Search

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]

We consider the Kisin variety associated to an n-dimensional absolutely irreducible mod p Galois representation ρ ¯ {\bar{\rho}} of a p-adic field K together with a cocharacter μ. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if K is totally ramified with n = 3 {n=3} or μ is of a very particular form. As an application, we get a connectedness result for the deformation ring associated to ρ ¯ {\bar{\rho}} of given Hodge–Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. An element r ∈ R is said to be absolutely irreducible in R if for all natural numbers n > 1, rn has essentially only one factorization namely r n = r ⋯ r. If r ∈ R is irreducible in R but for some n > 1, rn has other factorizations distinct from r n = r ⋯ r , then r is called non-absolutely irreducible. In this paper, we construct non-absolutely irreducible elements in the ring Int (Z) = { f ∈ Q [ x ] | f (Z) ⊆ Z } of integer-valued polynomials. We also give generalizations of these constructions. [ABSTRACT FROM AUTHOR]

We consider the arithmetic background of integral representations of finite groups over -adic and algebraic number rings. Some infinite series of integral pairwise inequivalent absolutely irreducible representations of finite -groups with the extra congruence conditions are constructed, and some applications are given. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed. [ABSTRACT FROM AUTHOR]

This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented by two items: first by a parametric representative system, i.e., the equations that define the component and second by a parametric effective generic point which gives a parametric rational univariate representation of the elements of the component. The number of absolutely irreducible components is constant in each constructible set. The complexity bound of this algorithm is $${\delta^{O(r^4)}d^{r^4d^{O(n^3)}}}$$, being double exponential in n, where d (resp. δ) is an upper bound on the degrees of the input parametric polynomials w.r.t. the main n variables (resp. w.r.t. r parameters). [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]

It is known that for finite simple groups it is possible for a faithful absolutely irreducible module to have 1-cohomology of dimension at least 3. However, until now no explicit examples have been found. We present two explicit examples where the dimension is exactly 3. It remains an open question as to whether the dimension can be bigger than 3. [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]

If in a given rank r, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number \ell, there is an absolutely irreducible \ell-adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank r if once there is an irreducible rank r complex linear representation with determinant isomorphic to this torsion character, then for any \ell, there is an absolutely irreducible one of rank r and determinant this given character, which is defined over \bar { \mathbb {Z}}_\ell. We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth quasi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne's companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong's conjecture (Gaitsgory for \ell \ge 3). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of the Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, which shows the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple. [ABSTRACT FROM AUTHOR]

We introduce a notion of absolute irreducibility for finitary skew linear groups consistent with the existing notions for finite-dimensional linear groups, finite-dimensional skew linear groups and finitary linear groups and then prove some relatively strong results about such absolutely irreducible groups. [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]

It is shown that strong atoms are rather abundant in monadically Krull monoids. An application to rings of integer-valued polynomials on Krull domains yields new results on strong atoms in these rings. [ABSTRACT FROM AUTHOR]

The subject of this paper is the study of small complete arcs in PG(2,q), for q odd, with at least (q+1)∕2 points on a conic. We give a short comprehensive proof of the completeness problem left open by Segre in his seminal work. This gives an alternative to Pellegrino's long proof which was obtained in a series of papers in the 1980s. As a corollary of our analysis, we obtain a counterexample to a misconception in the literature. [ABSTRACT FROM AUTHOR]

We prove that in either the convergent or overconvergent setting, an absolutely irreducible |$F$| -isocrystal on the absolute product of two or more smooth schemes over perfect fields of characteristic |$p$|⁠ , further equipped with actions of the partial Frobenius maps, is an external product of |$F$| -isocrystals over the multiplicands. The corresponding statement for lisse |$\overline{{\mathbb{Q}}}_{\ell }$| -sheaves, for |$\ell \neq p$| a prime, is a consequence of Drinfeld's lemma on the fundamental groups of absolute products of schemes in characteristic |$p$|⁠. The latter plays a key role in V. Lafforgue's approach to the Langlands correspondence for reductive groups with |$\ell $| -adic coefficients; the |$p$| -adic analogue will be considered in subsequent work with Daxin Xu. [ABSTRACT FROM AUTHOR]

It is shown that strong atoms are rather abundant in Krull monoids. For the proof extraction methods are used. An application of this result yields a positive answer to a question of D. D. Anderson, D. F. Anderson and J. Park about Dedekind domains. [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]

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]

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]

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]

