Search

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.

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

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

A theorem of N. Katz \cite{Ka} p.45, states that an irreducible differential operator $L$ over a suitable differential field $k$, which has an isotypical decomposition over the algebraic closure of $k$, is a tensor product $L=M\otimes_k N$ of an absolutely irreducible operator $M$ over $k$ and an irreducible operator $N$ over $k$ having a finite differential Galois group. Using the existence of the tensor decomposition $L=M\otimes N$, an algorithm is given in \cite{C-W}, which computes an absolutely irreducible factor $F$ of $L$ over a finite extension of $k$. Here, an algorithmic approach to finding $M$ and $N$ is given, based on the knowledge of $F$. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields $k$ which are $C_1$-fields., 21 pages

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 $t$ 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.

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

Let $q$ be a prime power and let $G$ be an absolutely irreducible subgroup of $GL_d(F)$, where $F$ is a finite field of the same characteristic as $\F_q$, the field of $q$ elements. Assume that $G \cong G(q)$, a quasisimple group of exceptional Lie type over $\F_q$ which is neither a Suzuki nor a Ree group. We present a Las Vegas algorithm that constructs an isomorphism from $G$ to the standard copy of $G(q)$. If $G \not\cong {}^3 D_4(q)$ with $q$ even, then the algorithm runs in polynomial time, subject to the existence of a discrete log oracle.

We study intertwining relations for matrix one-dimensional, in general, non-Hermitian Hamiltonians by matrix differential operators of arbitrary order. It is established that for any matrix intertwining operator Q_N^- of minimal order N there is a matrix operator Q_{N'}^+ of different, in general, order N' that intertwines the same Hamiltonians as Q_N^- in the opposite direction and such that the products Q_{N'}^+Q_N^- and Q_N^-Q_{N'}^+ are identical polynomials of the corresponding Hamiltonians. The related polynomial algebra of supersymmetry is constructed. The problems of minimization and of reducibility of a matrix intertwining operator are considered and the criteria of minimizability and of reducibility are presented. It is shown that there are absolutely irreducible matrix intertwining operators, in contrast to the scalar case., 9 pages

In this paper we present various algorithms for multiplying multivariate polynomials and series. All algorithms have been implemented in the C++ libraries of the Mathemagix system. We describe naive and softly optimal variants for various types of coefficients and supports and compare their relative performances. For the first time, under the assumption that a tight superset of the support of the product is known, we are able to observe the benefit of asymptotically fast arithmetic for sparse multivariate polynomials and power series, which might lead to speed-ups in several areas of symbolic and numeric computation. For the sparse representation, we present new softly linear algorithms for the product whenever the destination support is known, together with a detailed bit-complexity analysis for the usual coefficient types. As an application, we are able to count the number of the absolutely irreducible factors of a multivariate polynomial with a cost that is essentially quadratic in the number of the integral points in the convex hull of the support of the given polynomial. We report on examples that were previously out of reach.

Building on recent work of Ardakov and Wadsley, we prove Schur's lemma for absolutely irreducible admissible p-adic Banach space (respectively locally analytic) representations of p-adic Lie groups. We also prove finiteness results for the endomorphism algebra of an irreducible admissible representation., 11 pages

It has been an open question whether any bifurcation problem with absolutely irreducible group action would lead to bifurcation of steady states. A positive proposal is also known as the “Ize-conjecture”. Algebraically speaking, this is to ask whether every absolutely irreducible real representation has an odd dimensional fixed point subspace corresponding to some subgroups. Recently, Reiner Lauterbach and Paul Matthews have found counter examples to this conjecture and interestingly, all of the representations are of dimension $4k$, for $k\in\mathbb{N}$. A natural question arises: what about the case $4k+2$? ¶ In this paper, we give a partial answer to this question and prove that in any $6$-dimensional absolutely irreducible real representation of a finite solvable group, there exists an odd dimensional fixed point subspace with respect to subgroups.

Let $\cC$ be a smooth absolutely irreducible curve of genus $g \ge 1$ defined over $\F_q$, the finite field of $q$ elements. Let $# \cC(\F_{q^n})$ be the number of $\F_{q^n}$-rational points on $\cC$. Under a certain multiplicative independence condition on the roots of the zeta-function of $\cC$, we derive an asymptotic formula for the number of $n =1, ..., N$ such that $(# \cC(\F_{q^n}) - q^n -1)/2gq^{n/2}$ belongs to a given interval $\cI \subseteq [-1,1]$. This can be considered as an analogue of the Sato-Tate distribution which covers the case when the curve $\E$ is defined over $\Q$ and considered modulo consecutive primes $p$, although in our scenario the distribution function is different. The above multiplicative independence condition has, recently, been considered by E. Kowalski in statistical settings. It is trivially satisfied for ordinary elliptic curves and we also establish it for a natural family of curves of genus $g=2$., 14 pages

