相关论文: Newton polygons for $L$-functions of generalized K…

The aim of this study is to provide a perspective to help understand the singular average operator over polynomial hypersurfaces. In particular, this perspective will provide brevity and the possibility of generalizing previous results…

We study p-adic hyper-Kloosterman sums, a generalization of the Kloosterman sum with a parameter k that recovers the classical Kloosterman sum when k=2, over general p-adic rings and even equal characteristic local rings. These can be…

The purpose of this article is to prove a "Newton over Hodge" result for finite characters on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve.…

We consider arbitrary algebraic families of lower order deformations of nondegenerate toric exponential sums over a finite field. We construct a relative polytope with the aid of which we define a ring of coefficients consisting of p-adic…

We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field of characteristic $p$ of degree relatively prime to $p$. Let $a\mapsto \omega(a)$ be the Teichm\"uller lift of $\mathbb F_q$, and let $\chi:\mathbb{Z}\to \mathbb…

We construct p-adic relative cohomology for a family of toric exponential sums which generalize the classical Kloosterman sums. Under natural hypotheses such as quasi-homogeneity and nondegeneracy, this cohomology is acyclic except in the…

We consider the distribution of polygonal paths joining the partial sums of normalized Kloosterman sums modulo an increasingly high power p^n of a fixed odd prime p, a pure depth-aspect analogue of theorems of Kowalski-Sawin and…

Let $L$ be an even lattice of odd rank with discriminant group $L'/L$, and let $\alpha,\beta \in L'/L$. We prove the Weil bound for the Kloosterman sums $S_{\alpha,\beta}(m,n,c)$ of half-integral weight for the Weil Representation attached…

The symmetric power L-function of the hyper-Kloosterman family is a rational function over the integers. Its degree and complex absolute values of its zeros and poles are now known through the work of Fu and Wan. The purpose of this paper…

As a consequence of Wan's theorem about Dwork's conjecture, the unit root $L$-functions of the $n$-dimensional Kloosterman family are $p$-adic meromorphic. By studying the symmetric power $L$-functions associated to the Kloosterman family,…

The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the $A$-polynomial…

We develop a framework that systematically casts the solvability and uniqueness conditions of linearized geometric boundary-value problems into cohomological terms. The theory is designed to be applicable without assumptions on the…

The classical $n$-variable Kloosterman sums over the finite field ${\bf F}_p$ give rise to a lisse $\bar {\bf Q}_l$-sheaf ${\rm Kl}_{n+1}$ on ${\bf G}_{m, {\bf F}_p}={\bf P}^1_{{\bf F}_p}-\{0,\infty\}$, which we call the Kloosterman sheaf.…

We return to some past studies of hyperkloosterman sums ([9,10]) via $p$-adic cohomology with an aim to improve earlier results. In particular, we work here with Dwork's $\theta_\infty$-splitting function and a better choice of basis for…

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…

We introduce the notion of "hypergeometric" polynomials with respect to Newtonian bases. These polynomials are eigenfunctions ($L P_n(x) = \lambda_n P_n(x)$) of some abstract operator $L$ which is 2-diagonal in the Newtonian basis…

We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods…

In this paper we study Newton's method for solving the generalized equation $F(x)+T(x)\ni 0$ in Hilbert spaces, where $F$ is a Fr\'echet differentiable function and $T$ is set-valued and maximal monotone. We show that this method is local…

The main purpose of this paper is to study the hybrid mean value problem involving generalized Dedekind sums, generalized Hardy sums and Kloosterman sums, and give some exact computational formulae for them by using the properties of Gauss…

We introduce the class of \emph{Log-Noetherian} (LN) functions. These are holomorphic solutions to algebraic differential equations (in several variables) with logarithmic singularities. We prove an upper bound on the number of solutions…