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Let G = SLn(q), where n ≥ 2 and q is a power of a prime p. A Singer cycle of the group G is its cyclic subgroup of order (qn − 1)/(q − 1). We classify absolutely irreducible G-modules over a field of characteristic p where an element of fixed prime order m from a Singer cycle of G acts freely in the following three cases: (a) the residue of q modulo m generates the multiplicative group of the field of order m (in particular, this holds for m = 3); (b) m = 5; (c) n = 2.

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.

Jedlicka, Hernando and McGuire proved that Gold and Kasami functions are the only power mappings which are APN on infinitely many extensions of $$\mathbb {F}_2$$ F 2 . For $$p$$ p an odd prime, we prove that the only power mappings $$x\mapsto x^m$$ x ? x m such that $$m\equiv 1\mod p$$ m ? 1 mod p which are PN on infinitely many extensions of $$\mathbb {F}_p$$ F p are those such that $$m=1+p^l$$ m = 1 + p l , l positive integer. As Jedlicka, Hernando and McGuire, we prove that $$\frac{(x+1)^m-x^m-(y+1)^m+y^m}{x-y}$$ ( x + 1 ) m - x m - ( y + 1 ) m + y m x - y has an absolutely irreducible factor by using Bezout's theorem.

We study Hopf bifurcation with SN-symmetry for the standard absolutely irreducible action of SN obtained from the action of SN by permutation of N coordinates. Stewart (1996 Symmetry methods in collisionless many-body problems, J. Nonlinear Sci. 6 543–63) obtains a classification theorem for the C-axial subgroups of SN × S1. We use this classification to prove the existence of branches of periodic solutions with C-axial symmetry in systems of ordinary differential equations with SN-symmetry posed on a direct sum of two such SN-absolutely irreducible representations, as a result of a Hopf bifurcation occurring as a real parameter is varied. We determine the (generic) conditions on the coefficients of the fifth order SN × S1-equivariant vector field that describe the stability and criticality of those solution branches. We finish this paper with an application to the cases N = 4 and N = 5.

We want to characterize in PG(2, 2) and in PG(2, 3) the absolutely irreducible elliptic cubics containing the maximum number of rational points and all the absolutely irreducible singular cubics.

I present some results towards a complete classification of monomials that are Almost Perfect Nonlinear (APN), or equivalently differentially 2-uniform, over F"2"^"n for infinitely many positive integers n. APN functions are useful in constructing S-boxes in AES-like cryptosystems. An application of a theorem by Weil [A. Weil, Sur les courbes algebriques et les varietes qui s'en deduisent, in: Actualites Sci. Ind., vol. 1041, Hermann, Paris, 1948] on absolutely irreducible curves shows that a monomial x^m is not APN over F"2"^"n for all sufficiently large n if a related two variable polynomial has an absolutely irreducible factor defined over F"2. I will show that the latter polynomial's singularities imply that except in three specific, narrowly defined cases, all monomials have such a factor over a finite field of characteristic 2. Two of these cases, those with exponents of the form 2^k+1 or 4^k-2^k+1 for any integer k, are already known to be APN for infinitely many fields. The last, relatively rare case when a certain gcd is maximal is still unproven; my method fails. Some specific, special cases of power functions have already been known to be APN over only finitely many fields, but they also follow from the results below.

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”.

Let $t$ be an integer $\ge 5$. 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. If $t \equiv 5 \bmod{8}$, we give a criterion that ensures that $\phi_t(x, y)$ is absolutely irreducible. We prove that if $\phi_t(x, y)$ is not absolutely irreducible, then it is divisible by $\phi_{13}(x, y)$. We also exhibit an infinite family of integers $t$ such that $\phi_t(x, y)$ is not absolutely irreducible.

We study the problem of bounding a polynomial which is absolutely irreducible, away from polynomials which are not absolutely irreducible. These separation bounds are useful for testing whether an empirical polynomial is absolutely irreducible or not, for the given tolerance or error bound of its coefficients. Kaltofen and May studied a method which finds applicable separation bounds using an absolute irreducibility criterion due to Ruppert. In this paper, we study some improvements on their method, by which we are able to find more accurate separation bounds, for bivariate polynomials.

