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Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (absolutely irreducibles) and irreducible elements where some power has a factorization different from the trivial one. In this paper, we study irreducible polynomials F ∈ Int (R) where R is a discrete valuation domain with finite residue field and show that it is possible to explicitly determine a number S ∈ N that reduces the absolute irreducibility of F to the unique factorization of F S. To this end, we establish a connection between the factors of powers of F and the kernel of a certain linear map that we associate to F. This connection yields a characterization of absolute irreducibility in terms of this so-called fixed divisor kernel. Given a non-trivial element v of this kernel, we explicitly construct non-trivial factorizations of F k , provided that k ≥ L , where L depends on F as well as the choice of v . We further show that this bound cannot be improved in general. Additionally, we provide other (larger) lower bounds for k , one of which only depends on the valuation of the denominator of F and the size of the residue class field of R. [ABSTRACT FROM AUTHOR]

Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain (R , M) with finite residue field, it is known that there exist absolutely irreducible elements, that is, irreducible elements all of whose powers factor uniquely, and non-absolutely irreducible elements. We completely and constructively characterize the absolutely irreducible elements among split integer-valued polynomials. They correspond bijectively to finite sets, which we call balanced , characterized by a combinatorial property regarding the distribution of their elements among residue classes of powers of M. For each such balanced set as the set of roots of a split polynomial, there exists a unique vector of multiplicities and a unique constant so that the corresponding product of monic linear factors times the constant is an absolutely irreducible integer-valued polynomial. This also yields sufficient criteria for integer-valued polynomials over Dedekind domains to be absolutely irreducible. [ABSTRACT FROM AUTHOR]

Let V be an irreducible module for a finite group G over an algebraically closed field k ‾. We prove that the algebra Hom k ‾ (V , V) has a proper G -invariant subalgebra if and only if V is either imprimitive or tensor decomposable. We give an algorithm to determine whether an absolutely irreducible matrix representation defined over a finite field has one of these properties. Our algorithm reduces the computation to an instance of the Pure Tensor problem. This asks whether a subspace of a tensor product of vector spaces X and Y contains an element of the form x ⊗ y with x ∈ X and y ∈ Y. We show that the Pure Tensor Problem reduces to the calculation of an appropriate Gröbner basis. [ABSTRACT FROM AUTHOR]

In this article we define and study a zeta function ζ G —similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value ζ G (k) - 1 at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that ζ G is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζ G . We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro- C groups, where C is a class of finite groups with prescribed composition factors. We prove that every real number a ≥ 1 is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζ G are rational functions in p - s if G is virtually abelian. For finite groups G we calculate ζ G using the rational representation theory of G. [ABSTRACT FROM AUTHOR]

Weil-type zeta functions defined by the numbers of absolutely irreducible $ \mathrm{SL}_2$ knot group over finite fields are computed explicitly. They are expressed in terms of the congruence zeta functions of reductions of a certain elliptic curve defined over the rational number field. Then the Hasse-Weil type zeta function of the figure $ 8$ knot and a certain family of elliptic curves. [ABSTRACT FROM AUTHOR]

We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain low genus curves. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization. [ABSTRACT FROM AUTHOR]

Let p>3 and F be a non-archimedean local field with residue field a proper finite extension of \mathbb {F}_p. We construct smooth absolutely irreducible non-admissible representations of \mathrm {GL}_2(F) defined over the residue field of F extending the earlier results of the authors for F unramified over \mathbb {Q}_{p}. This construction uses the theory of diagrams of Breuil and Pašk\bar {\mathrm {u}}nas. By parabolic induction, we obtain smooth absolutely irreducible non-admissible representations of \mathrm {GL}_n(F) for n>2. [ABSTRACT FROM AUTHOR]

Planar functions are mappings from a finite field F q to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even. We classify polynomials of degree at most q 1 / 4 that induce planar functions on F q , by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over F q that induce planar functions on infinitely many extensions of F q. The proof strategy is to study the number of F q -rational points of an algebraic curve attached to a putative planar function. Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions. [ABSTRACT FROM AUTHOR]

Let t be an integer ≥ 3 such that t ≡ 1 mod 4. The absolute irreducibility of the polynomial $\phi _{t}(x, y) = \frac {x^{t} + y^{t} + 1 + (x + y + 1)^{t}}{(x + y)(x + 1)(y + 1)}$ (over $\mathbb {F}_{2}$ ) plays an important role in the study of APN functions. We prove that this polynomial is absolutely irreducible under the assumptions that the largest odd integer which divides t − 1 is large enough and can not be written in a specific form. [ABSTRACT FROM AUTHOR]