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., Dedicated to Prof. W. Narkiewicz on the occasion of his 70th birthday. To appear in Functiones et Approximatio. 4 pages

Let K be a global field, V an infinite proper subset of the set of all primes of K, and S a finite subset of V. Denote the maximal Galois extension of K in which each p ∈ S totally splits by Ktot,S . Let M be an algebraic extension of K. Let VM (resp. SM ) be the set of primes of M which lie over primes in V (resp. S). For each q ∈ VM let OM,q = {x ∈ Mq | |x|q ≤ 1}, where Mq is a completion of M at q, and let OM,V = {x ∈M | |x|q ≤ 1 for each q ∈ VM}. For σ = (σ1, . . . , σe) ∈ Gal(K), let Ks(σ) = {x ∈ Ks | σi(x) = x, i = 1, . . . , e}. Then, for almost all σ ∈ Gal(K) (with respect to the Haar measure), the field M = Ks(σ) ∩ Ktot,S satisfies the following local global principle: Let V ⊆ A be an affine absolutely irreducible variety defined over M . Suppose that there exist xq ∈ V (OM,q) for each q ∈ VM rSM and xq ∈ Vsimp(OM,q) for each q ∈ SM such that |xi,q|q < 1, i = 1, . . . , n, for each archimedean prime q ∈ VM . Then V (OM,V) 6= ∅. MR Classification: 12E30 6 September, 2008 * Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. Introduction Hilbert’s tenth problem asked for the existence of an algorithm to solve diophantine equations, that is equations with coefficients in Z whose solutions are sought in Z. The developement of recursion theory since 1930 and works of Martin Davis, Hilary Putnam, and Julia Robinson finally led Juri Matijasevich in 1972 to a negative answer to that problem. This invoked Julia Robinson to ask whether Hilbert’s tenth problem has a positive solution over the ring Z of all algebraic integers. Indeed, on page 367 of her joint paper [DMR84] with Davis and Matijasevich she guessed that there should be one. Using capacity theory, Rumely [Rum86, Rum89] proved in 1987 a local global principle for Z: If an absolutely irreducible affine variety V over Q has an integral point over every completion of Q, then V has a point with coordinates in Z. This led Rumely to an algorithm for solving diophantine problems over Z. In 1988-89 Moret-Bailly [MoB88, MoB89a, MoB89b] reproved Rumely’s theorem with rationality conditions using methods of algebraic geometry. Green, Pop, and Roquette [GPR95] consider a global field K, a set V of primes of K which does not include all of the primes of K, and a finite subset S of V. Each p ∈ V is an equivalence class of absolute values of K. Let | |p be an absolute value representing p. If p is archimedean and complex, let Kp be the algebraic closure K of K. If p is archimedean and real, let Kp be a real closure of K at p. If p is nonarchimedean, let Kp be a Henselian closure of K at p. Now let N = Ktot,S = ⋂ p∈S ⋂ τ∈Gal(K)K τ p be the field of totally S-adic numbers. It is the maximal Galois extension of K in which each p ∈ S totally decomposes. Here Gal(K) = Gal(Ks/K) is the absolute Galois group of K. Consider the subset ON,V of N consisting of all x ∈ N such that |x|q ≤ 1 for each prime q of N whose restriction to K lies in V. The main result of [GPR95] is a local-global principle for ON,V : If an affine absolutely irreducible variety V defined over K has a Kp-rational point xp with |xp|p ≤ 1 for each p ∈ V such that xp is simple if p ∈ S and |xp|p < 1 if p is archimedean, then V has a simple ON,V -rational point. The language of proof of [GPR95] is that of the theory of algebraic function fields of one variable.

In this paper we generalize a result of Urban on the structure of residually reducible representations on local Artinian rings from the case that the semi-simplification of the residual representation splits into 2 absolutely irreducible representations to the case where it splits into m ≥ 2 absolutely irreducible representations.

We determine the sign of the polarization of any polarized irreducible factor of a Galois representation attached to a polarized cohomological cuspidal automorphic form of Gl_n of a CM field: it is always +1, as was conjectured by Gross., 16 pages

Let K be a field of characteristic p>0, and let q be a power of p. We determine all polynomials f in K[t]\K[t^p] of degree q(q-1)/2 such that the Galois group of f(t)-u over K(u) has a transitive normal subgroup isomorphic to PSL_2(q), subject to a certain ramification hypothesis. As a consequence, we describe all polynomials f in K[t], of degree not a power of p, such that f is functionally indecomposable over K but f decomposes over an extension of K. Moreover, except for one ramification setup (which is treated in the companion paper arxiv:0707.1837), we describe all indecomposable polynomials f in K[t] of non-p-power degree which are exceptional, in the sense that x-y is the only absolutely irreducible factor of f(x)-f(y) which lies in K[x,y]. It is known that, when K is finite, a polynomial f is exceptional if and only if it induces a bijection on infinitely many finite extensions of K., 44 pages; changed notation throughout and made various minor changes

For a linear group $G$ acting on an absolutely irreducible variety $X$ over the rationals $\QQ$, we describe the orbits of $X(\QQ_p)$ under $G(\QQ_p)$ and of $X(\FF_p((t)))$ under $G(\FF_p((t)))$ for $p$ big enough. This allows us to show that the degree of a wide class of orbital integrals over $\QQ_p$ or $\FF_p((t))$ is $\leq 0$ for $p$ big enough, and similarly for all finite field extensions of $\QQ_p$ and $\FF_p((t))$.

