6,220 results on '"Absolutely irreducible"'
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102. Applications of the Hasse–Weil bound to permutation polynomials
- Author
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Xiang-dong Hou
- Subjects
Polynomial ,Algebra and Number Theory ,Absolutely irreducible ,Applied Mathematics ,010102 general mathematics ,General Engineering ,0102 computer and information sciences ,Function (mathematics) ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,Permutation ,Riemann hypothesis ,symbols.namesake ,Finite field ,010201 computation theory & mathematics ,symbols ,Irreducibility ,0101 mathematics ,Mathematics - Abstract
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.
- Published
- 2018
103. Irreducible integer-valued polynomials with prescribed minimal power that factors non-uniquely
- Author
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Nakato, Sarah and Rissner, Roswitha
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Mathematics - Commutative Algebra ,13A05, 11R09, 13B25, 13F20, 11C08 - Abstract
We study the question up to which power an irreducible integer-valued polynomial that is not absolutely irreducible can factor uniquely. For example, for integer-valued polynomials over principal ideal domains with square-free denominator, already the third power has to factor non-uniquely or the element is absolutely irreducible. Recently, it has been shown that for any $N\in\mathbb{N}$, there exists a discrete valuation domain $D$ and a polynomial $F\in\operatorname{Int}(D)$ such that the minimal $k$ for which $F^k$ factors non-uniquely is greater than $N$. In this paper, we show that, over principal ideal domains with infinitely many maximal ideals of finite index, the minimal power for which an irreducible but not absolutely irreducible element has to factor non-uniquely depends on the $p$-adic valuations of the denominator and cannot be bounded by a constant.
- Published
- 2024
104. Algebraic curves with automorphism groups of large prime order
- Author
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Nazar Arakelian and Pietro Speziali
- Subjects
Absolutely irreducible ,FUNÇÕES AUTOMORFAS ,General Mathematics ,010102 general mathematics ,Automorphism ,01 natural sciences ,Prime (order theory) ,Combinatorics ,Mathematics - Algebraic Geometry ,Genus (mathematics) ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Classification of finite simple groups ,Algebraic curve ,0101 mathematics ,Algebraically closed field ,Abelian group ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
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 $$q = g, g-1$$ .
- Published
- 2021
105. Sur la reéduction modulo p des polynoômes absolument irreéductibles
- Author
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Poulakis, Dimitrios
- Published
- 2000
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106. Locally Finite Normal Subgroups of Absolutely Irreducible Skew Linear Groups
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Wehrfritz, B. A. F., primary
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- 1985
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107. A General Theory of Algebraic Geometry Over Dedekind Domains, III: Absolutely Irreducible Models, Simple Spots
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- 1959
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108. An Equation for the Degrees of the Absolutely Irreducible Representations of a Group of Finite Order
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Zassenhaus, Hans, primary
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- 1950
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109. The Distribution of Absolutely Irreducible Polynomials in Several Indeterminates
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- 1972
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110. NORMAL SUBGROUPS OF ABSOLUTELY IRREDUCIBLE GROUPS.
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Shirvani and Wehrfritz, B. A. F.
- Published
- 1986
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111. Correction to: Non-absolutely irreducible elements in the ring of integer-valued polynomials.
