On the semisimplicity conjecture and Galois representations

We obtain a criteria for a pure sheaf to be semisimple. As a corollary, we prove the following: Let X 0 X_0 and S 0 S_0 be two schemes over a finite field F q \mathbf {F}_q , and let f 0 : X 0 → S 0 f_0: X_0\rightarrow S_0 be a proper smooth morphism. Assume S 0 S_0 is normal and geometrically connected, and assume there exists a closed point s s in S 0 S_0 such that the Frobenius automorphism F s F_s acts semisimply on H i ( X s ¯ , Q l ¯ ) H^i(X_{\bar s}, {\overline {\mathbf {Q}_l}}) , where X s ¯ X_{\bar s} is the geometric fiber of f 0 f_0 at s s (this last assumption is unnecessary if the semisimplicity conjecture is true). Then R i f 0 ∗ Q l ¯ R^if_{0\ast } {\overline {\mathbf {Q}_l}} is a semisimple sheaf on S 0 S_0 . This verifies a conjecture of Grothendieck and Serre provided the semisimplicity conjecture holds. As an application, we prove that the galois representations of function fields associated to the l l -adic cohomologies of K 3 K3 surfaces are semisimple. We also get a theorem of Zarhin about the semisimplicity of the Galois representations of function fields arising from abelian varieties. The proof relies heavily on Deligne’s work on Weil conjectures.

Number of Points on the Projective Curves aYl=bXl+cZl and aY2l=bX2l+cZ2l Defined over Finite Fields, l an Odd Prime

In this paper, we consider a tower of function fields [Formula: see text] over a finite field 𝔽qand a finite extension E/F0such that the sequence [Formula: see text] is a tower over the field 𝔽q. Then we study invariants of [Formula: see text], that is, the asymptotic number of the places of degree r in [Formula: see text], for any r ≥ 1, if those of [Formula: see text] are known. We first give a method for constructing towers of function fields over any finite field 𝔽qwith finitely many prescribed invariants being positive. For q a square, we prove that with the same method one can also construct towers with at least one positive invariant and certain prescribed invariants being zero. Our method is based on explicit extensions. Moreover, we show the existence of towers over a finite field 𝔽qattaining the Drinfeld–Vladut bound of order r, for any r ≥ 1 with qra square (see [1, Problem-2]). Finally, we give some examples of non-optimal recursive towers with all but one invariants equal to zero.

We study the distribution of the numbers of \({F_{{q^r}}}\)-rational points of hyperelliptic curves over a finite field Fq in odd characteristic. This extends the result of Kurlberg and Rudnick [4]. We also study the distribution of the number of \({F_{{q^r}}}\)-rational points and the trace of high powers of the Frobenius class of real hyperelliptic curves over a finite field Fq in even characteristic.

Intersection distribution, non-hitting index and Kakeya sets in affine planes

Polarization phenomenon over any finite field Fq with size q being a power of a prime is considered. This problem is a generalization of the original proposal of channel polarization by Arikan for the binary field, as well as its extension to a prime field by Sasoglu, Telatar, and Arikan. In this paper, a necessary and sufficient condition of a matrix over a finite field Fq is shown under which any source and channel are polarized. Furthermore, the result of the speed of polarization for the binary alphabet obtained by Arikan and Telatar is generalized to arbitrary finite field. It is also shown that the asymptotic error probability of polar codes is improved by using the Reed-Solomon matrices, which can be regarded as a natural generalization of the 2 × 2 binary matrix used in the original proposal by Arikan.

There exists a fixed rule in classical dynamical systems that describes a point in geometric space over time. In this paper, based on the algebraic structure perspective, the dynamical system is defined as a category characterized by ordered state projections, and the dynamical system is inscribed using the algebraic structure, covering the phase space, continuous self-maps containing a single parametric variable, and the dynamical system itself. Meanwhile, two types of self-isomorphisms of algebraic maps are explored. One is the self-isomorphism of ideal inclusion maps on an algebra Mn(Fq) consisting of full matrices of order n over a finite field Fq . The second is the self-isomorphism of ideal relational graphs on a finite field Fq . It is proved that any self-isomorphism problem of graph Mn (Fq ) when n >3 can be used with both criteria on it. Finally, a classical model of a dynamical system obtained from f(x) = cos x iterations is studied and its global convergence is discussed. Numerical experimental results show that the discrete dynamical system generated by function f(x) = cos x iteration has a unique ω limit point of 0.735, indicating that the stability and predictability of classical dynamical systems can be achieved using algebraic structures, as well as revealing the complexity, instability, and chaos of the system.

