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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-Nekovař's arithmetic local constants in terms of certain local Iwasawa invariants.

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.

Let $X/\mathbb{F}_{q}$ be a smooth geometrically connected variety. Inspired by work of Corlette-Simpson over $\mathbb{C}$, we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on $X$ come from families of abelian varieties. When $X$ is a projective variety, we prove a Lefschetz-style theorem for abelian schemes of $\text{GL}_2$-type on $X$, modeled after a theorem of Simpson. If one assumes a strong form of Deligne's ($p$-adic) \emph{companions conjecture} from Weil II, this implies that our conjecture for projective varieties also reduces to the case of projective curves. We also answer affirmitavely a question of Grothendieck on extending abelian schemes via their $p$-divisible groups., Comment: 29 pages, comments very welcome. v3: completely reorganized, minor errors fixed. v4: final version

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.

We consider vectorial maps F ( x 1 , … , x n ) = ( f 1 ( x 1 , … , x n ) , … , f n ( x 1 , … , x n ) ) : F q n ↦ F q n , which induce permutations of F q n . We show that the degrees of the components f 1 , f 2 , … , f n ∈ F q [ x 1 , … , x n ] are at least 2 when 2 ≤ deg ( F ) = d q and d | ( q − 1 ) . Our proof uses an absolutely irreducible curve over F q and the number of rational points on it that we relate to the cardinality of the value set of a polynomial.

We provide an infinite sequence of upper bounds for the number of rational points of absolutely irreducible smooth projective curves X X over a finite field, starting from the Weil classical bound, continuing to the Ihara bound, passing through infinitely many n n -th order Weil bounds, and ending asymptotically at the Drinfeld-Vlăduţ bound. We relate this set of bounds to those of Oesterlé, proving that these are inverse functions in some sense. We explain how the Riemann hypothesis for the curve X X can be merely seen as a euclidean property coming from the Toeplitz shape of some intersection matrix on the surface X × X X\times 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 F X \mathcal {F}_X of the numerical group Num ⁡ ( X × X ) \operatorname {Num}(X\times 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 F X \mathcal {F}_X .

We define the notion of a smooth pseudo-Riemannian algebraic variety (X, g) over a field k of characteristic 0, which is an algebraic analogue of the notion of Riemannian manifold and we study, from a model-theoretic perspective, the algebraic differential equation describing the geodesics on (X, g). When k is the field of real numbers, we prove that if the real points of X are Zariski-dense in X and if the real analytification of (X, g) is a compact Riemannian manifold with negative curvature, then the algebraic differential equation describing the geodesics on (X, g) is absolutely irreducible and its generic type is orthogonal to the constants.

We consider the polynomials of the form P(x) = x(k) - gamma Tr(x) over F-qn for n >= 2. We show that P(x) is not a permutation of F-qn in the case gcd(k,q(n) - 1) > 1. Our proof uses an absolutely irreducible curve over( )F(qn) and the number of rational points on it.

This paper deals with properties of the algebraic variety defined as the set of zeros of a "typical" sequence of polynomials. We consider various types of "nice" varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero "obstruction" polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define a degenerate variety, namely an absolutely irreducible nonsingular hypersurface in some linear projective subspace., 31 pages

Let ${\bf G}$ be a connected reductive group over $\bar{\mathbb{F}}_q$, the algebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$ elements), with the standard Frobenius map $F$. Let ${\bf B}$ be an $F$-stable Borel subgroup. Let $\Bbbk$ be a field of characteristic $r\neq p$. In this paper, we completely determine the composition factors of the induced module $Ind_{B}^{G}{tr}=\Bbbk{G}\otimes_{\Bbbk{\bf B}}$ tr (here $\Bbbk{H}$ is the group algebra of the group ${H}$, and tr is the trivial $B$-module). In particular, we find a new family of infinite dimensional irreducible abstract representations of $G$., Comment: Accepted by Mathematische Zeitschrift

Riemann's hypothesis on function fields over a finite field implies the Hasse–Weil bound for the number of zeros of an absolutely irreducible bi-variate polynomial over a finite field. The Hasse–Weil bound has extensive applications in the arithmetic of finite fields. In this paper, we use the Hasse–Weil bound to prove two results on permutation polynomials over F q where q is sufficiently large. To facilitate these applications, the absolute irreducibility of certain polynomials in F q [ X , Y ] is established.

