Search

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

The Suzuki simple group Sz(8) has an automorphism group 3. Using the electronic Atlas [22], the group Sz(8) : 3 has an absolutely irreducible module of dimension 12 over F 2. Therefore a split extension group of the form 2 12 : (S z (8) : 3) : = G ¯ exists. In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford-Fischer Theory. We determined the inertia factor groups of G ¯ by analysing the maximal subgroups of Sz(8) : 3 and maximal of the maximal subgroups of Sz(8) : 3 together with various other information. It turns out that the character table of G ¯ is a 43 × 43 complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 7. [ABSTRACT FROM AUTHOR]

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

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

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

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

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

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

In this article, we prove that for a convergent sequence of residually absolutely irreducible representations of the absolute Galois group of a number field F with coefficients in a domain, which admits a finite monomorphism from a power series ring over a p -adic integer ring, the set of places of F where some of the representations ramifies has density zero. Using this, we extend a result of Das–Rajan to such convergent sequences. We also establish a strong multiplicity one theorem for big Galois representations. [ABSTRACT FROM AUTHOR]

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

Planar functions are mappings from a finite field F q to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even. We classify polynomials of degree at most q 1 / 4 that induce planar functions on F q , by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over F q that induce planar functions on infinitely many extensions of F q. The proof strategy is to study the number of F q -rational points of an algebraic curve attached to a putative planar function. Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions. [ABSTRACT FROM AUTHOR]

This paper describes the chief factors of the commutator subgroups of finite nonsolvable groups G with disconnected Gruenberg-Kegel graph having exactly 5 vertices in the case where G/F(G) is an almost simple n-primary group for n ≤ 4. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

An ordinary character χ of a finite group is called orthogonally stable, if all non-degenerate invariant quadratic forms on any module affording the character χ have the same discriminant. This is the orthogonal discriminant, disc(χ), of χ, a square class of the character field. Based on experimental evidence we conjecture that the orthogonal discriminant is always an odd square class in the sense of Definition 1.4. This note proves this conjecture for finite solvable groups. For p-groups there is an explicit formula for disc(χ) that reads disc(χ) = (−p) χ(1)/2 if p ≡ 3 (mod 4) and disc(χ) = (−1)χ(1)/2 for p = 2. For rational characters χ and p ≡ 1 (mod 4) the discriminant is disc(χ) = p χ(1)/(p−1) . [ABSTRACT FROM AUTHOR]

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

We are given a finite group H, an automorphism τ of H of order r, a Galois extension L/K of fields of characteristic zero with cyclic Galois group ⟨ σ ⟩ of order r, and an absolutely irreducible representation ρ : H → GL (n , L) such that the action of τ on the character of ρ is the same as the action of σ . Then the following are equivalent. ∙ ρ is equivalent to a representation ρ ′ : H → GL (n , L) such that the action of σ on the entries of the matrices corresponds to the action of τ on H, and ∙ the induced representation ind H , H ⋊ ⟨ τ ⟩ (ρ) has Schur index one; that is, it is similar to a representation over K. As examples, we discuss a three dimensional irreducible representation of A 5 over Q [ 5 ] and a four dimensional irreducible representation of the double cover of A 7 over Q [ - 7 ] . [ABSTRACT FROM AUTHOR]

Let p ≥ 5 be a prime, and p a prime of Q ¯ above p. Let g 1 and g 2 be p -ordinary, p -distinguished and p-stabilized cuspidal newforms of nebentype characters ϵ 1 , ϵ 2 , respectively, and even weight k ≥ 2 , whose associated newforms have level prime to p. Assume that the residual representations at p associated to g 1 and g 2 are absolutely irreducible and isomorphic. Then, the imprimitive p-adic L-functions associated with the symmetric square representations are shown to exhibit a congruence modulo p . Furthermore, the analytic and algebraic Iwasawa invariants associated to these representations of the g i are shown to be related. Along the way, we give a complete proof of the integrality of the p -adic L-function, normalized with Hida's canonical period. This fills a gap in the literature, since, despite the result being widely accepted, no complete proof seems to ever have been written down. On the algebraic side, we establish the corresponding congruence for Greenberg's Selmer groups and verify that the Iwasawa main conjectures for the twisted symmetric square representations for g 1 and g 2 are compatible with the congruences. [ABSTRACT FROM AUTHOR]

