-adic analytic spaces.
The goal of this paper is to study certain -adic differential operators on automorphic forms on . These operators are a generalization to the higher-dimensional, vector-valued situation of the -adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the -adic case of the -differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. The operators should be useful in the construction of certain -adic...
For a variety over a local field, Bloch proposed a conjectural formula for the alternating sum of Artin conductor of -adic cohomology. We prove that the formula is valid modulo 2 if the variety has even dimension.
We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the non-existence of non-trivial primitive integer solutions for the signatures (5,5,11), (5,5,13), and (7,7,11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.
Soit une courbe elliptique sur par un modèle de Weierstrass généralisé :Soit avec , un point rationnel sur cette courbe. Pour tout entier , on exprime les coordonnées de sous la forme :où et , , sont déduits par multiplication par des puissances convenables de .Soit un nombre premier impair et supposons que est non singulier et que le rang d’apparition de dans la suite d’entiers est supérieur ou égal à trois. Notons ce rang par et soit . Nous montrons que la suite ...
We determine explicitly the set of algebraic points of degree at most 12 over ℚ on the Fermat quintic. This extends a previous result given by M. Klassen and P. Tzermias (1997), who described the set of algebraic points of degree at most 6 over ℚ.