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POLYNOMIAL rings ,IRREDUCIBLE polynomials - Published
- 2020
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112. Absolutely irreducible equations f(x1,...,xn)=0
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Wolfgang M. Schmidt
- Published
- 1976
113. Images de représentations galoisiennes associées à certaines formes modulaires de Siegel de genre 2
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Salim Tayou, École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL), Département de Mathématiques et Applications - ENS Paris (DMA), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)
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Algebra and Number Theory ,Mathematics - Number Theory ,Absolutely irreducible ,Mathematics::Number Theory ,11F46, 11F33, 11F80, 11E57 ,010102 general mathematics ,16. Peace & justice ,Galois module ,01 natural sciences ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics ,Siegel modular form - Abstract
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
- Published
- 2017
114. Adequate subgroups and indecomposable modules
- Author
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Pham Huu Tiep, Robert M. Guralnick, and Florian Herzig
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Classical group ,Pure mathematics ,Artin–Wedderburn theorem ,Absolutely irreducible ,General Mathematics ,Dimension (graph theory) ,Automorphic form ,Group Theory (math.GR) ,01 natural sciences ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics ,Mathematics - Number Theory ,Applied Mathematics ,010102 general mathematics ,16. Peace & justice ,Galois module ,Field of definition ,010307 mathematical physics ,20C20, 11F80 ,Indecomposable module ,Mathematics - Group Theory ,Mathematics - Representation Theory - Abstract
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
- Published
- 2017
115. Absolute irreducibility of the binomial polynomials
- Author
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Roswitha Rissner and Daniel Windisch
- Subjects
Ring (mathematics) ,Algebra and Number Theory ,Binomial (polynomial) ,Absolutely irreducible ,010102 general mathematics ,Prime number ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Combinatorics ,Number theory ,Factorization ,0103 physical sciences ,FOS: Mathematics ,Rank (graph theory) ,Irreducibility ,010307 mathematical physics ,0101 mathematics ,13F20, 13A05, 11C08, 11C20 ,Mathematics - Abstract
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
- Published
- 2020
- Full Text
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116. Regular orbits of quasisimple linear groups II
- Author
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Melissa Lee
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Pointwise ,Algebra and Number Theory ,Group (mathematics) ,Absolutely irreducible ,PSL ,Regular orbit ,Combinatorics ,Base (group theory) ,Mathematics::Group Theory ,Finite field ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics - Representation Theory ,Vector space ,Mathematics - Abstract
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
- Published
- 2020
- Full Text
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117. Flat manifolds with holonomy representation of quaternionic type
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Rafał Lutowski, Gerhard Hiss, and Andrzej Szczepański
- Subjects
Flat manifold ,Pure mathematics ,Algebra and Number Theory ,Absolutely irreducible ,Holonomy ,Representation (systemics) ,Group Theory (math.GR) ,Type (model theory) ,Frobenius–Schur indicator ,Quaternionic representation ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Mathematics::Differential Geometry ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Group Theory ,Hyperkähler manifold ,Mathematics - Representation Theory ,Mathematics ,Primary: 20H15, Secondary: 20C15, 53C26, 57N16 - Abstract
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.
- Published
- 2020
- Full Text
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118. A Projective Two-Weight Code Related to the Simple Group $$\mathrm{Co}_1$$ Co 1 of Conway
- Author
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Bernardo Gabriel Rodrigues
- Subjects
Absolutely irreducible ,010102 general mathematics ,Code word ,Minimum weight ,0102 computer and information sciences ,01 natural sciences ,Theoretical Computer Science ,Dual code ,Combinatorics ,Maximal subgroup ,010201 computation theory & mathematics ,Simple group ,Discrete Mathematics and Combinatorics ,Coset ,0101 mathematics ,Leech lattice ,Mathematics - Abstract
A binary $$[98280, 24, 47104]_2$$ projective two-weight code related to the sporadic simple group $$\mathrm{Co}_1$$ of Conway is constructed as a faithful and absolutely irreducible submodule of the permutation module induced by the primitive action of $$\mathrm{Co}_1$$ on the cosets of $$\mathrm{Co}_2$$ . The dual code of this code is a uniformly packed $$[98280, 98256,3]_2$$ code. The geometric significance of the codewords of the code can be traced to the vectors in the Leech lattice, thus revealing that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of $$\mathrm{Co}_1$$ . Similarly, the stabilizer of the codewords of minimum weight in the dual code is a maximal subgroup of $$\mathrm{Co}_1$$ . As by-product, a new strongly regular graph on 16777216 vertices and valency 98280 is constructed using the codewords of the code.
- Published
- 2018
119. An application of random plane slicing to counting Fq-points on hypersurfaces
- Author
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Kaloyan Slavov
- Subjects
Discrete mathematics ,Algebra and Number Theory ,Degree (graph theory) ,Plane (geometry) ,Absolutely irreducible ,Applied Mathematics ,010102 general mathematics ,General Engineering ,0102 computer and information sciences ,01 natural sciences ,Upper and lower bounds ,Slicing ,Theoretical Computer Science ,Combinatorics ,Hypersurface ,Finite field ,010201 computation theory & mathematics ,Interval (graph theory) ,0101 mathematics ,Mathematics - Abstract
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.