The classical generalized Reed-Muller codes introduced by Kasami, Lin and Peterson [5], and studied also by Delsarte, Goethals and Mac Williams [2], are defined over the affine space An(Fq) over the finite field Fq with q elements. Moreover Lachaud [6], following Manin and Vladut [7], has considered projective Reed-Muller codes, i.e. defined over the projective space Pn(Fq).In this paper, the evaluation of the forms with coefficients in the finite field Fq is made on the points of a projective algebraic variety V over the projective space Pn(Fq). Firstly, we consider the case where V is a quadric hypersurface, singular or not, Parabolic, Hyperbolic or Elliptic. Some results about the number of points in a (possibly degenerate) quadric and in the hyperplane sections are given, and also is given an upper bound of the number of points in the intersection of two quadrics.In application of these results, we obtain Reed-Muller codes of order 1 associated to quadrics with three weights and we give their parameters, as well as Reed-Muller codes of order 2 with their parameters.Secondly, we take V as a hypersurface, which is the union of hyperplanes containing a linear variety of codimension 2 (these hypersurfaces reach the Serre bound). If V is of degree h, we give parameters of Reed-Muller codes of order d < h, associated to V.

Construction of quasi-cyclic self-dual codes

0.1. This paper should be regarded as a sequel to [7]. There it was shown that the geometric Langlands conjecture for GLn follows from a certain vanishing conjecture. The goal of the present paper is to prove this vanishing conjecture. Let X be a smooth projective curve over a ground field k. Let E be an m-dimensional local system on X, and let Bunm be the moduli stack of rank m vector bundles on X. The geometric Langlands conjecture says that to E we can associate a perverse sheaf FE on Bunm, which is a Hecke eigensheaf with respect to E. The vanishing conjecture of [7] says that for all integers n < m, a certain functor AvE , depending on E and a parameter d ∈ Z+, which maps the category D(Bunn) to itself, vanishes identically, when d is large enough. The fact that the vanishing conjecture implies the geometric Langlands conjecture may be regarded as a geometric version of the converse theorem. Moreover, as will be explained in the sequel, the vanishing of the functor AvE is analogous to the condition that the Rankin-Selberg convolution of E, viewed as an m-dimensional Galois representation, and an automorphic form on GLn with n < m is well-behaved. Both the geometric Langlands conjecture and the vanishing conjecture can be formulated in any of the sheaf-theoretic situations, e.g., Q -adic sheaves (when char(k) = ), D-modules (when char(k) = 0), and sheaves with coefficients in a finite field F (again, when char(k) = ). When the ground field is the finite field Fq and we are working with -adic coefficients, it was shown in [7] that the vanishing conjecture can be deduced from Lafforgue’s theorem that establishes the full Langlands correspondence for global fields of positive characteristic; cf. [9].

0. Introduction. Algebraic curves over finite fields with many rational points have been of increasing interest in the last two decades. The question of explicitly determining the maximal number of points on a curve of given genus was initiated and in some special cases solved by Serre [34, 35, 36] around 1982. Since then there have been attempts to attack the problem by means of algebraic geometry as well as field arithmetic. Constructions by explicit equations have been carried out by van der Geer and van der Vlugt [7, 8]. The present paper, which makes use of class field theory, has its immediate predecessors in work by Lauter [13, 14, 15] and Niederreiter and Xing [19, 20, 21, 25, 26, 38]. The numerical results obtained improve several entries of the tables given in [9], [17] and [27]. As we are looking from the field theoretic point of view, with an algebraic curveX (smooth, projective, absolutely irreducible) defined over a finite field Fq we associate its field K = Fq(X) of algebraic functions, a global function field with full constant field Fq. Its genus is that of X, and coverings of X correspond to field extensions of K, the degree of the covering being the degree of the extension. A place of K, by which we mean the maximal ideal p in some discrete valuation ring of K, with (residue field) degree d = deg p, corresponds to (a Galois conjugacy class of) d points on X(Fqd), and each point on X having Fqd as its minimal field of definition over Fq lies in such a conjugacy class. In particular the rational places, i.e. the places of degree 1, of K|Fq are in 1-1 correspondence with the Fq-rational points on X. The (normalized) discrete valuation associated with a place p of K will be denoted by vp.

Divisibility on point counting over finite Witt rings

In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

For a fixed mod $p$ automorphic Galois representation, $p$-adic automorphic Galois representations lifting it determine points in universal deformation space. In the case of modular forms and under some technical conditions, Bockle showed that every component of deformation space contains a smooth modular point, which then implies their Zariski density when coupled with the infinite fern of Gouvea-Mazur. We generalize Bockle's result to the context of polarized Galois representations for CM fields, and to two dimensional Galois representations for totally real fields. More specifically, under assumptions necessary to apply a small $R = \mathbb{T}$ theorem and an assumption on the local mod $p$ representation, we prove that every irreducible component of the universal polarized deformation space contains an automorphic point. When combined with work of Chenevier, this implies new results on the Zariski density of automorphic points in polarized deformation space in dimension three.