Finding minimal fields of definition for representations is one of the most important unsolved problems of computational representation theory. While good methods exist for representations over finite fields, it is still an open question in the case of number fields. In this paper we give a practical method for finding minimal defining subfields for absolutely irreducible representations. We illustrate the new algorithm by determining a minimal field for each absolutely irreducible representation of Sz(8).

We classify all pairs ( V , H ) , where H is a proper subgroup of G = S p 6 ( q ) , q even, and V is an l -modular representation of G for l ≠ 2 which is absolutely irreducible as a representation of H . This problem is motivated by the Aschbacher–Scott program on classifying maximal subgroups of finite classical groups.

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 B = Z U $B=ZU$ . 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 ℋ ( G , K , V ) $\mathcal {H}(G,K,V)$ for the intertwining Hecke algebra of V in G, we define a natural algebra homomorphism from ℋ ( G , K , V ) $\mathcal {H}(G,K,V)$ to ℋ ( Z , Z ∩ K , V U ∩ K ) $\mathcal {H}(Z,Z\cap K,V^{U\cap K})$ , 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 𝔽 p . We show that in the general case ℋ ( G , K , V ) $\mathcal {H}(G,K,V)$ 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.

We determine the l-modular decomposition matrices of Steinberg's triality groups 3 D 4(2 n ) for all primes l ≠ 2 except for some entries in the unipotent characters. As an application, we classify all absolutely irreducible representations of 3 D 4(2 n ) in non-defining characteristic up to a certain degree.

Given an absolutely irreducible ternary form $F$, the purpose of this paper is to produce better upper bounds for the number of integer solutions to the equation F=0, that are restricted to lie in very lopsided boxes. As an application of the main result, a new paucity estimate is obtained for equal sums of two like powers., 15 pages; to appear in Math. Zeit

Suppose ρ 1 and ρ 2 are two pure l-adic degree n representations of the absolute Galois group of a number field K of weights k 1 and k 2 respectively, having equal normalized Frobenius traces T r ( ρ 1 ( σ v ) ) / N v k 1 / 2 and T r ( ρ 2 ( σ v ) ) / N v k 2 / 2 at a set of primes v of K with positive upper density. Assume further that the algebraic monodromy group of ρ 1 is connected and ρ 1 is absolutely irreducible. We prove that ρ 1 and ρ 2 are twists of each other by a power of the l-adic cyclotomic character times a character of finite order. As a corollary, we deduce a theorem of Murty and Pujahari proving a refinement of the strong multiplicity one theorem for normalized eigenvalues of newforms.

This paper contains corrections to the tables of low-dimensional representations of quasi-simple groups published in the paper, ‘Lowdimensional representations of quasi-simple groups’, LMS Journal of Computation and Mathematics4 (2001) 22‐63. In our paper ‘Low-dimensional representations of quasi-simple groups’ , we determine all the absolutely irreducible representations of quasi-simple groups of dimension at most 250, excluding those of groups of Lie type in their defining characteristic. Martin Liebeck has kindly pointed out to us three omissions in our tables: the 12- and 13dimensional representations of the group L3.3/, and the 248-dimensional representations of L4.5/ in characteristic 2. When checking our arguments and calculations we realized that in fact all the representations of L3.3/ were missing, as well as the representations of L4.5/ of dimension exceeding 247. The absolutely irreducible representations of L 3.3/ can be found in the modular Atlas [7]. This leads to the first part of Table 1 below.

We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X,A) = B, where w is a finite word in the alphabet {X,A}. We think of A,B in G as fixed coefficients, and X in G as the unknown. Certain word equations, such as XAXAX=B, have solutions in terms of radicals, while others such as XXAX = B do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification. To a word w we associate a polynomial P_w in Z[x,y] in two commuting variables, which factors whenever w is a composition of smaller words. We prove that if P_w(x^2,y^2) has an absolutely irreducible factor in Z[x,y], then the equation w(X,A)=B is not solvable in terms of radicals., Comment: 18 pages, added Lemma 5.2. To appear in Bull. Lon. Math. Soc

A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein criterion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field, but keeping a certain collection of them nonzero.