We address the problem of determining when a plane algebraic cubic curve is complete as an ( n, 3)-arc in $$\mathrm {PG}(2,q)$$ . Theoretical results are given for absolutely irreducible singular cubic curves, while computer based results are given for $$q\le 81$$ . [ABSTRACT FROM AUTHOR]

Let K be a complete, discretely valued field with finite residue field and GK its absolute Galois group. The subject of this note is the study of the set of positive integers d for which there exists an absolutely irreducible ℓ‐adic representation of GK of dimension d with rational traces on inertia. Our main result is that non‐Sophie Germain primes are not in this set when the residue characteristic of K is >3. The result stated in the title is a special case. [ABSTRACT FROM AUTHOR]

For a given genus $g \geq 1$ , we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$. As a consequence of Katz–Sarnak theory, we first get for any given $g>0$ , any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q. [ABSTRACT FROM AUTHOR]

In Section 8, we saw the role of absolute irreducibility in determining the number of zeroes of a bivariate polynomial over a finite field. Let us consider the following algorithmic question: given a bivariate polynomial f(x,y) how do we determine whether or not it is absolutely irreducible? In this section, we will see how partial derivatives help in converting this apparently highly nonlinear problem into a simple task in linear algebra. This then gives an efficient algorithm for determining absolute irreducibility. [ABSTRACT FROM AUTHOR]

Let n be a positive integer, q = 2 n , and let F q be the finite field with q elements. For each positive integer m , let D m ( X ) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m > 1 is a divisor of q + 1 . We study the existence of α ∈ F q ⁎ such that D m ( α ) = D m ( α − 1 ) = 0 . We also explore the connections of this question to an open question by Wiedemann and a game called “Button Madness”. [ABSTRACT FROM AUTHOR]

We study irreducible |$\mathop{\textrm{SL}}\nolimits _2$| -representations of twist knots. We first determine all non-acyclic |$\mathop{\textrm{SL}}\nolimits _2({\mathbb{C}})$| -representations, which turn out to lie on a line denoted as |$x=y$| in |${\mathbb{R}}^2$|⁠. Our main tools are character variety, Reidemeister torsion, and Chebyshev polynomials. We also verify a certain common tangent property, which yields a result on |$L$| -functions, that is, the orders of the knot modules associated to the universal deformations. Secondly, we prove that a representation is on the line |$x=y$| if and only if it factors through the |$-3$| -Dehn surgery, and is non-acyclic if and only if the image of a certain element is of order 3. Finally, we study absolutely irreducible non-acyclic representations |$\overline{\rho }$| over a finite field with characteristic |$p>2$|⁠ , to concretely determine all non-trivial |$L$| -functions |$L_{{\boldsymbol{\rho }}}$| of the universal deformations over complete discrete valuation rings. We show among other things that |$L_{{\boldsymbol{\rho }}}$|   |$\dot{=}$|   |$k_n(x)^2$| holds for a certain series |$k_n(x)$| of polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we completely determine all necessary and sufficient conditions such that the polynomial f (x) = x 3 + a x q + 2 + b x 2 q + 1 + c x 3 q , where a , b , c ∈ F q ⁎ , is a permutation quadrinomial of F q 2 over any finite field of odd characteristic. This quadrinomial has been studied first in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, where c h a r (F q) = 2 and finally, in [16] , Li, Qu, Li and Chen proved that the sufficient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial x 3 + a x q + 2 + b x 2 q + 1 + c x 3 q , where c h a r (F q) = 3 , 5 and a , b , c ∈ F q ⁎ and proposed some new classes of permutation quadrinomials of F q 2 . In particular, in this paper we classify all permutation polynomials of F q 2 of the form f (x) = x 3 + a x q + 2 + b x 2 q + 1 + c x 3 q , where a , b , c ∈ F q ⁎ , over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials. [ABSTRACT FROM AUTHOR]

In the area of symbolic-numerical computation within computer algebra, an interesting question is how "close" a random input is to the "critical" ones. Examples are the singular matrices in linear algebra or the polynomials with multiple roots for Newton's root-finding method. Bounds, sometimes very precise, are known for the volumes over R or C of such neighborhoods of the varieties of "critical" inputs; see the references below. This paper deals with the discrete version of this question: over a finite field, how many points lie in a certain type of neighborhood around a given variety? A trivial upper bound on this number is given by the product (size of the variety) ⋅ (size of a neighborhood of a point). It turns out that this bound is usually asymptotically tight, in particular for the singular matrices, polynomials with multiple roots, and pairs of non-coprime polynomials. The interesting question then is: for which varieties is this bound not asymptotically tight? We show that these are precisely those that admit a shift, that is, where one absolutely irreducible component of maximal dimension is a shift (translation by a fixed nonzero point) of another such component. Furthermore, the shift-invariant absolutely irreducible varieties are characterized as being cylinders over some base variety. Computationally, determining whether a given variety is shift-invariant turns out to be intractable, namely NP-hard even in simple cases. [ABSTRACT FROM AUTHOR]

