Almost Perfect Nonlinear (APN) functions are very useful in cryptography, when they are used as S-Boxes, because of their good resistance to differential cryptanalysis. An APN function $f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$ is called exceptional APN if it is APN on infinitely many extensions of $\mathbb{F}_{2^n}$. 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 $(2^k+1)$ or a Kasami-Welch number $(2^{2k}-2^k+1)$. When the degree of the polynomial function is a Gold number, several partial results have been obtained [1, 7, 8, 10, 17]. One of the results in this article is a proof of the relatively primeness of the multivariate APN polynomial conjecture, in the Gold degree case. This helps us extend substantially previous results. We prove that Gold degree polynomials of the form $x^{2^k+1}+h(x)$, where $deg(h)$ is any odd integer (with the natural exceptions), can not be exceptional APN.
We also show absolute irreducibility of several classes of multivariate polynomials over finite fields and discuss their applications.
We show that many quadratic binomial functions on a finite field of characteristic 2 are not APN infinitely often. This is of interest in the light of recent discoveries of new families of quadratic binomial APN functions. The proof uses the Weil bound from algebraic geometry.
Let $f$ be a function on a finite field $F$. The decomposition of the generalized Fermat variety $X$ defined by the multivariate polynomial of degree $n$, $ϕ(x,y,z)=f(x)+f(y)+f(z)$ in $\Bbb{P}^3(\overline{\mathbb{F}}_2)$, plays a crucial role in the study of almost perfect non-linear (APN) functions and exceptional APN functions. Their structure depends fundamentally on the Fermat varieties corresponding to the monomial functions of exceptional degrees $n=2^k+1$ and $n=2^{2k}-2^k+1$ (Gold and Kasami-Welch numbers, respectively). Very important results for these have been obtained by Janwa, McG…
We consider the Kisin variety associated to a $n$-dimensional absolutely irreducible mod $p$ Galois representation $\barρ$ of a $p$-adic field $K$ and 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 ramfied with $n=3$ or $μ$ is of a very particular form. As an application, we also get a connectedness result for the deformation ring associated to $\barρ$ of given Hodge-Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general.
We consider absolutely irreducible polynomials $f \in Z[x,y]$ with $°_x(f)=m$, $°_y(f)=n$ and height $H$. We show that for any prime $p$ with $p>c_{mn} H^{2mn+n-1}$ the reduction $f \bmod p$ is also absolutely irreducible. Furthermore if the Bouniakowsky conjecture is true we show that there are infinitely many absolutely irreducible polynomials $f \in Z[x,y]$ which are reducible mod $p$ where $p$ is a prime with $p>H^{2m}$.
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…
Let $p>2$ be a prime. We give examples of smooth absolutely irreducible representations of $\mathrm{GL}_2(\mathbb{Q}_{p^3})$ over $\mathbb{F}_{p^3}$ which are not admissible.
Let $F$ be a finite extension of $\mathbb{Q}_p$. We determine the Lubin-Tate $(φ,Γ)$-modules associated to the absolutely irreducible mod $p$ representations of the absolute Galois group ${\rm Gal}(\bar{F}/F)$.
Let $k$ be a totally real field, and let $A/k$ be an absolutely irreducible, polarized Abelian variety of odd, prime dimension whose endomorphisms are all defined over $k$. Then the only strictly compatible families of abstract, absolutely irreducible representations of $\gal(\ov{k}/k)$ coming from $A$ are tensor products of Tate twists of symmetric powers of two-dimensional $λ$-adic representations plus field automorphisms. The main ingredients of the proofs are the work of Borel and Tits on the `abstract' homomorphisms of almost simple algebraic groups, plus the work of Shimura on the fields…
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.
The p-adic local Langlands correspondence for GL_2(Q_p) is given by an exact functor from unitary Banach representations of GL_2(Q_p) to representations of the absolute Galois group G_{Q_p} of Q_p. We prove, using characteristic 0 methods, that this correspondence induces a bijection between absolutely irreducible non-ordinary representations of GL_2(Q_p) and absolutely irreducible 2-dimensional representations of G_{Q_p}. This had already been proved, by characteristic p methods, but only for p\geq 5.
