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Generalised Weber functions

Andreas Enge, François Morain (2014)

Acta Arithmetica

A generalised Weber function is given by N ( z ) = η ( z / N ) / η ( z ) , where η(z) is the Dedekind function and N is any integer; the original function corresponds to N=2. We classify the cases where some power N e evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating N ( z ) and j(z). Our ultimate goal is the use of these invariants in constructing...

Loi de réciprocité quadratique dans les corps quadratiques imaginaires

Abdelmejid Bayad (1995)

Annales de l'institut Fourier

À partir d’une courbe elliptique définie sur le corps des classes de Hilbert d’un corps quadratique imaginaire K et à multiplicité complexe par l’anneau des entiers de K , on construit des fonctions elliptiques. Nous établissons des formules produits relatives à ces fonctions. De ce fait, nous obtenons une formulation analytique du lemme de Gauss généralisé ainsi qu’une expression explicite pour le symbole quadratique de Legendre défini sur l’anneau des entiers du corps quadratique imaginaire. Comme...

Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs

Ilwoo Cho, Palle E. T. Jorgensen (2015)

Special Matrices

In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we...

Non-existence of points rational over number fields on Shimura curves

Keisuke Arai (2016)

Acta Arithmetica

Jordan, Rotger and de Vera-Piquero proved that Shimura curves have no points rational over imaginary quadratic fields under a certain assumption. In this article, we extend their results to the case of number fields of higher degree. We also give counterexamples to the Hasse principle on Shimura curves.

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