Let $${\mathcal {X}}$$ be a (projective, algebraic, non-singular, absolutely irreducible) curve of genus g defined over an algebraically closed field K of characteristic $$p \ge 0$$ , and let q be a prime dividing the cardinality of $$\text{ Aut }({\mathcal {X}})$$ . We say that $${\mathcal {X}}$$ is a q-curve. Homma proved that either $$q \le g+1$$ or $$q = 2g+1$$ , and classified $$(2g+1)$$ -curves up to birational equivalence. In this note, we give the analogous classification for $$(g+1)$$ -curves, including a characterization of hyperelliptic $$(g+1)$$ -curves. Also, we provide the characterization of the full automorphism groups of q-curves for $$q= 2g+1, g+1$$ . Here, we make use of two different techniques: the former case is handled via a result by Vdovin bounding the size of abelian subgroups of finite simple groups, the second via the classification by Giulietti and Korchmaros of automorphism groups of curves of even genus. Finally, we give some partial results on the classification of q-curves for $$q = g, g-1$$ .

In this paper we investigate the factorization behaviour of the binomial polynomials $\binom{x}{n} = \frac{x(x-1)\cdots (x-n+1)}{n!}$ and their powers in the ring of integer-valued polynomials $\operatorname{Int}(\mathbb{Z})$. While it is well-known that the binomial polynomials are irreducible elements in $\operatorname{Int}(\mathbb{Z})$, 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 $\operatorname{Int}(\mathbb{Z})$, that is, $\binom{x}{n}^m$ factors uniquely into irreducible elements in $\operatorname{Int}(\mathbb{Z})$ for all $m\in \mathbb{N}$. 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 $n$, $n-1$, \ldots, $n-(k-1)$ are composite integers, then there exists a prime number $p > 2k$ that divides one of these integers., This is an update to the journal version to include more references; all results and proofs remain unchanged

Let $V$ be a finite-dimensional vector space over a finite field, and suppose $G \leq \Gamma \mathrm{L}(V)$ 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 $B\subseteq V$ 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 $E(G)/Z(E(G)) \cong \mathrm{PSL}_n(q)$ in defining characteristic, with an aim of classifying those with a regular orbit on $V$., Comment: 45 pages

We are interested in the question of the existence of flat manifolds for which all $\mathbb R$-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.

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 $\text{Int}(D)=\{f\in K[x]\mid f(D)\subseteq D\}$, 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., Comment: To appear in Comm. Algebra

Let X be an absolutely irreducible hypersurface of degree d in A n , defined over a finite field F q . The Lang–Weil bound gives an interval that contains # X ( F q ) . We exhibit an explicit interval, which does not contain # X ( F q ) , and which overlaps with the Lang–Weil interval. In particular, we sharpen the best known nontrivial lower bound for # X ( F q ) . The proof uses a combinatorial probabilistic technique.

We study the image of the $\ell$-adic Galois representations associated to the four vector valued Siegel modular forms appearing in the work of Chenevier and Lannes. These representations are symplectic of dimension $4$. Following a method of Dieulefait, we determine the primes $\ell$ for which these representations are absolutely irreducible. In addition, we show that their image is "full" for all primes $\ell$ such that the associated residual representation is absolutely irreducible, except in two cases., Comment: in French, 18 pages, 6 tables

Let p > 3 p>3 be a prime number and let G Q p G_{\mathbb {Q}_{p}} be the absolute Galois group of Q p \mathbb {Q}_{p} . In this paper, we find Galois stable lattices in the 3 3 -dimensional irreducible semi-stable non-crystalline representations of G Q p G_{\mathbb {Q}_{p}} with Hodge–Tate weights ( 0 , 1 , 2 ) (0,1,2) by constructing the corresponding strongly divisible modules. We also compute the Breuil modules corresponding to the mod p p reductions of these strongly divisible modules and determine which of the original representations has an absolutely irreducible mod p p reduction.

The notion of adequate subgroups was introduced by Jack Thorne [59]. 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 [60], which allows p to divide the dimension of the module. We prove adequacy for almost all irreducible representations of SL_2(p^a) 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., Comment: Final version. 58 pages

We partially answer, in terms of monodromy, Murty and Patankar's question: Given an absolutely simple abelian variety over a number field, does it have simple specializations at a set of places of positive Dirichlet density? The answer is based on the classification of pairs (G,V) consisting of a semi-simple algebraic group G over a non-archimedean local field and an absolutely irreducible representation V of G such that G admits a maximal torus acting irreducibly on V.

