Modularity of an odd icosahedral representation

Arnaud Jehanne; Michael Müller

Displaying similar documents to “Modularity of an odd icosahedral representation”

Newforms, inner twists, and the inverse Galois problem for projective linear groups

Luis V. Dieulefait (2001)

Journal de théorie des nombres de Bordeaux

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We reformulate more explicitly the results of Momose, Ribet and Papier concerning the images of the Galois representations attached to newforms without complex multiplication, restricted to the case of weight $2$ and trivial nebentypus. We compute two examples of these newforms, with a single inner twist, and we prove that for every inert prime greater than $3$ the image is as large as possible. As a consequence, we prove that the groups $PGL (2, 𝔽_{ℓ^{2}})$ for every prime $(ℓ \equiv 3, 5 (mod 8), ℓ > 3)$ , and $PGL (2, 𝔽_{ℓ^{5}})$ for every prime $ℓ \neg \equiv 0 \pm 1 (mod 11); ℓ > 3)$ , are...

Galois representations, embedding problems and modular forms.

Teresa Crespo (1997)

Collectanea Mathematica

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To an odd irreducible 2-dimensional complex linear representation of the absolute Galois group of the field Q of rational numbers, a modular form of weight 1 is associated (modulo Artin's conjecture on the L-series of the representation in the icosahedral case). In addition, linear liftings of 2-dimensional projective Galois representations are related to solutions of certain Galois embedding problems. In this paper we present some recent results on the existence of liftings of projective...

On the order of vanishing of modular $L$ -functions at the critical point

Henryk Iwaniec (1990)

Journal de théorie des nombres de Bordeaux

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Weber's class invariants revisited

Reinhard Schertz (2002)

Journal de théorie des nombres de Bordeaux

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Let $K$ be a quadratic imaginary number field of discriminant $d$ . For $t \in ℕ$ let $𝔒_{t}$ denote the order of conductor $t$ in $K$ and $j (𝔒_{t})$ its modular invariant which is known to generate the ring class field modulo $t$ over $K$ . The coefficients of the minimal equation of $j (𝔒_{t})$ being quite large Weber considered in [We] the functions $f, f_{1}, f_{2}, γ_{2}, γ_{3}$ defined below and thereby obtained simpler generators of the ring class fields. Later on the singular values of these functions played a crucial role in Heegner’s solution [He] of...

Special values of twisted symmetric square $L$ -functions and the trace formula

Jeffrey Stopple (1996)

Compositio Mathematica

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On integral representations by totally positive ternary quadratic forms

Elise Björkholdt (2000)

Journal de théorie des nombres de Bordeaux

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Let $K$ be a totally real algebraic number field whose ring of integers $R$ is a principal ideal domain. Let $f (x_{1}, x_{2}, x_{3})$ be a totally definite ternary quadratic form with coefficients in $R$ . We shall study representations of totally positive elements $N \in R$ by $f$ . We prove a quantitative formula relating the number of representations of $N$ by different classes in the genus of $f$ to the class number of $R [\sqrt{- c_{f} N}]$ , where $c_{f} \in R$ is a constant depending only on $f$ . We give an algebraic proof of a classical result of H. Maass...

An analytic series of irreducible representations of the free group

Ryszard Szwarc (1988)

Annales de l'institut Fourier

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Let $F_{k}$ be a free group on $k$ generators. We construct the series of uniformly bounded representations $\prod_{z}$ of $F_{k}$ acting on the common Hilbert space, depending analytically on the complex parameter z, $1 / (2 k - 1) < | z | < 1$ , such that each representation $\prod_{z}$ is irreducible. If $z$ is real or $| z | = 1 / (\sqrt{2 k - 1})$ then $\prod_{z}$ is unitary; in other cases $\prod_{z}$ cannot be made unitary. For $z \neq z^{'}$ representations $\prod_{z}$ and $\prod_{z^{'}}$ are congruent modulo compact operators.

The diophantine equation $a x^{2} + b x y + c y^{2} = N, D = b^{2} - 4 a c > 0$

Keith Matthews (2002)

Journal de théorie des nombres de Bordeaux

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We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of $a x^{2} + b x y + c y^{2} = N$ in relatively prime integers $x, y$ , where $N \neq 0$ , gcd $(a, b, c) = gcd (a, N) = 1 et D = b^{2} - 4 a c > 0$ is not a perfect square. In the case of solubility, solutions with least positive y, from each equivalence class, are also constructed. Our paper is a generalisation of an earlier paper by the author on the equation $x^{2} - D y^{2} = N$ . As in that paper, we use a lemma on unimodular matrices that gives a much simpler proof than Lagrange’s...

On an approximation property of Pisot numbers II

Toufik Zaïmi (2004)

Journal de Théorie des Nombres de Bordeaux

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Let $q$ be a complex number, $m$ be a positive rational integer and $l_{m} (q) = inf {|P (q)|, P \in ℤ_{m} [X], P (q) \neq 0}$ , where $ℤ_{m} [X]$ denotes the set of polynomials with rational integer coefficients of absolute value $\leq m$ . We determine in this note the maximum of the quantities $l_{m} (q)$ when $q$ runs through the interval $] m, m + 1 [$ . We also show that if $q$ is a non-real number of modulus $> 1$ , then $q$ is a complex Pisot number if and only if $l_{m} (q) > 0$ for all $m$ .