For generalized Reed-Solomon codes, it has been proved \cite{GuruswamiVa05} that the problem of determining if a received word is a deep hole is co-NP-complete. The reduction relies on the fact that the evaluation set of the code can be exponential in the length of the code -- a property that practical codes do not usually possess. In this paper, we first presented a much simpler proof of the same result. We then consider the problem for standard Reed-Solomon codes, i.e. the evaluation set consists of all the nonzero elements in the field. We reduce the problem of identifying deep holes to deciding whether an absolutely irreducible hypersurface over a finite field contains a rational point whose coordinates are pairwise distinct and nonzero. By applying Schmidt and Cafure-Matera estimation of rational points on algebraic varieties, we prove that the received vector $(f(\alpha))_{\alpha \in \F_q}$ for Reed-Solomon $[q,k]_q$, $k < q^{1/7 - \epsilon}$, cannot be a deep hole, whenever $f(x)$ is a polynomial of degree $k+d$ for $1\leq d < q^{3/13 -\epsilon}$.

In [P] R. Pellikaan introduced a two variable zeta-function for a curve over a finite field and proved that it is a rational function. Here we show that its denominator is absolutely irreducible. This is motivated by work of J. Lagarias and E. Rains on an analogous two variable zeta-funtion for number fields., 7 pages

Let F be a field and, for i = 1,2, let G i be a group and V i an irreducible, primitive, finite dimensional FG i -module. Set G = G 1 \times G 2 and $V=V_1\otimes _F V_2$ . The main aim of this paper is to determine sufficient conditions for V to be primitive as a G-module. In particular this turns out to be the case if V 1 and V 2 are absolutely irreducible and V 1 is absolutely quasi-primitive. Thus we extend a result of N.S. Heckster, who has shown that V is primitive whenever G is finite and F is the complex field. We also give a characterization of absolutely quasi-primitive modules. Ultimately, our results rely on the classification of finite simple groups.

We consider systems of partial differential equations equivariant under the Euclidean group E ( n ) \mathbf {E}(n) and undergoing steady-state bifurcation (with nonzero critical wavenumber) from a fully symmetric equilibrium. A rigorous reduction procedure is presented that leads locally to an optimally small system of equations. In particular, when n = 1 n=1 and n = 2 n=2 and for reaction-diffusion equations with general n n , reduction leads to a single equation. (Our results are valid generically, with perturbations consisting of relatively bounded partial differential operators.) In analogy with equivariant bifurcation theory for compact groups, we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of E ( n ) \mathbf {E}(n) . The representation theory of E ( n ) \mathbf {E}(n) is driven by the irreducible representations of O ( n − 1 ) \mathbf {O}(n-1) . For n = 1 n=1 , this constitutes a mathematical statement of the ‘universality’ of the Ginzburg-Landau equation on the line. (In recent work, we addressed the validity of this equation using related techniques.) When n = 2 n=2 , there are precisely two significantly different types of reduced equation: scalar and pseudoscalar, corresponding to the trivial and nontrivial one-dimensional representations of O ( 1 ) \mathbf {O}(1) . There are infinitely many possibilities for each n ≥ 3 n\ge 3 .

International audience; We prove that if $f : Y\longrightarrow X$ is a finite fiat morphism between two reduced absolutely irreducible algebraic projective curves defined over the finite field ${\sb F}_q$, then $$\mid \sharp Y({\sb F}_q) - \sharp X({\sb F}_q)\mid \leq 2({\pi}_Y - {\pi}_X)\sqrt q,$$ where $\pi_C$ is the arithmetic genus of a curve $C$. As application, we give some character sum estimation on singular curves.

We consider symmetry-breaking bifurcations at non-positive, radially symmetric solutions of semilinear elliptic equations on a ball with Dirichlet boundary conditions. For nonlinearities which are asymptotically affine linear, we find solutions at which the symmetry breaks. The kernel of the linearized equation at these solutions is an absolutely irreducible representation of the group O(n). For this kind of equation a transversality condition is satisfied if the perturbation of the affine linear problem is small enough. Thus we obtain, by the equivariant branching lemma, a large variety of isotropy subgroups of O(n) which occur as symmetries of the bifurcating solution branches.