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Uniformisation des variétés de Laumon-Rapoport-Stuhler et conjecture de Drinfeld-Carayol

Thomas Hausberger (2005)

Annales de l’institut Fourier

Considérons les variétés de “ D -faisceaux elliptiques” introduites par Laumon, Rapoport et Stuhler, définies sur un corps de fonctions F d’une variable sur un corps fini, où D est une algèbre de division de dimension d 2 sur F . Nous montrons que ces variétés admettent, en une place o de F D o est un corps gauche d’invariant 1 / d , une uniformisation rigide-analytique par l’espace de Drinfeld Ω d , ou par les revêtements Σ n d de Ω d (selon la structure de niveau). Ce résultat constitue l’analogue du théorème...

Uniformization of certain Shimura curves

Pilar Bayer (2002)

Banach Center Publications

We present an approach to the uniformization of certain Shimura curves by means of automorphic functions, obtained by integration of non-linear differential equations. The method takes as its starting point a differential construction of the modular j-function, first worked out by R. Dedekind in 1877, and makes use of a differential operator of the third order, introduced by H. A. Schwarz in 1873.

Uniformly counting rational points on conics

Efthymios Sofos (2014)

Acta Arithmetica

We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over ℚ. The estimate is uniform in the coefficients of the underlying quadratic form.

Unimodular rows over Laurent polynomial rings

Abdessalem Mnif, Morou Amidou (2022)

Czechoslovak Mathematical Journal

We prove that for any ring 𝐑 of Krull dimension not greater than 1 and n 3 , the group E n ( 𝐑 [ X , X - 1 ] ) acts transitively on Um n ( 𝐑 [ X , X - 1 ] ) . In particular, we obtain that for any ring 𝐑 with Krull dimension not greater than 1, all finitely generated stably free modules over 𝐑 [ X , X - 1 ] are free. All the obtained results are proved constructively.

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