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Charles Hermite’s stroll through the Galois fields

Catherine Goldstein (2011)

Revue d'histoire des mathématiques

Although everything seems to oppose the two mathematicians, Charles Hermite’s role was crucial in the study and diffusion of Évariste Galois’s results in France during the second half of the nineteenth century. The present article examines that part of Hermite’s work explicitly linked to Galois, the reduction of modular equations in particular. It shows how Hermite’s mathematical convictions—concerning effectiveness or the unity of algebra, analysis and arithmetic—shaped his interpretation of Galois...

Class invariants by Shimura's reciprocity law

Alice Gee (1999)

Journal de théorie des nombres de Bordeaux

We apply the Shimura reciprocity law to determine when values of modular functions of higher level can be used to generate the Hilbert class field of an imaginary quadratic field. In addition, we show how to find the corresponding polynomial in these cases. This yields a proof for conjectural formulas of Morain and Zagier concerning such polynomials.

Classes d'idéaux et classes de diviseurs

Serge Lang (1976/1977)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Completely normal elements in some finite abelian extensions

Ja Koo, Dong Shin (2013)

Open Mathematics

We present some completely normal elements in the maximal real subfields of cyclotomic fields over the field of rational numbers, relying on the criterion for normal element developed in [Jung H.Y., Koo J.K., Shin D.H., Normal bases of ray class fields over imaginary quadratic fields, Math. Z., 2012, 271(1–2), 109–116]. And, we further find completely normal elements in certain abelian extensions of modular function fields in terms of Siegel functions.

Computations of Galois representations associated to modular forms of level one

Peng Tian (2014)

Acta Arithmetica

We propose an improved algorithm for computing mod ℓ Galois representations associated to a cusp form f of level one. The proposed method allows us to explicitly compute the case with ℓ = 29 and f of weight k = 16, and the cases with ℓ = 31 and f of weight k = 12,20,22. All the results are rigorously proved to be correct. As an example, we will compute the values modulo 31 of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved uper bound on...

Concernant la relation de distribution satisfaite par la fonction ...associé à un réseau complexe.

G. Robert (1990)

Inventiones mathematicae

Congruence subgroups of groups commensurable with $PSL (2, ℤ)$ of genus 0 and 1.

Cummins, C.J. (2004)

Experimental Mathematics

Congruences et formes modulaires

Jean-Pierre Serre (1971/1972)

Séminaire Bourbaki

Construction of entire modular forms of weights 5 and 6 for the congruence group $Γ_{0} (4 N)$ .

Lomadze, G. (1995)

Georgian Mathematical Journal

Construire un noyau de la fonctorialité ? Le cas de l’induction automorphe sans ramification de GL $_{1}$ à GL $_{2}$

Laurent Lafforgue (2010)

Annales de l’institut Fourier

Le but de cet article est de présenter une nouvelle méthode purement adélique pour réaliser le principe de fonctorialité de Langlands dans le cas de l’induction automorphe sans ramification de GL $_{1}$ à GL $_{2}$ sur les corps de fonctions. On construit sur le produit des groupes adéliques GL $_{1}$ et GL $_{2}$ un noyau de la fonctorialité. C’est une version “en famille” et locale de la construction par les modèles de Whittaker globaux, utilisée classiquement dans les “théorèmes réciproques” de Weil et Piatetski-Shapiro....

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