We consider the polynomials of the form P(x) = xk - γTr(x) over 픽 qn for n ≥ 2. We show that P(x) is not a permutation of 픽qn in the case gcd(k, qn - 1) > 1. Our proof uses an absolutely irreducible curve over 픽qn and the number of rational points on it. [ABSTRACT FROM AUTHOR]

An alternative proof of the necessary conditions on A , B ∈ q 2 * for f (X) = X + A X 1 + q 2 (q − 1) / 4 + B X 1 + 3 q 2 (q − 1) / 4 to be a permutation polynomial in q 2 , q even, is given. This proof involves standard arguments from algebraic geometry over finite fields and fast symbolic computations. [ABSTRACT FROM AUTHOR]

Let (𝕋 f , 𝔪 f) denote the mod p local Hecke algebra attached to a normalized Hecke eigenform f , which is a commutative algebra over some finite field 𝔽 q of characteristic p and with residue field 𝔽 q . By a result of Carayol we know that, if the residual Galois representation ρ ¯ f : G ℚ → GL 2 (𝔽 q) is absolutely irreducible, then one can attach to this algebra a Galois representation ρ f : G ℚ → GL 2 (𝕋 f) that is a lift of ρ ¯ f . We will show how one can determine the image of ρ f under the assumptions that (i) the image of the residual representation contains SL 2 (𝔽 q) , (ii) 𝔪 f 2 = 0 and (iii) the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow to deduce the existence of certain p -elementary abelian extensions of big non-solvable number fields. [ABSTRACT FROM AUTHOR]

We provide an infinite sequence of upper bounds for the number of rational points of absolutely irreducible smooth projective curves X over a finite field, starting from the Weil classical bound, continuing to the Ihara bound, passing through infinitely many n-th order Weil bounds, and ending asymptotically at the Drinfeld-Vlădut, bound. We relate this set of bounds to those of Oesterl'e, proving that these are inverse functions in some sense. We explain how the Riemann hypothesis for the curve X can be merely seen as a euclidean property coming from the Toeplitz shape of some intersection matrix on the surface X × X together with the general theory of symmetric Toeplitz matrices. We also give some interpretation for the defect of asymptotically exact towers. This is achieved by pushing further the classical Weil proof in terms of euclidean relationships between classes in the euclidean part FX of the numerical group Num(X × X) generated by classes of graphs of iterations of the Frobenius morphism. The noteworthy Toeplitz shape of their intersection matrix takes a central place by implying a very strong cyclic structure on FX. [ABSTRACT FROM AUTHOR]

Ovoids of the parabolic quadric Q(6, q) of PG (6 , q) have been largely studied in the last 40 years. They can only occur if q is an odd prime power and there are two known families of ovoids of Q(6, q), the Thas-Kantor ovoids and the Ree-Tits ovoids, both for q a power of 3. It is well known that to any ovoid of Q(6, q) two polynomials f 1 (X , Y , Z) , f 2 (X , Y , Z) can be associated. In this paper we classify ovoids of Q(6, q) with max { deg (f 1) , deg (f 2) } < (1 6.3 q) 3 13 - 1 . [ABSTRACT FROM AUTHOR]

This paper provides an algorithmic approach to some basic algebraic and combinatorial properties of algebraic curves over finite fields: the number of points on a curve or a projection, its number of absolutely irreducible components, and the property of being "exceptional". [ABSTRACT FROM AUTHOR]

We propose new results on low weight codewords of affine and projective generalized Reed-Muller (GRM) codes. In the affine case we prove that if the cardinality of the ground field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then, without this assumption on the cardinality of the field, we study codewords associated to an irreducible but not absolutely irreducible polynomial, and prove that they cannot be second, third or fourth weight depending on the hypothesis. In the projective case the second distance of GRM codes is estimated, namely a lower bound and an upper bound on this weight are given. [ABSTRACT FROM AUTHOR]

We prove that the discrete logarithm problem can be solved in quasi-polynomial expected time in the multiplicative group of finite fields of fixed characteristic. More generally, we prove that it can be solved in the field of cardinality p^n in expected time (pn)^{2\log _2(n) + O(1)}. [ABSTRACT FROM AUTHOR]

