Arithmétique des Algèbres de Quaternions

A generalised triangle group has a presentation of the form where R is a cyclically reduced word involving both x and y. When R=xy, these classical triangle groups have representations as discrete groups of isometries of S 2 , R 2 , H 2 depending on In this paper, for other words R, faithful discrete representations of these groups in Isom + H 3 = PSL(2,C) are considered with particular emphasis on the case /? = [x, y] and also on the relationship between the Euler characteristic x and finite covolume representations.1991 Mathematics subject classification: 20H15.

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“…[25,1]. Let k be a number field with one complex place and A a quarternion algebra over k which is ramified at all real places.…”

Section: Arithmeticitymentioning

confidence: 99%

On discrete generalised triangle groups

Hagelberg

MacLachlan

Rosenberger

1995

Proceedings of the Edinburgh Mathematical Society

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“…The theorem of the norm and the strong approximation theorem (Theorems 4.1 and 4.3 in Chapter 3 of [34]) gives a decomposition…”

Section: Another Description Of Quaternionic Formsmentioning

confidence: 99%

Heegner points and $p$-adic $L$-functions for elliptic curves over certain totally real fields

Mok¹

2011

Comment. Math. Helv.

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Abstract. For an elliptic curve E over Q satisfying suitable hypotheses, Bertolini and Darmon have derived a formula for the Heegner point on E in terms of the central derivative of the two variable p-adic L-function associated to E. In this paper, we generalize their work to the setting of totally real fields in which p is inert. We also use this generalization to improve the results obtained by Bertolini-Darmon in the case of an elliptic curve defined over the field of rational numbers. Mathematics Subject Classification (2010). 11G40, 11F33.

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“…By the Jacquet-Langlands correspondence (Theorem 3.7), there exists (unique up to multiples) a modular form f : R × \ B × /B × → Z with the same eigenvalues as φ. Since all primes dividing n − are inert in K, it follows by [47,III.3.8], that there exists a monomorphism Ψ : K → B. Assume that Ψ is an optimal embedding of the integers O K of K into the Eichler order R, that is, Ψ(O K ) = Ψ(K)∩R.…”

Section: The Gross-zhang Formula For the Special Value Of L-seriesmentioning

confidence: 99%

On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

Longo¹

2006

Ann. inst. Fourier

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Let E/F be a modular elliptic curve defined over a totally real number field F and let f be its associated eigenform. This article presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of E over suitable quadratic imaginary extensions K/F. In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachev, that is, when [F : Q] is even and phi not new at any prime

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Arithmétique des Algèbres de Quaternions

Cited by 602 publications

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On discrete generalised triangle groups

On discrete generalised triangle groups

Heegner points and $p$-adic $L$-functions for elliptic curves over certain totally real fields

On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

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