In the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building $\mathcal{B}(\mathrm{G},k)$ of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding $\mathcal{B}(\mathrm{G},k)$ in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

Let Q(x, y) be a quadratic form with discriminant D ≠ 0. We obtain non-trivial upper bound estimates for the number of solutions of the congruence Q(x, y) ≡ λ ( mod p), where p is a prime and x, y lie in certain intervals of length M, under the assumption that Q(x, y) - λ is an absolutely irreducible polynomial modulo p. In particular, we prove that the number of solutions to this congruence is Mo(1) when M ≪ p1/4. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence xy ≡ λ( mod p).

Shuhong Gao (2003) [6] has proposed an efficient algorithm to factor a bivariate polynomial f over a field F. This algorithm is based on a simple partial differential equation and depends on a crucial fact: the dimension of the polynomial solution space G associated with this differential equation is equal to the number r of absolutely irreducible factors of f. However, this holds only when the characteristic of F is either zero or sufficiently large in terms of the degree of f. In this paper we characterize a vector subspace of G for which the dimension is r, regardless of the characteristic of F, and the properties of Gao’s construction hold. Moreover, we identify a second vector subspace of G that leads to an analogous theory for the rational factorization of f.

Let K be a number field and F(X,Y) an absolutely irreducible polynomial of K[X,Y] such that the algebraic curve defined by the equation F(X,Y)=0 is rational. In this paper we present practical algorithms for the determination of all solutions of the Diophantine equation F(X,Y)=0 in algebraic integers of K.

In this note, we use the deformation theory of an absolutely irreducible Galois representation to study certain Hecke rings. Specifically, we investigate the variation in the rank of the new part of a completed Hecke ring as we vary the set of primes at which the deformations are allowed to ramify.

The notion of adequate subgroups was introduced by Jack Thorne [42]. It is a weakening of the notion of big subgroups used in generalizations of the Taylor-Wiles method for proving the automorphy of certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] that if the dimension is small compared to the characteristic then all absolutely irreducible representations are adequate. Here we extend the result by showing that, in almost all cases, absolutely irreducible kG-modules in characteristic p, whose irreducible G+-summands have dimension less than p (where G+ denotes the subgroup of G generated by all p-elements of G), are adequate., 60 pages

On the reduction modulopof absolutely irreducible polynomials. Let K be a number field and F(X,Y) be an absolutely irreducible polynomial of K[X,Y]. In this note, using an effective version of Riemann-Roch theorem and a version of the implicit functions theorem, we calculate a positive number A such that if ℘ is prime ideal of the ring of integers of K with norm \(\), then the reduction of F(X,Y) modulo ℘ is an absolutely irreducible polynomial.

Let F(X) be an absolutely irreducible polynomial in \({\mathbb{Z} [X_{1},\dots, X_{n}]}\), with degree d. We prove that, for any δ < 4/3, for any sufficiently large x, there exists a positive density of integral n-tuples m = (m1, . . . , mn) in the hypercube max |mi| ≤ x such that every prime divisor of F(m) is smaller than xd–δ. This result is improved when F satisfies some geometrical hypotheses.

Let $f=a_0(x)+a_1(x)y+a_2(x)y^2\in {\Bbb Z}[x,y]$ be an absolutely irreducible polynomial of degree m in x. We show that the reduction f mod p will also be absolutely irreducible if $p\ge c_m\cdot H(f)^{e_m}$ where H (f) is the height of f and e 1 = 4,e 2 = 6, e 3 = 6 ${2}\over{3}$ and e m = 2 m for m≥ 4. We also show that the exponents e m are best possible for $m\ne 3$ if a plausible number theoretic conjecture is true.

We prove that a k-set K of PG(2, 3) of class [m, n]1 having some trisecant lines, is a non singular absolutely irreducible quartic, containing nine points of PG(2, 3) or a 9-set of the same type otherwise it is the complementary of a point of PG(2, 3) or, a 12-set of the same type and with the same characters. We consider moreover all the other sets of class [m, n]1 of PG(2, 3).