- Published
- 2017
120. The permutation module on flag varieties in cross characteristic
- Author
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Junbin Dong and Xiaoyu Chen
- Subjects
20G05 ,Mathematics::Commutative Algebra ,Absolutely irreducible ,General Mathematics ,Flag (linear algebra) ,010102 general mathematics ,Prime number ,Field (mathematics) ,Group algebra ,Reductive group ,01 natural sciences ,Combinatorics ,Borel subgroup ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Algebraically closed field ,Mathematics - Representation Theory ,Mathematics - Abstract
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
- Published
- 2018
121. Proof of a conjecture of Segre and Bartocci on monomial hyperovals in projective planes
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Hernando, Fernando and McGuire, Gary
- Published
- 2012
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122. On the Determination of Irreducible Modules by Restriction to a Subalgebra
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Lepowsky, J. and McCollum, G. W.
- Published
- 1973
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123. Abelian F.P.F. Operator Groups of Type $(p, p)$
- Author
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Richards, James W.
- Published
- 1971
- Full Text
- View/download PDF
124. Reduction modulo 𝑝 of certain semi-stable representations
- Author
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Chol Park
- Subjects
Reduction (recursion theory) ,Absolutely irreducible ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Modulo ,010102 general mathematics ,Prime number ,Absolute Galois group ,01 natural sciences ,Combinatorics ,Mod ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
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.
- Published
- 2017
125. Elliptic minuscule pairs and splitting abelian varieties
- Author
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Ying Zong and V. Kumar Murty
- Subjects
Abelian variety ,Pure mathematics ,Monodromy ,Absolutely irreducible ,Applied Mathematics ,General Mathematics ,Algebraic group ,Dirichlet density ,Maximal torus ,Abelian group ,Algebraic number field ,Mathematics - Abstract
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.
- Published
- 2017
126. Free subgroups in maximal subgroups ofGLn(D)
- Author
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R. Fallah-Moghaddam and M. Mahdavi-Hezavehi
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Normal subgroup ,Discrete mathematics ,Algebra and Number Theory ,Absolutely irreducible ,Mathematics::Number Theory ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Noncommutative geometry ,Combinatorics ,Mathematics::Group Theory ,Maximal subgroup ,Simple group ,Division algebra ,Division ring ,0101 mathematics ,Characteristic subgroup ,Mathematics - Abstract
Let D be a noncommutative finite dimensional F-central division algebra and M a noncommutative maximal subgroup of GLn(D). It is shown that either M contains a noncyclic free subgroup or M is absolutely irreducible and there exists a unique maximal subfield K of Mn(D) such that K*M, K∕F is Galois with Gal(K∕F)≅M∕K* and Gal(K∕F) is a finite simple group.
- Published
- 2016
127. On the structure of locally potentially equivalent Galois representations
- Author
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Vijay M. Patankar and C. S. Rajan
- Subjects
Pure mathematics ,Algebra and Number Theory ,11F80, 11G05, 11G15 ,Mathematics - Number Theory ,Absolutely irreducible ,Group (mathematics) ,010102 general mathematics ,Galois group ,010103 numerical & computational mathematics ,Absolute Galois group ,Algebraic number field ,Galois module ,01 natural sciences ,Monodromy ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Abelian group ,Mathematics - Abstract
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
- Published
- 2019
- Full Text
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128. On the Dickson–Guralnick–Zieve curve
- Author
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Massimo Giulietti, Marco Timpanella, and Gábor Korchmáros
- Subjects
Algebra and Number Theory ,Absolutely irreducible ,Plane curve ,Algebraic curves ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Finite field ,Finite fields ,Automorphism groups ,Fermat curve ,Projective linear group ,Algebraic curve ,0101 mathematics ,Invariant (mathematics) ,Quotient ,Mathematics - Abstract
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.
- Published
- 2019
129. Low-degree planar polynomials over finite fields of characteristic two
- Author
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Kai-Uwe Schmidt and Daniele Bartoli
- Subjects
Pure mathematics ,Monomial ,APN function ,Power of two ,01 natural sciences ,Exceptional ,Mathematics - Algebraic Geometry ,Planar ,Corollary ,Planar function ,Algebraic curve ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,Absolutely irreducible polynomial, Algebraic curve, APN function, Exceptional, Planar function ,Algebra and Number Theory ,51E20, 11T71 ,010102 general mathematics ,Nonlinear system ,Finite field ,Absolutely irreducible polynomial ,010307 mathematical physics ,Projective plane ,Combinatorics (math.CO) - Abstract
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.
