On the equation
Imin Chen (2010)
Acta Arithmetica
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Imin Chen (2010)
Acta Arithmetica
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Magali Bouffet (2003)
Bulletin de la Société Mathématique de France
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In this paper we study the formal differential Galois group of linear differential equations with coefficients in an extension of by an exponential of integral. We use results of factorization of differential operators with coefficients in such a field to give explicit generators of the Galois group. We show that we have very similar results to the case of .
Bin Zhao (2014)
Annales de l’institut Fourier
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We prove the indecomposability of the Galois representation restricted to the -decomposition group attached to a non CM nearly -ordinary weight two Hilbert modular form over a totally real field under the assumption that either the degree of over is odd or the automorphic representation attached to the Hilbert modular form is square integrable at some finite place of .
Lior Bary-Soroker, Arno Fehm (2013)
Journal de Théorie des Nombres de Bordeaux
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Let be a Galois extension of a countable Hilbertian field . Although need not be Hilbertian, we prove that an abundance of large Galois subextensions of are.
Xavier Caruso, Tong Liu (2009)
Bulletin de la Société Mathématique de France
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Fix a -adic field and denote by its absolute Galois group. Let be the extension of obtained by adding -th roots of a fixed uniformizer, and its absolute Galois group. In this article, we define a class of -adic torsion representations of , called. We prove that these representations are “explicitly” described by a certain category of linear algebraic objects. The results of this note should be considered as a first step in the understanding of the structure of quotient...
Frank Calegari, Toby Gee (2013)
Annales de l’institut Fourier
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Let be a regular, algebraic, essentially self-dual cuspidal automorphic representation of , where is a totally real field and is at most . We show that for all primes , the -adic Galois representations associated to are irreducible, and for all but finitely many primes , the mod Galois representations associated to are also irreducible. We also show that the Lie algebras of the Zariski closures of the -adic representations are independent of .
B. Bensebaa, A. Movahhedi, A. Salinier (2008)
Acta Arithmetica
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Andrzej J. Maciejewski, Maria Przybylska (2011)
Banach Center Publications
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We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form , where the potential V is a finite sum of homogeneous components. In this paper we show how to find, in the differential Galois framework, computable necessary conditions for the integrability of such systems. Our main result concerns potentials of the form , where and are homogeneous functions of integer degrees k and K > k, respectively. We present examples...
James E. Carter (2007)
Colloquium Mathematicae
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Let p be a rational prime, G a group of order p, and K a number field containing a primitive pth root of unity. We show that every tamely ramified Galois extension of K with Galois group isomorphic to G has a normal integral basis if and only if for every Galois extension L/K with Galois group isomorphic to G, the ring of integers in L is free as a module over the associated order . We also give examples, some of which show that this result can still hold without the assumption that...
Michael A. Bennett, Vandita Patel, Samir Siksek (2016)
Acta Arithmetica
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Using only elementary arguments, Cassels solved the Diophantine equation (x-1)³ + x³ + (x+1)³ = z² (with x, z ∈ ℤ). The generalization (with x, z, n ∈ ℤ and n ≥ 2) was considered by Zhongfeng Zhang who solved it for k ∈ 2,3,4 using Frey-Hellegouarch curves and their corresponding Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solutions for k = 5 have x = z = 0, and that there are no solutions for k = 6. The...
Henri Cohen, Francisco Diaz y Diaz, Michel Olivier (2006)
Journal de Théorie des Nombres de Bordeaux
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For each transitive permutation group on letters with , we give without proof results, conjectures, and numerical computations on discriminants of number fields of degree over such that the Galois group of the Galois closure of is isomorphic to .