Let φ be a Drinfeld A-module of arbitrary rank and arbitrary characteristic over a finitely generated field K, and set GK=Gal(Ksep/K). Let E=EndK(φ). We show that for almost all primes p of A the image of the group ring A[GK] in EndA(Tp(φ)) is the commutant of E. In the special case E=A it follows that the representation of GK on the p-torsion points φ[p](Ksep) of φ is absolutely irreducible for almost all p.

Let [math] be a smooth projective absolutely irreducible curve of genus [math] over a number field [math] , and denote its Jacobian by [math] . Let [math] be an integer and denote the [math] -th symmetric power of [math] by [math] . In this paper we adapt the classic Chabauty–Coleman method to study the [math] -rational points of [math] . Suppose that [math] has Mordell–Weil rank at most [math] . We give an explicit and practical criterion for showing that a given subset [math] is in fact equal to [math] .

We study certain comodule structures on spaces of linear morphisms between H-comodules, where H is a Hopf algebra over the field k. We apply the results to show that H has non-zero integrals if and only if there exists a non-zero finite dimensional injective right H-comodule. Using this approach, we prove an extension of a result of Sullivan, by showing that if H is involutory and has non-zero integrals, and there exists an injective indecomposable right comodule whose dimension is not a multiple of char(k), then H is cosemisimple. Also we prove without using character theory that if H is cosemisimple and M is an absolutely irreducible right H-comodule, then char(k) does not divide dim(M).

We prove the existence in many cases of minimally ramified p-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod p representations of the absolute Galois group of Q. It is predicted by Serre's conjecture that such representations arise from newforms of optimal level and weight.

Dans cette note, nous etablissons le lien entre le nombre de facteurs absolument irreductibles d'un polynome et la dimension d'un espace vectoriel de formes differentielles fermees. Notre resultat generalise un theoreme de Gao. In this note, we establish the relationship between the number of absolutely irreducible factors of a polynomial and the dimension of a space of closed differentials forms. Our result generalizes a theorem of Gao.

We prove that in PG(2, 3) two absolutely irreducible non singular quartics, containing the maximum number of rational points, exist.

Let G2(q) be the Chevalley group of type G2 defined over a finite field with q=pn elements, where p is a prime number and n is a positive integer. In this paper, we determine when the restriction of an absolutely irreducible representation of G in characteristic other than p to a maximal subgroup of G2(q) is still irreducible. Similar results are obtained for B22(q) and G22(q).

It is proved that the generating function defined by the numbers of isomorphism classes of absolutely irreducible representations of a finitely generated group G into GL d ( F q n ) for n ⩾ 1 is a rational function. It is also proved that the generating function defined by weighted sums over isomorphism classes of representations of G into GL d ( F q n ) for n ⩾ 1 is meromorphic over both the complex numbers and the p-adic complex numbers.

Let G be a finite group, \(\Lambda\) an absolutely irreducible \({\mathbb{Z}}[G]\) -module and w a weight of \(\Lambda\) . To any Galois covering with group G we associate two correspondences, the Schur and the Kanev correspondence. We work out their relation and compute their invariants. Using this, we give some new examples of Prym–Tyurin varieties.

We compute the 2-modular decomposition numbers of Steinberg's triality groups D 4 3 ( q ) , where q is a power of an odd prime, except for two decomposition numbers in the Steinberg character and classify all absolutely irreducible representations of D 4 3 ( q ) in non-defining characteristic up to a certain degree.

For a prime p and an absolutely irreducible modulo p polynomial f(U, V ) ∈ Z[U, V ] we obtain an asymptotic formulas for the number of solutions to the congruence f(x, y) ≡ a (mod p) in positive integers x 6 X, y 6 Y , with the additional condition gcd(x, y) = 1. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over a for a fixed prime p, and also on average over p for a fixed integer a.

International audience; We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus π.