- Published
- 2019
130. Equivariant Bifurcation and Absolute Irreducibility in $$\mathbb {R}^8$$ R 8 : A Contribution to Ize Conjecture and Related Bifurcations
- Author
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Reiner Lauterbach
- Subjects
Discrete mathematics ,Pure mathematics ,Group (mathematics) ,Absolutely irreducible ,Fixed-point space ,Irreducibility ,Lie group ,Equivariant map ,Context (language use) ,Fixed point ,Analysis ,Mathematics - Abstract
We refer to the hypotheses that for an absolutely irreducible representation of a compact Lie group there exists at least one subgroup with an odd dimensional fixed point space as the Ize conjecture (IC). If the IC is true, then it follows that loss of stability through an absolutely irreducible representation of a compact Lie group leads to bifurcation of steady states. Lauterbach and Matthews have shown that the (IC) is in general not true and have constructed three infinite families of finite subgroups of \(\mathop {\mathbf{SO}(4)}\) which act absolutely irreducibly on \(\mathbb {R}^4\) and have no odd dimensional fixed point space. They also have shown that in spite of this failure of the (IC) the nontrivial isotropy types are generically symmetry breaking at least for the groups in two of these three families. In this paper we show a similar bifurcation result for the third family defined by Lauterbach and Matthews. We go on and construct a family of groups acting absolutely irreducibly on \(\mathbb {R}^8\) which have only even dimensional fixed point spaces. Then we discuss the steady state bifurcations in this case. Key ingredients are an abstract group theoretic construction and a kind of inductive step reducing the issue of bifurcations to a problem in \(\mathbb {R}^4\). We end this paper with a discussion on how to extend the results by Lauterbach and Matthews to larger sets of groups which act on \(\mathbb {R}^{4}\) and \(\mathbb {R}^8\). In this context we point out, that the inductive step, which is important for our arguments, does not work in general and this gives rise to interesting new questions.
- Published
- 2014
131. Uniformization of p-adic curves via Higgs–de Rham flows
- Author
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Kang Zuo, Guitang Lan, Mao Sheng, and Yanhong Yang
- Subjects
Ring (mathematics) ,Pure mathematics ,Absolutely irreducible ,Applied Mathematics ,General Mathematics ,Riemann surface ,Algebraic closure ,Higgs bundle ,Higgs field ,symbols.namesake ,symbols ,Uniformization (set theory) ,Witt vector ,Mathematics - Abstract
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.
- Published
- 2016
132. Density of potentially crystalline representations of fixed weight
- Author
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Benjamin Schraen, Eugen Hellmann, Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), and ANR-11-BS01-0005,ThéHopaD,Théorie de Hodge p-adique et développements(2011)
- Subjects
Pure mathematics ,Deformation ring ,Zariski topology ,Algebra and Number Theory ,Mathematics - Number Theory ,Absolutely irreducible ,010102 general mathematics ,Diagonalizable matrix ,Absolute Galois group ,16. Peace & justice ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Finite field ,p-adic Hodge theory ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,[MATH]Mathematics [math] ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics ,Vector space - Abstract
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
- Published
- 2016
133. On the completeness of plane cubic curves over finite fields
- Author
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Fernanda Pambianco, Stefano Marcugini, and Daniele Bartoli
- Subjects
3)-arcs ,Pure mathematics ,Plane algebraic cubic curves ,Cubic surface ,Absolutely irreducible ,Plane curve ,Plane (geometry) ,Applied Mathematics ,(n, 3)-arcs, Near-MDS codes, Plane algebraic cubic curves ,020206 networking & telecommunications ,(n ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Cubic plane curve ,Computer Science Applications ,Combinatorics ,Finite field ,010201 computation theory & mathematics ,Near-MDS codes ,0202 electrical engineering, electronic engineering, information engineering ,Cubic form ,Algebraic number ,Mathematics - Abstract
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.
- Published
- 2016
134. Projective Crystalline Representations of \'Etale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow
- Author
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Jinbang Yang, Ruiran Sun, and Kang Zuo
- Subjects
Pure mathematics ,Degree (graph theory) ,Coprime integers ,Mathematics - Number Theory ,Absolutely irreducible ,Applied Mathematics ,General Mathematics ,Image (category theory) ,Order (ring theory) ,Higgs bundle ,Higgs field ,Mathematics - Algebraic Geometry ,p-adic Hodge theory ,Mathematics - Abstract