Let V be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on V is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on V are obtained, and the distribution of visible points into different congruence classes is investigated. [ABSTRACT FROM PUBLISHER]

We study the class of univariate polynomials β k (X) , introduced by Carlitz, with coefficients in the algebraic function field F q (t) over the finite field F q with q elements. It is implicit in the work of Carlitz that these polynomials form an F q [ t ] -module basis of the ring Int (F q [ t ]) = { f ∈ F q (t) [ X ] | f (F q [ t ]) ⊆ F q [ t ] } of integer-valued polynomials on the polynomial ring F q [ t ]. This stands in close analogy to the famous fact that a Z -module basis of the ring Int (Z) is given by the binomial polynomials ( X k ). We prove, for k = q s , where s is a non-negative integer, that β k is irreducible in Int (F q [ t ]) and that it is even absolutely irreducible, that is, all of its powers β k m with m > 0 factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that β k is not even irreducible if k is not a power of q. [ABSTRACT FROM AUTHOR]

In this paper, we study the permutation properties of the class of trinomials of the form f(x) = x4q+1 + λ1xq+4 + λ2x2q+3 ∈ Fq²[x] where λ1, λ2 ∈ Fq and they are not simultaneously zero. We find all necessary and sufficient conditions on λ1 and λ2 such that f(x) permutes Fq², where q is odd and q = 22k+1, k ∈ N. [ABSTRACT FROM AUTHOR]

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on extensions of finite fields, especially for the cases of quadratic and cubic extensions. Our achievements are obtained by investigating absolute irreducibility of some polynomials in two indeterminates. [ABSTRACT FROM AUTHOR]

An elementary and self-contained argument for the complete determination of maximal plane curves of degree 3 over F4 will be given, which complements Hirschfeld-Storme-Thas-Voloch’s theorem on a characterization of Hermitian curves in P². This complementary part should be understood as the classification of Sziklai’s example of maximal plane curves of degree q − 1 over Fq. Although two maximal plane curves of degree 3 over F4 up to projective equivalence over F4 appear, they are birationally equivalent over F4 each other. [ABSTRACT FROM AUTHOR]

Given a smooth curve of genus g ≥ 1 which admits a smooth projective embedding of dimension m over the ground field $$ \mathbb{F}_q $$ of q elements, we obtain the asymptotic formula q g+ o( g) for the size of set of the $$ \mathbb{F}_q $$ -rational points on its Jacobian in the case when m and q are bounded and g → ∞. We also obtain a similar result for curves of bounded gonality. For example, this applies to the Jacobian of a hyperelliptic curve of genus g → ∞. [ABSTRACT FROM AUTHOR]

Let V be a finite-dimensional vector space over a finite field, and suppose G ≤ Γ L (V) is a group with a unique subnormal quasisimple subgroup E (G) that is absolutely irreducible on V. A base for G is a set of vectors B ⊆ V with pointwise stabiliser G B = 1. If G has a base of size 1, we say that it has a regular orbit on V. In this paper we investigate the minimal base size of groups G with E (G) / Z (E (G)) ≅ PSL n (q) in defining characteristic, with an aim of classifying those with a regular orbit on V. [ABSTRACT FROM AUTHOR]

In this paper we consider a family F of 2n-dimensional F q -linear rank metric codes in F q n × n arising from polynomials of the form x q s + δ x q n 2 + s ∈ F q n [ x ] . The family F was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8 , and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8 , providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F q -rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far. [ABSTRACT FROM AUTHOR]

Some ten years ago we managed to take a first step in the classification of nonsingular 2 × 2 × 2 × 2 hypercubes over a finite field by resolving the special case where the hypercubes can be written as a product of two 2 × 2 × 2 hypercubes, i.e., nonsingular 2 × 2 × 2 × 2 hypercubes of 12-rank two. We have now been able to extend this classification to hypercubes of 12-rank three, based on the connection between nonsingular hypercubes and bundles of nonintersecting quadrics in 3 dimensions. A bit surprisingly, the number of inequivalent nonsingular hypercubes of 12-rank three is only of the same order of magnitude as in the case of 12-rank two. We also made some headway into the remaining case of 12-rank four. In particular, we prove that there are essentially only 2 nonsingular 2 × 2 × 2 × 2 hypercubes that correspond to a hyperbolic fibration of 3-dimensional projective space. [ABSTRACT FROM AUTHOR]