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+-sum…
We show that if $\{ρ_{\ell}\}$ is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation $\overlineρ_{\ell}$ is absolutely irreducible for $\ell$ in a density 1 set of primes. The key technical result is the following theorem: the image of $ρ_{\ell}$ 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…
We prove that if $J$ is the limit set of an irreducible conformal iterated function system (with either finite or countably infinite alphabet), then the badly approximable vectors form a set of full Hausdorff dimension in $J$. The same is true if $J$ is the radial Julia set of an irreducible meromorphic function (either rational or transcendental). The method of proof is to find subsets of $J$ that support absolutely friendly and Ahlfors regular measures of large dimension. In the appendix to this paper, we answer a question of Broderick, Kleinbock, Reich, Weiss, and the second-named author ('…
Let $p>3$ be a prime number and let $G_{\mathbb{Q}_p}$ be the absolute Galois group of $\mathbb{Q}_p$. In this paper, we find Galois stable lattices in the irreducible $3$-dimensional semi-stable and non-crystalline representations of $G_{\mathbb{Q}_p}$ with Hodge--Tate weights $(0,1,2)$ by constructing their strongly divisible modules. We also compute the Breuil modules corresponding to the mod $p$ reductions of the strongly divisible modules, and determine which of the semi-stable representations has an absolutely irreducible mod $p$ reduction.
Let $p$ be an odd prime number. We prove that for $m\equiv1\mod p$, $x^m$ is perfectly nonlinear over $\mathbb{F}_{p^n}$ for infinitely many $n$ if and only if $m$ is of the form $p^l+1$, $l\in\mathbb{N}$. First, we study singularities of $f(x,y)=\frac{(x+1)^m-x^m-(y+1)^m+y^m}{x-y}$ and we use Bezout theorem to show that for $m\neq 1+p^l$, $f(x,y)$ has an absolutely irreducible factor. Then by Weil theorem, f(x,y) has rationnal points such that $x\neq y$ which means that $x^m$ is not PN.
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.
For a prime $p$ and an absolutely irreducible modulo $p$ polynomial $f(U,V) \in \Z[U,V]$ we obtain an asymptotic formulas for the number of solutions to the congruence $f(x,y) \equiv a \pmod p$ in positive integers $x \le X$, $y \le Y$, 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 $a$.
Let $p$ be a prime number, $X$ be an absolutely irreducible affine plane curve over $\mathbb{F}_p$, and $g,f\in\mathbb{F}_p(x,y)$. We study the distribution of the values of the hybrid exponential sums S_n on $n\in\mathcal{I}$ for some short interval $\mathcal{I}$. We show that under some natural conditions the limiting distribution of the projections of the sum $S_n$, $n\in\mathcal{I}$ on any straight line through the origin is Gaussian as $p$ tends to infinity.
A well-known result of von zur Gathen asserts that a non-exceptional permutation polynomial of degree $n$ over $\mathbb{F}_{q}$ exists only if $q<n^{4}$. With the help of the Weil bound for the number of $\mathbb{F}_{q}$-points on an absolutely irreducible (possibly singular) affine plane curve, Chahal and Ghorpade improved von zur Gathen's proof to replace $n^{4}$ by a bound less than $n^{2}(n-2)^{2}$. Also based on the Weil bound, we further refine the upper bound for $q$ with respect to $n$, by a more concise and direct proof following Wan's arguments.
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 …
Fix a prime $p > 2$. Let $ρ: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{I})$ be the Galois representation coming from a non-CM irreducible component $\mathbb{I}$ of Hida's $p$-ordinary Hecke algebra. Assume the residual representation $\barρ$ is absolutely irreducible. Under a minor technical condition we identify a subring $\mathbb{I}_0$ of $\mathbb{I}$ containing $\mathbb{Z}_p[[T]]$ such that the image of $ρ$ is large with respect to $\mathbb{I}_0$. That is, $\text{Im} ρ$ contains $\text{ker}(\text{SL}_2(\mathbb{I}_0) \to \text{SL}_2(\mathbb{I}_0/\mathfrak{a}))$ for…
Let f in Z[X,Y,Z] be a non-constant, absolutely irreducible, homogeneous polynomial with integer coefficients, such that the projective curve given by f=0 has a function field isomorphic to the rational function field Q(t). We show that all integral solutions of the Diophantine equation f=0 (up to those corresponding to some singular points) can be parametrized by a single triple of integer-valued polynomials. In general, it is not possible to parametrize this set of solutions by a single triple of polynomials with integer coefficients.
We study irreducible ${\rm SL}_2$-representations of twist knots. We first determine all non-acyclic ${\rm SL}_2(\mathbb{C})$-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 the $L$-functions of universal deformations, that is, the orders of the associated knot modules. 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 no…
Let $F$ be a finite unramified extension of $\mathbb Q\_p$ and $\barρ$ be an absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We conjecture that the deformation space parametrizing the potentially Barsotti--Tate liftings of $\barρ$ having type $t$ depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into $(\mathbb P^1)^f$ and its shape stratification. We give evidences towards this conjecture by proving that the Kisin variety determines the cardinality of the …