In this article, we construct zeta morphisms for the universal deformations of odd absolutely irreducible two dimensional mod p Galois representations satisfying some mild assumptions, and prove that our zeta morphisms interpolate Kato's zeta morphisms for Galois representations associated to Hecke eigen cusp newforms. The existence of such morphisms was predicted by Kato's generalized Iwasawa main conjecture. Based on Kato's original construction, we construct our zeta morphisms using many deep results in the theory of p -adic (local and global) Langlands correspondence for GL 2 / Q . As an application of our zeta morphisms and the recent article (Kim et al. in On the Iwasawa invariants of Kato's zeta elements for modular forms, 2019, arXiv:1909.01764v2), we prove a theorem which roughly states that, under some μ = 0 assumption, Iwasawa main conjecture without p -adic L -function for f holds if this conjecture holds for one g which is congruent to f . [ABSTRACT FROM AUTHOR]

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring Int (D) = { f ∈ K [ x ] | f (D) ⊆ D } , of integer-valued polynomials on a principal ideal domain D with quotient field K, we give an easy to verify graph-theoretic sufficient condition for an element to be absolutely irreducible and show a partial converse: the condition is necessary and sufficient for polynomials with square-free denominator. [ABSTRACT FROM AUTHOR]

This paper contains three new results. (1) We introduce new notions of projective crystalline representations and twisted periodic Higgs--de Rham flows. These new notions generalize crystalline representations of étale fundamental groups introduced by Faltings [Algebraic Analysis, Geometry, and Number Theory (1989)] and Fontaine and Laffaille [Ann. Sci. Éc. Norm. Supér. (4) 15 (1983)] and periodic Higgs--de Rham flows introduced by Lan, Sheng and Zuo [J. Eur. Math. Soc. (JEMS) 21 (2019)]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs--de Rham flows via the category of twisted Fontaine--Faltings module which is also introduced in this paper. (2) We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. (3) We investigate the dynamic of self-maps induced by the Higgs--de Rham flow on the moduli spaces of rank-2 stable Higgs bundles of degree 1 on P1 with logarithmic structure on marked points D : = {x1,..., xn} for n ≥ 4 and construct infinitely many geometrically absolutely irreducible PGL2 (Zpur)-crystalline representations of π1ét (PQp1ur) \ D). We find an explicit formula of the self-map for the case {0, 1, ∞ λ} conjecture that a Higgs bundle is periodic if and only if the zero of the Higgs field is the image of a torsion point in the associated elliptic curve Cλ defined by y² = x(x -- 1)(x - λ) with the order coprime to p. [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]

We study the vanishing of Massey products of order at least 3 for absolutely irreducible smooth projective curves over a field with coefficients in Z/ℓ$\mathbb {Z}/\ell$. We mainly focus on elliptic curves, for which we obtain a complete characterization of when triple Massey products do not vanish. [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]

Persistence of nondegeneracy is a phenomenon that appears in the theory of |$\overline{\mathbb{Q}}_l$| -representations of the linear group: every irreducible submodule of the restriction to the mirabolic sub-representation of a nondegenerate irreducible representation is nondegenerate. This is not true anymore in general, if we look at the modulo |$l$| reduction of some stable lattice. As in the Clozel–Harris–Taylor generalization of global Ihara's lemma, we show that this property, called nondegeneracy persistence and related to the notion of essentially absolutely irreducible and generic representations in the work of Emerton and Helm, remains true for lattices given by the cohomology of Lubin–Tate spaces. As a global application, we give a new construction of automorphic congruences in the Ribet spirit. Résumé. La persistence de la non dégénérescence est un phénomène qui apparait dans la théorie des |$\overline{\mathbb{Q}}_l$| -représentations du groupe linéaire: toute sous-représentation irréductible de la restriction au groupe mirabolique d'une représentation irréductible non dégénérée, est non dégénérée. Ce n'est plus le cas en général pour la réduction modulo |$l$| d'un réseau stable. Comme dans la généralisation par Clozel-Harris-Taylor du lemme d'Ihara, nous montrons que cette propriété de non dégénérescence, qui est reliée à la notion de représentation essentiellement absolument générique de Emerton-Helm, reste valide pour les réseaux donnés par la cohomologie des espaces de Lubin-Tate. Nous donnons une application de nature globale en construisant des congruences automorphes dans l'esprit du travail de Ribet. [ABSTRACT FROM AUTHOR]

Let K be a finite extension of the p-adic field Q𝑝 of degree d, let F be a finite field of characteristic p and let 𝐷 be an n-dimensional pseudocharacter in the sense of Chenevier of the absolute Galois group of K over the field F. For the universal mod p pseudodeformation ring 𝑅 univ 𝐷 of 𝐷, we prove the following: The ring 𝑅ps 𝐷 is equidimensional of dimension 𝑑𝑛2 +1. Its reduced quotient 𝑅 univ 𝐷, red contains a dense open subset of regular points x whose associated pseudocharacter 𝐷𝑥 is absolutely irreducible and nonspecial in a certain technical sense that we shall define. Moreover, we will characterize in most cases when K does not contain a p-th root of unity the singular locus of Spec 𝑅 univ 𝐷. Similar results were proved by Chenevier for the generic fiber of the universal pseudodeformation ring 𝑅univ 𝐷 of 𝐷. [ABSTRACT FROM AUTHOR]

