theorem that allows one to count solutions to determinant equations twisted by a periodic weight with high uniformity in the modulus. It is obtained by using spectral methods of $\operatorname{SL}_2(\mathbb{R})$ automorphic forms to study Poincaré series over congruence subgroups. By keeping track of interactions between multiple orbits we get advantages over the widely used sums of Kloosterman sums techniques. We showcase this with applications to correlations of the divisor functions twisted by periodic functions and the fourth moment of Dirichlet $L$-functions on the critical lin…
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.
We prove an exact formula for the second moment of Rankin-Selberg $L$-functions $L(1/2,f \times g)$ twisted by $λ_f(p)$, where $g$ is a fixed holomorphic cusp form and $f$ is summed over automorphic forms of a given level $q$. The formula is a reciprocity relation that exchanges the twist parameter $p$ and the level $q$. The method involves the Bruggeman/Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.
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.
We give two quantum algorithms for computing (twisted) Kloosterman sums attached to a finite field $\mathbf{F}$ of $q$ elements. The first algorithm computes a quantum state containing, as its coefficients with respect to the standard basis, all Kloosterman sums for $\mathbf{F}$ twisted by a given multiplicative character, and runs in time polynomial in $\log q$. The second algorithm computes a single Kloosterman sum to a prescribed precision, and runs in time quasi-linear in $\sqrt{q}$.
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 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.