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

We give a description of finite 4-primary groups with disconnected Gruenberg–Kegel graph containing a triangle. As a corollary, finite groups whose Gruenberg–Kegel graph coincides with the Gruenberg–Kegel graph of 3D4(2) are exemplified, which generalizes V. D. Mazurov' description of finite groups isospectral to the group 3D4(2). [ABSTRACT FROM AUTHOR]

A well known result of 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 Mazur. In the general case the question about equality is the subject of Gouvêa'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. [ABSTRACT FROM AUTHOR]

Let K be a complete, discretely valued field with finite residue field and GK its absolute Galois group. The subject of this note is the study of the set of positive integers d for which there exists an absolutely irreducible ℓ‐adic representation of GK of dimension d with rational traces on inertia. Our main result is that non‐Sophie Germain primes are not in this set when the residue characteristic of K is >3. The result stated in the title is a special case. [ABSTRACT FROM AUTHOR]

We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two Q ¯ ℓ -smooth sheaves on X - S , where X is a smooth projective absolutely irreducible curve of genus g over a finite field F q and S is a reduced divisor, with pre-specified tamely ramified ramification data at S. Properties of this cardinality are studied. The approach is based on using a relatively elementary explicit form of the trace formula for GL (2) , and introducing new types of almost pseudo-coefficients of principal series and discrete series representations. [ABSTRACT FROM AUTHOR]

We prove that the sequence 1!, 2!, 3!, . . . produces at least (√ 2+o(1))√p distinct residues modulo prime p. Moreover, the factorials within an interval I⊆ {0, 1, . . ., p - 1} of length N > p7/8+ε produce at least (1 + o(1)) √p distinct residues modulo p. As a corollary, we prove that every non-zero residue class can be expressed as a product of seven factorials n1! ... n7! modulo p, where ni = O(p6/7+ε) for all i = 1, . . ., 7, which provides a polynomial improvement upon the preceding results. [ABSTRACT FROM AUTHOR]

For a given genus $g \geq 1$ , we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$. As a consequence of Katz–Sarnak theory, we first get for any given $g>0$ , any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q. [ABSTRACT FROM AUTHOR]

In Section 8, we saw the role of absolute irreducibility in determining the number of zeroes of a bivariate polynomial over a finite field. Let us consider the following algorithmic question: given a bivariate polynomial f(x,y) how do we determine whether or not it is absolutely irreducible? In this section, we will see how partial derivatives help in converting this apparently highly nonlinear problem into a simple task in linear algebra. This then gives an efficient algorithm for determining absolute irreducibility. [ABSTRACT FROM AUTHOR]

Let n be a positive integer, q = 2 n , and let F q be the finite field with q elements. For each positive integer m , let D m ( X ) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m > 1 is a divisor of q + 1 . We study the existence of α ∈ F q ⁎ such that D m ( α ) = D m ( α − 1 ) = 0 . We also explore the connections of this question to an open question by Wiedemann and a game called “Button Madness”. [ABSTRACT FROM AUTHOR]

We study irreducible |$\mathop{\textrm{SL}}\nolimits _2$| -representations of twist knots. We first determine all non-acyclic |$\mathop{\textrm{SL}}\nolimits _2({\mathbb{C}})$| -representations, which turn out to lie on a line denoted as |$x=y$| in |${\mathbb{R}}^2$|⁠. Our main tools are character variety, Reidemeister torsion, and Chebyshev polynomials. We also verify a certain common tangent property, which yields a result on |$L$| -functions, that is, the orders of the knot modules associated to the universal deformations. Secondly, we prove that a representation is on the line |$x=y$| if and only if it factors through the |$-3$| -Dehn surgery, and is non-acyclic if and only if the image of a certain element is of order 3. Finally, we study absolutely irreducible non-acyclic representations |$\overline{\rho }$| over a finite field with characteristic |$p>2$|⁠ , to concretely determine all non-trivial |$L$| -functions |$L_{{\boldsymbol{\rho }}}$| of the universal deformations over complete discrete valuation rings. We show among other things that |$L_{{\boldsymbol{\rho }}}$|   |$\dot{=}$|   |$k_n(x)^2$| holds for a certain series |$k_n(x)$| of polynomials. [ABSTRACT FROM AUTHOR]