We study short crystalline, minimal, essentially self-dual deformations of a mod p nonsemisimple Galois representation ̅σ with ̅σss = χk-2 ⊕ρ⊕χk-1, where χ is the mod p cyclotomic character and ρ is an absolutely irreducible reduction of the Galois representation ρf attached to a cusp form f of weight 2k-2. We show that if the Bloch-Kato Selmer groups H¹f (Q,ρf (1-k)⊗Qp/Zp) and H¹f (Q,ρ(2-k)) have order p, and there exists a characteristic zero absolutely irreducible deformation of ̅σ then the universal deformation ring is a dvr. When k = 2 this allows us to establish the modularity part of the Paramodular Conjecture in cases when one can find a suitable congruence of Siegel modular forms. As an example we prove the modularity of an abelian surface of conductor 731. When k >2, we obtain an Rred = T theorem showing modularity of all such deformations of ̅σ. [ABSTRACT FROM AUTHOR]

Abstract: We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer is said to be exceptional if is APN (Almost Perfect Nonlinear) over for infinitely many values of n. Equivalently, t is exceptional if the binary cyclic code of length with two zeros has minimum distance 5 for infinitely many values of n. The conjecture we prove states that every exceptional number has the form or . [Copyright &y& Elsevier]

Let K / F be a finite Galois extension of number fields and let σ be an absolutely irreducible, self-dual, complex valued representation of Gal (K / F). Let p be an odd prime and consider two elliptic curves E 1 , E 2 defined over Q with good, ordinary reduction at primes above p and equivalent mod- p Galois representations. In this article, we study the variation of the parity of the multiplicities of σ in the representation space associated to the p ∞ -Selmer groups of E 1 and E 2 over K. We also compare the root numbers for the twists of E 1 and E 2 over F by σ and show that the p -parity conjecture holds for the twist of E 1 / F by σ if and only if it holds for the twist of E 2 / F by σ. We also express Mazur-Rubin-Nekovář's arithmetic local constants in terms of certain local Iwasawa invariants. [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]

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]

Over fields of characteristic zero, we determine all absolutely irreducible Yetter–Drinfeld modules over groups that have prime dimension and yield a finite-dimensional Nichols algebra. To achieve our goal, we introduce orders of braided vector spaces and study their degenerations and specializations. [ABSTRACT FROM AUTHOR]

A binary triply-even [98280; 25; 47104]2 code invariant under the sporadic simple group Co1 is constructed by adjoining the all-ones vector to the faithful and absolutely irreducible 24-dimensional code of length 98280. Using the action of Co1 on the code we give a description of the nature of the codewords of any non-zero weight relating these to vectors of types 2, 3 and 4, respectively of the Leech lattice. We show that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of Co1. Moreover, we give a partial description of the nature of the codewords of minimum weight of the dual code. [ABSTRACT FROM AUTHOR]

We show that if {ρℓ} is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation ¯ρℓ is absolutely irreducible for ℓ in a density 1 set of primes. The key technical result is the following theorem: the image of ρℓ is an open subgroup of a hyperspecial maximal compact subgroup of its Zariski closure with bounded index (as ℓ varies). This result combines a theorem of Larsen on the semi-simple part of the image with an analogous result for the central torus that was recently proved by Barnet-Lamb, Gee, Geraghty, and Taylor, and for which we give a new proof. [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]

Let X / F q be a smooth, geometrically connected variety. For X projective, we prove a Lefschetz-style theorem for abelian schemes of GL 2 -type on X, modeled after a theorem of Simpson. Inspired by work of Corlette-Simpson over C , we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on X come from families of abelian varieties. We have the following application of our main result. If one assumes a strong form of Deligne's (p-adic) companions conjecture from Weil II, then our conjecture for projective varieties reduces to the conjecture for projective curves. We also answer affirmitavely a question of Grothendieck on extending abelian schemes via their p-divisible groups. [ABSTRACT FROM AUTHOR]

In this paper we investigate the factorization behaviour of the binomial polynomials ( x n ) = x (x − 1) ⋯ (x − n + 1) n ! and their powers in the ring of integer-valued polynomials Int (Z). While it is well-known that the binomial polynomials are irreducible elements in Int (Z) , the factorization behaviour of their powers has not yet been fully understood. We fill this gap and show that the binomial polynomials are absolutely irreducible in Int (Z) , that is, ( x n ) m factors uniquely into irreducible elements in Int (Z) for all m ∈ N. By reformulating the problem in terms of linear algebra and number theory, we show that the question can be reduced to determining the rank of, what we call, the valuation matrix of n. A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof: If n > 10 and n , n − 1 , ..., n − (k − 1) are composite integers, then there exists a prime number p > 2 k that divides one of these integers. [ABSTRACT FROM AUTHOR]

