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

Imin Chen

Displaying similar documents to “On the equation $a^{2} + b^{2 p} = c^{5}$ ”

Local Indecomposability of Hilbert Modular Galois Representations

Bin Zhao (2014)

Annales de l’institut Fourier

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We prove the indecomposability of the Galois representation restricted to the $p$ -decomposition group attached to a non CM nearly $p$ -ordinary weight two Hilbert modular form over a totally real field $F$ under the assumption that either the degree of $F$ over $ℚ$ is odd or the automorphic representation attached to the Hilbert modular form is square integrable at some finite place of $F$ .

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...

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.

Irreducibility of automorphic Galois representations of $G L (n)$ , $n$ at most $5$

Frank Calegari, Toby Gee (2013)

Annales de l’institut Fourier

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Let $π$ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of ${GL}_{n} (𝔸_{F})$ , where $F$ is a totally real field and $n$ is at most $5$ . We show that for all primes $l$ , the $l$ -adic Galois representations associated to $π$ are irreducible, and for all but finitely many primes $l$ , the mod $l$ Galois representations associated to $π$ are also irreducible. We also show that the Lie algebras of the Zariski closures of the $l$ -adic representations are independent of $l$ .

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...

Preperiodic dynatomic curves for $z \mapsto z^{d} + c$

Yan Gao (2016)

Fundamenta Mathematicae

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The preperiodic dynatomic curve $_{n, p}$ is the closure in ℂ² of the set of (c,z) such that z is a preperiodic point of the polynomial $z \mapsto z^{d} + c$ with preperiod n and period p (n,p ≥ 1). We prove that each $_{n, p}$ has exactly d-1 irreducible components, which are all smooth and have pairwise transverse intersections at the singular points of $_{n, p}$ . We also compute the genus of each component and the Galois group of the defining polynomial of $_{n, p}$ .

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 $G$ .

On the Euler Function on Differences Between the Coordinates of Points on Modular Hyperbolas

Igor E. Shparlinski (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola $_{a, p} (X, Y) = (x, y) : x y \equiv a (m o d p), 1 \leq x \leq X, 1 \leq y \leq Y$ . We give asymptotic formulas for the average values $\sum_{\begin{matrix} ( \end{matrix} x, y) \in_{a, p} (X, Y) x \neq y *} φ (| x - y |) / | x - y |$ and $\sum_{\begin{matrix} ( \end{matrix} x, y) \in_{a, p} (X, X) x \neq y *} φ (| x - y |)$ with the Euler function φ(k) on the differences between the components of points of $_{a, p} (X, Y)$ .

Stacks of group representations

Paul Balmer (2015)

Journal of the European Mathematical Society

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We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$ , the derived and the stable categories of representations of a subgroup $H$ can be constructed out of the corresponding category for $G$ by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods...

Kronecker’s solution of Pell’s equation for CM fields

Riad Masri (2013)

Annales de l’institut Fourier

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We generalize Kronecker’s solution of Pell’s equation to CM fields $K$ whose Galois group over $ℚ$ is an elementary abelian 2-group. This is an identity which relates CM values of a certain Hilbert modular function to products of logarithms of fundamental units. When $K$ is imaginary quadratic, these CM values are algebraic numbers related to elliptic units in the Hilbert class field of $K$ . Assuming Schanuel’s conjecture, we show that when $K$ has degree greater than 2 over $ℚ$ these CM values...

Self-intersection of the relative dualizing sheaf on modular curves $X_{1} (N)$

Hartwig Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than $4$ . Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves $X_{1} (N) / ℚ$ . From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian $J_{1} (N) / ℚ$ of $X_{1} (N) / ℚ$ , and, for sufficiently large $N$ , an effective version of Bogomolov’s conjecture for $X_{1} (N) / ℚ$ . ...

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].

Displaying similar documents to “On the equation a 2 + b 2 p = c 5 ”

Displaying similar documents to “On the equation $a^{2} + b^{2 p} = c^{5}$ ”