The number of rank two $\overline{\Bbb{Q}}_\ell$-local systems, or $\overline{\Bbb{Q}}_\ell$-smooth sheaves, on $(X-\{u\})\otimes_{\Bbb{F}_q}\Bbb{F}$, where $X$ is a smooth projective absolutely irreducible curve over $\Bbb{F}_q$, $\Bbb{F}$ an algebraic closure of $\Bbb{F}_q$ and $u$ is a closed point of $X$, with principal unipotent monodromy at $u$, and fixed by ${\rm Gal}(\Bbb{F}/\Bbb{F}_q)$, is computed. It is expressed as the trace of the Frobenius on the virtual $\overline{\Bbb{Q}}_\ell$-smooth sheaf found in the author's work with Deligne on the moduli stack of curves with \'etale divisors of degree $M\ge 1$. This completes the work with Deligne in rank two. This number is the same as that of representations of the fundamental group $\pi_1((X-\{u\})\otimes_{\Bbb{F}_q}\Bbb{F})$ invariant under the Frobenius ${\rm Fr}_q$ with principal unipotent monodromy at $u$, or cuspidal representations of ${\rm GL}(2)$ over the function field $F=\Bbb{F}_q(X)$ of $X$ over $\Bbb{F}_q$ with Steinberg component twisted by an unramified character at $u$ and unramified elsewhere, trivial at the fixed id\`ele $\alpha$ of degree 1. This number is computed in Theorem 4.1 using the trace formula evaluated at $f_u\prod_{v\not=u}\chi_{K_v}$, with an Iwahori component $f_u=\chi_{I_u}/|I_u|$, hence also the pseudo-coefficient $\chi_{I_u}/|I_u|-2\chi_{K_u}$ of the Steinberg representation twisted by any unramified character, at $u$. Theorem 2.1 records the trace formula for ${\rm GL}(2)$ over the function field $F$. The proof of the trace formula of Theorem 2.1 recently appeared elsewhere. Theorem 3.1 computes, following Drinfeld, the number of $\overline{\Bbb{Q}}_\ell$-local systems, or $\overline{\Bbb{Q}}_\ell$-smooth sheaves, on $X\otimes_{\Bbb{F}_q}\Bbb{F}$, fixed by ${\rm Fr}_q$, namely $\overline{\Bbb{Q}}_\ell$-representations of the absolute fundamental group $\pi(X\otimes_{\Bbb{F}_q}\Bbb{F})$ invariant under the Frobenius, by counting the nowhere ramified cuspidal representations of ${\rm GL}(2)$ trivial at a fixed id\`ele $\alpha$ of degree 1. This number is expressed as the trace of the Frobenius of a virtual $\overline{\Bbb{Q}}_\ell$-smooth sheaf on a moduli stack. This number is obtained on evaluating the trace formula at the characteristic function $\prod_v\chi_{K_v}$ of the maximal compact subgroup, with volume normalized by $|K_v|=1$. Section 5, based on a letter of P. Deligne to the author dated August 8, 2012, computes the number of such objects with any unipotent monodromy, principal or trivial, in our rank two case. Surprisingly, this number depends only on $X$ and ${\rm deg}(S)$, and not on the degrees of the points in $S_1$.

Let C be a smooth, absolutely irreducible genus 3 curve over a number field M. Suppose that the Jacobian of C has complex multiplication by a sextic CM-field K. Suppose further that K contains no imaginary quadratic subfield. We give a bound on the primes \(\mathfrak{p}\) of M such that the stable reduction of C at \(\mathfrak{p}\) contains three irreducible components of genus 1.

The chief aim of this paper is to describe a procedure which, given a d-dimensional absolutely irreducible matrix representation of a finite group over a finite field E, produces an equivalent representation such that all matrix entries lie in a subfield F of E which is as small as possible. The algorithm relies on a matrix version of Hilbert's Theorem 90, and is probabilistic with expected running time O(|E:F|d3) when |F| is bounded. Using similar methods we then describe an algorithm which takes as input a prime number and a power-conjugate presentation for a finite soluble group, and as output produces a full set of absolutely irreducible representations of the group over fields whose characteristic is the specified prime, each representation being written over its minimal field.

We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time--space complexity is roughly quadratic in the logarithm of the cardinality of the field and a geometric invariant of the input system (called its degree), which is always bounded by the Bezout number of the system. Our algorithm works for fields of any characteristic, but requires the cardinality of the field to be greater than a quantity which is roughly the fourth power of the degree of the input variety., 36 pages