We study the image of the -adic Galois representations associated to the four vector valued Siegel modular forms appearing in the work of Chenevier and Lannes [3]. These representations are symplectic of dimension 4. Following methods used by Dieulefait in [4], we determine the primes for which these representations are absolutely irreducible. In addition, we show that their image is 'full' for all primes such that the associated residual representation is absolutely irreducible, except in two cases. [ABSTRACT FROM AUTHOR]

The notion of adequate subgroups was introduced by Jack Thorne [60]. It is a weakening of the notion of big subgroups used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] and [23] that if the dimension is smaller than the characteristic then almost all absolutely irreducible representations are adequate. We extend the results by considering all absolutely irreducible modules in characteristic p of dimension p. This relies on a modified definition of adequacy, provided by Thorne in [61], which allows p to divide the dimension of the module. We prove adequacy for almost all irreducible representations of SL2(pa) in the natural characteristic and for finite groups of Lie type as long as the field of definition is sufficiently large.We also essentially classify indecomposable modules in characteristic p of dimension less than 2p - 2 and answer a question of Serre concerning complete reducibility of subgroups in classical groups of low dimension. [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]

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]

Suppose ρ 1 , ρ 2 are two ℓ -adic Galois representations of the absolute Galois group of a number field, such that the algebraic monodromy group of one of the representations is connected and the representations are locally potentially equivalent at a set of places of positive upper density. We classify such pairs of representations and show that up to twisting by some representation, it is given by a pair of representations one of which is trivial and the other abelian. Consequently, assuming that the first representation has connected algebraic monodromy group, we obtain that the representations are potentially equivalent, provided one of the following conditions hold: (a) the first representation is absolutely irreducible; (b) the ranks of the algebraic monodromy groups are equal; (c) the algebraic monodromy group of the second representation is also connected and (d) the commutant of the image of the second representation remains the same upon restriction to subgroups of finite index of the Galois group. [ABSTRACT FROM AUTHOR]

We are interested in the question of the existence of flat manifolds for which all R -irreducible components of the holonomy representation are either absolutely irreducible, of complex or of quaternionic type. In the first two cases such examples are well known. But the existence of the third type of flat manifolds was unknown to the authors. In this article, we construct such an example. Moreover, we present a list of finite groups for which a construction of manifolds of quaternionic type is impossible. [ABSTRACT FROM AUTHOR]

We refer to the hypotheses that for an absolutely irreducible representation of a compact Lie group there exists at least one subgroup with an odd dimensional fixed point space as the Ize conjecture (IC). If the IC is true, then it follows that loss of stability through an absolutely irreducible representation of a compact Lie group leads to bifurcation of steady states. Lauterbach and Matthews have shown that the (IC) is in general not true and have constructed three infinite families of finite subgroups of $$\mathop {\mathbf{SO}(4)}$$ which act absolutely irreducibly on $$\mathbb {R}^4$$ and have no odd dimensional fixed point space. They also have shown that in spite of this failure of the (IC) the nontrivial isotropy types are generically symmetry breaking at least for the groups in two of these three families. In this paper we show a similar bifurcation result for the third family defined by Lauterbach and Matthews. We go on and construct a family of groups acting absolutely irreducibly on $$\mathbb {R}^8$$ which have only even dimensional fixed point spaces. Then we discuss the steady state bifurcations in this case. Key ingredients are an abstract group theoretic construction and a kind of inductive step reducing the issue of bifurcations to a problem in $$\mathbb {R}^4$$ . We end this paper with a discussion on how to extend the results by Lauterbach and Matthews to larger sets of groups which act on $$\mathbb {R}^{4}$$ and $$\mathbb {R}^8$$ . In this context we point out, that the inductive step, which is important for our arguments, does not work in general and this gives rise to interesting new questions. [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]

Let K be an extension of the field Q p of p -adic rationals, let O K be its ring of integers, and let G be a finite group. According to the classical Jordan–Zassenhaus Theorem, if K ∕ Q p is finite, every isomorphism class of K G -representation modules splits in a finite number of isomorphism classes of O K G -representation modules. We consider a p -group G of a given nilpotency class k > 1 and the extension K ∕ Q p where K = Q p (ζ p ∞ ) obtained by adjoining all roots ζ p i , i = 1 , 2 , 3 ,... of unity, and we use an explicit combinatorial construction of a faithful absolutely irreducible K G -module M to show that the number of O K G -isomorphism classes of O K G -representation modules, which are K G -equivalent to M , is infinite, and there are extra congruence properties for these O K G -modules. [ABSTRACT FROM AUTHOR]