Let C be a linear code of length n and dimension k over the finite field F q m . The trace code Tr (C) is a linear code of the same length n over the subfield F q . The obvious upper bound for the dimension of the trace code over F q is m k. If equality holds, then we say that C has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let C a denote the code obtained from C and a multiplier vector a ∈ ( F q m ) n . In this study, we give a lower bound for the probability that a random multiplier vector produces a code C a of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever n ≥ m (k + h) , where h ≥ 0 is the Singleton defect of C. For the extremal case n = m (h + k) , numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank. [ABSTRACT FROM AUTHOR]

In this article, Algebraic-Geometric (AG) codes and quantum codes associated with a family of curves that includes the famous Suzuki curve are investigated. The Weierstrass semigroup at some rational point is computed. Notably, each curve in the family turns out to be a Castle curve over some finite field and a weak Castle curve over its extensions. This is a relevant feature when codes constructed from the curve are considered. [ABSTRACT FROM AUTHOR]

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\bar{\unicode[STIX]{x1D70C}}$ be a continuous, absolutely irreducible representation of its absolute Galois group with values in a finite field of characteristic $p$. We prove that the Galois representations that become crystalline of a fixed regular weight after an abelian extension are Zariski-dense in the generic fiber of the universal deformation ring of $\bar{\unicode[STIX]{x1D70C}}$. In fact we deduce this from a similar density result for the space of trianguline representations. This uses an embedding of eigenvarieties for unitary groups into the spaces of trianguline representations as well as the corresponding density claim for eigenvarieties as a global input. [ABSTRACT FROM AUTHOR]

In this paper, we determine the finite classical simple groups of dimension n = 3, 5 which are (2, 3)-generated (the cases n = 2, 4 are known). If n = 3, they are 3(q), q ≠ 4, and 3(q2), q2 ≠ 9, 25. If n = 5 they are 5(q), for all q, and 5(q2), q2 ≥ 9. Also, the soluble group 3(4) is not (2, 3)-generated. We give explicit (2, 3)-generators of the linear preimages, in the special linear groups, of the (2, 3)-generated simple groups. [ABSTRACT FROM AUTHOR]

Let F be a local field with finite residue field of characteristic p. Let G be a connected reductive group over F and B a minimal parabolic subgroup of G with Levi decomposition . Let K be a special parahoric subgroup of G, in good position relative to ( Z, U). Fix an absolutely irreducible smooth representation of K on a vector space V over some field C of characteristic p. Writing for the intertwining Hecke algebra of V in G, we define a natural algebra homomorphism from to , we show it is injective and identify its image. We thus generalize work of F. Herzig, who assumed F of characteristic 0, G unramified and K hyperspecial, and took for C an algebraic closure of the prime field 픽 p. We show that in the general case need not be commutative; that is in contrast with the cases Herzig considers and with the more classical situation where V is trivial and the field of coefficients is the field of complex numbers. [ABSTRACT FROM AUTHOR]

Let X be a normal geometrically connected variety over a finite field κ of characteristic p. Let (ρλ : π1(X) → GLn(Eλ))λ be any semisimple E-rational compatible system, where E is a number field and λ ranges over the finite places of E not above p. We derive new properties on the monodromy groups of such systems for almost all λ and give natural criteria for the corresponding geometric adelic representation to have open image in an appropriate sense. Key inputs to our results are automorphic methods and the Langlands correspondence over global function fields proved by L. Lafforgue. To say more, let (̅ρλ : π1(X) → GLn(kλ))λ be the corresponding mod-λ system, where for every λ by Oλ and kλ we denote the valuation ring and the residue field of Eλ, and where the reduction is done with respect to some π1(X)-stable Oλ-lattice Æ'©λ of En λ. Let also Ggeo λ be the Zariski closure of ρλ(π1(X̅κ)) in GLn,E, and let Ggeo λ be its schematic closure in AutOλ(Æ'©λ). Assume in the following that the algebraic groups Ggeo λ are connected. We prove that for almost all λ the group scheme Ggeo λ is semisimple over Oλ, and its special fiber agrees with the Nori envelope of ρλ(π1(X̅κ)). A comparable result under different hypotheses was proved by A. Cadoret, C.-Y. Hui, and A. Tamagawa using other methods. As an intermediate result, we show for X a curve that any potentially tame compatible system of mod-λ representations can be lifted to a compatible system over a number field; this implies for almost all λ the semisimplicity of the restriction ρλ | π1(Xκ). Finally, we establish adelic openness for (ρλ| π1(Xκ))λ in the sense of C. Y. Hui and M. Larsen, for E = Q in general, and for E ⊋ Q under additional hypotheses. [ABSTRACT FROM AUTHOR]