An almost perfect nonlinear (APN) function \begin{document}$f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$\end{document} (necessarily polynomial) is called exceptional APN if it is APN on infinitely many extensions of \begin{document}$\mathbb{F}_{2^n}$\end{document} . Aubry, McGuire and Rodier conjectured that the only exceptional APN functions are the Gold and the Kasami-Welch monomial functions. They established that a polynomial function of odd degree is not exceptional APN provided the degree is not a Gold number \begin{document}$(2^k+1)$\end{document} or a Kasami-Welch number \begin{document}$(2^{2k}-2^k+1)$\end{document} . When the degree of the polynomial function is a Gold number or a Kasami-Welch number, several partial results have been obtained by several authors including us. In this article we address these exceptions. We almost prove the exceptional APN conjecture in the Gold degree case when \begin{document}$\deg{(h(x))}$\end{document} is odd. We also show exactly when the corresponding multivariate polynomial \begin{document}$φ(x, y, z)$\end{document} is absolutely irreducible. Also, there is only one result known when \begin{document}$f(x)=x^{2^{k}+1} + h(x)$\end{document} , and \begin{document}$\deg(h(x))$\end{document} is even. Here, we extend this result as well, thus making progress in this case that seems more difficult.

International audience; 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.

Suppose $\rho_1, \rho_2$ are two $\ell$-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., Comment: Revised version; 19 pages; corrects the main theorem of Vijay M. Patankar and C. S. Rajan, Locally potentially equivalent Galois representations, J. of Ramanujan Math. Soc., Vol. 27, No. 1, (2012), 77-90. In this revised version, the application to abelian varieties has been modified. To appear in J. Number Theory

The Dickson–Guralnick–Zieve curve, briefly DGZ curve, defined over the finite field F q arises naturally from the classical Dickson invariant of the projective linear group P G L ( 3 , F q ) . The DGZ curve is an (absolutely irreducible, singular) plane curve of degree q 3 − q 2 and genus 1 2 q ( q − 1 ) ( q 3 − 2 q − 2 ) + 1 . In this paper we show that the DGZ curve has several remarkable features, those appearing most interesting are: the DGZ curve has a large automorphism group compared to its genus albeit its Hasse–Witt invariant is positive; the Fermat curve of degree q − 1 is a quotient curve of the DGZ curve; among the plane curves with the same degree and genus of the DGZ curve and defined over F q 3 , the DGZ curve is optimal with respect the number of its F q 3 -rational points.

Let k be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve X 1 {X_{1}} defined over k, there exists a lifting X of the curve to the ring W ⁢ ( k ) {W(k)} of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over X / W ⁢ ( k ) {X/W(k)} . These liftings give rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group π 1 ⁢ ( X K ) {\pi_{1}(X_{K})} of the generic fiber of X. This curve X and its associated representation is in close relation to the canonical curve and its associated canonical crystalline representation in the p-adic Teichmüller theory for curves due to S. Mochizuki. Our result may be viewed as an analogue of the Hitchin–Simpson’s uniformization theory of hyperbolic Riemann surfaces via Higgs bundles.

Let K be a finite extension of Qp. We fix a continuous absolutely irreducible representation of the absolute Galois group of K over a finite dimensional vector space with coefficient in a finite field of characteristic p and consider its universal deformation ring R. If we fix a regular set of Hodge-Tate weights k, we prove, under some hypothesis, that the closed points of Spec(R[1/p]) corresponding to potentially crystalline representations of fixed Hodge-Tate weights k are dense in Spec(R[1/p]) for the Zariski topology., Comment: We fixed a gap in the proof of previous Cor 3.7, now Theorem 4.11, and fixed some sign errors

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

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 G be a simple linear algebraic group over a field F, and V an absolutely irreducible representation of G. We show that under some mild hypotheses there exists an invariant homogeneous polynomial f for the action of G on V defined over F, such that twisted forms of f up to a scalar multiple classify twisted forms of G for which the representation V is defined over F. This result extends the classical case of a quadratic form q and its orthogonal group O ( q ) .

Let ${\rm\Delta}:G\rightarrow \text{GL}(n,K)$ be an absolutely irreducible representation of an arbitrary group $G$ over an arbitrary field $K$; let ${\it\chi}:G\rightarrow K:g\mapsto \text{tr}({\rm\Delta}(g))$ be its character. In this paper, we assume knowledge of ${\it\chi}$ only, and study which properties of ${\rm\Delta}$ can be inferred. We prove criteria to decide whether ${\rm\Delta}$ preserves a form, is realizable over a subfield, or acts imprimitively on $K^{n\times 1}$. If $K$ is finite, we can decide whether the image of ${\rm\Delta}$ belongs to certain Aschbacher classes.

