Let be an algebraic variety defined over a field of characteristic , and let be an -torsor under a torus. We compute the Brauer group of . In the case of a number field we deduce results concerning the arithmetic of .
This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result due to É. Fouvry and J. Klüners which proves that a conjecture provided by the Cohen-Lenstra philosophy implies another such conjecture. As a consequence of our work, we can deduce, for example, a conjecture for the probability laws of -ranks of Selmer groups...
Let be a prime, be the non-singular projective curve defined over by the affine model , the point of at infinity on this model, the Jacobian of , and the albanese embedding with as base point. Let be an algebraic closure of . Taking care of a case not covered in [12], we show that consists only of the image under of the Weierstrass points of and the points and , where denotes the torsion points of .
This paper is concerned with the density of rational points on the graph of a non-algebraic pfaffian function.
Let k be a real quadratic field and let and be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v (, ) is developed.