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We provide an effective version of Katz’ criterion for finiteness of the monodromy group of a lisse, pure of weight zero, ℓ-adic sheaf on a normal variety over a finite field, depending on the numerical complexity of the sheaf.

We first develop some basic facts about certain sorts of rigid local systems on the affine line in characteristic p > 0 . We then apply them to exhibit a number of rigid local systems of rank 23 on the affine line in characteristic p = 3 whose arithmetic and geometric monodromy groups are the Conway group Co 3 in its orthogonal irreducible representation of degree 23.

We prove an estimate for multi-variable multiplicative character sums over affine subspaces of A k n , which generalizes the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character sums.

The main purpose of this article is to study higher power mean values of generalized quadratic Gauss sums using estimates for character sums, analytic method and algebraic geometric methods. In this article, we prove two conjectures which were proposed in \cite{BLZ}., Comment: 12 Pages, no figures, comments are welcome. arXiv admin note: text overlap with arXiv:2104.09888

We give explicit formulas for the local multiplicative convolution functors, which express the local monodromies of the convolution of two l-adic sheaves on the torus Gm over the algebraic closure of a finite field in terms of the local monodromies of the factors. As a particular case, we recover Fu's formulas for the local Fourier transform.

Given a prime p and an integer d > 1, we give a numerical criterion to decide whether the ℓ-adic sheaf associated to the one-parameter exponential sums t 7→P x ψ(xd + tx) over Fp has finite monodromy or not, and work out some explicit cases where this is computable. Ministerio de Economía y Competitividad Fondo Europeo de Desarrollo Regional

We first develop some basic facts about hypergeometric sheaves on the multiplicative group [Formula: see text] in characteristic [Formula: see text]. Certain of their Kummer pullbacks extend to irreducible local systems on the affine line in characteristic [Formula: see text]. One of these, of rank [Formula: see text] in characteristic [Formula: see text], turns out to have the Conway group [Formula: see text], in its irreducible orthogonal representation of degree [Formula: see text], as its arithmetic and geometric monodromy groups.

For K and L two \(\ell \)-adic perverse sheaves on the one-dimensional torus \({\mathbb{G }_{m,\bar{k}}}\) over the algebraic closure of a finite field, we show that the local monodromies of their convolution \(K*L\) at its points of non-smoothness is completely determined by the local monodromies of K and L. We define local convolution bi-exact functors \(\rho _{(s,t)}^{(u)}\) for every \(s,t,u\in \mathbb{P }^1_{\bar{k}}\) that map continuous \(\ell \)-adic representations of the inertia groups at s and t to a representation of the inertia group at u, and show that the local monodromy of \(K*L\) at u is the direct sum of the \(\rho _{(s,t)}^{(u)}\) applied to the local monodromies of K and L. This generalizes a previous result of Katz for the case where K and L are smooth, tame at 0 and totally wild at infinity. As a corollary we obtain an estimate for some exponential sums associated to homothety invariant polynomials.

This paper looks at the L-function of the kth symmetric power of the [Formula: see text]-sheaf Ai f over the affine line [Formula: see text] associated to the generalized Airy family of exponential sums. Using ℓ-adic techniques, we compute the degree of this rational function as well as the local factors at infinity. Using p-adic techniques, we study the q-adic Newton polygon of the L-function.

We prove a simple explicit formula for the local Katz-Radon transform of an l-adic representation of the Galois group of the fraction field of a strictly henselian discrete valuation ring with positive residual characteristic, which can be defined as the local additive convolution with a fixed tame character. The formula is similar to one proved by D. Arinkin in the D-module setting, and answers a question posed by N. Katz.

For the Artin–Schreier curve yq − y = f(x) defined over a finite field \({{\mathbb F}_q}\) of q elements, the celebrated Weil bound for the number of \({{\mathbb F}_{q^r}}\)-rational points can be sharp, especially in super-singular cases and when r is divisible. In this paper, we show how the Weil bound can be significantly improved, using ideas from moment L-functions and Katz’s work on l-adic monodromy calculations. Roughly speaking, we show that in favorable cases (which happens quite often), one can remove an extra \({\sqrt{q}}\) factor in the error term.

We give a purity result for two kinds of exponential sums of the type $\sum_{x\in k^n}\psi(f(x))$ , where k is a finite field of characteristic p and $\psi:k\to{\mathbb C}^\star$ is a non-trivial additive character. In the first case, $f\in k[x_1,\dots,x_n]$ is a polynomial of degree divisible by p whose highest-degree homogeneous form defines a non-singular projective hypersurface, and in the second case, f is a polynomial of degree prime to p whose highest-degree homogeneous form defines a projective hypersurface with isolated singularities.

We consider the issue of when the L-polynomial of one curve over $\F_q$ divides the L-polynomial of another curve. We prove a theorem which shows that divisibility follows from a hypothesis that two curves have the same number of points over infinitely many extensions of a certain type, and one other assumption. We also present an application to a family of curves arising from a conjecture about exponential sums. We make our own conjecture about L-polynomials, and prove that this is equivalent to the exponential sums conjecture., Comment: 20 pages

We prove some improvements of the classical Weil bound for one variable additive and multiplicative character sums associated to a polynomial over a finite field k = Fq for two classes of polynomials which are invariant under a large abelian group of automorphisms of the affine line A1 k: those invariant under translation by elements of k and those invariant under homotheties with ratios in a large subgroup of the multiplicative group of k. In both cases, we are able to improve the bound by a factor of √q over an extension.f k of cardinality sufficiently large compared to the degree of f. Junta de Andalucía Ministerio de Educación y Ciencia Fondo Europeo de Desarrollo Regional

For a given l-adic sheaf F on a commutative algebraic group G over a finite field k and an integer r ≥ 1 we define the r-th local norm Lfunction of F at a point t ∈ G(k) and prove its rationality. This function gives information on the sum of the local Frobenius traces of F over the points of G(kr) (where kr is the extension of degree r of k) with norm t. For G the onedimensional affine line or the torus, these sums can in turn be used to estimate the number of rational points on curves or the absolute value of exponential sums which are invariant under a large group of translations or homotheties. Junta de Andalucía Ministerio de Educación y Ciencia Fondo Europeo de Desarrollo Regional

We give a purity result for exponential sums of the type x2kn ˆ(f(x)), where k is a flnite fleld of characteristic p, ˆ : k !C ? is a non-trivial additive character and f 2 k[x1;:::;xn] is a polynomial whose highest degree homogeneous form splits as a product of factors deflning a divisor with normal crossings in P ni1 .

Moment zeta functions provide a diophantine formulation for the distribution of rational points on a family of algebraic varieties over finite fields. They also form algebraic approximations to Dwork's p-adic unit root zeta functions. In this paper, we use l-adic cohomology to calculate all the higher moment zeta functions for the mirror family of the Calabi-Yau family of smooth projective hypersurfaces over finite fields. Our main result is a complete determination of the purity decomposition and the trivial factors for the moment zeta functions.

We give some estimates for multiplicative character sums on quasiprojective varieties over finite fields depending on the severity of the singularities of the variety at infinity. We also remove the hypothesis of non-divisibility by the characteristic of the base field in the known estimates for the non-singular case. Ministerio de Educación y Ciencia Fondo Europeo de Desarrollo Regional

We give some estimates for multiplicative character sums on quasiprojective varieties over flnite flelds depending on the severity of the singularities of the variety at inflnity. We also remove the hypothesis of non-divisibility by the characteristic of the base fleld in the known estimates for the non-singular case.