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
- Published
- 2017
135. Local Ihara's Lemma and Applications.
- Author
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Boyer, Pascal
- Subjects
GENERALIZATION ,COHOMOLOGY theory - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
136. Residual irreducibility of compatible systems
- Author
-
Andrew Wiles, Andrew Snowden, and Stefan Patrikis
- Subjects
Mathematics - Number Theory ,Absolutely irreducible ,General Mathematics ,Image (category theory) ,010102 general mathematics ,Algebraic number field ,Galois module ,01 natural sciences ,Combinatorics ,Closure (mathematics) ,Bounded function ,0103 physical sciences ,FOS: Mathematics ,Irreducibility ,010307 mathematical physics ,Number Theory (math.NT) ,0101 mathematics ,Maximal compact subgroup ,Mathematics - Abstract
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
- Published
- 2016
137. Asymptotics of Reduced Algebraic Curves Over Finite Fields
- Author
-
J. I. Farrán
- Subjects
Pure mathematics ,Finite field ,Absolutely irreducible ,Genus (mathematics) ,Point (geometry) ,Coding theory ,Algebraic curve ,Connection (algebraic framework) ,Quotient ,Mathematics - Abstract
The number A(q) shows the asymptotic behaviour of the quotient of the number of rational points over the genus of non-singular absolutely irreducible curves over \(\mathbb {F}_{q}\,\). Research on bounds for A(q) is closely connected with the so-called asymptotic main problem in Coding Theory. In this paper, we study some generalizations of this number for non-irreducible curves, their connection with A(q) and their application in Coding Theory. We also discuss the possibility of constructing codes from non-irreducible curves, both from theoretical and practical point of view.
- Published
- 2018
138. On the irreducibility of the hyperplane sections of Fermat varieties in ℙ 3 $\mathbb {P}^{3}$ in characteristic 2. II
- Author
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Eric Férard, Laboratoire de Géométrie Algébrique et Applications à la Théorie de l'Information (GAATI), and Université de la Polynésie Française (UPF)
- Subjects
Polynomial (hyperelastic model) ,Discrete mathematics ,Fermat's Last Theorem ,Computer Networks and Communications ,Absolutely irreducible ,Applied Mathematics ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Computational Theory and Mathematics ,Hyperplane ,Integer ,010201 computation theory & mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Irreducibility ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
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.
- Published
- 2017
139. On a group of the form 214:Sp(6, 2)
- Author
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Ayoub B.M. Basheer and Thekiso T. Seretlo
- Subjects
Discrete mathematics ,Symplectic group ,Group (mathematics) ,Absolutely irreducible ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Matrix (mathematics) ,Mathematics (miscellaneous) ,Clifford theory ,Conjugacy class ,Character table ,Coset ,0101 mathematics ,Mathematics - Abstract
The symplectic group Sp(6, 2) has a 14−dimensional absolutely irreducible module over . Hence a split extension group of the form Ḡ = 214:Sp(6, 2) does exist. In this paper we first determine the conjugacy classes of Ḡ using the coset analysis technique. The structures of inertia factor groups were determined. The inertia factor groups are Sp(6, 2), (21+4 × 22):(S3 × S3), S3 × S6, PSL(2, 8), (((22 ×Q8):3):2):2, S3 ×A5,and 2×S4 ×S3.We then determine the Fischer matrices and apply the Clifford-Fischer theory to compute the ordinary character table of . The Fischer matrices of are all integer valued, with size ranging from 4 to 16. The full character table of is a 186 × 186 complex valued matrix.
- Published
- 2015
140. Classifying forms of simple groups via their invariant polynomials
- Author
-
Anthony Ruozzi and H. Bermudez
- Subjects
Discrete mathematics ,Linear algebraic group ,Pure mathematics ,Algebra and Number Theory ,Absolutely irreducible ,Quadratic form ,Homogeneous polynomial ,Orthogonal group ,ε-quadratic form ,Isotropic quadratic form ,Invariant theory ,Mathematics - Abstract
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 ) .
- Published
- 2015
141. On a group extension involving the Suzuki group Sz(8)
- Author
-
Basheer, Ayoub B. M.
- Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
142. Zeta morphisms for rank two universal deformations.
- Author
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Nakamura, Kentaro
- Subjects
ZETA functions ,MODULAR forms ,LOGICAL prediction - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
143. DETERMINING ASCHBACHER CLASSES USING CHARACTERS
- Author
-
Sebastian Jambor
- Subjects
Combinatorics ,Character (mathematics) ,Absolutely irreducible ,Group (mathematics) ,General Mathematics ,Character theory ,Field (mathematics) ,Representation (mathematics) ,Mathematics ,Image (mathematics) - Abstract
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.
