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Register a new userNewton polygons for $L$-functions of generalized Kloosterman sums
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In this paper, we study the Newton polygons for the $L$-functions of $n$-variable generalized Kloosterman sums. Generally, the Newton polygon has a topological lower bound, called the Hodge polygon. In order to determine the Hodge polygon, we explicitly construct a basis of the top dimensional Dwork cohomology. Using Wans decomposition theorem and diagonal local theory, we obtain when the Newton polygon coincides with the Hodge polygon. In particular, we concretely get the slope sequence for $L$-function of $bar{F}(bar{lambda},x):=sum_{i=1}^nx_i^{a_i}+bar{lambda}prod_{i=1}^nx_i^{-1}$.
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In this paper, we consider the following $(A, B)$-polynomial $f$ over finite field: $$f(x_0,x_1,cdots,x_n)=x_0^Ah(x_1,cdots,x_n)+g(x_1,cdots,x_n)+P_B(1/x_0),$$ where $h$ is a Deligne polynomial of degree $d$, $g$ is an arbitrary polynomial of degree $< dB/(A+B)$ and $P_B(y)$ is a one-variable polynomial of degree $le B$. Let $Delta$ be the Newton polyhedron of $f$ at infinity. We show that $Delta$ is generically ordinary if $pequiv 1 mod D$, where $D$ is a constant only determined by $Delta$. In other words, we prove that the Adolphson--Sperber conjecture is true for $Delta$.
In this paper, we focus on a family of generalized Kloosterman sums over the torus. With a few changes to Haessig and Sperbers construction, we derive some relative $p$-adic cohomologies corresponding to the $L$-functions. We present explicit forms of bases of top dimensional cohomology spaces, so to obtain a concrete method to compute lower bounds of Newton polygons of the $L$-functions. Using the theory of GKZ system, we derive the Dworks deformation equation for our family. Furthermore, with the help of Dworks dual theory and deformation theory, the strong Frobenius structure of this equation is established. Our work adds some new evidences for Dworks conjecture.
A conjecture of Le says that the Deligne polytope $Delta_d$ is generically ordinary if $pequiv 1 (!!bmod D(Delta_d))$, where $D(Delta_d)$ is a combinatorial constant determined by $Delta_d$. In this paper a counterexample is given to show that the conjecture is not true in general.
Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet $L$-functions. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square $L$-functions and an asymptotic expansion for the average of central values of generalized Dirichlet $L$-functions.
We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.
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