Nonvanishing of L-functions and structure of Mordell-Weil groups.
Let be a modular elliptic curve over
In this paper, we give a numerical characterization of nef arithmetic -Cartier divisors of -type on an arithmetic surface. Namely an arithmetic -Cartier divisor of -type is nef if and only if is pseudo-effective and .
Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion . In the first part, we apply ideas from the proof of by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if are such that , then . This allows us to conclude, among other things, that and .
Si un système d’équations polynomiales à coefficients entiers admet une solution dans , il en admet sur tout complété -adique ou réel de . La réciproque a été démontrée par Hasse pour les quadriques, mais elle est fausse en général. Une grande partie des contre-exemples connus peuvent être expliqués à l’aide de l’obstruction de Brauer-Manin, basée sur la théorie du corps de classe. Il est donc naturel de se demander si, pour certaines classes de variétés, cette obstruction est la seule. Le but...
Watkins has conjectured that if is the rank of the group of rational points of an elliptic curve over the rationals, then divides the modular parametrisation degree. We show, for a certain class of , chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain -adic deformation ring is isomorphic to a certain Hecke ring, and is a complete intersection. However, we show also that the method developed by Taylor, Wiles and others, to prove...
Let be a morphism of a variety defined over a number field , let be a -subvariety, and let be the orbit of a point . We describe a local-global principle for the intersection . This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of are Brauer–Manin unobstructed for power maps on in two cases: (1) is a translate of a torus. (2) is a line and has a preperiodic coordinate. A key tool in the proofs is the classical...
Let be a family of elliptic curves over , where is a positive integer and , are distinct odd primes. We study the torsion part and the rank of . More specifically, we prove that the torsion subgroup of is trivial and the -rank of this family is at least 2, whenever , and with neither nor dividing .
A well known theorem of Mestre and Schoof implies that the order of an elliptic curve over a prime field can be uniquely determined by computing the orders of a few points on and its quadratic twist, provided that . We extend this result to all finite fields with , and all prime fields with .