- Published
- 2014
144. Splitting of abelian varieties
- Author
-
Ying Zong and V. Kumar Murty
- Subjects
Abelian variety ,Pure mathematics ,Algebra and Number Theory ,Conjecture ,Computer Networks and Communications ,Absolutely irreducible ,Applied Mathematics ,Algebraic number field ,Simple (abstract algebra) ,Algebraic group ,Discrete Mathematics and Combinatorics ,Maximal torus ,Abelian group ,Mathematics - Abstract
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 $V$.
- Published
- 2014
145. Recipes to Fermat-type equations of the form $$x^r + y^r =Cz^p$$ x r + y r = C z p
- Author
-
Nuno Freitas
- Subjects
Fermat's Last Theorem ,Discrete mathematics ,Elliptic curve ,Absolutely irreducible ,General Mathematics ,Type (model theory) ,Signature (topology) ,Constant (mathematics) ,Prime (order theory) ,Mathematics ,Frey curve - Abstract
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.
- Published
- 2014
146. Maps to weight space in Hida families
- Author
-
Ravi Ramakrishna
- Subjects
Rational number ,Degree (graph theory) ,Absolutely irreducible ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Modular form ,Weight space ,Absolute Galois group ,Galois module ,Algebra ,Combinatorics ,lcsh:Q ,lcsh:Science ,Irreducible component ,Mathematics - Abstract
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.
- Published
- 2014
147. A universal deformation ring with unexpected Krull dimension
- Author
-
Johannes Sprang
- Subjects
Deformation ring ,Pure mathematics ,Conjecture ,Mathematics - Number Theory ,Absolutely irreducible ,Mathematics::Number Theory ,General Mathematics ,Group cohomology ,Adjoint representation ,Dimension (vector space) ,Mathematik ,FOS: Mathematics ,Number Theory (math.NT) ,Krull dimension ,Mathematics ,Counterexample - Abstract
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
- Published
- 2013
148. On planar arcs of size (q+3)∕2.
- Author
-
Günay, Gülizar and Lavrauw, Michel
- Subjects
- *
PROJECTIVE planes , *FINITE geometries , *EVIDENCE , *SIZE - Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
149. ABSOLUTE IRREDUCIBILITY OF BIVARIATE POLYNOMIALS VIA POLYTOPE METHOD
- Author
-
Fatih Koyuncu and MÜ
- Subjects
Polytope Method ,Birkhoff polytope ,Absolutely irreducible ,General Mathematics ,Algebraic extension ,Field (mathematics) ,Polytope ,Uniform k 21 polytope ,Integral Indecomposability ,Integral Polygons ,Combinatorics ,Absolute Irreducibility ,Bivariate Polynomials ,Mathematics::Metric Geometry ,Mathematics::Representation Theory ,Indecomposable module ,Ehrhart polynomial ,Mathematics - Abstract
WOS: 000294590700013 For any field F, a polynomial f is an element of F[x(1), x(2),., x(k)] can be associated with a polytope, called its Newton polytope. If the polynomial f has integrally indecomposable Newton polytope, in the sense of Minkowski sum, then it is absolutely irreducible over F, i.e., irreducible over every algebraic extension of F. We present some results giving new integrally indecomposable classes of polygons. Consequently, we have some criteria giving many types of absolutely irreducible bivariate polynomials over arbitrary fields.
- Published
- 2011
150. Effective descent for differential operators
- Author
-
Elie Compoint, Marius van der Put, Jacques-Arthur Weil, University of Groningen [Groningen], DMI, XLIM (XLIM), Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS)-Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS), and Algebra
- Subjects
Pure mathematics ,Absolutely irreducible ,010103 numerical & computational mathematics ,Differential Galois theory ,01 natural sciences ,Algebraic element ,Mathematics - Algebraic Geometry ,Symbolic computation ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Genus field ,Galois extension ,0101 mathematics ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,EQUATIONS ,Mathematics ,Discrete mathematics ,Algebra and Number Theory ,010102 general mathematics ,Galois cohomology ,Differential operator ,34M15, 20Gxx, 12G05, 33F10, 68W30 ,Linear algebraic groups ,Mathematics - Classical Analysis and ODEs ,Differential algebraic geometry ,Algebraic differential equation - Abstract
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
- Published
- 2010
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