It is possible that a simple (or absolutely simple) Abelian variety defined over a number field splits modulo every prime of good reduction. This is a new problem that arises in designing crypto systems using Abelian varieties of dimension larger than $1$. We discuss what is behind this phenomenon. In particular, we discuss the question of given an absolutely simple abelian variety over a number field, whether it has simple specializations at a set of places of positive Dirichlet density? A conjectural answer to this question was given by Murty and Patankar, and we explain some recent progress towards proving the conjecture. Our result ([14], Theorem 1.1) is based on the classification of pairs $(G,V)$ consisting of a semi-simple algebraic group $G$ over a non-archimedean local field and an absolutely irreducible representation $V$ of $G$ such that $G$ admits a maximal torus acting irreducibly on $V$.

We describe a strategy to attack infinitely many Fermat-type equations of signature $$(r,r,p)$$ , where $$r \ge 7$$ is a fixed prime and $$p$$ is a prime allowed to vary. Indeed, to a solution $$(a,b,c)$$ of $$x^r + y^r =Cz^p$$ we will attach several Frey curves $$E=E_{(a,b)}$$ defined over totally real subfields of $$\mathbb {Q}(\zeta _r)$$ . We prove modularity of all the Frey curves and the exsitence of a constant constant $$M_r$$ , depending only on $$r$$ , such that for all $$p>M_r$$ the representations $$\bar{\rho }_{E,p}$$ are absolutely irreducible. Along the way, we also prove modularity of certain elliptic curves that are semistable at all $$v \mid 3$$ . Finally, we illustrate our methods by proving arithmetic statements about equations of signature $$(7,7,p)$$ . Among which we emphasize that, using a multi-Frey technique, we show there is some constant $$M$$ such that if $$p > M$$ then the equation $$x^7 + y^7 = 3z^p$$ has no non-trivial primitive solutions.

Let \(\bar \rho\) be a two-dimensional Fp-valued representation of the absolute Galois group of the rationals. Suppose \(\bar \rho\) is odd, absolutely irreducible and ordinary at p. Then we show that \(\bar \rho\) arises from the irreducible component of a Hida family (of necessarily greater level than that of \(\bar \rho\)) whose map to weight space, including conjugate forms, has degree at least 4.

This paper contains three new results. {\bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of \'etale fundamental groups introduced in [7,10] and periodic Higgs-de Rham flows introduced in [19]. 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. {\bf 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. {\bf 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 $\mathbb{P}^1$ with logarithmic structure on marked points $D:=\{x_1,\,...,x_n\}$ for $n\geq 4$ and construct infinitely many geometrically absolutely irreducible $\mathrm{PGL_2}(\mathbb Z_p^{\mathrm{ur}})$-crystalline representations of $\pi_1^\text{et}(\mathbb{P}^1_{{\mathbb{Q}}_p^\text{ur}}\setminus D)$. We find an explicit formula of the self-map for the case $\{0,\,1,\,\infty,\,\lambda\}$ and 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 $\mathcal{C}_\lambda$ defined by $ y^2=x(x-1)(x-\lambda)$ with the order coprime to $p$., Comment: 84 pages

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 show that if $\{\rho_{\ell}\}$ is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation $\overline{\rho}_{\ell}$ is absolutely irreducible for $\ell$ in a density 1 set of primes. The key technical result is the following theorem: the image of $\rho_{\ell}$ is an open subgroup of a hyperspecial maximal compact subgroup of its Zariski closure with bounded index (as $\ell$ 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., Comment: 11 pages

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.

A well known result of B. Mazur gives a lower bound for the Krull dimension of the universal deformation ring associated to an absolutely irreducible residual representation in terms of the group cohomology of the adjoint representation. The question about equality - at least in the Galois case - also goes back to B. Mazur. In the general case the question about equality is the subject of Gouv\^{e}a's "Dimension conjecture". In this note we provide a counterexample to this conjecture. More precisely, we construct an absolutely irreducible residual representation with smooth universal deformation ring of strict greater Krull dimension as expected., Comment: small corrections; final version

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.

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

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.

