Superelliptic equations arising from sums of consecutive powers

Michael A. Bennett; Vandita Patel; Samir Siksek

Displaying similar documents to “Superelliptic equations arising from sums of consecutive powers”

Differential Galois Theory for an Exponential Extension of $ℂ ((z))$

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 $ℂ ((z))$ 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 $ℂ ((z))$ .

Automorphic realization of residual Galois representations

Robert Guralnick, Michael Harris, Nicholas M. Katz (2010)

Journal of the European Mathematical Society

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We show that it is possible in rather general situations to obtain a finite-dimensional modular representation $ρ$ of the Galois group of a number field $F$ as a constituent of one of the modular Galois representations attached to automorphic representations of a general linear group over $F$ , provided one works “potentially.” The proof is based on a close study of the monodromy of the Dwork family of Calabi–Yau hypersurfaces; this in turn makes use of properties of rigid local systems and...

On the equation $a^{2} + b^{2 p} = c^{5}$

Imin Chen (2010)

Acta Arithmetica

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Polynomials over $Q$ solving an embedding problem

Nuria Vila (1985)

Annales de l'institut Fourier

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The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group $A_{n}$ , can be embedded in any central extension of $A_{n}$ if and only if $n \equiv 0 (m o d 8)$ , or $n \equiv 2 (m o d 8)$ and $n$ is a sum of two squares. Consequently, for theses values of $n$ , every central extension of $A_{n}$ occurs as a Galois group over $Q$ .

Some remarks on Hilbert-Speiser and Leopoldt fields of given type

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 $O_{L}$ in L is free as a module over the associated order $_{L / K}$ . We also give examples, some of which show that this result can still hold without the assumption that...

The Galois group of $X^{p} + a X^{s} + a$

B. Bensebaa, A. Movahhedi, A. Salinier (2008)

Acta Arithmetica

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Random Galois extensions of Hilbertian fields

Lior Bary-Soroker, Arno Fehm (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $L$ be a Galois extension of a countable Hilbertian field $K$ . Although $L$ need not be Hilbertian, we prove that an abundance of large Galois subextensions of $L / K$ are.

On generalized Fermat equations of signature (p,p,3)

Karolina Krawciów (2011)

Colloquium Mathematicae

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This paper focuses on the Diophantine equation $x ⁿ + p^{α} y ⁿ = M z ³$ , with fixed α, p, and M. We prove that, under certain conditions on M, this equation has no non-trivial integer solutions if $n \geq ℱ (M, p^{α})$ , where $ℱ (M, p^{α})$ is an effective constant. This generalizes Theorem 1.4 of the paper by Bennett, Vatsal and Yazdani [Compos. Math. 140 (2004), 1399-1416].

Quaternion extensions with restricted ramification

Peter Schmid (2014)

Acta Arithmetica

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In any normal number field having Q₈, the quaternion group of order 8, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes 2 and 3 are ramified. In this note we describe in detail all Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime $p \equiv 1 (m o d 2^{n - 1})$ , there exist unique real and complex...

Cubic forms, powers of primes and the Kraus method

Andrzej Dąbrowski, Tomasz Jędrzejak, Karolina Krawciów (2012)

Colloquium Mathematicae

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We consider the Diophantine equation $(x + y) (x ² + B x y + y ²) = D z^{p}$ , where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D’s, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).

Overview of the differential Galois integrability conditions for non-homogeneous potentials

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 $H = 1 / 2 \sum_{i = 1}^{n} p ²_{i} + V (q)$ , 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 $V = V_{k} + V_{K}$ , where $V_{k}$ and $V_{K}$ are homogeneous functions of integer degrees k and K > k, respectively. We present examples...

Two classes of almost Galois coverings for algebras

Piotr Dowbor, Adam Hajduk (2012)

Colloquium Mathematicae

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We prove that for any representation-finite algebra A (in fact, finite locally bounded k-category), the universal covering F: Ã → A is either a Galois covering or an almost Galois covering of integral type, and F admits a degeneration to the standard Galois covering F̅: Ã→ Ã/G, where $G = Π (Γ_{A})$ is the fundamental group of $Γ_{A}$ . It is shown that the class of almost Galois coverings F: R → R’ of integral type, containing the series of examples from our earlier paper [Bol. Soc. Mat. Mexicana 17 (2011)],...

Quasi-semi-stable representations

Xavier Caruso, Tong Liu (2009)

Bulletin de la Société Mathématique de France

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Fix $K$ a $p$ -adic field and denote by $G_{K}$ its absolute Galois group. Let $K_{\infty}$ be the extension of $K$ obtained by adding $p^{n}$ -th roots of a fixed uniformizer, and $G_{\infty} \subset G_{K}$ its absolute Galois group. In this article, we define a class of $p$ -adic torsion representations of $G_{\infty}$ , 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...

Counting discriminants of number fields

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 $G$ on $n$ letters with $n \leq 4$ , we give without proof results, conjectures, and numerical computations on discriminants of number fields $L$ of degree $n$ over $ℚ$ such that the Galois group of the Galois closure of $L$ is isomorphic to $G$ .