The topic of this thesis is the p-adic Langlands correspondence, imagined by Breuil and established by Colmez for GL_2(Q_p). Let L be a finite extension of Q_p and let V be an irreducible, two-dimensional L-representation of the absolute Galois group of Q_p. Using Fontaine's theory of (phi,Gamma)-modules, Colmez associates to V a GL_2(Q_p)-Banach space representation Pi(V), which is unitary, admissible and topologically irreducible. We give a new proof, much easier, of a theorem of Colmez, which describes the locally analytic vectors Pi(V)^an of Pi(V) in terms of the overconvergent (phi,Gamma)-module attached to V. The main result of this thesis is a simple description of the infinitesimal action of GL_2(Q_p) on Pi(V)^an. In particular, we show that Pi(V)^an has an infinitesimal character, which can be computed in terms of the Hodge-Tate weights of V, answering therefore a question of Harris. We show that there is no p-adic analogue of a classical theorem of Saito and Tunnell, answering another question of Harris. We extend results of Colmez concerning the Kirillov model of the U-finite vectors of Pi(V) (U is the upper unipotent of GL_2(Q_p)). Combining this study with the description of the infinitesimal action, we obtain a simple proof of one of the main results of Colmez, characterizing the representations V such that Pi(V) has nonzero locally algebraic vectors. This result is the first step in making the connection with the classical Langlands correspondence, and it is also a key ingredient in Emerton's proof of the Fontaine-Mazur conjecture in dimension two. We extend our methods to prove the analogous result for infinitesimal deformations of V. This answers a question of Paskunas and has applications to the Breuil-Mézard conjecture. We apply differential methods to study the Jacquet module of Pi(V)^an, proving for instance that it is nonzero if and only if V is trianguline and giving a new and direct proof of conjectures of Berger, Breuil and Emerton. Finally, in joint work with Benjamin Schraen we prove Schur's lemma for topologically irreducible Banach and locally analytic representations of p-adic Lie groups. This basic result was previously known only for commutative Lie groups and for GL_2(Q_p)., Cette thèse s'inscrit dans le cadre de la correspondance de Langlands locale $p$-adique, imaginée par Breuil et établie par Colmez pour GL_2(Q_p). Soit L une extension finie de Q_p et soit V une L-représentation irréductible du groupe de Galois absolu de Q_p, de dimension 2. En utilisant la théorie des (phi,Gamma)-modules de Fontaine, Colmez associe à V une GL_2(Q_p)-représentation de Banach Pi(V), unitaire, admissible, topologiquement irréductible. On donne une nouvelle preuve, nettement plus simple, d'un théorème de Colmez, qui permet de décrire les vecteurs localement analytiques Pi(V)^an de Pi(V) en fonction du (phi,\Gamma)-module surconvergent attaché à V. Le résultat principal de cette thèse est une description simple de l'action infinitésimale de GL_2(Q_p) sur Pi(V)^an. En particulier, on montre que Pi(V)^an admet un caractère infinitésimal, que l'on peut calculer en fonction des poids de Hodge-Tate de V, ce qui répond à une question de Harris. En utilisant ces résultats, on montre aussi l'absence d'un analogue p-adique d'un théorème classique de Tunnell et Saito, répondant à une autre question de Harris. Nous étendons et précisons certains résultats de Colmez concernant le modèle de Kirillov des vecteurs U-finis de Pi(V) (U est l'unipotent supérieur de GL_2(Q_p)). En combinant cette étude avec la description de l'action infinitésimale, on obtient une démonstration simple d'un des résultats principaux de Colmez, caractérisant les représentations V telles que Pi(V) possède des vecteurs localement algébriques non nuls. Ce résultat permet de faire le pont avec la correspondance classique et est un des ingrédients clés de la preuve d'Emerton de la conjecture de Fontaine-Mazur en dimension 2. On étend nos méthodes pour démontrer l'analogue de ce résultat pour les déformations infinitésimales de V. Cela répond à une question de Paskunas et a des applications à la conjecture de Breuil-Mézard. Une autre application est l'étude du module de Jacquet de Pi(V)^an. On montre qu'il est non nul si et seulement si V est trianguline, ce qui permet de donner une preuve simple des conjectures de Berger, Breuil et Emerton. Enfin, dans un travail en collaboration avec Benjamin Schraen, nous démontrons le lemme de Schur pour les représentations de Banach et localement analytiques topologiquement irréductibles d'un groupe de Lie p-adique. Ce résultat basique n'était connu que pour des groupes de Lie commutatifs et pour GL_2(Q_p).